Leo-III---1.8.0.UNS-Ref.s

%------------------------------------------------------------------------------
% File     : Leo-III---1.8.0
% Problem  : LCL836-1 : TPTP v9.3.1. Released v4.1.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p  --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39

% Computer : n012.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8046.5625MB
% OS       : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Sun Sep 27 07:53:09 AM UTC 2026

% Result   : Unsatisfiable 18.67s 5.15s
% Output   : Refutation 19.48s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    2
%            Number of leaves      :  625
% Syntax   : Number of formulae    : 1251 ( 343 unt;   0 typ;   0 def)
%            Number of atoms       : 2755 ( 812 equ;   0 cnn)
%            Maximal formula atoms :    6 (   2 avg)
%            Number of connectives : 17230 (1450   ~;1504   |;   0   &;14276   @)
%                                         (   0 <=>;   0  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   23 (   9 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    0 (   0   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  105 ( 103 usr;  21 con; 0-6 aty)
%            Number of variables   : 4048 (   0   ^;4048   !;   0   ?;4048   :)

% Comments : 
%------------------------------------------------------------------------------
thf(hBOOL_decl,type,
    hBOOL: $i > $o ).

thf(hAPP_decl,type,
    hAPP: $i > $i > $i ).

thf(c_InductTermi_OIT_decl,type,
    c_InductTermi_OIT: $i ).

thf(c_Lambda_Osubst_decl,type,
    c_Lambda_Osubst: $i ).

thf(c_List_Ofoldl_decl,type,
    c_List_Ofoldl: $i > $i > $i > $i > $i > $i ).

thf(c_Lambda_OdB_OApp_decl,type,
    c_Lambda_OdB_OApp: $i ).

thf(v_r_____decl,type,
    v_r____: $i ).

thf(v_a_____decl,type,
    v_a____: $i ).

thf(c_HOL_Ozero__class_Ozero_decl,type,
    c_HOL_Ozero__class_Ozero: $i > $i ).

thf(tc_nat_decl,type,
    tc_nat: $i ).

thf(v_as_____decl,type,
    v_as____: $i ).

thf(tc_Lambda_OdB_decl,type,
    tc_Lambda_OdB: $i ).

thf(v_u_____decl,type,
    v_u____: $i ).

thf(v_i_____decl,type,
    v_i____: $i ).

thf(c_Lambda_OdB_OVar_decl,type,
    c_Lambda_OdB_OVar: $i > $i ).

thf(class_Orderings_Olinorder_decl,type,
    class_Orderings_Olinorder: $i > $o ).

thf(c_HOL_Oord__class_Oless_decl,type,
    c_HOL_Oord__class_Oless: $i > $i > $i > $o ).

thf(class_Orderings_Opreorder_decl,type,
    class_Orderings_Opreorder: $i > $o ).

thf(c_lessequals_decl,type,
    c_lessequals: $i > $i > $i > $o ).

thf(class_Ring__and__Field_Oordered__idom_decl,type,
    class_Ring__and__Field_Oordered__idom: $i > $o ).

thf(c_List_Oappend_decl,type,
    c_List_Oappend: $i > $i > $i > $i ).

thf(c_List_Olist_ONil_decl,type,
    c_List_Olist_ONil: $i > $i ).

thf(c_List_Olist_OCons_decl,type,
    c_List_Olist_OCons: $i > $i > $i > $i ).

thf(class_Orderings_Oorder_decl,type,
    class_Orderings_Oorder: $i > $o ).

thf(class_OrderedGroup_Opordered__ab__group__add_decl,type,
    class_OrderedGroup_Opordered__ab__group__add: $i > $o ).

thf(c_HOL_Ominus__class_Ominus_decl,type,
    c_HOL_Ominus__class_Ominus: $i > $i > $i > $i ).

thf(tc_fun_decl,type,
    tc_fun: $i > $i > $i ).

thf(c_List_Omap_decl,type,
    c_List_Omap: $i > $i > $i > $i > $i ).

thf(c_Orderings_Oord__class_Omax_decl,type,
    c_Orderings_Oord__class_Omax: $i > $i > $i > $i ).

thf(class_Orderings_Obot_decl,type,
    class_Orderings_Obot: $i > $o ).

thf(c_Orderings_Obot__class_Obot_decl,type,
    c_Orderings_Obot__class_Obot: $i > $i ).

thf(c_ListOrder_Ostep1_decl,type,
    c_ListOrder_Ostep1: $i > $i > $i > $i > $o ).

thf(tc_bool_decl,type,
    tc_bool: $i ).

thf(c_List_Ofoldr_decl,type,
    c_List_Ofoldr: $i > $i > $i > $i > $i > $i ).

thf(c_List_Olistsp_decl,type,
    c_List_Olistsp: $i > $i > $i > $o ).

thf(c_Lambda_OdB_OAbs_decl,type,
    c_Lambda_OdB_OAbs: $i > $i ).

thf(class_OrderedGroup_Oab__group__add_decl,type,
    class_OrderedGroup_Oab__group__add: $i > $o ).

thf(c_Predicate_Oconversep_decl,type,
    c_Predicate_Oconversep: $i > $i > $i > $i ).

thf(c_fequal_decl,type,
    c_fequal: $i > $i ).

thf(class_Nat_Osemiring__char__0_decl,type,
    class_Nat_Osemiring__char__0: $i > $o ).

thf(c_Nat_Osemiring__1__class_Oof__nat_decl,type,
    c_Nat_Osemiring__1__class_Oof__nat: $i > $i > $i ).

thf(c_List_Oreplicate_decl,type,
    c_List_Oreplicate: $i > $i > $i > $i ).

thf(class_OrderedGroup_Osemigroup__add_decl,type,
    class_OrderedGroup_Osemigroup__add: $i > $o ).

thf(c_OrderedGroup_Osemigroup__add_decl,type,
    c_OrderedGroup_Osemigroup__add: $i > $i > $o ).

thf(c_HOL_Oplus__class_Oplus_decl,type,
    c_HOL_Oplus__class_Oplus: $i > $i ).

thf(c_List_Odrop_decl,type,
    c_List_Odrop: $i > $i > $i > $i ).

thf(c_Transitive__Closure_Ortranclp_decl,type,
    c_Transitive__Closure_Ortranclp: $i > $i > $i ).

thf(class_Ring__and__Field_Ocomm__semiring__1_decl,type,
    class_Ring__and__Field_Ocomm__semiring__1: $i > $o ).

thf(class_OrderedGroup_Ogroup__add_decl,type,
    class_OrderedGroup_Ogroup__add: $i > $o ).

thf(c_Groebner__Basis_Ogb__semiring_decl,type,
    c_Groebner__Basis_Ogb__semiring: $i > $i > $i > $i > $i > $i > $o ).

thf(c_Type_Otyping_decl,type,
    c_Type_Otyping: $i > $i > $i > $o ).

thf(v_T_____decl,type,
    v_T____: $i ).

thf(c_Type_Oshift_decl,type,
    c_Type_Oshift: $i > $i > $i > $i > $i ).

thf(tc_Type_Otype_decl,type,
    tc_Type_Otype: $i ).

thf(class_Ring__and__Field_Ozero__neq__one_decl,type,
    class_Ring__and__Field_Ozero__neq__one: $i > $o ).

thf(class_Ring__and__Field_Ono__zero__divisors_decl,type,
    class_Ring__and__Field_Ono__zero__divisors: $i > $o ).

thf(class_Ring__and__Field_Omult__zero_decl,type,
    class_Ring__and__Field_Omult__zero: $i > $o ).

thf(class_Power_Opower_decl,type,
    class_Power_Opower: $i > $o ).

thf(c_Power_Opower__class_Opower_decl,type,
    c_Power_Opower__class_Opower: $i > $i > $i > $i ).

thf(class_OrderedGroup_Opordered__comm__monoid__add_decl,type,
    class_OrderedGroup_Opordered__comm__monoid__add: $i > $o ).

thf(class_OrderedGroup_Ocancel__ab__semigroup__add_decl,type,
    class_OrderedGroup_Ocancel__ab__semigroup__add: $i > $o ).

thf(c_ATP__Linkup_Osko__Nat__Xinfinite__descent0__1__1_decl,type,
    c_ATP__Linkup_Osko__Nat__Xinfinite__descent0__1__1: $i > $i ).

thf(c_OrderedGroup_Omonoid__add_decl,type,
    c_OrderedGroup_Omonoid__add: $i > $i > $i > $o ).

thf(c_Nitpick_Onat__gcd_decl,type,
    c_Nitpick_Onat__gcd: $i > $i > $i ).

thf(class_OrderedGroup_Opordered__ab__semigroup__add__imp__le_decl,type,
    class_OrderedGroup_Opordered__ab__semigroup__add__imp__le: $i > $o ).

thf(class_Ring__and__Field_Oordered__semidom_decl,type,
    class_Ring__and__Field_Oordered__semidom: $i > $o ).

thf(class_Ring__and__Field_Oidom_decl,type,
    class_Ring__and__Field_Oidom: $i > $o ).

thf(class_Int_Onumber__ring_decl,type,
    class_Int_Onumber__ring: $i > $o ).

thf(c_Lambda_Obeta_decl,type,
    c_Lambda_Obeta: $i ).

thf(c_COMBC_decl,type,
    c_COMBC: $i > $i > $i > $i > $i > $i ).

thf(c_Lambda_Olift_decl,type,
    c_Lambda_Olift: $i ).

thf(class_OrderedGroup_Omonoid__add_decl,type,
    class_OrderedGroup_Omonoid__add: $i > $o ).

thf(class_OrderedGroup_Opordered__cancel__ab__semigroup__add_decl,type,
    class_OrderedGroup_Opordered__cancel__ab__semigroup__add: $i > $o ).

thf(c_Lambda_OdB_OdB__case_decl,type,
    c_Lambda_OdB_OdB__case: $i > $i > $i > $i > $i > $i ).

thf(class_Ring__and__Field_Osemiring__1_decl,type,
    class_Ring__and__Field_Osemiring__1: $i > $o ).

thf(c_Lambda_Osubstn_decl,type,
    c_Lambda_Osubstn: $i > $i > $i > $i ).

thf(class_OrderedGroup_Ocomm__monoid__add_decl,type,
    class_OrderedGroup_Ocomm__monoid__add: $i > $o ).

thf(c_ListApplication_Osko__ListApplication__Xex__head__tail__1__2_decl,type,
    c_ListApplication_Osko__ListApplication__Xex__head__tail__1__2: $i > $i ).

thf(c_ListApplication_Osko__ListApplication__Xex__head__tail__1__1_decl,type,
    c_ListApplication_Osko__ListApplication__Xex__head__tail__1__1: $i > $i ).

thf(v_sko__Nat__Xnat__Xinducts__1_decl,type,
    v_sko__Nat__Xnat__Xinducts__1: $i > $i ).

thf(c_List_Olistsum_decl,type,
    c_List_Olistsum: $i > $i > $i ).

thf(class_OrderedGroup_Olordered__ab__group__add_decl,type,
    class_OrderedGroup_Olordered__ab__group__add: $i > $o ).

thf(class_OrderedGroup_Oab__semigroup__add_decl,type,
    class_OrderedGroup_Oab__semigroup__add: $i > $o ).

thf(c_Wellfounded_Oaccp_decl,type,
    c_Wellfounded_Oaccp: $i > $i > $i > $o ).

thf(c_Lambda_OdB_OdB__rec_decl,type,
    c_Lambda_OdB_OdB__rec: $i > $i > $i > $i > $i > $i ).

thf(class_OrderedGroup_Opordered__ab__semigroup__add_decl,type,
    class_OrderedGroup_Opordered__ab__semigroup__add: $i > $o ).

thf(c_ListApplication_Osko__ListApplication__XApp__eq__foldl__conv__1__1_decl,type,
    c_ListApplication_Osko__ListApplication__XApp__eq__foldl__conv__1__1: $i > $i > $i > $i > $i ).

thf(c_Nat_Onat_Onat__case_decl,type,
    c_Nat_Onat_Onat__case: $i > $i > $i > $i > $i ).

thf(t_a_decl,type,
    t_a: $i ).

thf(c_Lambda_Oliftn_decl,type,
    c_Lambda_Oliftn: $i > $i > $i > $i ).

thf(class_OrderedGroup_Ocancel__semigroup__add_decl,type,
    class_OrderedGroup_Ocancel__semigroup__add: $i > $o ).

thf(v_e_____decl,type,
    v_e____: $i ).

thf(v_T_H_____decl,type,
    v_T_H____: $i ).

thf(v_sko__List__Xupt__Xinduct__1_decl,type,
    v_sko__List__Xupt__Xinduct__1: $i > $i ).

thf(class_Ring__and__Field_Oring__1_decl,type,
    class_Ring__and__Field_Oring__1: $i > $o ).

thf(c_nat_Onat__size_decl,type,
    c_nat_Onat__size: $i > $i ).

thf(v_ua_______decl,type,
    v_ua______: $i ).

thf(class_Ring__and__Field_Oring__1__no__zero__divisors_decl,type,
    class_Ring__and__Field_Oring__1__no__zero__divisors: $i > $o ).

thf(class_Ring__and__Field_Osemiring__0_decl,type,
    class_Ring__and__Field_Osemiring__0: $i > $o ).

thf(v_sko__Nat__Xnat__Xinduct__1_decl,type,
    v_sko__Nat__Xnat__Xinduct__1: $i > $i ).

thf(c_Nat_Osize__class_Osize_decl,type,
    c_Nat_Osize__class_Osize: $i > $i > $i ).

thf(tc_String_Ochar_decl,type,
    tc_String_Ochar: $i ).

thf(c_String_Ochar_Ochar__size_decl,type,
    c_String_Ochar_Ochar__size: $i > $i ).

thf(109,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__trans_0) ).

thf(892,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(168,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( v_a____ @ ( tc_Type_Otype @ ( v_T____ @ ( B @ ( C @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) )
      | ~ ( A @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL )
      | ~ ( v_T____ @ ( A @ ( C @ c_Type_Otyping ) ) )
      | ( B @ ( A @ ( v_a____ @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_SI2_0) ).

thf(1036,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( v_a____ @ ( tc_Type_Otype @ ( v_T____ @ ( B @ ( C @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) )
      | ~ ( A @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL )
      | ~ ( v_T____ @ ( A @ ( C @ c_Type_Otyping ) ) )
      | ( B @ ( A @ ( v_a____ @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).

thf(75,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( A
     != ( C @ ( A @ ( B @ c_List_Olist_OCons ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__Cons__self_0) ).

thf(809,plain,
    ! [C: $i,B: $i,A: $i] :
      ( A
     != ( C @ ( A @ ( B @ c_List_Olist_OCons ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(471,axiom,
    tc_nat @ class_OrderedGroup_Opordered__comm__monoid__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Opordered__comm__monoid__add) ).

thf(1718,plain,
    tc_nat @ class_OrderedGroup_Opordered__comm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[471]) ).

thf(329,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__add1_0) ).

thf(1411,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[329]) ).

thf(519,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__le__cancel__left_1) ).

thf(1823,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[519]) ).

thf(347,axiom,
    ! [A: $i] :
      ( ( ( A @ ( A @ c_List_Olist_ONil @ c_List_Olistsum ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_listsum_Osimps_I1_J_0) ).

thf(1454,plain,
    ! [A: $i] :
      ( ( ( A @ ( A @ c_List_Olist_ONil @ c_List_Olistsum ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[347]) ).

thf(433,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__iff__diff__less__0_0) ).

thf(1633,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[433]) ).

thf(624,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__diff__cancel_0) ).

thf(2072,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[624]) ).

thf(476,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__iff__add_1) ).

thf(1729,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[476]) ).

thf(88,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C @ ( C @ ( B @ ( D @ c_List_Oreplicate ) ) @ ( C @ ( B @ ( A @ c_List_Oreplicate ) ) @ c_List_Oappend ) ) )
      = ( C @ ( C @ ( B @ ( A @ c_List_Oreplicate ) ) @ ( C @ ( B @ ( D @ c_List_Oreplicate ) ) @ c_List_Oappend ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append__replicate__commute_0) ).

thf(846,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C @ ( C @ ( B @ ( D @ c_List_Oreplicate ) ) @ ( C @ ( B @ ( A @ c_List_Oreplicate ) ) @ c_List_Oappend ) ) )
      = ( C @ ( C @ ( B @ ( A @ c_List_Oreplicate ) ) @ ( C @ ( B @ ( D @ c_List_Oreplicate ) ) @ c_List_Oappend ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(551,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ c_COMBC ) ) ) ) @ hAPP ) )
      = ( B @ ( F @ ( A @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_COMBC__def_0) ).

thf(1899,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ c_COMBC ) ) ) ) @ hAPP ) )
      = ( B @ ( F @ ( A @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[551]) ).

thf(526,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) )
      = ( D @ ( C @ ( B @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subst__App_0) ).

thf(1840,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) )
      = ( D @ ( C @ ( B @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[526]) ).

thf(531,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
     != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Abs__App__neq__Var__apps_0) ).

thf(1850,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
     != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[531]) ).

thf(128,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__not__le_1) ).

thf(941,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(63,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__antisym__conv3_0) ).

thf(777,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(17,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = D )
      | ( ( C @ ( B @ ( A @ c_List_Oappend ) ) )
       != ( C @ ( B @ ( D @ c_List_Oappend ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append__same__eq_0) ).

thf(666,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = D )
      | ( ( C @ ( B @ ( A @ c_List_Oappend ) ) )
       != ( C @ ( B @ ( D @ c_List_Oappend ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(396,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( A @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( D @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subst__preserves__beta_0) ).

thf(1554,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( A @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( D @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[396]) ).

thf(456,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( tc_Lambda_OdB @ c_List_Olist_ONil ) )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( C @ c_Lambda_OdB_OAbs ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_apps__eq__Abs__conv_1) ).

thf(1686,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( tc_Lambda_OdB @ c_List_Olist_ONil ) )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( C @ c_Lambda_OdB_OAbs ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[456]) ).

thf(116,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( B = E )
      | ( ( C @ ( B @ ( A @ c_List_Olist_OCons ) ) )
       != ( C @ ( E @ ( D @ c_List_Olist_OCons ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_list_Oinject_1) ).

thf(909,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( B = E )
      | ( ( C @ ( B @ ( A @ c_List_Olist_OCons ) ) )
       != ( C @ ( E @ ( D @ c_List_Olist_OCons ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(2,axiom,
    ! [B: $i,A: $i] :
      ( ( A = B )
      | ( ( A @ c_Lambda_OdB_OVar )
       != ( B @ c_Lambda_OdB_OVar ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I1_J_0) ).

thf(628,plain,
    ! [B: $i,A: $i] :
      ( ( A = B )
      | ( ( A @ c_Lambda_OdB_OVar )
       != ( B @ c_Lambda_OdB_OVar ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(166,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I17_J_0) ).

thf(1031,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(189,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__iff__diff__le__0_0) ).

thf(1083,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).

thf(138,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( E @ ( D @ ( C @ ( B @ c_List_Omap ) ) ) @ ( A @ c_List_Odrop ) ) )
      = ( E @ ( D @ ( D @ ( C @ ( A @ c_List_Odrop ) ) @ ( B @ c_List_Omap ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_drop__map_0) ).

thf(962,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( E @ ( D @ ( C @ ( B @ c_List_Omap ) ) ) @ ( A @ c_List_Odrop ) ) )
      = ( E @ ( D @ ( D @ ( C @ ( A @ c_List_Odrop ) ) @ ( B @ c_List_Omap ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(574,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( B @ ( A @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ c_Lambda_OdB_OAbs @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_abs_0) ).

thf(1950,plain,
    ! [B: $i,A: $i] :
      ( ~ ( B @ ( A @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ c_Lambda_OdB_OAbs @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[574]) ).

thf(394,axiom,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_double__eq__0__iff_1) ).

thf(1549,plain,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[394]) ).

thf(533,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_right__minus__eq_1) ).

thf(1855,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[533]) ).

thf(144,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( C @ c_List_Olist_ONil ) )
      | ( A
       != ( C @ ( B @ ( A @ c_List_Oappend ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_self__append__conv_0) ).

thf(977,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( C @ c_List_Olist_ONil ) )
      | ( A
       != ( C @ ( B @ ( A @ c_List_Oappend ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(223,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( A @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( C @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_apps__preserves__beta2_0) ).

thf(1158,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( A @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( C @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).

thf(156,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__less__irrefl_0) ).

thf(1005,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(183,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
       != ( D @ ( C @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I2_J_0) ).

thf(1068,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
       != ( D @ ( C @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).

thf(151,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( C @ c_List_Olist_ONil ) )
      | ( ( C @ ( B @ ( A @ c_List_Oappend ) ) )
       != A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append__self__conv_0) ).

thf(994,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( C @ c_List_Olist_ONil ) )
      | ( ( C @ ( B @ ( A @ c_List_Oappend ) ) )
       != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).

thf(334,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__le__cancel__left_0) ).

thf(1422,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[334]) ).

thf(243,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__diff__iff_1) ).

thf(1205,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).

thf(270,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( B @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( D @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subst__preserves__beta2_H_0) ).

thf(1269,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( B @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( D @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).

thf(170,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__neg__neg_0) ).

thf(1041,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(343,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I18_J_0) ).

thf(1443,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[343]) ).

thf(513,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__diff__iff_0) ).

thf(1808,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[513]) ).

thf(47,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( D @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_eq__eqI_1) ).

thf(741,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( D @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(185,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__diff__conv_1) ).

thf(1073,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

thf(583,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( A @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( C @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_rtrancl__beta__AppL_0) ).

thf(1971,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( A @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( C @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[583]) ).

thf(506,axiom,
    ! [A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_plus__nat_Oadd__0_0) ).

thf(1793,plain,
    ! [A: $i] :
      ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[506]) ).

thf(443,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__cancel__21_1) ).

thf(1655,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[443]) ).

