ITP001 Axioms: ITP089^5.ax
%------------------------------------------------------------------------------
% File : ITP089^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : toto^2.ax [Gau20]
% : HL4089^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 226 ( 27 unt; 52 typ; 0 def)
% Number of atoms : 3030 ( 249 equ; 0 cnn)
% Maximal formula atoms : 145 ( 13 avg)
% Number of connectives : 5129 ( 18 ~; 7 |; 116 &;4584 @)
% ( 33 <=>; 371 =>; 0 <=; 0 <~>)
% Maximal formula depth : 43 ( 9 avg;4584 nst)
% Number of types : 5 ( 4 usr)
% Number of type conns : 40 ( 40 >; 0 *; 0 +; 0 <<)
% Number of symbols : 127 ( 124 usr; 93 con; 0-2 aty)
% Number of variables : 534 ( 19 ^ 504 !; 11 ?; 534 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Etoto_2Enum__dt,type,
ty_2Etoto_2Enum__dt: del ).
thf(tp_ty_2Etoto_2Etoto,type,
ty_2Etoto_2Etoto: del > del ).
thf(tp_c_2Etoto_2EListOrd,type,
c_2Etoto_2EListOrd: del > $i ).
thf(mem_c_2Etoto_2EListOrd,axiom,
! [A_27a: del] : ( mem @ ( c_2Etoto_2EListOrd @ A_27a ) @ ( arr @ ( ty_2Etoto_2Etoto @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ty_2EternaryComparisons_2Eordering ) ) ) ) ).
thf(tp_c_2Etoto_2EStrongLinearOrder__of__TO,type,
c_2Etoto_2EStrongLinearOrder__of__TO: del > $i ).
thf(mem_c_2Etoto_2EStrongLinearOrder__of__TO,axiom,
! [A_27a: del] : ( mem @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Etoto_2ETO,type,
c_2Etoto_2ETO: del > $i ).
thf(mem_c_2Etoto_2ETO,axiom,
! [A_27a: del] : ( mem @ ( c_2Etoto_2ETO @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) @ ( ty_2Etoto_2Etoto @ A_27a ) ) ) ).
thf(tp_c_2Etoto_2ETO__inv,type,
c_2Etoto_2ETO__inv: del > $i ).
thf(mem_c_2Etoto_2ETO__inv,axiom,
! [A_27a: del] : ( mem @ ( c_2Etoto_2ETO__inv @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) ) ) ).
thf(tp_c_2Etoto_2ETO__of__LinearOrder,type,
c_2Etoto_2ETO__of__LinearOrder: del > $i ).
thf(mem_c_2Etoto_2ETO__of__LinearOrder,axiom,
! [A_27a: del] : ( mem @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) ) ) ).
thf(tp_c_2Etoto_2ETotOrd,type,
c_2Etoto_2ETotOrd: del > $i ).
thf(mem_c_2Etoto_2ETotOrd,axiom,
! [A_27a: del] : ( mem @ ( c_2Etoto_2ETotOrd @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) @ bool ) ) ).
thf(tp_c_2Etoto_2EWeakLinearOrder__of__TO,type,
c_2Etoto_2EWeakLinearOrder__of__TO: del > $i ).
thf(mem_c_2Etoto_2EWeakLinearOrder__of__TO,axiom,
! [A_27a: del] : ( mem @ ( c_2Etoto_2EWeakLinearOrder__of__TO @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Etoto_2Eapto,type,
c_2Etoto_2Eapto: del > $i ).
thf(mem_c_2Etoto_2Eapto,axiom,
! [A_27a: del] : ( mem @ ( c_2Etoto_2Eapto @ A_27a ) @ ( arr @ ( ty_2Etoto_2Etoto @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) ) ) ).
thf(stp_ty_2Etoto_2Enum__dt,type,
tp__ty_2Etoto_2Enum__dt: $tType ).
thf(stp_inj_ty_2Etoto_2Enum__dt,type,
inj__ty_2Etoto_2Enum__dt: tp__ty_2Etoto_2Enum