thf(214,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( F @ ( F @ ( F @ ( F @ c_List_Olist_ONil @ ( C @ c_List_Olist_OCons ) ) @ ( E @ c_List_Oappend ) ) @ ( F @ ( F @ ( F @ c_List_Olist_ONil @ ( B @ c_List_Olist_OCons ) ) @ ( D @ c_List_Oappend ) ) @ ( A @ c_ListOrder_Ostep1 ) ) ) )
      | ( F @ ( E @ ( D @ ( A @ c_ListOrder_Ostep1 ) ) ) )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Snoc__step1__SnocD_1) ).

thf(1137,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( F @ ( F @ ( F @ ( F @ c_List_Olist_ONil @ ( C @ c_List_Olist_OCons ) ) @ ( E @ c_List_Oappend ) ) @ ( F @ ( F @ ( F @ c_List_Olist_ONil @ ( B @ c_List_Olist_OCons ) ) @ ( D @ c_List_Oappend ) ) @ ( A @ c_ListOrder_Ostep1 ) ) ) )
      | ( F @ ( E @ ( D @ ( A @ c_ListOrder_Ostep1 ) ) ) )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).

thf(479,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( B @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__add__assoc2_0) ).

thf(1735,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( B @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[479]) ).

thf(71,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) )
       != ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) ) )
      | ~ ( A @ class_Nat_Osemiring__char__0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__eq__iff_0) ).

thf(797,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) )
       != ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) ) )
      | ~ ( A @ class_Nat_Osemiring__char__0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(590,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__ringb_Osubr0__iff_1) ).

thf(1988,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[590]) ).

thf(49,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ ( A @ c_fequal @ c_Predicate_Oconversep ) ) )
      = ( A @ c_fequal ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_conversep__eq_0) ).

thf(747,plain,
    ! [A: $i] :
      ( ( A @ ( A @ ( A @ c_fequal @ c_Predicate_Oconversep ) ) )
      = ( A @ c_fequal ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(436,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( C @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( F @ ( E @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osplit_1) ).

thf(1640,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( C @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( F @ ( E @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[436]) ).

thf(130,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( B @ c_List_Olist_ONil @ ( A @ c_List_Odrop ) ) )
      = ( B @ c_List_Olist_ONil ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_drop__Nil_0) ).

thf(945,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( B @ c_List_Olist_ONil @ ( A @ c_List_Odrop ) ) )
      = ( B @ c_List_Olist_ONil ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(576,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I16_J_0) ).

thf(1954,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[576]) ).

thf(612,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( E @ ( G @ ( F @ ( B @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I2_J_0) ).

thf(2042,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( E @ ( G @ ( F @ ( B @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[612]) ).

thf(305,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__less__le_0) ).

thf(1354,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[305]) ).

thf(368,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__le__linear_0) ).

thf(1497,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[368]) ).

thf(465,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__less__cancel__right_1) ).

thf(1705,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[465]) ).

thf(187,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( B @ ( A @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ c_Lambda_OdB_OAbs @ ( A @ c_Lambda_OdB_OAbs @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_rtrancl__beta__Abs_0) ).

thf(1078,plain,
    ! [B: $i,A: $i] :
      ( ~ ( B @ ( A @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ c_Lambda_OdB_OAbs @ ( A @ c_Lambda_OdB_OAbs @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).

thf(336,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__nonneg__eq__0__iff_1) ).

thf(1426,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[336]) ).

thf(524,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__add__left__cancel__less_1) ).

thf(1835,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[524]) ).

thf(154,axiom,
    ! [C: $i,B: $i,A: $i] :
      ~ ( B @ ( C @ ( B @ c_List_Olist_ONil @ ( A @ c_ListOrder_Ostep1 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__Nil__step1_0) ).

thf(1001,plain,
    ! [C: $i,B: $i,A: $i] :
      ~ ( B @ ( C @ ( B @ c_List_Olist_ONil @ ( A @ c_ListOrder_Ostep1 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(259,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Nat_Odiff__diff__eq_0) ).

thf(1243,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).

thf(35,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( ( C @ ( B @ ( A @ c_List_Oappend ) ) )
       != ( C @ ( D @ ( A @ c_List_Oappend ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_same__append__eq_0) ).

thf(709,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( ( C @ ( B @ ( A @ c_List_Oappend ) ) )
       != ( C @ ( D @ ( A @ c_List_Oappend ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(173,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( C @ ( B @ ( A @ c_List_Olist_OCons ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_List_Odrop ) ) )
      = ( C @ ( B @ ( A @ c_List_Olist_OCons ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_drop__Cons_H_0) ).

thf(1046,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( C @ ( B @ ( A @ c_List_Olist_OCons ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_List_Odrop ) ) )
      = ( C @ ( B @ ( A @ c_List_Olist_OCons ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).

thf(424,axiom,
    v_T____ @ ( v_u____ @ ( v_e____ @ c_Type_Otyping ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_uT_0) ).

thf(1613,plain,
    v_T____ @ ( v_u____ @ ( v_e____ @ c_Type_Otyping ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[424]) ).

thf(474,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Var__apps__eq__Var__apps__conv_0) ).

thf(1724,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[474]) ).

thf(310,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__add__inverse2_0) ).

thf(1366,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).

thf(264,axiom,
    ! [B: $i,A: $i] :
      ~ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_beta__cases_I1_J_0) ).

thf(1254,plain,
    ! [B: $i,A: $i] :
      ~ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).

thf(10,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__antisym__conv1_1) ).

thf(651,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(81,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( C @ c_List_Olist_ONil ) )
      | ( A
       != ( C @ ( A @ ( B @ c_List_Oappend ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_self__append__conv2_0) ).

thf(829,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( C @ c_List_Olist_ONil ) )
      | ( A
       != ( C @ ( A @ ( B @ c_List_Oappend ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(195,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( E @ ( G @ ( F @ ( C @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( B @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I3_J_0) ).

thf(1095,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( E @ ( G @ ( F @ ( C @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( B @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).

thf(65,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ c_List_Olist_OCons ) ) )
     != B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__Cons__self2_0) ).

thf(782,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ c_List_Olist_OCons ) ) )
     != B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(342,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__less__diff_1) ).

thf(1441,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[342]) ).

thf(598,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( v_as____ @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( v_a____ @ ( v_r____ @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( tc_Type_Otype @ ( v_T____ @ ( B @ ( C @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) )
      | ~ ( A @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL )
      | ~ ( v_T____ @ ( A @ ( C @ c_Type_Otyping ) ) )
      | ( B @ ( A @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( v_as____ @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( v_a____ @ ( v_r____ @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_SI1_0) ).

thf(2010,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( v_as____ @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( v_a____ @ ( v_r____ @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( tc_Type_Otype @ ( v_T____ @ ( B @ ( C @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) )
      | ~ ( A @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL )
      | ~ ( v_T____ @ ( A @ ( C @ c_Type_Otyping ) ) )
      | ( B @ ( A @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( v_as____ @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( v_a____ @ ( v_r____ @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[598]) ).

thf(567,axiom,
    tc_nat @ class_Ring__and__Field_Omult__zero,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Omult__zero) ).

thf(1935,plain,
    tc_nat @ class_Ring__and__Field_Omult__zero,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[567]) ).

thf(440,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__right__mono_0) ).

thf(1650,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[440]) ).

thf(438,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__nonpos__nonpos_0) ).

thf(1645,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[438]) ).

thf(560,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( A @ v_sko__Nat__Xnat__Xinduct__1 @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat_Oinduct_0) ).

thf(1921,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( A @ v_sko__Nat__Xnat__Xinduct__1 @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[560]) ).

thf(622,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( E @ ( E @ ( E @ c_List_Olist_ONil @ ( C @ c_List_Olist_OCons ) ) @ ( F @ c_List_Oappend ) ) @ ( E @ ( E @ ( E @ c_List_Olist_ONil @ ( B @ c_List_Olist_OCons ) ) @ ( D @ c_List_Oappend ) ) @ ( A @ c_ListOrder_Ostep1 ) ) ) )
      | ( B = C )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Snoc__step1__SnocD_3) ).

thf(2067,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( E @ ( E @ ( E @ c_List_Olist_ONil @ ( C @ c_List_Olist_OCons ) ) @ ( F @ c_List_Oappend ) ) @ ( E @ ( E @ ( E @ c_List_Olist_ONil @ ( B @ c_List_Olist_OCons ) ) @ ( D @ c_List_Oappend ) ) @ ( A @ c_ListOrder_Ostep1 ) ) ) )
      | ( B = C )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[622]) ).

thf(385,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__iff__diff__le__0_1) ).

thf(1529,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[385]) ).

thf(431,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__le__double__add__iff__zero__le__single__add_1) ).

thf(1628,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[431]) ).

thf(611,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_ab__semigroup__add__class_Oadd__ac_I1_J_0) ).

thf(2039,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[611]) ).

thf(490,axiom,
    ! [A: $i] :
      ( ( A @ c_Lambda_OdB_OVar )
      = ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ c_List_Olist_ONil @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Var__eq__apps__conv_2) ).

thf(1760,plain,
    ! [A: $i] :
      ( ( A @ c_Lambda_OdB_OVar )
      = ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ c_List_Olist_ONil @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[490]) ).

thf(66,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__less__trans_0) ).

thf(785,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(555,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Power_Opower__class_Opower ) ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__eq__0__iff_1) ).

thf(1907,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ c_Power_Opower__class_Opower ) ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[555]) ).

thf(292,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( D @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__le__mono_0) ).

thf(1323,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( D @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).

thf(132,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( E @ ( C @ ( A @ c_ListOrder_Ostep1 ) ) ) )
      | ( D @ ( D @ ( E @ ( B @ c_List_Olist_OCons ) ) @ ( D @ ( C @ ( B @ c_List_Olist_OCons ) ) @ ( A @ c_ListOrder_Ostep1 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Cons__step1__Cons_5) ).

thf(949,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( E @ ( C @ ( A @ c_ListOrder_Ostep1 ) ) ) )
      | ( D @ ( D @ ( E @ ( B @ c_List_Olist_OCons ) ) @ ( D @ ( C @ ( B @ c_List_Olist_OCons ) ) @ ( A @ c_ListOrder_Ostep1 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(328,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( D @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( C @ ( B @ ( B @ ( B @ ( A @ c_Predicate_Oconversep ) ) @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_rtranclp__converseI_0) ).

thf(1409,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( D @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( C @ ( B @ ( B @ ( B @ ( A @ c_Predicate_Oconversep ) ) @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[328]) ).

thf(41,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__antisym__conv_0) ).

thf(724,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(295,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) )
        = ( A @ ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Osemigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_foldl__assoc_0) ).

thf(1330,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) )
        = ( A @ ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Osemigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).

thf(535,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__mono_0) ).

thf(1860,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[535]) ).

thf(268,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__strict__right__mono_0) ).

thf(1264,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).

thf(32,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( D @ ( E @ ( D @ ( C @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Ofoldr ) ) ) ) )
      = ( F @ ( D @ ( F @ ( D @ ( E @ ( C @ ( A @ c_List_Ofoldr ) ) ) ) @ ( B @ ( A @ c_List_Ofoldr ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_foldr__append_0) ).

thf(702,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( D @ ( E @ ( D @ ( C @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Ofoldr ) ) ) ) )
      = ( F @ ( D @ ( F @ ( D @ ( E @ ( C @ ( A @ c_List_Ofoldr ) ) ) ) @ ( B @ ( A @ c_List_Ofoldr ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(51,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__less__le_1) ).

thf(751,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(288,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D
        = ( tc_Lambda_OdB @ c_List_Olist_ONil ) )
      | ( A
        = ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ ( B @ ( A @ c_ListApplication_Osko__ListApplication__XApp__eq__foldl__conv__1__1 ) ) ) @ ( C @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) )
      | ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_App__eq__foldl__conv_2) ).

thf(1313,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D
        = ( tc_Lambda_OdB @ c_List_Olist_ONil ) )
      | ( A
        = ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ ( B @ ( A @ c_ListApplication_Osko__ListApplication__XApp__eq__foldl__conv__1__1 ) ) ) @ ( C @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) )
      | ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).

thf(274,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( E @ ( D @ ( C @ ( B @ ( A @ c_Predicate_Oconversep ) ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_conversep_Ointros_0) ).

thf(1279,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( E @ ( D @ ( C @ ( B @ ( A @ c_Predicate_Oconversep ) ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).

thf(523,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocancel__ab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__imp__eq_0) ).

thf(1832,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocancel__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[523]) ).

thf(125,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__le__less_1) ).

thf(934,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(392,axiom,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_nat_Onat__size )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat_Osize_I1_J_0) ).

thf(1545,plain,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_nat_Onat__size )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[392]) ).

thf(77,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_max__less__iff__conj_2) ).

thf(817,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(509,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__iff__diff__less__0_1) ).

thf(1800,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[509]) ).

thf(315,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_trans__le__add1_0) ).

thf(1379,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[315]) ).

thf(191,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__pos__nonneg_0) ).

thf(1087,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(290,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__less__diff_0) ).

thf(1319,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).

thf(422,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( A @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( C @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_lift__preserves__beta_0) ).

thf(1609,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( A @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( C @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[422]) ).

thf(141,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__less__asym_H_0) ).

thf(969,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(404,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__diff__iff_1) ).

thf(1568,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[404]) ).

thf(604,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( D @ ( C @ ( B @ c_List_Odrop ) ) @ ( A @ c_List_Odrop ) ) )
      = ( D @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_List_Odrop ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_drop__drop_0) ).

thf(2024,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( D @ ( C @ ( B @ c_List_Odrop ) ) @ ( A @ c_List_Odrop ) ) )
      = ( D @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_List_Odrop ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[604]) ).

thf(366,axiom,
    tc_nat @ class_Nat_Osemiring__char__0,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Nat_Osemiring__char__0) ).

thf(1494,plain,
    tc_nat @ class_Nat_Osemiring__char__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[366]) ).

thf(579,axiom,
    tc_nat @ class_OrderedGroup_Oab__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Oab__semigroup__add) ).

thf(1963,plain,
    tc_nat @ class_OrderedGroup_Oab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[579]) ).

thf(499,axiom,
    ! [C: $i,B: $i,A: $i] : ( C @ ( C @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_rtranclp_Ortrancl__refl_0) ).

thf(1778,plain,
    ! [C: $i,B: $i,A: $i] : ( C @ ( C @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[499]) ).

thf(254,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( A @ v_sko__Nat__Xnat__Xinducts__1 @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat_Oinducts_0) ).

thf(1232,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( A @ v_sko__Nat__Xnat__Xinducts__1 @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).

thf(5,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C = B )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__neqE__ordered__idom_0) ).

thf(638,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C = B )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(91,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( C @ c_List_Olist_ONil ) )
      | ( ( C @ ( B @ ( A @ c_List_Oappend ) ) )
       != ( C @ c_List_Olist_ONil ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append__is__Nil__conv_1) ).

thf(853,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( C @ c_List_Olist_ONil ) )
      | ( ( C @ ( B @ ( A @ c_List_Oappend ) ) )
       != ( C @ c_List_Olist_ONil ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(597,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I23_J_0) ).

thf(2007,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[597]) ).

thf(363,axiom,
    ! [A: $i] :
      ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ c_List_Olist_ONil @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
      = ( A @ c_Lambda_OdB_OAbs ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_apps__eq__Abs__conv_2) ).

thf(1489,plain,
    ! [A: $i] :
      ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ c_List_Olist_ONil @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
      = ( A @ c_Lambda_OdB_OAbs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[363]) ).

thf(121,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__max__iff__disj_1) ).

thf(924,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(352,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__diff__conv_0) ).

thf(1465,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[352]) ).

thf(202,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_minus__nat_Odiff__0_0) ).

thf(1109,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(8,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ c_List_Olist_ONil )
     != ( A @ ( C @ ( B @ c_List_Olist_OCons ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_list_Osimps_I2_J_0) ).

thf(646,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ c_List_Olist_ONil )
     != ( A @ ( C @ ( B @ c_List_Olist_OCons ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(534,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__add__assoc_0) ).

thf(1858,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[534]) ).

thf(220,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( D @ ( B @ ( E @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( A @ c_Type_Otyping ) ) )
      | ~ ( F @ ( E @ ( tc_Type_Otype @ ( C @ ( D @ ( A @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) )
      | ~ ( C @ ( B @ ( A @ c_Type_Otyping ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subst__lemma_0) ).

thf(1151,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( D @ ( B @ ( E @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( A @ c_Type_Otyping ) ) )
      | ~ ( F @ ( E @ ( tc_Type_Otype @ ( C @ ( D @ ( A @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) )
      | ~ ( C @ ( B @ ( A @ c_Type_Otyping ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).

thf(608,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__nonneg__nonneg_0) ).

thf(2034,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[608]) ).

thf(409,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL )
      | ( B @ c_Lambda_OdB_OVar @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_app__Var__IT_0) ).

thf(1581,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL )
      | ( B @ c_Lambda_OdB_OVar @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[409]) ).

thf(276,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I22_J_0) ).

thf(1283,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).

thf(175,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( D @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__add__eq__less_0) ).

thf(1051,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( D @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).

thf(283,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__le__imp__of__nat_0) ).

thf(1302,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).

thf(54,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_Orderings_Oord__class_Omax ) ) )
        = ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Osup__assoc_0) ).

thf(756,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_Orderings_Oord__class_Omax ) ) )
        = ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(446,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__strict__left__mono_0) ).

thf(1664,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[446]) ).

thf(454,axiom,
    tc_nat @ class_OrderedGroup_Opordered__ab__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Opordered__ab__semigroup__add) ).

thf(1682,plain,
    tc_nat @ class_OrderedGroup_Opordered__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[454]) ).

thf(83,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ c_List_Olist_OCons ) ) )
     != ( C @ c_List_Olist_ONil ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_neq__Nil__conv_1) ).

thf(834,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ c_List_Olist_OCons ) ) )
     != ( C @ c_List_Olist_ONil ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(521,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__diff__conv_0) ).

thf(1828,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[521]) ).

thf(586,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Omul__c_0) ).

thf(1978,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[586]) ).

thf(317,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__imp__diff__less_0) ).

thf(1383,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[317]) ).

thf(162,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) )
        = B )
      | ( tc_nat @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subst__Var_1) ).

thf(1020,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) )
        = B )
      | ( tc_nat @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).

thf(578,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_comm__monoid__add_Omult__commute_0) ).

thf(1960,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[578]) ).

thf(74,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( C = B )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__less__linear_0) ).

thf(806,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( C = B )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(228,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I25_J_0) ).

thf(1169,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(564,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_double__add__le__zero__iff__single__add__le__zero_0) ).

thf(1928,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[564]) ).

thf(247,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Nat_Osemiring__1__class_Oof__nat ) )
        = ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__add_0) ).

thf(1215,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Nat_Osemiring__1__class_Oof__nat ) )
        = ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).

thf(45,axiom,
    ! [B: $i,A: $i] :
      ( ( A = B )
      | ( ( A @ c_Lambda_OdB_OAbs )
       != ( B @ c_Lambda_OdB_OAbs ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I3_J_0) ).

thf(734,plain,
    ! [B: $i,A: $i] :
      ( ( A = B )
      | ( ( A @ c_Lambda_OdB_OAbs )
       != ( B @ c_Lambda_OdB_OAbs ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(135,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_xt1_I9_J_0) ).

thf(956,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(546,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__less__mono2_0) ).

thf(1886,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[546]) ).

thf(428,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__less__cancel__right_0) ).

thf(1622,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[428]) ).

thf(383,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_Orderings_Oord__class_Omax ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_max__0R_0) ).

thf(1526,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_Orderings_Oord__class_Omax ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[383]) ).

thf(14,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ ( D @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__sup__iff_1) ).

thf(660,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ ( D @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(301,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__ringb_Oadd__r0__iff_1) ).

thf(1345,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).

thf(339,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_double__add__le__zero__iff__single__add__le__zero_1) ).

thf(1433,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[339]) ).

thf(326,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ ( D @ ( C @ ( B @ c_List_Olist_OCons ) ) @ ( A @ c_List_Omap ) ) ) )
      = ( E @ ( E @ ( D @ ( C @ ( A @ c_List_Omap ) ) ) @ ( B @ ( A @ hAPP ) @ c_List_Olist_OCons ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_map__eq__Cons__conv_3) ).

thf(1405,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ ( D @ ( C @ ( B @ c_List_Olist_OCons ) ) @ ( A @ c_List_Omap ) ) ) )
      = ( E @ ( E @ ( D @ ( C @ ( A @ c_List_Omap ) ) ) @ ( B @ ( A @ hAPP ) @ c_List_Olist_OCons ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[326]) ).

thf(417,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( D @ ( A @ c_Type_Otyping ) ) )
      | ~ ( B @ ( D @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ c_Type_Otyping ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subject__reduction_0) ).

thf(1600,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( D @ ( A @ c_Type_Otyping ) ) )
      | ~ ( B @ ( D @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ c_Type_Otyping ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[417]) ).

thf(550,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower__class_Opower ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Osemiring__0 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__0__left_1) ).

thf(1896,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower__class_Opower ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Osemiring__0 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[550]) ).

thf(102,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__iff__sup_0) ).

thf(877,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(200,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) )
        = ( A @ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldr ) ) ) ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_foldl__foldr1_0) ).

thf(1104,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) )
        = ( A @ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldr ) ) ) ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).

thf(461,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__add__left__commute_0) ).

thf(1697,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[461]) ).

thf(585,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( tc_Lambda_OdB @ c_List_Olist_ONil ) )
      | ( ( A @ c_Lambda_OdB_OVar )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( B @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Var__eq__apps__conv_1) ).

thf(1975,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( tc_Lambda_OdB @ c_List_Olist_ONil ) )
      | ( ( A @ c_Lambda_OdB_OVar )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( B @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[585]) ).

thf(401,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__le__self_0) ).

thf(1564,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[401]) ).

thf(419,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__nonneg__pos_0) ).

thf(1604,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[419]) ).

thf(458,axiom,
    ! [A: $i] :
      ( ~ ( A @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL )
      | ( tc_Lambda_OdB @ ( A @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Predicate_Oconversep ) ) @ c_Wellfounded_Oaccp ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_IT__implies__termi_0) ).

thf(1691,plain,
    ! [A: $i] :
      ( ~ ( A @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL )
      | ( tc_Lambda_OdB @ ( A @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Predicate_Oconversep ) ) @ c_Wellfounded_Oaccp ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[458]) ).

thf(341,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_monoid__add__class_Oadd__0__right_0) ).

thf(1438,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[341]) ).

thf(69,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__sup__iff_0) ).

thf(793,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(241,axiom,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I14_J_0) ).

thf(1199,plain,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(298,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__less__mono_0) ).

thf(1338,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( D @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).

thf(614,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A
        = ( C @ c_Lambda_OdB_OAbs ) )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( C @ c_Lambda_OdB_OAbs ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_apps__eq__Abs__conv_0) ).

thf(2047,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A
        = ( C @ c_Lambda_OdB_OAbs ) )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( C @ c_Lambda_OdB_OAbs ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[614]) ).

thf(558,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I24_J_0) ).

thf(1915,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[558]) ).

thf(502,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( E @ ( D @ ( C @ ( B @ ( A @ c_Predicate_Oconversep ) ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_conversep__iff_1) ).

thf(1783,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( E @ ( D @ ( C @ ( B @ ( A @ c_Predicate_Oconversep ) ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[502]) ).

thf(287,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Orderings_Oord__class_Omax ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_max__0L_0) ).

thf(1311,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Orderings_Oord__class_Omax ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).

thf(182,axiom,
    tc_nat @ class_Ring__and__Field_Oordered__semidom,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Oordered__semidom) ).

thf(1067,plain,
    tc_nat @ class_Ring__and__Field_Oordered__semidom,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).

thf(477,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( A @ c_ATP__Linkup_Osko__Nat__Xinfinite__descent0__1__1 @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_infinite__descent0_1) ).

thf(1730,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( A @ c_ATP__Linkup_Osko__Nat__Xinfinite__descent0__1__1 @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[477]) ).

thf(221,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__commute_0) ).

thf(1153,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( B @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).

thf(56,axiom,
    ! [B: $i,A: $i] :
      ( A
      = ( B @ ( A @ ( B @ c_List_Olist_ONil @ c_List_Oappend ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_self__append__conv2_1) ).

thf(761,plain,
    ! [B: $i,A: $i] :
      ( A
      = ( B @ ( A @ ( B @ c_List_Olist_ONil @ c_List_Oappend ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(494,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ c_Lambda_Osubstn ) ) )
      = ( D @ ( C @ ( B @ c_Lambda_Osubstn ) ) @ ( D @ ( C @ ( A @ c_Lambda_Osubstn ) ) @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_substn_Osimps_I2_J_0) ).

thf(1768,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ c_Lambda_Osubstn ) ) )
      = ( D @ ( C @ ( B @ c_Lambda_Osubstn ) ) @ ( D @ ( C @ ( A @ c_Lambda_Osubstn ) ) @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[494]) ).

thf(36,axiom,
    ! [B: $i,A: $i] :
      ( A
      = ( B @ ( B @ c_List_Olist_ONil @ ( A @ c_List_Oappend ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_self__append__conv_1) ).

thf(714,plain,
    ! [B: $i,A: $i] :
      ( A
      = ( B @ ( B @ c_List_Olist_ONil @ ( A @ c_List_Oappend ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(538,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( B @ ( D @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( C @ ( D @ ( A @ c_Type_Otyping ) ) )
      | ( C @ ( B @ ( A @ c_Type_Otyping ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subject__reduction_H_0) ).

thf(1868,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( B @ ( D @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( C @ ( D @ ( A @ c_Type_Otyping ) ) )
      | ( C @ ( B @ ( A @ c_Type_Otyping ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[538]) ).

thf(272,axiom,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_lessequals ) ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__0__eq_0) ).

thf(1274,plain,
    ! [A: $i] :
      ( ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_lessequals ) ) )
      | ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).

thf(319,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( A @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( D @ ( B @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( C @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_rtrancl__beta__App_0) ).

thf(1387,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( A @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( D @ ( B @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( C @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[319]) ).

thf(110,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Osup__left__idem_0) ).

thf(894,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(193,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__le__iff_1) ).

thf(1091,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).

thf(300,axiom,
    ! [A: $i] :
      ( ~ ( A @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL )
      | ( A @ c_Lambda_OdB_OAbs @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Lambda_0) ).

thf(1343,plain,
    ! [A: $i] :
      ( ~ ( A @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL )
      | ( A @ c_Lambda_OdB_OAbs @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).

thf(371,axiom,
    tc_nat @ class_Orderings_Opreorder,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Orderings_Opreorder) ).

thf(1502,plain,
    tc_nat @ class_Orderings_Opreorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[371]) ).

thf(447,axiom,
    v_ua______ @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Beta_I6_J_0) ).

thf(1666,plain,
    v_ua______ @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[447]) ).

thf(362,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__less__double__add__iff__zero__less__single__add_1) ).

thf(1487,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[362]) ).

thf(27,axiom,
    ! [C: $i,B: $i,A: $i] :
      ~ ( C @ ( C @ c_List_Olist_ONil @ ( B @ ( A @ c_ListOrder_Ostep1 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__step1__Nil_0) ).

thf(693,plain,
    ! [C: $i,B: $i,A: $i] :
      ~ ( C @ ( C @ c_List_Olist_ONil @ ( B @ ( A @ c_ListOrder_Ostep1 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(607,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__less__iff_1) ).

thf(2032,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[607]) ).

thf(43,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( F @ ( F @ ( C @ ( A @ c_List_Olist_OCons ) ) @ ( F @ ( D @ ( B @ c_List_Olist_OCons ) ) @ ( E @ c_ListOrder_Ostep1 ) ) ) )
      | ( C = D )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Cons__step1__Cons_2) ).

thf(729,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( F @ ( F @ ( C @ ( A @ c_List_Olist_OCons ) ) @ ( F @ ( D @ ( B @ c_List_Olist_OCons ) ) @ ( E @ c_ListOrder_Ostep1 ) ) ) )
      | ( C = D )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(257,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__diff__iff_0) ).

thf(1238,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).

thf(208,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( B @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Omul__1_0) ).

thf(1123,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( B @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).

thf(95,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ c_List_Olist_OCons ) ) )
      = ( C @ ( C @ ( B @ ( A @ c_List_Olist_OCons ) ) @ ( C @ c_List_Olist_ONil @ c_List_Oappend ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Cons__eq__append__conv_4) ).

thf(862,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ c_List_Olist_OCons ) ) )
      = ( C @ ( C @ ( B @ ( A @ c_List_Olist_OCons ) ) @ ( C @ c_List_Olist_ONil @ c_List_Oappend ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(405,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Oadd__c_0) ).

thf(1570,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[405]) ).

thf(230,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__nonneg__eq__0__iff_0) ).

thf(1174,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).

thf(545,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
        = C )
      | ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ c_List_Olist_ONil @ ( C @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_App__eq__foldl__conv_0) ).

thf(1883,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
        = C )
      | ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ c_List_Olist_ONil @ ( C @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[545]) ).

thf(399,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__cancel_0) ).

thf(1560,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[399]) ).

thf(118,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = B )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Osup__absorb1_0) ).

thf(916,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = B )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(569,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( E @ ( D @ ( C @ ( A @ c_List_Omap ) ) ) @ ( B @ ( A @ hAPP ) @ c_List_Olist_OCons ) ) )
      = ( E @ ( D @ ( D @ ( C @ ( B @ c_List_Olist_OCons ) ) @ ( A @ c_List_Omap ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Cons__eq__map__conv_3) ).

thf(1938,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( E @ ( D @ ( C @ ( A @ c_List_Omap ) ) ) @ ( B @ ( A @ hAPP ) @ c_List_Olist_OCons ) ) )
      = ( E @ ( D @ ( D @ ( C @ ( B @ c_List_Olist_OCons ) ) @ ( A @ c_List_Omap ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[569]) ).

thf(232,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Lambda_Osubstn ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_substn__subst__0_0) ).

thf(1178,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Lambda_Osubstn ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(547,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( E @ c_HOL_Oord__class_Oless ) ) )
      | ( ( E @ ( D @ ( C @ ( B @ ( A @ c_Type_Oshift ) ) ) @ hAPP ) )
        = ( E @ ( A @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_shift__gt_0) ).

thf(1888,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( E @ c_HOL_Oord__class_Oless ) ) )
      | ( ( E @ ( D @ ( C @ ( B @ ( A @ c_Type_Oshift ) ) ) @ hAPP ) )
        = ( E @ ( A @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[547]) ).

thf(348,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ c_Lambda_Osubstn ) ) )
      = ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( C @ c_Lambda_Oliftn ) ) @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_substn__subst__n_0) ).

thf(1457,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ c_Lambda_Osubstn ) ) )
      = ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( C @ c_Lambda_Oliftn ) ) @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[348]) ).

thf(429,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_even__less__0__iff_1) ).

thf(1624,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[429]) ).

thf(504,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__add__assoc_0) ).

thf(1787,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[504]) ).

thf(245,axiom,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I15_J_0) ).

thf(1210,plain,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).

thf(210,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ c_Lambda_OdB_OVar @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__case ) ) ) ) )
      = ( D @ ( A @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I10_J_0) ).

thf(1128,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ c_Lambda_OdB_OVar @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__case ) ) ) ) )
      = ( D @ ( A @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(275,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__less__0__iff_0) ).

thf(1281,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).

thf(321,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( B @ ( A @ ( C @ c_fequal @ hAPP ) @ hAPP ) @ hBOOL )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_ATP__Linkup_Ofequal__imp__equal_0) ).

thf(1392,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( B @ ( A @ ( C @ c_fequal @ hAPP ) @ hAPP ) @ hBOOL )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[321]) ).

thf(279,axiom,
    ! [A: $i] :
      ( ~ ( tc_Lambda_OdB @ ( A @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Predicate_Oconversep ) ) @ c_Wellfounded_Oaccp ) ) )
      | ( A @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_termi__implies__IT_0) ).

thf(1291,plain,
    ! [A: $i] :
      ( ~ ( tc_Lambda_OdB @ ( A @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Predicate_Oconversep ) ) @ c_Wellfounded_Oaccp ) ) )
      | ( A @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).

thf(159,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) )
       != C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__iff__sup_1) ).

thf(1012,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) )
       != C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(525,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( B @ c_Power_Opower__class_Opower ) ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oring__1__no__zero__divisors ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_field__power__not__zero_0) ).

thf(1837,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( B @ c_Power_Opower__class_Opower ) ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Oring__1__no__zero__divisors ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[525]) ).

thf(418,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ c_Nat_Onat_Onat__case ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat_Osplit_0) ).

thf(1602,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ c_Nat_Onat_Onat__case ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[418]) ).

thf(239,axiom,
    ! [A: $i] :
      ( A
      = ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( A @ c_ListApplication_Osko__ListApplication__Xex__head__tail__1__1 @ ( A @ c_ListApplication_Osko__ListApplication__Xex__head__tail__1__2 @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_ex__head__tail_0) ).

thf(1195,plain,
    ! [A: $i] :
      ( A
      = ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( A @ c_ListApplication_Osko__ListApplication__Xex__head__tail__1__1 @ ( A @ c_ListApplication_Osko__ListApplication__Xex__head__tail__1__2 @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(327,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__0__le__iff_0) ).

thf(1407,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[327]) ).

thf(481,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL )
      | ~ ( B @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL )
      | ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( B @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_IT_OBeta_0) ).

thf(1740,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL )
      | ~ ( B @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL )
      | ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( B @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[481]) ).

thf(556,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( C @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I12_J_0) ).

thf(1910,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( C @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[556]) ).

thf(86,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Oless__supI2_0) ).

thf(842,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(406,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( D @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) )
      = ( F @ ( E @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) @ ( F @ ( D @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) @ ( E @ ( D @ ( B @ hAPP ) @ hAPP ) @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Orecs_I2_J_0) ).

thf(1573,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( D @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) )
      = ( F @ ( E @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) @ ( F @ ( D @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) @ ( E @ ( D @ ( B @ hAPP ) @ hAPP ) @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[406]) ).

thf(143,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ c_List_Olist_OCons ) ) )
     != ( C @ c_List_Olist_ONil ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_list_Osimps_I3_J_0) ).

thf(974,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ c_List_Olist_OCons ) ) )
     != ( C @ c_List_Olist_ONil ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

thf(414,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__add__right__cancel_0) ).

thf(1592,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[414]) ).

thf(588,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_eq__iff__diff__eq__0_0) ).

thf(1983,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[588]) ).

thf(491,axiom,
    ! [B: $i,A: $i] : ( B @ ( B @ ( A @ c_fequal @ hAPP ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_ATP__Linkup_Oequal__imp__fequal_0) ).

thf(1762,plain,
    ! [B: $i,A: $i] : ( B @ ( B @ ( A @ c_fequal @ hAPP ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[491]) ).

thf(388,axiom,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( F @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I20_J_0) ).

thf(1534,plain,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( F @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[388]) ).

thf(294,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ ( D @ ( C @ ( B @ c_List_Olist_OCons ) ) @ ( A @ c_List_Omap ) ) ) )
      = ( E @ ( E @ ( D @ ( C @ ( A @ c_List_Omap ) ) ) @ ( B @ ( A @ hAPP ) @ c_List_Olist_OCons ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_map_Osimps_I2_J_0) ).

thf(1328,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ ( D @ ( C @ ( B @ c_List_Olist_OCons ) ) @ ( A @ c_List_Omap ) ) ) )
      = ( E @ ( E @ ( D @ ( C @ ( A @ c_List_Omap ) ) ) @ ( B @ ( A @ hAPP ) @ c_List_Olist_OCons ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).

thf(181,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__diff__inverse_0) ).

thf(1064,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).

thf(450,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Oadd__a_0) ).

thf(1672,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[450]) ).

thf(390,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__add__left__cancel_0) ).

thf(1540,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[390]) ).

thf(543,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( C @ ( B @ c_Lambda_OdB_OVar @ ( A @ c_Lambda_Oliftn ) ) )
        = ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Lambda_OdB_OVar ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_liftn_Osimps_I1_J_1) ).

thf(1878,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( C @ ( B @ c_Lambda_OdB_OVar @ ( A @ c_Lambda_Oliftn ) ) )
        = ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Lambda_OdB_OVar ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[543]) ).

thf(472,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
     != ( C @ c_Lambda_OdB_OVar ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I5_J_0) ).

thf(1719,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
     != ( C @ c_Lambda_OdB_OVar ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[472]) ).

thf(99,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ ( D @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__supE_1) ).

thf(869,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ ( D @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(488,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ ( B @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( A @ c_Lambda_Oliftn ) ) )
      = ( D @ ( C @ ( A @ c_Lambda_Oliftn ) ) @ ( D @ ( B @ ( A @ c_Lambda_Oliftn ) ) @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_liftn_Osimps_I2_J_0) ).

thf(1755,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ ( B @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( A @ c_Lambda_Oliftn ) ) )
      = ( D @ ( C @ ( A @ c_Lambda_Oliftn ) ) @ ( D @ ( B @ ( A @ c_Lambda_Oliftn ) ) @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[488]) ).

thf(149,axiom,
    ! [A: $i] :
      ( ( A @ c_List_Olist_ONil )
      = ( A @ ( A @ c_List_Olist_ONil @ ( A @ c_List_Olist_ONil @ c_List_Oappend ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Nil__is__append__conv_2) ).

thf(989,plain,
    ! [A: $i] :
      ( ( A @ c_List_Olist_ONil )
      = ( A @ ( A @ c_List_Olist_ONil @ ( A @ c_List_Olist_ONil @ c_List_Oappend ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(323,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( D @ ( F @ ( E @ ( C @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( B @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( D @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I4_J_0) ).

thf(1397,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( D @ ( F @ ( E @ ( C @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( B @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( D @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[323]) ).

thf(308,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I21_J_0) ).

thf(1360,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).

thf(303,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_List_Oreplicate ) ) )
      = ( B @ c_List_Olist_ONil ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_replicate__empty_1) ).

thf(1350,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_List_Oreplicate ) ) )
      = ( B @ c_List_Olist_ONil ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[303]) ).

thf(568,axiom,
    ! [A: $i] :
      ( ( tc_String_Ochar @ ( A @ c_Nat_Osize__class_Osize ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_size__char_0) ).

thf(1936,plain,
    ! [A: $i] :
      ( ( tc_String_Ochar @ ( A @ c_Nat_Osize__class_Osize ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[568]) ).

thf(29,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( B @ class_Orderings_Obot )
      | ( B @ ( A @ tc_fun ) @ class_Orderings_Obot ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_fun__Orderings_Obot) ).

thf(697,plain,
    ! [B: $i,A: $i] :
      ( ~ ( B @ class_Orderings_Obot )
      | ( B @ ( A @ tc_fun ) @ class_Orderings_Obot ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(192,axiom,
    ! [B: $i,A: $i] :
      ~ ( tc_nat @ ( B @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__add__less2_0) ).

thf(1089,plain,
    ! [B: $i,A: $i] :
      ~ ( tc_nat @ ( B @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(255,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_List_Oreplicate ) ) )
      = ( B @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_List_Oreplicate ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_replicate__eq__replicate_2) ).

thf(1234,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_List_Oreplicate ) ) )
      = ( B @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_List_Oreplicate ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).

thf(101,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__antisym__conv1_0) ).

thf(874,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(217,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__add__diff__inverse_0) ).

thf(1143,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).

thf(496,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_double__add__less__zero__iff__single__less__zero_0) ).

thf(1772,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[496]) ).

thf(266,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldr ) ) ) ) )
        = ( A @ ( A @ ( B @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_foldr__conv__foldl_0) ).

thf(1259,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldr ) ) ) ) )
        = ( A @ ( A @ ( B @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).

thf(203,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__add__diff__inverse2_0) ).

thf(1111,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(251,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__increasing2_0) ).

thf(1227,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).

thf(145,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__leE_0) ).

thf(980,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(412,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( B @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__diff__split__asm_1) ).

thf(1587,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( B @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[412]) ).

thf(403,axiom,
    tc_nat @ class_OrderedGroup_Ocancel__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Ocancel__semigroup__add) ).

thf(1567,plain,
    tc_nat @ class_OrderedGroup_Ocancel__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[403]) ).

thf(105,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__max__iff__disj_1) ).

thf(884,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(224,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( A @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( D @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subst__preserves__beta_H_0) ).

thf(1160,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( A @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( D @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).

thf(52,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( B @ ( C @ ( B @ c_List_Olist_ONil @ ( A @ c_List_Ofoldr ) ) ) ) )
      = C ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_foldr_Osimps_I1_J_0) ).

thf(753,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( B @ ( C @ ( B @ c_List_Olist_ONil @ ( A @ c_List_Ofoldr ) ) ) ) )
      = C ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(234,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Oadd__c_0) ).

thf(1182,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).

thf(350,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__add__left__cancel__le_1) ).

thf(1461,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[350]) ).

thf(335,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__leE_1) ).

thf(1424,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[335]) ).

thf(411,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( A
       != ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_natgb_Oadd__r0__iff_0) ).

thf(1584,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( A
       != ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[411]) ).

thf(514,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ ( D @ ( C @ ( B @ c_List_Oreplicate ) ) @ ( A @ c_List_Omap ) ) ) )
      = ( E @ ( C @ ( A @ hAPP ) @ ( B @ c_List_Oreplicate ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_map__replicate_0) ).

thf(1810,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ ( D @ ( C @ ( B @ c_List_Oreplicate ) ) @ ( A @ c_List_Omap ) ) ) )
      = ( E @ ( C @ ( A @ hAPP ) @ ( B @ c_List_Oreplicate ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[514]) ).

thf(344,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__add__commute_0) ).

thf(1446,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[344]) ).

thf(244,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) )
        = ( A @ c_Lambda_OdB_OVar ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_lift_Osimps_I1_J_0) ).

thf(1207,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) )
        = ( A @ c_Lambda_OdB_OVar ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).

thf(21,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__not__less_1) ).

thf(679,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(333,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_comm__monoid__add_Omult__1_0) ).

thf(1419,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[333]) ).

thf(365,axiom,
    v_T_H____ @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( v_as____ @ ( v_a____ @ ( v_r____ @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( tc_Type_Otype @ ( v_T____ @ ( v_i____ @ ( v_e____ @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_T_0) ).

thf(1493,plain,
    v_T_H____ @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( v_as____ @ ( v_a____ @ ( v_r____ @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( tc_Type_Otype @ ( v_T____ @ ( v_i____ @ ( v_e____ @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[365]) ).

thf(455,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = E )
      | ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( E @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__cancel__21_0) ).

thf(1683,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = E )
      | ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( E @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[455]) ).

thf(580,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ ( B @ c_Power_Opower__class_Opower ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__less__power_0) ).

thf(1964,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ ( B @ c_Power_Opower__class_Opower ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[580]) ).

thf(571,axiom,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( A @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = B )
      | ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__imp__diff__is__add_1) ).

thf(1942,plain,
    ! [B: $i,A: $i] :
      ( ( ( tc_nat @ ( A @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = B )
      | ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[571]) ).

thf(113,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A
        = ( C @ c_List_Olist_ONil ) )
      | ( ( C @ ( B @ ( A @ c_List_Oappend ) ) )
       != ( C @ c_List_Olist_ONil ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append__is__Nil__conv_0) ).

thf(901,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A
        = ( C @ c_List_Olist_ONil ) )
      | ( ( C @ ( B @ ( A @ c_List_Oappend ) ) )
       != ( C @ c_List_Olist_ONil ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(24,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__antisym__conv2_1) ).

thf(686,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(186,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__ringb_Oadd__cancel_0) ).

thf(1075,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(331,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( C @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( F @ ( E @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osplit__asm_1) ).

thf(1414,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( C @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( F @ ( E @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[331]) ).

thf(616,axiom,
    ! [B: $i,A: $i] :
      ( ( B = A )
      | ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__neq__iff_0) ).

thf(2053,plain,
    ! [B: $i,A: $i] :
      ( ( B = A )
      | ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[616]) ).

thf(527,axiom,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__is__0_2) ).

thf(1842,plain,
    ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[527]) ).

thf(222,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I6_J_0) ).

thf(1155,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).

thf(483,axiom,
    tc_nat @ class_Orderings_Olinorder,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Orderings_Olinorder) ).

thf(1744,plain,
    tc_nat @ class_Orderings_Olinorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[483]) ).

thf(84,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_eq__eqI_0) ).

thf(837,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(152,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = ( A @ ( B @ ( C @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Osup__commute_0) ).

thf(997,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = ( A @ ( B @ ( C @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(197,axiom,
    ! [A: $i] :
      ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gr__implies__not0_0) ).

thf(1099,plain,
    ! [A: $i] :
      ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(31,axiom,
    tc_bool @ class_Orderings_Obot,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_bool__Orderings_Obot) ).

thf(701,plain,
    tc_bool @ class_Orderings_Obot,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(281,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I9_J_0) ).

thf(1297,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).

thf(346,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ c_Lambda_OdB_OVar )
     != ( C @ ( B @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I4_J_0) ).

thf(1451,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ c_Lambda_OdB_OVar )
     != ( C @ ( B @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[346]) ).

thf(395,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( E @ c_Lambda_OdB_OAbs @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osplit__asm_2) ).

thf(1552,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( E @ c_Lambda_OdB_OAbs @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[395]) ).

thf(238,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I21_J_0) ).

thf(1192,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).

thf(179,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_trans__less__add2_0) ).

thf(1060,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(147,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__antisym__conv2_0) ).

thf(984,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(573,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__diff__assoc_0) ).

thf(1947,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[573]) ).

thf(541,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_termination__basic__simps_I1_J_0) ).

thf(1874,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[541]) ).

thf(337,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( B @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__diff__split_1) ).

thf(1429,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( B @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[337]) ).

thf(260,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_rtrancl__beta__AppR_0) ).

thf(1246,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).

thf(469,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( E @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( E @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__add__right__mono_0) ).

thf(1715,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( E @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( E @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[469]) ).

thf(520,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__diff__cancel_0) ).

thf(1825,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( C @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[520]) ).

thf(204,axiom,
    ! [A: $i] :
      ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__less0_0) ).

thf(1114,plain,
    ! [A: $i] :
      ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(359,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocancel__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__left__cancel_0) ).

thf(1478,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C = D )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocancel__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[359]) ).

thf(262,axiom,
    ! [B: $i,A: $i] :
      ( ( B = A )
      | ( ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diffs0__imp__equal_0) ).

thf(1249,plain,
    ! [B: $i,A: $i] :
      ( ( B = A )
      | ( ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).

thf(269,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Omul__a_0) ).

thf(1266,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).

thf(253,axiom,
    ! [C: $i,B: $i,A: $i] : ( C @ ( C @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Nitpick_Ortranclp__def_1) ).

thf(1231,plain,
    ! [C: $i,B: $i,A: $i] : ( C @ ( C @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).

thf(296,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_natgb_Oadd__cancel_0) ).

thf(1333,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).

thf(591,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ c_Lambda_OdB_OVar )
        = B )
      | ( ( A @ c_Lambda_OdB_OVar )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( B @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Var__eq__apps__conv_0) ).

thf(1991,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ c_Lambda_OdB_OVar )
        = B )
      | ( ( A @ c_Lambda_OdB_OVar )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( B @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[591]) ).

thf(242,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_comm__monoid__add_Omult__left__commute_0) ).

thf(1202,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).

thf(277,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ c_Lambda_OdB_OVar @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) )
      = ( D @ ( A @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Orecs_I1_J_0) ).

thf(1286,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ c_Lambda_OdB_OVar @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) )
      = ( D @ ( A @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).

thf(184,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__self__eq__0_0) ).

thf(1071,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).

thf(378,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( D @ ( E @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( C @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_converse__rtranclp__into__rtranclp_0) ).

thf(1515,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( D @ ( E @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( C @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[378]) ).

thf(478,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A = C )
      | ( ( C @ ( B @ ( A @ c_Lambda_OdB_OVar @ c_Lambda_Osubstn ) ) )
        = ( A @ c_Lambda_OdB_OVar ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_substn_Osimps_I1_J_2) ).

thf(1732,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A = C )
      | ( ( C @ ( B @ ( A @ c_Lambda_OdB_OVar @ c_Lambda_Osubstn ) ) )
        = ( A @ c_Lambda_OdB_OVar ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[478]) ).

thf(122,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__supI2_0) ).

thf(926,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(516,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__is__0_1) ).

thf(1814,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[516]) ).

thf(127,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( C @ c_List_Olist_ONil ) )
      | ( ( D @ ( C @ ( B @ ( A @ c_List_Omap ) ) ) )
       != ( D @ c_List_Olist_ONil ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_map__is__Nil__conv_0) ).

thf(938,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B
        = ( C @ c_List_Olist_ONil ) )
      | ( ( D @ ( C @ ( B @ ( A @ c_List_Omap ) ) ) )
       != ( D @ c_List_Olist_ONil ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(3,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C = B )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__neqE_0) ).

thf(633,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C = B )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(500,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__add__0_0) ).

thf(1779,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[500]) ).

thf(58,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C = B )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_xt1_I12_J_0) ).

thf(765,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C = B )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(529,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__diff__conv2_0) ).

thf(1846,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[529]) ).

thf(176,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_termination__basic__simps_I2_J_0) ).

thf(1054,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).

thf(164,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__add__cancel_0) ).

thf(1026,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).

thf(227,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B @ ( D @ ( C @ ( B @ ( A @ c_Type_Oshift ) ) ) @ hAPP ) )
      = C ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_shift__eq_0) ).

thf(1167,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B @ ( D @ ( C @ ( B @ ( A @ c_Type_Oshift ) ) ) @ hAPP ) )
      = C ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).

thf(134,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( D @ ( D @ ( D @ c_List_Olist_ONil @ ( F @ c_List_Olist_OCons ) ) @ ( B @ c_List_Oappend ) ) @ ( D @ ( D @ ( D @ c_List_Olist_ONil @ ( E @ c_List_Olist_OCons ) ) @ ( A @ c_List_Oappend ) ) @ ( C @ c_ListOrder_Ostep1 ) ) ) )
      | ( D @ ( B @ ( A @ ( C @ c_ListOrder_Ostep1 ) ) ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Snoc__step1__SnocD_0) ).

thf(953,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( D @ ( D @ ( D @ c_List_Olist_ONil @ ( F @ c_List_Olist_OCons ) ) @ ( B @ c_List_Oappend ) ) @ ( D @ ( D @ ( D @ c_List_Olist_ONil @ ( E @ c_List_Olist_OCons ) ) @ ( A @ c_List_Oappend ) ) @ ( C @ c_ListOrder_Ostep1 ) ) ) )
      | ( D @ ( B @ ( A @ ( C @ c_ListOrder_Ostep1 ) ) ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(235,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_comm__monoid__add_Omult__assoc_0) ).

thf(1185,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).

thf(61,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ c_List_Olist_ONil )
      = ( A @ ( C @ ( C @ c_List_Olist_ONil @ ( B @ c_List_Omap ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Nil__is__map__conv_1) ).

thf(773,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ c_List_Olist_ONil )
      = ( A @ ( C @ ( C @ c_List_Olist_ONil @ ( B @ c_List_Omap ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(593,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__0__less__iff_0) ).

thf(1996,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[593]) ).

thf(15,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__less__le__trans_0) ).

thf(662,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(38,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_xt1_I7_J_0) ).

thf(718,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(107,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_max__less__iff__conj_0) ).

thf(888,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(213,axiom,
    ( ( tc_nat @ c_Orderings_Obot__class_Obot )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_bot__nat__def_0) ).

thf(1135,plain,
    ( ( tc_nat @ c_Orderings_Obot__class_Obot )
    = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).

thf(595,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Omul__0_0) ).

thf(2001,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[595]) ).

thf(218,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ c_Lambda_OdB_OAbs )
     != ( C @ ( B @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I9_J_0) ).

thf(1146,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ c_Lambda_OdB_OAbs )
     != ( C @ ( B @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).

thf(453,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( D @ ( A @ c_Wellfounded_Oaccp ) ) )
      | ~ ( D @ ( B @ ( C @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ c_Wellfounded_Oaccp ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_accp__downwards_0) ).

thf(1680,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( D @ ( A @ c_Wellfounded_Oaccp ) ) )
      | ~ ( D @ ( B @ ( C @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ c_Wellfounded_Oaccp ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[453]) ).

thf(79,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C @ ( C @ ( D @ ( B @ c_List_Olist_OCons ) ) @ ( C @ ( B @ ( A @ c_List_Oreplicate ) ) @ c_List_Oappend ) ) )
      = ( C @ ( C @ ( D @ ( C @ ( B @ ( A @ c_List_Oreplicate ) ) @ c_List_Oappend ) ) @ ( B @ c_List_Olist_OCons ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_replicate__app__Cons__same_0) ).

thf(824,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C @ ( C @ ( D @ ( B @ c_List_Olist_OCons ) ) @ ( C @ ( B @ ( A @ c_List_Oreplicate ) ) @ c_List_Oappend ) ) )
      = ( C @ ( C @ ( D @ ( C @ ( B @ ( A @ c_List_Oreplicate ) ) @ c_List_Oappend ) ) @ ( B @ c_List_Olist_OCons ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(563,axiom,
    v_T____ @ ( v_u____ @ ( v_e____ @ c_Type_Otyping ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_CHAINED_0) ).

thf(1927,plain,
    v_T____ @ ( v_u____ @ ( v_e____ @ c_Type_Otyping ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[563]) ).

thf(92,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__max__iff__disj_2) ).

thf(856,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(415,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I5_J_0) ).

thf(1595,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[415]) ).

thf(313,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C @ ( D @ ( C @ c_List_Olist_ONil @ ( B @ ( A @ c_List_Ofoldl ) ) ) ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_foldl__Nil_0) ).

thf(1374,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C @ ( D @ ( C @ c_List_Olist_ONil @ ( B @ ( A @ c_List_Ofoldl ) ) ) ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[313]) ).

thf(475,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__less__cancel__left_1) ).

thf(1727,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[475]) ).

thf(376,axiom,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I7_J_0) ).

thf(1510,plain,
    ! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( H @ ( G @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[376]) ).

thf(78,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( A = D )
      | ( ( C @ ( B @ ( A @ c_List_Olist_OCons ) ) )
       != ( C @ ( E @ ( D @ c_List_Olist_OCons ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_list_Oinject_0) ).

thf(819,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( A = D )
      | ( ( C @ ( B @ ( A @ c_List_Olist_OCons ) ) )
       != ( C @ ( E @ ( D @ c_List_Olist_OCons ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(114,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__eqI_0) ).

thf(904,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(157,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( B = E )
      | ( ( C @ ( C @ ( C @ c_List_Olist_ONil @ ( B @ c_List_Olist_OCons ) ) @ ( A @ c_List_Oappend ) ) )
       != ( C @ ( C @ ( C @ c_List_Olist_ONil @ ( E @ c_List_Olist_OCons ) ) @ ( D @ c_List_Oappend ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append1__eq__conv_1) ).

thf(1007,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( B = E )
      | ( ( C @ ( C @ ( C @ c_List_Olist_ONil @ ( B @ c_List_Olist_OCons ) ) @ ( A @ c_List_Oappend ) ) )
       != ( C @ ( C @ ( C @ c_List_Olist_ONil @ ( E @ c_List_Olist_OCons ) ) @ ( D @ c_List_Oappend ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(44,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__max__iff__disj_2) ).

thf(732,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(320,axiom,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__is__0_0) ).

thf(1389,plain,
    ! [B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[320]) ).

thf(427,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( C @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_r__into__rtranclp_0) ).

thf(1620,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( C @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[427]) ).

thf(487,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( A @ ( C @ c_OrderedGroup_Omonoid__add ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_monoid__add_Oadd__0__right_0) ).

thf(1752,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( A @ ( C @ c_OrderedGroup_Omonoid__add ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[487]) ).

thf(441,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__pos__pos_0) ).

thf(1652,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[441]) ).

thf(601,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( B = D )
      | ( ( A @ ( C @ ( B @ c_Power_Opower__class_Opower ) ) )
       != ( A @ ( C @ ( D @ c_Power_Opower__class_Opower ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__eq__imp__eq__base_0) ).

thf(2016,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( B = D )
      | ( ( A @ ( C @ ( B @ c_Power_Opower__class_Opower ) ) )
       != ( A @ ( C @ ( D @ c_Power_Opower__class_Opower ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[601]) ).

thf(80,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__antisym_0) ).

thf(826,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(380,axiom,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I13_J_0) ).

thf(1520,plain,
    ! [J: $i,I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( J @ ( I @ ( H @ ( G @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[380]) ).

thf(265,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I20_J_0) ).

thf(1256,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( E @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).

thf(136,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( C @ ( C @ c_List_Olist_ONil @ ( B @ c_List_Olist_OCons ) ) @ ( C @ ( B @ ( A @ c_List_Oreplicate ) ) @ c_List_Oappend ) ) )
      = ( C @ ( C @ ( B @ ( A @ c_List_Oreplicate ) ) @ ( B @ c_List_Olist_OCons ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_replicate__append__same_0) ).

thf(958,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( C @ ( C @ c_List_Olist_ONil @ ( B @ c_List_Olist_OCons ) ) @ ( C @ ( B @ ( A @ c_List_Oreplicate ) ) @ c_List_Oappend ) ) )
      = ( C @ ( C @ ( B @ ( A @ c_List_Oreplicate ) ) @ ( B @ c_List_Olist_OCons ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(325,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( D @ ( C @ ( F @ c_List_Olist_OCons ) ) @ ( D @ ( B @ ( E @ c_List_Olist_OCons ) ) @ ( A @ c_ListOrder_Ostep1 ) ) ) )
      | ( F @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( C @ ( B @ ( A @ c_ListOrder_Ostep1 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Cons__step1__Cons_1) ).

thf(1403,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( D @ ( C @ ( F @ c_List_Olist_OCons ) ) @ ( D @ ( B @ ( E @ c_List_Olist_OCons ) ) @ ( A @ c_ListOrder_Ostep1 ) ) ) )
      | ( F @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( C @ ( B @ ( A @ c_ListOrder_Ostep1 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[325]) ).

thf(507,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( A @ c_ATP__Linkup_Osko__Nat__Xinfinite__descent0__1__1 @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( C @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_infinite__descent0_2) ).

thf(1795,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( A @ c_ATP__Linkup_Osko__Nat__Xinfinite__descent0__1__1 @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( C @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[507]) ).

thf(463,axiom,
    tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( v_as____ @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( v_a____ @ ( v_r____ @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Beta_I1_J_0) ).

thf(1701,plain,
    tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( v_as____ @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( v_a____ @ ( v_r____ @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[463]) ).

thf(6,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( B @ c_List_Olist_ONil @ ( A @ c_List_Oappend ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append__Nil2_0) ).

thf(641,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( B @ c_List_Olist_ONil @ ( A @ c_List_Oappend ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(126,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Oplus__class_Oplus @ c_OrderedGroup_Osemigroup__add ) )
      | ~ ( A @ class_OrderedGroup_Osemigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_semigroup__add__class_Oaxioms_0) ).

thf(936,plain,
    ! [A: $i] :
      ( ( A @ ( A @ c_HOL_Oplus__class_Oplus @ c_OrderedGroup_Osemigroup__add ) )
      | ~ ( A @ class_OrderedGroup_Osemigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(606,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( A @ c_OrderedGroup_Osemigroup__add ) )
      | ( ( E @ ( E @ ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ c_List_Ofoldl ) ) ) ) )
        = ( E @ ( E @ ( D @ ( C @ ( A @ c_List_Ofoldl ) ) ) ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_semigroup__add_Ofoldl__assoc_0) ).

thf(2029,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( A @ c_OrderedGroup_Osemigroup__add ) )
      | ( ( E @ ( E @ ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ c_List_Ofoldl ) ) ) ) )
        = ( E @ ( E @ ( D @ ( C @ ( A @ c_List_Ofoldl ) ) ) ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[606]) ).

thf(466,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Oadd__0_0) ).

thf(1707,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[466]) ).

thf(67,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( F @ ( F @ ( F @ ( F @ c_List_Olist_ONil @ ( D @ c_List_Olist_OCons ) ) @ ( B @ c_List_Oappend ) ) @ ( F @ ( F @ ( F @ c_List_Olist_ONil @ ( C @ c_List_Olist_OCons ) ) @ ( A @ c_List_Oappend ) ) @ ( E @ c_ListOrder_Ostep1 ) ) ) )
      | ( C = D )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Snoc__step1__SnocD_2) ).

thf(787,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( F @ ( F @ ( F @ ( F @ c_List_Olist_ONil @ ( D @ c_List_Olist_OCons ) ) @ ( B @ c_List_Oappend ) ) @ ( F @ ( F @ ( F @ c_List_Olist_ONil @ ( C @ c_List_Olist_OCons ) ) @ ( A @ c_List_Oappend ) ) @ ( E @ c_ListOrder_Ostep1 ) ) ) )
      | ( C = D )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(537,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocancel__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__right__cancel_0) ).

thf(1865,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Ocancel__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[537]) ).

thf(434,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_List_Oreplicate ) ) )
      = ( D @ ( D @ ( C @ ( B @ c_List_Oreplicate ) ) @ ( D @ ( C @ ( A @ c_List_Oreplicate ) ) @ c_List_Oappend ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_replicate__add_0) ).

thf(1635,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_List_Oreplicate ) ) )
      = ( D @ ( D @ ( C @ ( B @ c_List_Oreplicate ) ) @ ( D @ ( C @ ( A @ c_List_Oreplicate ) ) @ c_List_Oappend ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[434]) ).

thf(119,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( E @ c_lessequals ) ) )
      | ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__eqI_1) ).

thf(919,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( E @ c_lessequals ) ) )
      | ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(33,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__le__less_0) ).

thf(704,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(131,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Oless__supI1_0) ).

thf(947,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(250,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__self_0) ).

thf(1224,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).

thf(306,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( B @ ( A @ c_Orderings_Oord__class_Omax ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__minus__add__max_0) ).

thf(1356,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = ( tc_nat @ ( B @ ( A @ c_Orderings_Oord__class_Omax ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[306]) ).

thf(451,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_List_Olistsum ) )
        = ( A @ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldr ) ) ) ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_listsum__foldr_0) ).

thf(1675,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_List_Olistsum ) )
        = ( A @ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldr ) ) ) ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[451]) ).

thf(215,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__add__inverse_0) ).

thf(1139,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).

thf(372,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__add__left__cancel__less_0) ).

thf(1503,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[372]) ).

thf(271,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__diff__right_0) ).

thf(1271,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).

thf(312,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ c_Lambda_OdB_OVar @ ( A @ c_Type_Otyping ) ) )
      | ( ( B @ ( A @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_typing__elims_I1_J_0) ).

thf(1371,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ c_Lambda_OdB_OVar @ ( A @ c_Type_Otyping ) ) )
      | ( ( B @ ( A @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[312]) ).

thf(178,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_List_Odrop ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_drop__0_0) ).

thf(1058,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_List_Odrop ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(201,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ c_List_Olist_ONil )
      = ( A @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_List_Oreplicate ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_empty__replicate_1) ).

thf(1107,plain,
    ! [B: $i,A: $i] :
      ( ( A @ c_List_Olist_ONil )
      = ( A @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_List_Oreplicate ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(206,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I19_J_0) ).

thf(1117,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( E @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( C @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(382,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL )
      | ( B @ ( A @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_lift__IT_0) ).

thf(1524,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL )
      | ( B @ ( A @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[382]) ).

thf(174,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( A @ ( C @ c_OrderedGroup_Omonoid__add ) ) )
      | ( ( E @ ( E @ ( D @ ( C @ ( A @ c_List_Ofoldl ) ) ) ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( E @ ( D @ ( B @ ( A @ c_List_Ofoldl ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_monoid__add_Ofoldl__absorb0_0) ).

thf(1048,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( A @ ( C @ c_OrderedGroup_Omonoid__add ) ) )
      | ( ( E @ ( E @ ( D @ ( C @ ( A @ c_List_Ofoldl ) ) ) ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( E @ ( D @ ( B @ ( A @ c_List_Ofoldl ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).

thf(473,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__less__mono_0) ).

thf(1722,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[473]) ).

thf(314,axiom,
    ! [A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gr0I_0) ).

thf(1376,plain,
    ! [A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[314]) ).

thf(621,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( F @ ( E @ ( D @ ( C @ c_List_Oappend ) ) @ ( B @ ( A @ c_List_Ofoldl ) ) ) ) )
      = ( E @ ( F @ ( D @ ( E @ ( F @ ( C @ ( B @ ( A @ c_List_Ofoldl ) ) ) ) @ ( A @ c_List_Ofoldl ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_foldl__append_0) ).

thf(2065,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( F @ ( E @ ( D @ ( C @ c_List_Oappend ) ) @ ( B @ ( A @ c_List_Ofoldl ) ) ) ) )
      = ( E @ ( F @ ( D @ ( E @ ( F @ ( C @ ( B @ ( A @ c_List_Ofoldl ) ) ) ) @ ( A @ c_List_Ofoldl ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[621]) ).

thf(22,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( A = D )
      | ( ( C @ ( C @ ( C @ c_List_Olist_ONil @ ( B @ c_List_Olist_OCons ) ) @ ( A @ c_List_Oappend ) ) )
       != ( C @ ( C @ ( C @ c_List_Olist_ONil @ ( E @ c_List_Olist_OCons ) ) @ ( D @ c_List_Oappend ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append1__eq__conv_0) ).

thf(681,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( A = D )
      | ( ( C @ ( C @ ( C @ c_List_Olist_ONil @ ( B @ c_List_Olist_OCons ) ) @ ( A @ c_List_Oappend ) ) )
       != ( C @ ( C @ ( C @ c_List_Olist_ONil @ ( E @ c_List_Olist_OCons ) ) @ ( D @ c_List_Oappend ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(532,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__increasing_0) ).

thf(1853,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[532]) ).

thf(413,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( C @ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) )
        = ( A @ c_Lambda_OdB_OVar ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subst__lt_0) ).

thf(1589,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( ( C @ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) )
        = ( A @ c_Lambda_OdB_OVar ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[413]) ).

thf(566,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( E @ c_Lambda_OdB_OAbs @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osplit_2) ).

thf(1933,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( E @ c_Lambda_OdB_OAbs @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[566]) ).

thf(103,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( B @ class_Orderings_Oorder )
      | ( B @ ( A @ tc_fun ) @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_fun__Orderings_Oorder) ).

thf(880,plain,
    ! [B: $i,A: $i] :
      ( ~ ( B @ class_Orderings_Oorder )
      | ( B @ ( A @ tc_fun ) @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(9,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_xt1_I8_J_0) ).

thf(649,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(171,axiom,
    tc_nat @ class_OrderedGroup_Ocancel__ab__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Ocancel__ab__semigroup__add) ).

thf(1043,plain,
    tc_nat @ class_OrderedGroup_Ocancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).

thf(582,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower ) ) @ ( A @ ( C @ ( B @ c_Power_Opower__class_Opower ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__strict__mono_0) ).

thf(1969,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower ) ) @ ( A @ ( C @ ( B @ c_Power_Opower__class_Opower ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[582]) ).

thf(285,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__left__mono_0) ).

thf(1307,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).

thf(355,axiom,
    tc_nat @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Opordered__ab__semigroup__add__imp__le) ).

thf(1471,plain,
    tc_nat @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[355]) ).

thf(188,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( C = B )
      | ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__less__cases_0) ).

thf(1080,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( C = B )
      | ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(93,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( D @ ( C @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Oappend ) ) )
      = ( D @ ( C @ ( D @ ( B @ ( A @ c_List_Oappend ) ) @ c_List_Oappend ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append__eq__appendI_0) ).

thf(858,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( D @ ( C @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Oappend ) ) )
      = ( D @ ( C @ ( D @ ( B @ ( A @ c_List_Oappend ) ) @ c_List_Oappend ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(432,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_foldl__absorb0_0) ).

thf(1630,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[432]) ).

thf(225,axiom,
    ! [B: $i,A: $i] :
      ( ( B = A )
      | ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__neqE__nat_0) ).

thf(1162,plain,
    ! [B: $i,A: $i] :
      ( ( B = A )
      | ( tc_nat @ ( A @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).

thf(536,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
     != ( C @ c_Lambda_OdB_OAbs ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I8_J_0) ).

thf(1862,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
     != ( C @ c_Lambda_OdB_OAbs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[536]) ).

thf(89,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ c_Lambda_OdB_OVar )
     != ( B @ c_Lambda_OdB_OAbs ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I6_J_0) ).

thf(848,plain,
    ! [B: $i,A: $i] :
      ( ( A @ c_Lambda_OdB_OVar )
     != ( B @ c_Lambda_OdB_OAbs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(293,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ c_Lambda_OdB_OAbs )
        = B )
      | ( ( A @ c_Lambda_OdB_OAbs )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( B @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Abs__eq__apps__conv_0) ).

thf(1325,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ c_Lambda_OdB_OAbs )
        = B )
      | ( ( A @ c_Lambda_OdB_OAbs )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( B @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).

thf(165,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__diff__split_0) ).

thf(1029,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).

thf(1,negated_conjecture,
    ~ ( v_i____ @ ( v_u____ @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( v_as____ @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( v_a____ @ ( v_r____ @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_conjecture_0) ).

thf(626,plain,
    ~ ( v_i____ @ ( v_u____ @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( v_as____ @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( v_a____ @ ( v_r____ @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1]) ).

thf(397,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__less__cancel__left_0) ).

thf(1556,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[397]) ).

thf(129,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__not__less_0) ).

thf(943,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(280,axiom,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__nonneg__eq__0__iff_2) ).

thf(1293,plain,
    ! [A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).

thf(575,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( D @ ( A @ c_Type_Otyping ) ) )
      | ( E @ ( B @ ( D @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) @ ( tc_Type_Otype @ ( C @ ( B @ ( A @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_lift__type_0) ).

thf(1952,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( D @ ( A @ c_Type_Otyping ) ) )
      | ( E @ ( B @ ( D @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) @ ( tc_Type_Otype @ ( C @ ( B @ ( A @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[575]) ).

thf(518,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__ringb_Osubr0__iff_0) ).

thf(1820,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[518]) ).

thf(108,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Osup__least_0) ).

thf(890,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(448,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( tc_Lambda_OdB @ c_List_Olist_ONil ) )
      | ( ( A @ c_Lambda_OdB_OAbs )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( B @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Abs__eq__apps__conv_1) ).

thf(1667,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( tc_Lambda_OdB @ c_List_Olist_ONil ) )
      | ( ( A @ c_Lambda_OdB_OAbs )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( B @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[448]) ).

thf(219,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( C @ ( B @ ( A @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subst__lift_0) ).

thf(1149,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( C @ ( B @ ( A @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).

thf(139,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = ( A @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omax ) ) @ ( C @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Osup__left__commute_0) ).

thf(964,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = ( A @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omax ) ) @ ( C @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).

thf(231,axiom,
    tc_nat @ class_Ring__and__Field_Osemiring__1,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Osemiring__1) ).

thf(1177,plain,
    tc_nat @ class_Ring__and__Field_Osemiring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).

thf(316,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_trans__le__add2_0) ).

thf(1381,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[316]) ).

thf(625,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Lambda_Oliftn ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_liftn__0_0) ).

thf(2075,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Lambda_Oliftn ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[625]) ).

thf(357,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__lessD1_0) ).

thf(1474,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[357]) ).

thf(462,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( B @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( D @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subst__preserves__beta2_0) ).

thf(1699,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( B @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( D @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[462]) ).

thf(384,axiom,
    tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__0__eq_1) ).

thf(1528,plain,
    tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[384]) ).

thf(64,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( D @ ( E @ ( C @ c_List_Oappend ) ) @ ( D @ ( E @ ( B @ c_List_Oappend ) ) @ ( A @ c_ListOrder_Ostep1 ) ) ) )
      | ~ ( D @ ( C @ ( B @ ( A @ c_ListOrder_Ostep1 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append__step1I_0) ).

thf(780,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( D @ ( E @ ( C @ c_List_Oappend ) ) @ ( D @ ( E @ ( B @ c_List_Oappend ) ) @ ( A @ c_ListOrder_Ostep1 ) ) ) )
      | ~ ( D @ ( C @ ( B @ ( A @ c_ListOrder_Ostep1 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(160,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A
        = ( C @ c_List_Olist_ONil ) )
      | ( ( C @ ( B @ ( A @ c_List_Oappend ) ) )
       != B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append__self__conv2_0) ).

thf(1015,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A
        = ( C @ c_List_Olist_ONil ) )
      | ( ( C @ ( B @ ( A @ c_List_Oappend ) ) )
       != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(11,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( E @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__eqI_0) ).

thf(653,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( E @ c_lessequals ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(13,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( B @ c_List_Olist_ONil @ ( A @ c_List_Omap ) ) ) )
      = ( C @ c_List_Olist_ONil ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_map__is__Nil__conv_1) ).

thf(658,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( B @ c_List_Olist_ONil @ ( A @ c_List_Omap ) ) ) )
      = ( C @ c_List_Olist_ONil ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(400,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_lem0_0) ).

thf(1562,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[400]) ).

thf(246,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__le__mono_0) ).

thf(1213,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).

thf(59,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( C = B )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_xt1_I11_J_0) ).

thf(768,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( C = B )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(28,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__max__iff__disj_0) ).

thf(695,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(4,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__less__imp__le_0) ).

thf(636,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(172,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( tc_nat @ ( A @ c_ATP__Linkup_Osko__Nat__Xinfinite__descent0__1__1 @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_infinite__descent0_0) ).

thf(1044,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( tc_nat @ ( A @ c_ATP__Linkup_Osko__Nat__Xinfinite__descent0__1__1 @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(153,axiom,
    tc_bool @ class_Orderings_Oorder,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_bool__Orderings_Oorder) ).

thf(1000,plain,
    tc_bool @ class_Orderings_Oorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(340,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( A = B )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__less__le_2) ).

thf(1435,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( A = B )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[340]) ).

thf(90,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__not__le_0) ).

thf(851,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(199,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( D @ ( C @ c_Power_Opower__class_Opower ) ) @ ( A @ ( D @ ( B @ c_Power_Opower__class_Opower ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__less__imp__less__base_0) ).

thf(1102,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( D @ ( C @ c_Power_Opower__class_Opower ) ) @ ( A @ ( D @ ( B @ c_Power_Opower__class_Opower ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(460,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( E @ ( F @ ( A @ c_List_Olist_OCons ) ) @ ( E @ ( D @ ( B @ c_List_Olist_OCons ) ) @ ( C @ c_ListOrder_Ostep1 ) ) ) )
      | ( A @ ( B @ ( C @ hAPP ) @ hAPP ) @ hBOOL )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Cons__step1__Cons_0) ).

thf(1694,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( E @ ( F @ ( A @ c_List_Olist_OCons ) ) @ ( E @ ( D @ ( B @ c_List_Olist_OCons ) ) @ ( C @ c_ListOrder_Ostep1 ) ) ) )
      | ( A @ ( B @ ( C @ hAPP ) @ hAPP ) @ hBOOL )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[460]) ).

thf(120,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__linear_0) ).

thf(922,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(584,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_termination__basic__simps_I5_J_0) ).

thf(1973,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[584]) ).

thf(511,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ ( B @ c_Power_Opower__class_Opower ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__le__power_0) ).

thf(1805,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ ( B @ c_Power_Opower__class_Opower ) ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[511]) ).

thf(430,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( B @ ( C @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( C @ ( D @ ( A @ c_Wellfounded_Oaccp ) ) )
      | ( C @ ( B @ ( A @ c_Wellfounded_Oaccp ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_accp__downwards__aux_0) ).

thf(1626,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( B @ ( C @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( C @ ( D @ ( A @ c_Wellfounded_Oaccp ) ) )
      | ( C @ ( B @ ( A @ c_Wellfounded_Oaccp ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[430]) ).

thf(40,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ ( D @ ( C @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Omap ) ) ) )
      = ( E @ ( E @ ( D @ ( C @ ( A @ c_List_Omap ) ) ) @ ( E @ ( D @ ( B @ ( A @ c_List_Omap ) ) ) @ c_List_Oappend ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_map__append_0) ).

thf(722,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ ( D @ ( C @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Omap ) ) ) )
      = ( E @ ( E @ ( D @ ( C @ ( A @ c_List_Omap ) ) ) @ ( E @ ( D @ ( B @ ( A @ c_List_Omap ) ) ) @ c_List_Oappend ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(167,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( D @ ( E @ ( D @ ( C @ ( B @ c_List_Olist_OCons ) ) @ ( A @ c_List_Ofoldr ) ) ) ) )
      = ( F @ ( D @ ( E @ ( C @ ( A @ c_List_Ofoldr ) ) ) ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_foldr_Osimps_I2_J_0) ).

thf(1034,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( D @ ( E @ ( D @ ( C @ ( B @ c_List_Olist_OCons ) ) @ ( A @ c_List_Ofoldr ) ) ) ) )
      = ( F @ ( D @ ( E @ ( C @ ( A @ c_List_Ofoldr ) ) ) ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).

thf(410,axiom,
    tc_nat @ class_Ring__and__Field_Ozero__neq__one,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Ozero__neq__one) ).

thf(1583,plain,
    tc_nat @ class_Ring__and__Field_Ozero__neq__one,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[410]) ).

thf(353,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ ( A @ c_lessequals ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__refl_0) ).

thf(1467,plain,
    ! [A: $i] : ( tc_nat @ ( A @ ( A @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[353]) ).

thf(70,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( A @ c_List_Olist_ONil @ c_List_Oappend ) ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append__Nil_0) ).

thf(795,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( A @ c_List_Olist_ONil @ c_List_Oappend ) ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(554,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( D @ ( A @ c_Wellfounded_Oaccp ) ) )
      | ~ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ c_Wellfounded_Oaccp ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_accp__downward_0) ).

thf(1905,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( D @ ( A @ c_Wellfounded_Oaccp ) ) )
      | ~ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ c_Wellfounded_Oaccp ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[554]) ).

thf(610,axiom,
    ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_neq0__conv_1) ).

thf(2037,plain,
    ~ ( tc_nat @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[610]) ).

thf(273,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( D @ ( A @ c_Wellfounded_Oaccp ) ) )
      | ~ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ c_Wellfounded_Oaccp ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_accp_Ocases_0) ).

thf(1277,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( D @ ( A @ c_Wellfounded_Oaccp ) ) )
      | ~ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ c_Wellfounded_Oaccp ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).

thf(76,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( A = D )
      | ( ( C @ ( B @ ( A @ c_List_Oreplicate ) ) )
       != ( C @ ( E @ ( D @ c_List_Oreplicate ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_replicate__eq__replicate_0) ).

thf(812,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( A = D )
      | ( ( C @ ( B @ ( A @ c_List_Oreplicate ) ) )
       != ( C @ ( E @ ( D @ c_List_Oreplicate ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(50,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( D @ ( C @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Olist_OCons ) ) )
      = ( D @ ( C @ ( D @ ( B @ ( A @ c_List_Olist_OCons ) ) @ c_List_Oappend ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Cons__eq__append__conv_5) ).

thf(749,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( D @ ( C @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Olist_OCons ) ) )
      = ( D @ ( C @ ( D @ ( B @ ( A @ c_List_Olist_OCons ) ) @ c_List_Oappend ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(299,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( A @ ( B @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Opwr__mul_0) ).

thf(1340,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( A @ ( B @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).

thf(356,axiom,
    ! [A: $i] :
      ~ ( tc_nat @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__not__refl_0) ).

thf(1472,plain,
    ! [A: $i] :
      ~ ( tc_nat @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[356]) ).

thf(161,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( D @ ( C @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Olist_OCons ) ) )
      = ( D @ ( C @ ( D @ ( B @ ( A @ c_List_Olist_OCons ) ) @ c_List_Oappend ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Cons__eq__appendI_0) ).

thf(1018,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( D @ ( C @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Olist_OCons ) ) )
      = ( D @ ( C @ ( D @ ( B @ ( A @ c_List_Olist_OCons ) ) @ c_List_Oappend ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(284,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ c_List_Olist_ONil )
       != ( A @ ( C @ ( B @ c_List_Oreplicate ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_empty__replicate_0) ).

thf(1304,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ c_List_Olist_ONil )
       != ( A @ ( C @ ( B @ c_List_Oreplicate ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).

thf(540,axiom,
    tc_nat @ class_OrderedGroup_Omonoid__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Omonoid__add) ).

thf(1873,plain,
    tc_nat @ class_OrderedGroup_Omonoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[540]) ).

thf(423,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Nat_Onat_Onat__case ) ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Nitpick_Onat__case__def_0) ).

thf(1611,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Nat_Onat_Onat__case ) ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[423]) ).

thf(142,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__neq__le__trans_0) ).

thf(971,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(599,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_beta_0) ).

thf(2012,plain,
    ! [B: $i,A: $i] : ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[599]) ).

thf(439,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Oadd__a_0) ).

thf(1647,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[439]) ).

thf(263,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_double__add__less__zero__iff__single__less__zero_1) ).

thf(1252,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).

thf(46,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__max__iff__disj_0) ).

thf(739,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ ( D @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(311,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_eq__diff__iff_0) ).

thf(1368,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ ( B @ ( C @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[311]) ).

thf(386,axiom,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__add2_0) ).

thf(1531,plain,
    ! [B: $i,A: $i] : ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[386]) ).

thf(361,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( B @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( E @ ( B @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I1_J_0) ).

thf(1484,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( B @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( E @ ( B @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[361]) ).

thf(196,axiom,
    ! [A: $i] : ( A @ c_Lambda_OdB_OVar @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Var__IT_0) ).

thf(1098,plain,
    ! [A: $i] : ( A @ c_Lambda_OdB_OVar @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(393,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subst__eq_0) ).

thf(1547,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[393]) ).

thf(485,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ ( A @ c_lessequals ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_eq__imp__le_0) ).

thf(1748,plain,
    ! [A: $i] : ( tc_nat @ ( A @ ( A @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[485]) ).

thf(155,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( D @ ( C @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Oappend ) ) )
      = ( D @ ( C @ ( D @ ( B @ ( A @ c_List_Oappend ) ) @ c_List_Oappend ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append__eq__append__conv2_5) ).

thf(1003,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( D @ ( C @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Oappend ) ) )
      = ( D @ ( C @ ( D @ ( B @ ( A @ c_List_Oappend ) ) @ c_List_Oappend ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(495,axiom,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__0__eq__0_0) ).

thf(1770,plain,
    ! [A: $i] :
      ( ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[495]) ).

thf(26,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = B )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Osup__idem_0) ).

thf(690,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( B @ c_Orderings_Oord__class_Omax ) ) )
        = B )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(416,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ c_Lambda_OdB_OAbs @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) )
      = ( E @ ( D @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) @ ( D @ ( C @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Orecs_I3_J_0) ).

thf(1598,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ c_Lambda_OdB_OAbs @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) )
      = ( E @ ( D @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__rec ) ) ) ) @ ( D @ ( C @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[416]) ).

thf(619,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( B
       != ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__ringb_Oadd__r0__iff_0) ).

thf(2060,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( B
       != ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Int_Onumber__ring )
      | ~ ( A @ class_Ring__and__Field_Oidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[619]) ).

thf(324,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( C @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I10_J_0) ).

thf(1400,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( C @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[324]) ).

thf(34,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__maxI2_0) ).

thf(707,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( B @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(369,axiom,
    tc_nat @ class_Orderings_Obot,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Orderings_Obot) ).

thf(1499,plain,
    tc_nat @ class_Orderings_Obot,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[369]) ).

thf(19,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__less__le_2) ).

thf(673,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(505,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__0__left_0) ).

thf(1790,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[505]) ).

thf(236,axiom,
    tc_nat @ class_Orderings_Oorder,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Orderings_Oorder) ).

thf(1188,plain,
    tc_nat @ class_Orderings_Oorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).

thf(169,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower__class_Opower ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__eq__0__iff_2) ).

thf(1038,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_Power_Opower__class_Opower ) ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).

thf(267,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__le__double__add__iff__zero__le__single__add_0) ).

thf(1262,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).

thf(552,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower ) ) @ ( A @ ( C @ ( B @ c_Power_Opower__class_Opower ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__mono_0) ).

thf(1901,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ( A @ ( A @ ( C @ ( D @ c_Power_Opower__class_Opower ) ) @ ( A @ ( C @ ( B @ c_Power_Opower__class_Opower ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[552]) ).

thf(133,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C @ ( D @ ( C @ ( B @ ( A @ c_List_Oappend ) ) @ c_List_Oappend ) ) )
      = ( C @ ( C @ ( D @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Oappend ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append__eq__append__conv2_4) ).

thf(951,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C @ ( D @ ( C @ ( B @ ( A @ c_List_Oappend ) ) @ c_List_Oappend ) ) )
      = ( C @ ( C @ ( D @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Oappend ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).

thf(211,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Abs__apps__eq__Abs__apps__conv_0) ).

thf(1130,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(115,axiom,
    ! [A: $i] :
      ( ( A @ ( A @ c_List_Olist_ONil @ ( A @ c_List_Olist_ONil @ c_List_Oappend ) ) )
      = ( A @ c_List_Olist_ONil ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append__is__Nil__conv_2) ).

thf(907,plain,
    ! [A: $i] :
      ( ( A @ ( A @ c_List_Olist_ONil @ ( A @ c_List_Olist_ONil @ c_List_Oappend ) ) )
      = ( A @ c_List_Olist_ONil ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(282,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__diff__left_0) ).

thf(1300,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).

thf(435,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( C @ ( B @ c_Lambda_OdB_OVar @ ( A @ c_Lambda_Oliftn ) ) )
        = ( B @ c_Lambda_OdB_OVar ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_liftn_Osimps_I1_J_0) ).

thf(1637,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( C @ ( B @ c_Lambda_OdB_OVar @ ( A @ c_Lambda_Oliftn ) ) )
        = ( B @ c_Lambda_OdB_OVar ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[435]) ).

thf(85,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_linorder__neq__iff_1) ).

thf(840,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(503,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__add__diff_0) ).

thf(1785,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[503]) ).

thf(522,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__leE_0) ).

thf(1830,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[522]) ).

thf(492,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
     != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Var__apps__neq__Abs__apps_0) ).

thf(1763,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
     != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[492]) ).

thf(190,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_termination__basic__simps_I4_J_0) ).

thf(1085,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).

thf(444,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A = C )
      | ( ( C @ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) )
        = ( A @ c_Lambda_OdB_OVar ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subst__Var_2) ).

thf(1658,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( A @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A = C )
      | ( ( C @ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) )
        = ( A @ c_Lambda_OdB_OVar ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[444]) ).

thf(480,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( D @ ( C @ ( B @ c_List_Oreplicate ) ) @ ( A @ c_List_Odrop ) ) )
      = ( D @ ( C @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_List_Oreplicate ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_drop__replicate_0) ).

thf(1738,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( D @ ( C @ ( B @ c_List_Oreplicate ) ) @ ( A @ c_List_Odrop ) ) )
      = ( D @ ( C @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_List_Oreplicate ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[480]) ).

thf(248,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I5_J_0) ).

thf(1218,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( B @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).

thf(297,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Nat_Onat_Onat__case ) ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__case__0_0) ).

thf(1336,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Nat_Onat_Onat__case ) ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).

thf(7,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__less__iff__gr__or__eq_0) ).

thf(643,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(48,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__le__neq__trans_0) ).

thf(744,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( B = C )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(464,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__imp__diff__is__add_0) ).

thf(1702,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ ( tc_nat @ ( A @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[464]) ).

thf(249,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( A = B )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__neq__implies__less_0) ).

thf(1221,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( A = B )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).

thf(374,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_Lambda_OdB @ ( C @ ( B @ ( c_Lambda_Obeta @ c_ListOrder_Ostep1 ) ) ) )
      | ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_apps__preserves__betas_0) ).

thf(1506,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_Lambda_OdB @ ( C @ ( B @ ( c_Lambda_Obeta @ c_ListOrder_Ostep1 ) ) ) )
      | ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[374]) ).

thf(96,axiom,
    ! [B: $i,A: $i] : ( B @ ( B @ c_List_Olist_ONil @ ( A @ c_List_Olistsp ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_listsp_ONil_0) ).

thf(864,plain,
    ! [B: $i,A: $i] : ( B @ ( B @ c_List_Olist_ONil @ ( A @ c_List_Olistsp ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(486,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( B @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I8_J_0) ).

thf(1749,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( B @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( A @ hAPP ) @ hAPP ) @ ( E @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[486]) ).

thf(42,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__le__less__trans_0) ).

thf(727,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(470,axiom,
    ! [A: $i] : ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le0_0) ).

thf(1717,plain,
    ! [A: $i] : ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[470]) ).

thf(544,axiom,
    ! [A: $i] :
      ( ( A @ c_Lambda_OdB_OAbs )
      = ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ c_List_Olist_ONil @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Abs__eq__apps__conv_2) ).

thf(1881,plain,
    ! [A: $i] :
      ( ( A @ c_Lambda_OdB_OAbs )
      = ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ c_List_Olist_ONil @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[544]) ).

thf(617,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_List_Olist_OCons ) ) @ c_List_Olistsum ) )
        = ( A @ ( C @ c_List_Olistsum ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_listsum_Osimps_I2_J_0) ).

thf(2056,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_List_Olist_OCons ) ) @ c_List_Olistsum ) )
        = ( A @ ( C @ c_List_Olistsum ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[617]) ).

thf(307,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( D @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__case ) ) ) ) )
      = ( E @ ( D @ ( B @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I11_J_0) ).

thf(1358,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( F @ ( E @ ( D @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__case ) ) ) ) )
      = ( E @ ( D @ ( B @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[307]) ).

thf(391,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__strict__increasing2_0) ).

thf(1543,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[391]) ).

thf(493,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__less_0) ).

thf(1766,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[493]) ).

thf(216,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( E @ ( D @ ( C @ c_List_Olist_OCons ) ) @ ( E @ ( D @ ( B @ c_List_Olist_OCons ) ) @ ( A @ c_ListOrder_Ostep1 ) ) ) )
      | ~ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Cons__step1__Cons_4) ).

thf(1141,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( E @ ( D @ ( C @ c_List_Olist_OCons ) ) @ ( E @ ( D @ ( B @ c_List_Olist_OCons ) ) @ ( A @ c_ListOrder_Ostep1 ) ) ) )
      | ~ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).

thf(587,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( B @ ( C @ ( E @ ( D @ ( A @ c_Predicate_Oconversep ) ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_conversepD_0) ).

thf(1981,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( B @ ( C @ ( E @ ( D @ ( A @ c_Predicate_Oconversep ) ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[587]) ).

thf(12,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( B @ class_Orderings_Opreorder )
      | ( B @ ( A @ tc_fun ) @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_fun__Orderings_Opreorder) ).

thf(656,plain,
    ! [B: $i,A: $i] :
      ( ~ ( B @ class_Orderings_Opreorder )
      | ( B @ ( A @ tc_fun ) @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(510,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( B @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I11_J_0) ).

thf(1802,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( B @ ( F @ ( E @ ( A @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[510]) ).

thf(163,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I24_J_0) ).

thf(1023,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( B @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).

thf(449,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__le__mono1_0) ).

thf(1670,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[449]) ).

thf(508,axiom,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Oadd__0_0) ).

thf(1797,plain,
    ! [B: $i,A: $i] :
      ( ( ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[508]) ).

thf(258,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_List_Olistsum ) )
        = ( A @ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_listsum_0) ).

thf(1240,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ c_List_Olistsum ) )
        = ( A @ ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).

thf(150,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( D @ ( C @ ( F @ c_List_Olist_OCons ) ) @ ( D @ ( B @ ( E @ c_List_Olist_OCons ) ) @ ( A @ c_ListOrder_Ostep1 ) ) ) )
      | ( C = B )
      | ( D @ ( C @ ( B @ ( A @ c_ListOrder_Ostep1 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Cons__step1__Cons_3) ).

thf(991,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( D @ ( C @ ( F @ c_List_Olist_OCons ) ) @ ( D @ ( B @ ( E @ c_List_Olist_OCons ) ) @ ( A @ c_ListOrder_Ostep1 ) ) ) )
      | ( C = B )
      | ( D @ ( C @ ( B @ ( A @ c_ListOrder_Ostep1 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(426,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_eq__iff__diff__eq__0_1) ).

thf(1617,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Oab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[426]) ).

thf(570,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__trans_0) ).

thf(1940,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[570]) ).

thf(600,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__eq__self__zero_0) ).

thf(2013,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[600]) ).

thf(233,axiom,
    ! [B: $i,A: $i] :
      ~ ( tc_nat @ ( A @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__add__less1_0) ).

thf(1180,plain,
    ! [B: $i,A: $i] :
      ~ ( tc_nat @ ( A @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(581,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Var__apps__eq__Var__apps__conv_1) ).

thf(1966,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[581]) ).

thf(124,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ c_Lambda_OdB_OAbs )
     != ( B @ c_Lambda_OdB_OVar ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I7_J_0) ).

thf(931,plain,
    ! [B: $i,A: $i] :
      ( ( A @ c_Lambda_OdB_OAbs )
     != ( B @ c_Lambda_OdB_OVar ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(379,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__antisym_0) ).

thf(1517,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[379]) ).

thf(55,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_xt1_I6_J_0) ).

thf(759,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(501,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__le__cancel__right_1) ).

thf(1781,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( C @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[501]) ).

thf(209,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__strict__mono_0) ).

thf(1126,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).

thf(87,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( B @ ( A @ c_List_Olistsp ) ) )
      | ~ ( D @ ( C @ ( A @ c_List_Olistsp ) ) )
      | ( D @ ( D @ ( C @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Olistsp ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append__in__listsp__conv_2) ).

thf(844,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( B @ ( A @ c_List_Olistsp ) ) )
      | ~ ( D @ ( C @ ( A @ c_List_Olistsp ) ) )
      | ( D @ ( D @ ( C @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Olistsp ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(594,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I22_J_0) ).

thf(1998,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[594]) ).

thf(559,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( B = E )
      | ( ( C @ ( B @ ( A @ c_List_Oreplicate ) ) )
       != ( C @ ( E @ ( D @ c_List_Oreplicate ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_replicate__eq__replicate_1) ).

thf(1918,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( B = E )
      | ( ( C @ ( B @ ( A @ c_List_Oreplicate ) ) )
       != ( C @ ( E @ ( D @ c_List_Oreplicate ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[559]) ).

thf(289,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( B @ c_Power_Opower__class_Opower ) ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_power__eq__0__iff_0) ).

thf(1316,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( A @ ( C @ ( B @ c_Power_Opower__class_Opower ) ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Power_Opower )
      | ~ ( A @ class_Ring__and__Field_Omult__zero )
      | ~ ( A @ class_Ring__and__Field_Ono__zero__divisors )
      | ~ ( A @ class_Ring__and__Field_Ozero__neq__one ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).

thf(349,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( A @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( C @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_lift__preserves__beta_H_0) ).

thf(1459,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( A @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( C @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[349]) ).

thf(613,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__diff__conv_1) ).

thf(2045,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[613]) ).

thf(18,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C @ ( D @ ( C @ ( B @ ( A @ c_List_Oappend ) ) @ c_List_Oappend ) ) )
      = ( C @ ( C @ ( D @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Oappend ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append__assoc_0) ).

thf(671,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C @ ( D @ ( C @ ( B @ ( A @ c_List_Oappend ) ) @ c_List_Oappend ) ) )
      = ( C @ ( C @ ( D @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Oappend ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(517,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_right__minus__eq_0) ).

thf(1817,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B = C )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[517]) ).

thf(291,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_appR_0) ).

thf(1321,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( B @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).

thf(557,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__le__iff_0) ).

thf(1913,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[557]) ).

thf(528,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( C @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( C @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_rtranclp_Ortrancl__into__rtrancl_0) ).

thf(1844,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( C @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( D @ ( E @ ( A @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( C @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[528]) ).

thf(25,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__less__asym_0) ).

thf(688,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(442,axiom,
    tc_nat @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Opordered__cancel__ab__semigroup__add) ).

thf(1654,plain,
    tc_nat @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[442]) ).

thf(330,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__nonpos__neg_0) ).

thf(1412,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[330]) ).

thf(345,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( C @ ( F @ ( A @ ( E @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Opwr__0_0) ).

thf(1448,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( C @ ( F @ ( A @ ( E @ ( D @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[345]) ).

thf(237,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( A @ c_OrderedGroup_Osemigroup__add ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_semigroup__add_Oadd__assoc_0) ).

thf(1189,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( A @ c_OrderedGroup_Osemigroup__add ) )
      | ( ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).

thf(158,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( C @ ( D @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Olistsp ) ) )
      | ( C @ ( B @ ( A @ c_List_Olistsp ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append__in__listsp__conv_0) ).

thf(1010,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( C @ ( D @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Olistsp ) ) )
      | ( C @ ( B @ ( A @ c_List_Olistsp ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(407,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__is__0__eq_1) ).

thf(1575,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[407]) ).

thf(512,axiom,
    tc_nat @ class_Ring__and__Field_Ono__zero__divisors,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Ono__zero__divisors) ).

thf(1807,plain,
    tc_nat @ class_Ring__and__Field_Ono__zero__divisors,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[512]) ).

thf(60,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ ( D @ c_Orderings_Oord__class_Omax ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_max__less__iff__conj_1) ).

thf(771,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( B @ ( D @ c_Orderings_Oord__class_Omax ) ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(30,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ ( B @ c_List_Olist_OCons ) ) @ ( A @ c_List_Olist_ONil @ c_List_Oappend ) ) )
      = ( A @ ( C @ ( B @ c_List_Olist_OCons ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append__eq__Cons__conv_4) ).

thf(699,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ ( B @ c_List_Olist_OCons ) ) @ ( A @ c_List_Olist_ONil @ c_List_Oappend ) ) )
      = ( A @ ( C @ ( B @ c_List_Olist_OCons ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(318,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__less__le__mono_0) ).

thf(1385,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( E @ ( C @ c_lessequals ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[318]) ).

thf(358,axiom,
    ! [A: $i] :
      ~ ( tc_nat @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__less__le_1) ).

thf(1476,plain,
    ! [A: $i] :
      ~ ( tc_nat @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[358]) ).

thf(553,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__neg__nonpos_0) ).

thf(1903,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ c_lessequals ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__comm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[553]) ).

thf(360,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__eq__less__or__eq_0) ).

thf(1481,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[360]) ).

thf(198,axiom,
    ! [B: $i,A: $i] : ( B @ ( A @ hAPP ) @ ( B @ c_Lambda_OdB_OVar @ ( A @ c_Type_Otyping ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_typing_OVar_0) ).

thf(1101,plain,
    ! [B: $i,A: $i] : ( B @ ( A @ hAPP ) @ ( B @ c_Lambda_OdB_OVar @ ( A @ c_Type_Otyping ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).

thf(596,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( B @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Omul__pwr_0) ).

thf(2004,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( B @ ( A @ ( F @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( D @ ( C @ ( B @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( D @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[596]) ).

thf(484,axiom,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_double__eq__0__iff_0) ).

thf(1745,plain,
    ! [B: $i,A: $i] :
      ( ( B
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ( ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
       != ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[484]) ).

thf(605,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
       != ( D @ ( C @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I2_J_1) ).

thf(2026,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
       != ( D @ ( C @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[605]) ).

thf(381,axiom,
    tc_nat @ class_OrderedGroup_Ocomm__monoid__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Ocomm__monoid__add) ).

thf(1523,plain,
    tc_nat @ class_OrderedGroup_Ocomm__monoid__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[381]) ).

thf(398,axiom,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( F @ ( E @ ( D @ ( C @ c_List_Olist_OCons ) ) @ ( B @ ( A @ c_List_Ofoldl ) ) ) ) )
      = ( E @ ( F @ ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ c_List_Ofoldl ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_foldl__Cons_0) ).

thf(1558,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( F @ ( E @ ( D @ ( C @ c_List_Olist_OCons ) ) @ ( B @ ( A @ c_List_Ofoldl ) ) ) ) )
      = ( E @ ( F @ ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ c_List_Ofoldl ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[398]) ).

thf(180,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__le__cancel__right_0) ).

thf(1062,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(375,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__gr__0_2) ).

thf(1508,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[375]) ).

thf(364,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( C @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( D @ ( E @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( C @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_rtranclp__trans_0) ).

thf(1491,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( E @ ( C @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ~ ( D @ ( E @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( C @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[364]) ).

thf(457,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__gr__0_0) ).

thf(1689,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[457]) ).

thf(117,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__le__not__le_1) ).

thf(914,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(498,axiom,
    v_u____ @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_CHAINED_0_01) ).

thf(1777,plain,
    v_u____ @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[498]) ).

thf(377,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__le__mono2_0) ).

thf(1513,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[377]) ).

thf(592,axiom,
    ! [A: $i] :
      ( ( A @ c_String_Ochar_Ochar__size )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_char__size_0) ).

thf(1994,plain,
    ! [A: $i] :
      ( ( A @ c_String_Ochar_Ochar__size )
      = ( tc_nat @ c_HOL_Ozero__class_Ozero ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[592]) ).

thf(602,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) )
      = ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( tc_Lambda_OdB @ ( tc_nat @ ( tc_Lambda_OdB @ ( D @ ( tc_Lambda_OdB @ ( tc_nat @ tc_fun ) @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( c_Lambda_Osubst @ c_COMBC ) ) ) ) @ c_COMBC ) ) ) ) @ c_List_Omap ) ) ) @ ( D @ ( C @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subst__map_0) ).

thf(2019,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( D @ ( C @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) )
      = ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( tc_Lambda_OdB @ ( tc_nat @ ( tc_Lambda_OdB @ ( D @ ( tc_Lambda_OdB @ ( tc_nat @ tc_fun ) @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( c_Lambda_Osubst @ c_COMBC ) ) ) ) @ c_COMBC ) ) ) ) @ c_List_Omap ) ) ) @ ( D @ ( C @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[602]) ).

thf(421,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Lambda_OdB @ ( B @ ( c_InductTermi_OIT @ c_List_Olistsp ) ) )
      | ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_IT_OVar_0) ).

thf(1607,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_Lambda_OdB @ ( B @ ( c_InductTermi_OIT @ c_List_Olistsp ) ) )
      | ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[421]) ).

thf(57,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__supE_0) ).

thf(763,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ ( D @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(112,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__eq__iff_0) ).

thf(899,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(445,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_comm__monoid__add_Omult__1__right_0) ).

thf(1661,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[445]) ).

thf(82,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C @ ( D @ ( C @ ( B @ ( A @ c_List_Olist_OCons ) ) @ c_List_Oappend ) ) )
      = ( C @ ( C @ ( D @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Olist_OCons ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append__eq__Cons__conv_5) ).

thf(832,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C @ ( D @ ( C @ ( B @ ( A @ c_List_Olist_OCons ) ) @ c_List_Oappend ) ) )
      = ( C @ ( C @ ( D @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Olist_OCons ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(23,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_xt1_I10_J_0) ).

thf(684,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(549,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( A @ ( B @ c_OrderedGroup_Omonoid__add ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_monoid__add_Oadd__0__left_0) ).

thf(1893,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( A @ ( B @ c_OrderedGroup_Omonoid__add ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[549]) ).

thf(609,axiom,
    tc_nat @ class_Ring__and__Field_Osemiring__0,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Osemiring__0) ).

thf(2036,plain,
    tc_nat @ class_Ring__and__Field_Osemiring__0,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[609]) ).

thf(286,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL )
      | ( C @ ( B @ c_Lambda_OdB_OVar @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_subst__Var__IT_0) ).

thf(1309,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL )
      | ( C @ ( B @ c_Lambda_OdB_OVar @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).

thf(37,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__eq__refl_0) ).

thf(716,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(425,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ c_Lambda_OdB_OVar @ c_Lambda_Osubstn ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Lambda_Oliftn ) ) ) )
      | ( tc_nat @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_substn_Osimps_I1_J_1) ).

thf(1614,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( B @ ( A @ c_Lambda_OdB_OVar @ c_Lambda_Osubstn ) ) )
        = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ c_Lambda_Oliftn ) ) ) )
      | ( tc_nat @ ( A @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[425]) ).

thf(72,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( D @ c_List_Olist_ONil ) )
      | ( ( A @ c_List_Olist_ONil )
       != ( A @ ( D @ ( C @ ( B @ c_List_Omap ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Nil__is__map__conv_0) ).

thf(800,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C
        = ( D @ c_List_Olist_ONil ) )
      | ( ( A @ c_List_Olist_ONil )
       != ( A @ ( D @ ( C @ ( B @ c_List_Omap ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(205,axiom,
    tc_nat @ class_Power_Opower,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Power_Opower) ).

thf(1116,plain,
    tc_nat @ class_Power_Opower,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(212,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( D @ ( B @ ( B @ ( B @ ( A @ c_Predicate_Oconversep ) ) @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( C @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_rtranclp__converseD_0) ).

thf(1133,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( D @ ( B @ ( B @ ( B @ ( A @ c_Predicate_Oconversep ) ) @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( D @ ( C @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).

thf(302,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__0__less__iff_1) ).

thf(1348,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[302]) ).

thf(437,axiom,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__0__right_0) ).

thf(1642,plain,
    ! [B: $i,A: $i] :
      ( ( ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
        = B )
      | ~ ( A @ class_OrderedGroup_Ogroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[437]) ).

thf(497,axiom,
    ! [A: $i] :
      ( ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Nat_Osemiring__1__class_Oof__nat ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__0_0) ).

thf(1774,plain,
    ! [A: $i] :
      ( ( ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_Nat_Osemiring__1__class_Oof__nat ) )
        = ( A @ c_HOL_Ozero__class_Ozero ) )
      | ~ ( A @ class_Ring__and__Field_Osemiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[497]) ).

thf(452,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ c_Lambda_OdB_OAbs @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__case ) ) ) ) )
      = ( D @ ( C @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osimps_I12_J_0) ).

thf(1678,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( D @ c_Lambda_OdB_OAbs @ ( C @ ( B @ ( A @ c_Lambda_OdB_OdB__case ) ) ) ) )
      = ( D @ ( C @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[452]) ).

thf(140,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( E @ ( C @ ( A @ c_ListOrder_Ostep1 ) ) ) )
      | ( D @ ( D @ ( E @ ( B @ c_List_Oappend ) ) @ ( D @ ( C @ ( B @ c_List_Oappend ) ) @ ( A @ c_ListOrder_Ostep1 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append__step1I_1) ).

thf(967,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( D @ ( E @ ( C @ ( A @ c_ListOrder_Ostep1 ) ) ) )
      | ( D @ ( D @ ( E @ ( B @ c_List_Oappend ) ) @ ( D @ ( C @ ( B @ c_List_Oappend ) ) @ ( A @ c_ListOrder_Ostep1 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(252,axiom,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Nat_Oadd__0__right_0) ).

thf(1229,plain,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).

thf(322,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__add__left__cancel__le_0) ).

thf(1395,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( B @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[322]) ).

thf(542,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__le__less__mono_0) ).

thf(1876,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( E @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( E @ ( D @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__cancel__ab__semigroup__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[542]) ).

thf(589,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
      = ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ c_List_Olist_ONil @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_App__eq__foldl__conv_4) ).

thf(1986,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) )
      = ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ c_List_Olist_ONil @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[589]) ).

thf(97,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C @ ( D @ ( C @ ( B @ ( A @ c_List_Olist_OCons ) ) @ c_List_Oappend ) ) )
      = ( C @ ( C @ ( D @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Olist_OCons ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append__Cons_0) ).

thf(865,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( C @ ( D @ ( C @ ( B @ ( A @ c_List_Olist_OCons ) ) @ c_List_Oappend ) ) )
      = ( C @ ( C @ ( D @ ( B @ c_List_Oappend ) ) @ ( A @ c_List_Olist_OCons ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(408,axiom,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__is__0__eq_0) ).

thf(1578,plain,
    ! [B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) )
       != ( tc_nat @ c_HOL_Ozero__class_Ozero ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[408]) ).

thf(577,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_List_Oreplicate ) ) )
       != ( C @ c_List_Olist_ONil ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_replicate__empty_0) ).

thf(1957,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A
        = ( tc_nat @ c_HOL_Ozero__class_Ozero ) )
      | ( ( C @ ( B @ ( A @ c_List_Oreplicate ) ) )
       != ( C @ c_List_Olist_ONil ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[577]) ).

thf(261,axiom,
    v_a____ @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Beta_I3_J_0) ).

thf(1248,plain,
    v_a____ @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).

thf(68,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_List_Olist_ONil ) )
      | ( ( A @ c_List_Olist_ONil )
       != ( A @ ( C @ ( B @ c_List_Oappend ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Nil__is__append__conv_1) ).

thf(790,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( A @ c_List_Olist_ONil ) )
      | ( ( A @ c_List_Olist_ONil )
       != ( A @ ( C @ ( B @ c_List_Oappend ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(615,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( C @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Omul__d_0) ).

thf(2050,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( A @ ( C @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( E @ ( D @ ( C @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( E @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( C @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[615]) ).

thf(98,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__supI1_0) ).

thf(867,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( A @ ( D @ ( C @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(489,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Abs__apps__eq__Abs__apps__conv_1) ).

thf(1757,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( B = D )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( D @ ( C @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[489]) ).

thf(309,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( E @ ( D @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( C @ c_Type_Otyping ) ) )
      | ~ ( B @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( E @ ( D @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( C @ c_Type_Otyping ) ) )
      | ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_var__app__type__eq_0) ).

thf(1363,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( E @ ( D @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( C @ c_Type_Otyping ) ) )
      | ~ ( B @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( E @ ( D @ c_Lambda_OdB_OVar @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( C @ c_Type_Otyping ) ) )
      | ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[309]) ).

thf(367,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__gr__0_1) ).

thf(1495,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[367]) ).

thf(548,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( B @ ( C @ ( E @ ( D @ ( A @ c_Predicate_Oconversep ) ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_conversep_Ocases_0) ).

thf(1891,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( B @ ( C @ ( E @ ( D @ ( A @ c_Predicate_Oconversep ) ) @ hAPP ) @ hAPP ) @ hBOOL )
      | ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[548]) ).

thf(137,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__sup__iff_2) ).

thf(960,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(459,axiom,
    v_u____ @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_uIT_0) ).

thf(1693,plain,
    v_u____ @ ( c_InductTermi_OIT @ hAPP ) @ hBOOL,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[459]) ).

thf(623,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__diff__conv2_1) ).

thf(2070,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ~ ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_lessequals ) ) )
      | ( tc_nat @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[623]) ).

thf(229,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_even__less__0__iff_0) ).

thf(1172,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__idom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).

thf(111,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_not__less__iff__gr__or__eq_1) ).

thf(897,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( C @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(177,axiom,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_Nitpick_Onat__gcd ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_nat__gcd_Osimps_0) ).

thf(1056,plain,
    ! [A: $i] :
      ( ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ c_Nitpick_Onat__gcd ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(16,axiom,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( A @ c_Orderings_Obot__class_Obot @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Obot ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_bot__least_0) ).

thf(664,plain,
    ! [B: $i,A: $i] :
      ( ( A @ ( B @ ( A @ c_Orderings_Obot__class_Obot @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Obot ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(278,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_class__semiring_Osemiring__rules_I23_J_0) ).

thf(1288,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( C @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_Ring__and__Field_Ocomm__semiring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).

thf(467,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__diff__assoc2_0) ).

thf(1710,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( A @ ( B @ c_lessequals ) ) )
      | ( ( C @ ( tc_nat @ ( B @ ( A @ c_HOL_Ominus__class_Ominus ) ) @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( tc_nat @ ( B @ ( C @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[467]) ).

thf(240,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) )
      = ( C @ ( B @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_lift_Osimps_I2_J_0) ).

thf(1197,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) )
      = ( C @ ( B @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) @ ( C @ ( A @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(39,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( C @ ( B @ ( D @ c_List_Oappend ) ) @ ( A @ c_List_Olistsp ) ) )
      | ( C @ ( B @ ( A @ c_List_Olistsp ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_append__in__listsp__conv_1) ).

thf(720,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( C @ ( B @ ( D @ c_List_Oappend ) ) @ ( A @ c_List_Olistsp ) ) )
      | ( C @ ( B @ ( A @ c_List_Olistsp ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(226,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( A @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL )
      | ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( C @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_apps__preserves__beta_0) ).

thf(1165,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( A @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL )
      | ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( C @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).

thf(603,axiom,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( C @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I6_J_0) ).

thf(2021,plain,
    ! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( G @ ( F @ ( C @ ( E @ ( D @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( C @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[603]) ).

thf(620,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__strict__increasing_0) ).

thf(2063,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Ocomm__monoid__add )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[620]) ).

thf(106,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( C @ ( C @ ( B @ ( A @ c_Predicate_Oconversep ) ) @ c_Predicate_Oconversep ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_conversep__conversep_0) ).

thf(886,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B @ ( C @ ( C @ ( B @ ( A @ c_Predicate_Oconversep ) ) @ c_Predicate_Oconversep ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(482,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_termination__basic__simps_I3_J_0) ).

thf(1742,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_lessequals ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_lessequals ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[482]) ).

thf(123,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( E @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__eqI_1) ).

thf(928,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( E @ ( D @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( ( A @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) )
       != ( A @ ( E @ ( D @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(304,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( A @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( C @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_appL_0) ).

thf(1352,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( C @ ( A @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL )
      | ( B @ ( C @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( c_Lambda_Obeta @ hAPP ) @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[304]) ).

thf(389,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( A @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Nat_Osemiring__1__class_Oof__nat ) )
        = ( A @ ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring__1 ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__diff_0) ).

thf(1537,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( C @ c_lessequals ) ) )
      | ( ( A @ ( tc_nat @ ( C @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Nat_Osemiring__1__class_Oof__nat ) )
        = ( A @ ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_HOL_Ominus__class_Ominus ) ) ) )
      | ~ ( A @ class_Ring__and__Field_Oring__1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[389]) ).

thf(562,axiom,
    ! [A: $i] :
      ( A
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_natgb_Oadd__r0__iff_1) ).

thf(1925,plain,
    ! [A: $i] :
      ( A
      = ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[562]) ).

thf(468,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( B @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( E @ c_Lambda_OdB_OVar @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osplit__asm_0) ).

thf(1713,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( B @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( E @ c_Lambda_OdB_OVar @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[468]) ).

thf(100,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( D @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_max__diff__distrib__left_0) ).

thf(871,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_HOL_Ominus__class_Ominus ) ) )
        = ( A @ ( A @ ( D @ ( C @ c_HOL_Ominus__class_Ominus ) ) @ ( A @ ( D @ ( B @ c_HOL_Ominus__class_Ominus ) ) @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(373,axiom,
    tc_nat @ class_Ring__and__Field_Ocomm__semiring__1,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__Ring__and__Field_Ocomm__semiring__1) ).

thf(1505,plain,
    tc_nat @ class_Ring__and__Field_Ocomm__semiring__1,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[373]) ).

thf(207,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) )
        = ( A @ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldr ) ) ) ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_foldl__foldr1__lemma_0) ).

thf(1120,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldl ) ) ) ) )
        = ( A @ ( A @ ( A @ c_HOL_Ozero__class_Ozero @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ c_List_Ofoldr ) ) ) ) @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).

thf(194,axiom,
    ! [A: $i] :
      ( ~ ( tc_Lambda_OdB @ ( A @ ( c_InductTermi_OIT @ c_List_Olistsp ) ) )
      | ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( A @ ( tc_Lambda_OdB @ ( tc_nat @ ( tc_Lambda_OdB @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Lambda_Olift @ c_COMBC ) ) ) ) @ c_List_Omap ) ) ) @ ( c_InductTermi_OIT @ c_List_Olistsp ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_lifts__IT_0) ).

thf(1093,plain,
    ! [A: $i] :
      ( ~ ( tc_Lambda_OdB @ ( A @ ( c_InductTermi_OIT @ c_List_Olistsp ) ) )
      | ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( A @ ( tc_Lambda_OdB @ ( tc_nat @ ( tc_Lambda_OdB @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( c_Lambda_Olift @ c_COMBC ) ) ) ) @ c_List_Omap ) ) ) @ ( c_InductTermi_OIT @ c_List_Olistsp ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).

thf(565,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( C @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_apps__eq__tail__conv_0) ).

thf(1930,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A = C )
      | ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) )
       != ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( C @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[565]) ).

thf(256,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_of__nat__less__iff_0) ).

thf(1236,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( A @ ( C @ c_Nat_Osemiring__1__class_Oof__nat ) @ ( A @ ( B @ c_Nat_Osemiring__1__class_Oof__nat ) @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).

thf(148,axiom,
    ! [B: $i,A: $i] :
      ( ( B @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ c_Transitive__Closure_Ortranclp ) )
      = ( B @ ( A @ c_Transitive__Closure_Ortranclp ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_rtranclp__idemp_0) ).

thf(987,plain,
    ! [B: $i,A: $i] :
      ( ( B @ ( B @ ( A @ c_Transitive__Closure_Ortranclp ) @ c_Transitive__Closure_Ortranclp ) )
      = ( B @ ( A @ c_Transitive__Closure_Ortranclp ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(387,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_diff__cancel2_0) ).

thf(1532,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Ominus__class_Ominus ) ) )
      = ( tc_nat @ ( C @ ( A @ c_HOL_Ominus__class_Ominus ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[387]) ).

thf(94,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_le__maxI1_0) ).

thf(860,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ ( B @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(370,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( A @ v_sko__List__Xupt__Xinduct__1 @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_upt_Oinduct_0) ).

thf(1500,plain,
    ! [B: $i,A: $i] :
      ( ~ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( A @ hAPP ) @ hBOOL )
      | ( A @ v_sko__List__Xupt__Xinduct__1 @ ( A @ hAPP ) @ hBOOL )
      | ( B @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[370]) ).

thf(561,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( B @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Predicate_Oconversep ) ) @ c_Wellfounded_Oaccp ) ) )
      | ~ ( tc_Lambda_OdB @ ( B @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Predicate_Oconversep ) ) @ c_Wellfounded_Oaccp ) ) )
      | ~ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Predicate_Oconversep ) ) @ c_Wellfounded_Oaccp ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_double__induction__lemma_0) ).

thf(1923,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( B @ ( A @ c_Lambda_OdB_OAbs @ ( c_Lambda_OdB_OApp @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Predicate_Oconversep ) ) @ c_Wellfounded_Oaccp ) ) )
      | ~ ( tc_Lambda_OdB @ ( B @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Predicate_Oconversep ) ) @ c_Wellfounded_Oaccp ) ) )
      | ~ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( C @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( B @ ( A @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( c_Lambda_Obeta @ c_Predicate_Oconversep ) ) @ c_Wellfounded_Oaccp ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[561]) ).

thf(146,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_min__max_Ole__supI_0) ).

thf(982,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( D @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( D @ ( C @ c_lessequals ) ) )
      | ( A @ ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ c_lessequals ) ) )
      | ~ ( A @ class_Orderings_Olinorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(338,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) )
      = ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( tc_Lambda_OdB @ ( tc_nat @ ( tc_Lambda_OdB @ ( C @ ( c_Lambda_Olift @ c_COMBC ) ) ) ) @ c_List_Omap ) ) ) @ ( C @ ( A @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_lift__map_0) ).

thf(1431,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( A @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) )
      = ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( B @ ( tc_Lambda_OdB @ ( tc_nat @ ( tc_Lambda_OdB @ ( C @ ( c_Lambda_Olift @ c_COMBC ) ) ) ) @ c_List_Omap ) ) ) @ ( C @ ( A @ ( c_Lambda_Olift @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[338]) ).

thf(420,axiom,
    tc_nat @ class_OrderedGroup_Osemigroup__add,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_nat__OrderedGroup_Osemigroup__add) ).

thf(1606,plain,
    tc_nat @ class_OrderedGroup_Osemigroup__add,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[420]) ).

thf(73,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_order__eq__iff_2) ).

thf(803,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ~ ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( B = C )
      | ~ ( A @ class_Orderings_Oorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(572,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_trans__less__add1_0) ).

thf(1945,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( B @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( C @ ( B @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[572]) ).

thf(62,axiom,
    ! [B: $i,A: $i] :
      ( A
      = ( B @ ( A @ ( B @ c_List_Olist_ONil @ c_List_Oappend ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_eq__Nil__appendI_0) ).

thf(775,plain,
    ! [B: $i,A: $i] :
      ( A
      = ( B @ ( A @ ( B @ c_List_Olist_ONil @ c_List_Oappend ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(332,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_List_Oappend ) ) @ c_List_Olistsum ) )
        = ( A @ ( C @ c_List_Olistsum ) @ ( A @ ( B @ c_List_Olistsum ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_listsum__append_0) ).

thf(1416,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( A @ ( A @ ( C @ ( B @ c_List_Oappend ) ) @ c_List_Olistsum ) )
        = ( A @ ( C @ c_List_Olistsum ) @ ( A @ ( B @ c_List_Olistsum ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) ) )
      | ~ ( A @ class_OrderedGroup_Omonoid__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[332]) ).

thf(515,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_pos__add__strict_0) ).

thf(1812,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ ( D @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Ring__and__Field_Oordered__semidom ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[515]) ).

thf(354,axiom,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_gb__semiring_Osemiring__rules_I25_J_0) ).

thf(1468,plain,
    ! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( I @ ( H @ ( G @ ( F @ ( E @ ( A @ c_Groebner__Basis_Ogb__semiring ) ) ) ) ) )
      | ( ( D @ ( C @ ( A @ hAPP ) @ hAPP ) @ ( B @ ( A @ hAPP ) @ hAPP ) )
        = ( D @ ( C @ ( B @ ( A @ hAPP ) @ hAPP ) @ ( A @ hAPP ) @ hAPP ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[354]) ).

thf(104,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_less__le__not__le_2) ).

thf(882,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( A @ ( C @ ( B @ c_lessequals ) ) )
      | ( A @ ( B @ ( C @ c_lessequals ) ) )
      | ( A @ ( C @ ( B @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_Orderings_Opreorder ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(539,axiom,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_max__add__distrib__left_0) ).

thf(1870,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( D @ ( A @ ( C @ ( B @ c_Orderings_Oord__class_Omax ) ) @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) )
        = ( A @ ( D @ ( C @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( D @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_Orderings_Oord__class_Omax ) ) ) )
      | ~ ( A @ class_OrderedGroup_Opordered__ab__semigroup__add__imp__le ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[539]) ).

thf(351,axiom,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_zero__less__double__add__iff__zero__less__single__add_0) ).

thf(1463,plain,
    ! [B: $i,A: $i] :
      ( ~ ( A @ ( B @ ( B @ ( A @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ( A @ ( B @ ( A @ c_HOL_Ozero__class_Ozero @ c_HOL_Oord__class_Oless ) ) )
      | ~ ( A @ class_OrderedGroup_Olordered__ab__group__add ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[351]) ).

thf(402,axiom,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( B @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( E @ c_Lambda_OdB_OVar @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_dB_Osplit_0) ).

thf(1565,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( E @ ( B @ hAPP ) @ ( A @ hAPP ) @ hBOOL )
      | ~ ( t_a @ ( E @ c_Lambda_OdB_OVar @ ( D @ ( C @ ( B @ c_Lambda_OdB_OdB__case ) ) ) ) @ ( A @ hAPP ) @ hBOOL ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[402]) ).

thf(530,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_add__less__mono1_0) ).

thf(1848,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( tc_nat @ ( C @ ( A @ c_HOL_Oord__class_Oless ) ) )
      | ( tc_nat @ ( B @ ( C @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ ( B @ ( A @ ( tc_nat @ c_HOL_Oplus__class_Oplus @ hAPP ) @ hAPP ) @ c_HOL_Oord__class_Oless ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[530]) ).

thf(618,axiom,
    v_T_H____ @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( v_as____ @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( v_a____ @ ( v_r____ @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( tc_Type_Otype @ ( v_T____ @ ( v_i____ @ ( v_e____ @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_CHAINED_0_02) ).

thf(2059,plain,
    v_T_H____ @ ( tc_Lambda_OdB @ ( tc_Lambda_OdB @ ( v_as____ @ ( tc_nat @ c_HOL_Ozero__class_Ozero @ ( v_a____ @ ( v_r____ @ ( c_Lambda_Osubst @ hAPP ) @ hAPP ) @ hAPP ) @ ( c_Lambda_OdB_OApp @ c_List_Ofoldl ) ) ) ) @ ( tc_Type_Otype @ ( v_T____ @ ( v_i____ @ ( v_e____ @ c_Type_Oshift ) ) ) @ c_Type_Otyping ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[618]) ).

thf(53,axiom,
    tc_bool @ class_Orderings_Opreorder,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clsarity_bool__Orderings_Opreorder) ).

thf(755,plain,
    tc_bool @ class_Orderings_Opreorder,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(20,axiom,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_List_Olist_ONil ) )
      | ( ( A @ c_List_Olist_ONil )
       != ( A @ ( C @ ( B @ c_List_Oappend ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cls_Nil__is__append__conv_0) ).

thf(676,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( B
        = ( A @ c_List_Olist_ONil ) )
      | ( ( A @ c_List_Olist_ONil )
       != ( A @ ( C @ ( B @ c_List_Oappend ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(3712,plain,
    $false,
    inference(e,[status(thm)],[892,1036,809,1718,1411,1823,1454,1633,2072,1729,846,1899,1840,1850,941,777,666,1554,1686,909,628,1031,1083,962,1950,1549,1855,977,1158,1005,1068,994,1422,1205,1269,1041,1443,1808,741,1073,1971,1793,1655,1137,1735,797,1988,747,1640,945,1954,2042,1354,1497,1705,1078,1426,1835,1001,1243,709,1046,1613,1724,1366,1254,651,829,1095,782,1441,2010,1935,1650,1645,1921,2067,1529,1628,2039,1760,785,1907,1323,949,1409,724,1330,1860,1264,702,751,1313,1279,1832,934,1545,817,1800,1379,1087,1319,1609,969,1568,2024,1494,1963,1778,1232,638,853,2007,1489,924,1465,1109,646,1858,1151,2034,1581,1283,1051,1302,756,1664,1682,834,1828,1978,1383,1020,1960,806,1169,1928,1215,734,956,1886,1622,1526,660,1345,1433,1405,1600,1896,877,1104,1697,1975,1564,1604,1691,1438,793,1199,1338,2047,1915,1783,1311,1067,1730,1153,761,1768,714,1868,1274,1387,894,1091,1343,1502,1666,1487,693,2032,729,1238,1123,862,1570,1174,1883,1560,916,1938,1178,1888,1457,1624,1787,1210,1128,1281,1392,1291,1012,1837,1602,1195,1407,1740,1910,842,1573,974,1592,1983,1762,1534,1328,1064,1672,1540,1878,1719,869,1755,989,1397,1360,1350,1936,697,1089,1234,874,1143,1772,1259,1111,1227,980,1587,1567,884,1160,753,1182,1461,1424,1584,1810,1446,1207,679,1419,1493,1683,1964,1942,901,686,1075,1414,2053,1842,1155,1744,837,997,1099,701,1297,1451,1552,1192,1060,984,1947,1874,1429,1246,1715,1825,1114,1478,1249,1266,1231,1333,1991,1202,1286,1071,1515,1732,926,1814,938,633,1779,765,1846,1054,1026,1167,953,1185,773,1996,662,718,888,1135,2001,1146,1680,824,1927,856,1595,1374,1727,1510,819,904,1007,732,1389,1620,1752,1652,2016,826,1520,1256,958,1403,1795,1701,641,936,2029,1707,787,1865,1635,919,704,947,1224,1356,1675,1139,1503,1271,1371,1058,1107,1117,1524,1048,1722,1376,2065,681,1853,1589,1933,880,649,1043,1969,1307,1471,1080,858,1630,1162,1862,848,1325,1029,626,1556,943,1293,1952,1820,890,1667,1149,964,1177,1381,2075,1474,1699,1528,780,1015,653,658,1562,1213,768,695,636,1044,1000,1435,851,1102,1694,922,1973,1805,1626,722,1034,1583,1467,795,1905,2037,1277,812,749,1340,1472,1018,1304,1873,1611,971,2012,1647,1252,739,1368,1531,1484,1098,1547,1748,1003,1770,690,1598,2060,1400,707,1499,673,1790,1188,1038,1262,1901,951,1130,907,1300,1637,840,1785,1830,1763,1085,1658,1738,1218,1336,643,744,1702,1221,1506,864,1749,727,1717,1881,2056,1358,1543,1766,1141,1981,656,1802,1023,1670,1797,1240,991,1617,1940,2013,1180,1966,931,1517,759,1781,1126,844,1998,1918,1316,1459,2045,671,1817,1321,1913,1844,688,1654,1412,1448,1189,1010,1575,1807,771,699,1385,1476,1903,1481,1101,2004,1745,2026,1523,1558,1062,1508,1491,1689,914,1777,1513,1994,2019,1607,763,899,1661,832,684,1893,2036,1309,716,1614,800,1116,1133,1348,1642,1774,1678,967,1229,1395,1876,1986,865,1578,1957,1248,790,2050,867,1757,1363,1495,1891,960,1693,2070,1172,897,1056,664,1288,1710,1197,720,1165,2021,2063,886,1742,928,1352,1537,1925,1713,871,1505,1120,1093,1930,1236,987,1532,860,1500,1923,982,1431,1606,803,1945,775,1416,1812,1468,882,1870,1463,1565,1848,2059,755,676]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.01  % Problem  : LCL836-1 : TPTP v9.3.1. Released v4.1.0.
% 0.00/0.04  % Command  : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p  --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.06/0.32  % Computer : n012.cluster.edu
% 0.06/0.32  % Model    : x86_64 x86_64
% 0.06/0.32  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.06/0.32  % Memory   : 8046.5625MB
% 0.06/0.32  % OS       : Linux 6.8.0-71-generic
% 0.06/0.32  % CPULimit : 300
% 0.06/0.32  % WCLimit  : 300
% 0.06/0.32  % DateTime : Fri Sep 25 23:41:22 UTC 2026
% 0.06/0.32  % CPUTime  : 
% 0.06/0.32  Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p  --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.39/0.60  % [INFO] 	 Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ... 
% 0.71/0.77  % [INFO] 	 Parsing done (176ms). 
% 0.71/0.78  % [INFO] 	 Running in sequential loop mode. 
% 1.40/0.96  % [INFO] 	 eprover registered as external prover. 
% 1.40/0.97  % [INFO] 	 Scanning for conjecture ... 
% 1.64/1.06  % [INFO] 	 Found a conjecture (or negated_conjecture) and 624 axioms. Running axiom selection ... 
% 2.08/1.18  % [INFO] 	 Axiom selection finished. Selected 624 axioms (removed 0 axioms). 
% 2.86/1.32  % [INFO] 	 Problem is propositional (TPTP CNF). 
% 2.86/1.33  % [INFO] 	 Type checking passed. 
% 2.86/1.33  % [CONFIG] 	 Using configuration: timeout(300) with strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>.  Searching for refutation ... 
% 18.67/5.15  % External prover 'e' found a proof!
% 18.67/5.15  % [INFO] 	 Killing All external provers ... 
% 18.67/5.15  % Time passed: 4752ms (effective reasoning time: 4365ms)
% 18.67/5.15  % Axioms used in derivation (624): cls_Nil__is__map__conv_1, cls_semigroup__add__class_Oaxioms_0, cls_dB_Osplit_0, cls_add__0__left_0, cls_xt1_I9_J_0, cls_linorder__antisym__conv2_1, cls_add__pos__nonneg_0, cls_subst__lt_0, cls_nat_Oinduct_0, cls_subst__map_0, cls_dB_Osimps_I6_J_0, clsarity_nat__OrderedGroup_Oab__semigroup__add, cls_less__diff__iff_1, cls_less__iff__diff__less__0_0, cls_dB_Orecs_I1_J_0, cls_le__0__eq_1, cls_natgb_Oadd__r0__iff_1, cls_apps__eq__Abs__conv_0, cls_order__le__less__trans_0, cls_of__nat__add_0, cls_add__less__cancel__left_1, cls_gb__semiring_Omul__0_0, cls_eq__imp__le_0, cls_Cons__eq__append__conv_5, cls_Var__eq__apps__conv_1, cls_le__diff__iff_0, cls_gb__semiring_Osemiring__rules_I12_J_0, cls_of__nat__0_0, cls_gb__semiring_Osemiring__rules_I7_J_0, cls_class__ringb_Oadd__r0__iff_1, cls_lift_Osimps_I2_J_0, cls_subst__App_0, cls_order__less__le__trans_0, cls_order__eq__iff_0, cls_Cons__step1__Cons_0, cls_le__add__diff__inverse2_0, cls_Nat_Odiff__diff__eq_0, cls_subst__eq_0, cls_le__diff__conv_1, cls_liftn_Osimps_I1_J_1, cls_xt1_I12_J_0, cls_COMBC__def_0, cls_map__replicate_0, cls_append__self__conv_0, cls_dB_Osimps_I8_J_0, cls_subject__reduction_0, cls_min__max_Ole__sup__iff_2, cls_termi__implies__IT_0, cls_nat__add__left__cancel__less_1, cls_power__eq__0__iff_1, cls_termination__basic__simps_I5_J_0, cls_eq__diff__iff_0, cls_append__step1I_0, cls_replicate__empty_0, cls_monoid__add_Ofoldl__absorb0_0, cls_conversepD_0, cls_liftn_Osimps_I2_J_0, cls_CHAINED_0, cls_of__nat__less__iff_0, cls_Cons__eq__map__conv_3, clsarity_fun__Orderings_Oorder, cls_class__semiring_Osemiring__rules_I20_J_0, cls_diff__add__cancel_0, cls_accp__downwards_0, cls_class__semiring_Osemiring__rules_I21_J_0, cls_list_Oinject_0, cls_add__increasing2_0, cls_linorder__neqE_0, cls_listsum_Osimps_I1_J_0, cls_add__strict__right__mono_0, cls_pos__add__strict_0, cls_monoid__add_Oadd__0__left_0, cls_add__le__cancel__left_1, cls_rtrancl__beta__Abs_0, cls_uIT_0, cls_Snoc__step1__SnocD_2, cls_add__is__0_0, cls_not__Cons__self2_0, clsarity_nat__OrderedGroup_Omonoid__add, cls_gb__semiring_Osemiring__rules_I3_J_0, cls_add__le__cancel__right_1, clsarity_nat__Orderings_Olinorder, cls_SI2_0, cls_termination__basic__simps_I1_J_0, cls_substn__subst__0_0, cls_diff__cancel_0, cls_add__imp__eq_0, cls_app__Var__IT_0, cls_nat__add__left__cancel__less_0, cls_dB_Osimps_I12_J_0, cls_not__step1__Nil_0, cls_appR_0, cls_listsp_ONil_0, cls_add__leE_0, cls_add__diff__assoc2_0, cls_dB_Osplit__asm_0, cls_char__size_0, cls_order__less__trans_0, cls_self__append__conv_0, cls_add__left__cancel_0, cls_gb__semiring_Osemiring__rules_I13_J_0, cls_comm__monoid__add_Omult__1__right_0, cls_drop__0_0, cls_append__eq__Cons__conv_4, cls_add__nonpos__neg_0, cls_le__maxI2_0, cls_eq__Nil__appendI_0, cls_comm__monoid__add_Omult__assoc_0, cls_substn_Osimps_I1_J_2, cls_of__nat__0__less__iff_0, cls_rtranclp__trans_0, cls_Cons__step1__Cons_5, cls_add__cancel__21_1, cls_same__append__eq_0, cls_beta_0, cls_append__Cons_0, cls_le__0__eq_0, cls_listsum__append_0, cls_dB_Osimps_I7_J_0, cls_Abs__apps__eq__Abs__apps__conv_1, cls_subst__preserves__beta_0, cls_less__max__iff__disj_2, cls_foldl__foldr1_0, cls_add__nonneg__pos_0, cls_gr0I_0, cls_infinite__descent0_0, cls_lift__type_0, cls_add__gr__0_0, cls_Nil__is__map__conv_0, cls_drop__map_0, cls_add__is__0_1, clsarity_nat__Ring__and__Field_Oordered__semidom, cls_less__max__iff__disj_0, cls_dB_Orecs_I3_J_0, cls_lem0_0, cls_replicate__eq__replicate_0, cls_gb__semiring_Oadd__c_0, clsarity_nat__OrderedGroup_Opordered__comm__monoid__add, cls_foldr_Osimps_I1_J_0, cls_nat__less__cases_0, cls_diff__diff__right_0, cls_diff__diff__left_0, cls_zero__less__double__add__iff__zero__less__single__add_0, cls_diff__le__mono2_0, cls_gr__implies__not0_0, cls_add__le__mono_0, cls_diff__diff__cancel_0, cls_nat__minus__add__max_0, cls_add__less__mono_0, cls_gb__semiring_Omul__c_0, cls_of__nat__le__iff_1, clsarity_nat__Ring__and__Field_Osemiring__0, cls_append__eq__append__conv2_4, cls_Var__apps__eq__Var__apps__conv_0, cls_field__power__not__zero_0, cls_infinite__descent0_1, cls_order__less__asym_0, cls_linorder__not__less_0, cls_double__add__less__zero__iff__single__less__zero_0, cls_le__antisym_0, cls_add__right__mono_0, cls_append1__eq__conv_1, cls_less__iff__diff__less__0_1, cls_listsum_Osimps_I2_J_0, cls_order__eq__refl_0, cls_order__le__less_0, cls_order__less__asym_H_0, cls_double__eq__0__iff_1, cls_Beta_I1_J_0, cls_of__nat__le__iff_0, cls_gb__semiring_Omul__d_0, cls_linorder__antisym__conv2_0, cls_shift__eq_0, cls_linorder__antisym__conv1_0, cls_right__minus__eq_0, cls_nat__gcd_Osimps_0, cls_semigroup__add_Oadd__assoc_0, cls_substn__subst__n_0, clsarity_fun__Orderings_Obot, cls_liftn__0_0, clsarity_nat__OrderedGroup_Opordered__ab__semigroup__add__imp__le, cls_gb__semiring_Osemiring__rules_I16_J_0, cls_gb__semiring_Osemiring__rules_I14_J_0, cls_replicate__append__same_0, cls_foldl__absorb0_0, cls_le__imp__diff__is__add_1, cls_dB_Osimps_I3_J_0, cls_linorder__neqE__nat_0, cls_not__add__less2_0, cls_max__0R_0, clsarity_bool__Orderings_Oorder, cls_bot__least_0, cls_subst__Var_2, cls_diff__add__inverse2_0, cls_bot__nat__def_0, cls_xt1_I10_J_0, cls_add__nonneg__eq__0__iff_0, cls_le__maxI1_0, cls_append__Nil2_0, cls_nat__add__left__commute_0, cls_trans__le__add2_0, cls_append__self__conv2_0, cls_min__max_Osup__least_0, cls_nat_Oinducts_0, cls_class__semiring_Osemiring__rules_I6_J_0, cls_zero__le__double__add__iff__zero__le__single__add_1, cls_less__le__not__le_2, cls_Var__eq__apps__conv_2, cls_ab__semigroup__add__class_Oadd__ac_I1_J_0, cls_dB_Osimps_I11_J_0, cls_Nil__is__append__conv_0, cls_add__left__mono_0, cls_add__less__cancel__right_1, cls_nat_Osplit_0, cls_Var__apps__eq__Var__apps__conv_1, cls_not__less__iff__gr__or__eq_0, cls_min__max_Ole__supE_0, clsarity_nat__OrderedGroup_Opordered__cancel__ab__semigroup__add, cls_min__max_Ole__sup__iff_1, 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clsarity_nat__OrderedGroup_Ocancel__semigroup__add, clsarity_nat__Ring__and__Field_Ocomm__semiring__1, cls_uT_0, cls_less__add__eq__less_0, cls_max__less__iff__conj_2, cls_append__step1I_1, cls_not__Cons__self_0, cls_conversep__conversep_0, cls_dB_Osimps_I1_J_0, cls_self__append__conv_1, cls_apps__eq__Abs__conv_2, cls_Abs__App__neq__Var__apps_0, cls_rtrancl__beta__App_0, cls_order__antisym_0, cls_infinite__descent0_2, cls_lift__map_0, cls_add__cancel__21_0, cls_linorder__antisym__conv1_1, cls_append__same__eq_0, clsarity_bool__Orderings_Obot, cls_subst__Var_1, clsarity_bool__Orderings_Opreorder, cls_Cons__step1__Cons_2, cls_rtrancl__beta__AppL_0, cls_Nil__is__append__conv_1, cls_Nitpick_Ortranclp__def_1, cls_double__add__le__zero__iff__single__add__le__zero_0, cls_class__semiring_Osemiring__rules_I5_J_0, cls_gb__semiring_Osemiring__rules_I8_J_0, cls_diff__self_0, cls_less__imp__diff__less_0, cls_diff__is__0__eq_0, cls_natgb_Oadd__cancel_0, cls_drop__Cons_H_0, cls_neq__Nil__conv_1, cls_diff__less_0, cls_Nitpick_Onat__case__def_0, cls_monoid__add_Oadd__0__right_0, cls_linorder__not__le_1, cls_append__assoc_0, cls_not__Nil__step1_0, cls_foldr_Osimps_I2_J_0, cls_add__less__cancel__right_0, cls_listsum_0, cls_lift__preserves__beta_0, cls_add__diff__cancel_0, cls_le__trans_0, cls_drop__Nil_0, cls_neq0__conv_1, cls_apps__eq__Abs__conv_1, cls_min__max_Ole__sup__iff_0, cls_lifts__IT_0, cls_not__less__iff__gr__or__eq_1, cls_add__diff__assoc_0, cls_diff__add__assoc_0, cls_order__le__neq__trans_0, cls_zero__less__double__add__iff__zero__less__single__add_1, cls_nat__add__right__cancel_0, cls_add__increasing_0, cls_Var__apps__neq__Abs__apps_0, cls_subst__lemma_0, clsarity_nat__Orderings_Opreorder, cls_nat__diff__split__asm_1, cls_SI1_0, cls_zero__le__double__add__iff__zero__le__single__add_0, cls_max__0L_0, cls_le__imp__diff__is__add_0, cls_gb__semiring_Osemiring__rules_I10_J_0, cls_foldl__assoc_0, cls_class__semiring_Osemiring__rules_I23_J_0, cls_Beta_I6_J_0, cls_order__eq__iff_2, cls_less__not__refl_0, cls_append__in__listsp__conv_2, cls_double__add__le__zero__iff__single__add__le__zero_1, cls_gb__semiring_Osemiring__rules_I2_J_0, cls_class__ringb_Oadd__cancel_0, cls_add__nonpos__nonpos_0, cls_less__le__not__le_1, cls_foldl__foldr1__lemma_0, cls_Cons__step1__Cons_4, cls_right__minus__eq_1, cls_linorder__antisym__conv3_0, cls_conversep_Ointros_0, cls_nat__diff__split_1, cls_linorder__neqE__ordered__idom_0, cls_map__append_0, cls_diff__cancel2_0, cls_dB_Osimps_I2_J_1, cls_add__nonneg__eq__0__iff_1, cls_accp__downwards__aux_0, cls_add__nonneg__nonneg_0, cls_empty__replicate_0, cls_dB_Osplit__asm_2, cls_min__max_Osup__absorb1_0, cls_le__diff__conv_0, cls_le__diff__conv2_0, cls_order__antisym__conv_0, cls_add__lessD1_0, cls_gb__semiring_Osemiring__rules_I6_J_0, cls_diff__less__mono_0, cls_lift__preserves__beta_H_0, cls_order__neq__le__trans_0, cls_power__eq__imp__eq__base_0, cls_less__diff__conv_1, cls_Beta_I3_J_0, cls_class__ringb_Oadd__r0__iff_0, cls_less__diff__conv_0, cls_foldr__append_0, cls_order__le__less_1, cls_linorder__neq__iff_1, cls_replicate__add_0, cls_gb__semiring_Omul__pwr_0, cls_gb__semiring_Osemiring__rules_I22_J_0, cls_drop__drop_0, cls_even__less__0__iff_1, cls_add__strict__increasing_0, cls_max__less__iff__conj_1, cls_substn_Osimps_I2_J_0, cls_linorder__less__linear_0, clsarity_nat__Power_Opower, cls_drop__replicate_0, cls_CHAINED_0_02, cls_rtranclp_Ortrancl__refl_0, cls_nat__add__left__cancel__le_1, cls_le__add__diff__inverse_0, cls_Nat_Oadd__0__right_0, cls_eq__eqI_0, cls_order__less__le_2, cls_gb__semiring_Osemiring__rules_I18_J_0, cls_map__is__Nil__conv_1, cls_list_Osimps_I3_J_0, cls_nat__neq__iff_0, cls_le0_0, cls_gb__semiring_Oadd__0_0, cls_natgb_Oadd__r0__iff_0, cls_min__max_Ole__supI1_0, cls_diff__self__eq__0_0, cls_le__add2_0, cls_xt1_I11_J_0, cls_map_Osimps_I2_J_0, cls_add__neg__neg_0, cls_double__eq__0__iff_0, cls_trans__less__add2_0, cls_zero__less__diff_0, cls_le__eqI_0, cls_App__eq__foldl__conv_2, cls_nat__add__left__cancel__le_0, cls_min__max_Osup__left__commute_0, cls_accp__downward_0, cls_Snoc__step1__SnocD_0, cls_Abs__eq__apps__conv_2, cls_dB_Orecs_I2_J_0, cls_rtranclp__idemp_0, cls_append__is__Nil__conv_1, cls_typing_OVar_0, cls_diff__le__self_0, cls_IT_OVar_0, cls_listsum__foldr_0, cls_abs_0, cls_comm__monoid__add_Omult__left__commute_0, cls_apps__eq__tail__conv_0, cls_diff__0__right_0, cls_xt1_I6_J_0, cls_le__add__right__mono_0, cls_min__max_Ole__iff__sup_1, clsarity_nat__OrderedGroup_Opordered__ab__semigroup__add, cls_gb__semiring_Oadd__a_0, cls_nat__less__le_0, cls_dB_Osplit_1, cls_size__char_0, cls_lift__IT_0, cls_replicate__eq__replicate_2, cls_le__max__iff__disj_2, cls_diff__add__assoc2_0, cls_le__diff__conv2_1, cls_apps__preserves__beta2_0, cls_plus__nat_Oadd__0_0, cls_min__max_Osup__assoc_0, cls_nat__less__le_2, cls_linorder__not__le_0, cls_Cons__eq__appendI_0, cls_double__add__less__zero__iff__single__less__zero_1, cls_accp_Ocases_0, cls_diff__less__mono2_0, cls_nat__le__linear_0, cls_trans__less__add1_0, cls_comm__monoid__add_Omult__1_0, cls_liftn_Osimps_I1_J_0, cls_apps__preserves__beta_0, cls_order__less__irrefl_0, cls_append__is__Nil__conv_0, cls_add__nonneg__eq__0__iff_2, cls_class__semiring_Oadd__c_0, cls_less__diff__iff_0, cls_Abs__eq__apps__conv_0, cls_append__eq__appendI_0, cls_min__max_Oless__supI2_0, cls_add__le__mono1_0, clsarity_nat__Ring__and__Field_Ono__zero__divisors, cls_minus__nat_Odiff__0_0, cls_foldr__conv__foldl_0, cls_le__iff__add_1, clsarity_nat__OrderedGroup_Ocancel__ab__semigroup__add, cls_Cons__eq__append__conv_4, cls_App__eq__foldl__conv_4, cls_diff__is__0__eq_1, cls_diff__0__eq__0_0, cls_add__is__0_2, cls_ATP__Linkup_Oequal__imp__fequal_0, cls_power__eq__0__iff_0, cls_eq__eqI_1, cls_list_Oinject_1, cls_zero__less__diff_1, cls_less__eqI_0, cls_nat__add__left__cancel_0, cls_foldl__append_0, cls_substn_Osimps_I1_J_1, cls_add__eq__self__zero_0, clsarity_fun__Orderings_Opreorder, cls_le__diff__iff_1, cls_add__pos__pos_0, cls_power__less__imp__less__base_0, cls_add__mono_0, cls_le__max__iff__disj_0, cls_semigroup__add_Ofoldl__assoc_0, cls_gb__semiring_Osemiring__rules_I24_J_0, cls_add__neg__nonpos_0, cls_append__replicate__commute_0, cls_not__less0_0, cls_diffs0__imp__equal_0, cls_gb__semiring_Osemiring__rules_I19_J_0, cls_class__semiring_Osemiring__rules_I24_J_0, cls_eq__iff__diff__eq__0_1, cls_conversep__eq_0, cls_add__less__cancel__left_0, cls_class__ringb_Osubr0__iff_1, cls_shift__gt_0, cls_gb__semiring_Omul__a_0, cls_beta__cases_I1_J_0, cls_Var__eq__apps__conv_0, cls_Nil__is__append__conv_2, cls_dB_Osimps_I2_J_0, cls_var__app__type__eq_0, cls_dB_Osimps_I5_J_0, cls_diff__add__inverse_0, cls_replicate__empty_1, cls_subst__Var__IT_0, cls_min__max_Ole__iff__sup_0, cls_less__max__iff__disj_1, cls_zero__le__power_0, cls_gb__semiring_Osemiring__rules_I20_J_0, cls_max__less__iff__conj_0, clsarity_nat__Ring__and__Field_Osemiring__1, cls_add__leE_1, cls_dB_Osplit__asm_1, cls_max__add__distrib__left_0, cls_class__semiring_Osemiring__rules_I25_J_0, cls_IT_OBeta_0, cls_gb__semiring_Opwr__0_0, cls_foldl__Nil_0, cls_le__eqI_1, cls_Lambda_0, cls_gb__semiring_Osemiring__rules_I4_J_0, cls_nat__case__0_0, cls_le__iff__diff__le__0_1, cls_add__gr__0_2, cls_diff__le__mono_0, cls_add__less__le__mono_0, cls_lift_Osimps_I1_J_0, cls_min__max_Ole__supI2_0, cls_order__less__imp__le_0, cls_rtranclp_Ortrancl__into__rtrancl_0, cls_add__gr__0_1, cls_min__max_Ole__supE_1, cls_diff__commute_0, cls_ex__head__tail_0, cls_upt_Oinduct_0, cls_subst__lift_0, cls_termination__basic__simps_I2_J_0, cls_class__ringb_Osubr0__iff_0, cls_le__eq__less__or__eq_0, clsarity_nat__Ring__and__Field_Omult__zero, clsarity_nat__OrderedGroup_Ocomm__monoid__add, cls_eq__iff__diff__eq__0_0, cls_Abs__eq__apps__conv_1, cls_subst__preserves__beta2_0, cls_gb__semiring_Opwr__mul_0, cls_monoid__add__class_Oadd__0__right_0, cls_of__nat__less__iff_1, cls_dB_Osplit_2, cls_le__add1_0, cls_add__strict__increasing2_0, cls_le__add__diff_0, clsarity_nat__Orderings_Oorder, cls_order__less__le_1, cls_T_0, cls_linorder__not__less_1, cls_append1__eq__conv_0, cls_appL_0, cls_nat__less__le_1, cls_subst__preserves__beta2_H_0, cls_min__max_Osup__commute_0, cls_add__le__cancel__left_0, cls_empty__replicate_1, cls_Snoc__step1__SnocD_1, cls_gb__semiring_Osemiring__rules_I11_J_0, cls_xt1_I7_J_0, cls_le__neq__implies__less_0, cls_conversep__iff_1, cls_list_Osimps_I2_J_0, cls_rtranclp__converseD_0, clsarity_nat__OrderedGroup_Osemigroup__add, cls_conversep_Ocases_0, cls_gb__semiring_Osemiring__rules_I23_J_0, cls_less__eqI_1, cls_dB_Osimps_I9_J_0, cls_zero__le__imp__of__nat_0, cls_le__refl_0, cls_even__less__0__iff_0, cls_gb__semiring_Osemiring__rules_I1_J_0, cls_rtranclp__converseI_0, cls_class__semiring_Osemiring__rules_I22_J_0, cls_power__mono_0, cls_nat_Osize_I1_J_0, cls_of__nat__less__0__iff_0, cls_Abs__apps__eq__Abs__apps__conv_0, cls_gb__semiring_Osemiring__rules_I15_J_0, cls_power__0__left_1, cls_add__le__cancel__right_0, cls_r__into__rtranclp_0, cls_subst__preserves__beta_H_0, cls_nat__add__commute_0, cls_double__induction__lemma_0, cls_add__diff__inverse_0, cls_rtrancl__beta__AppR_0
% 18.67/5.15  % No. of inferences in proof: 1251
% 18.67/5.15  % SZS status Unsatisfiable for /export/starexec/sandbox2/benchmark/theBenchmark.p : 4752 ms resp. 4365 ms w/o parsing
% 19.48/5.33  % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 19.48/5.33  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------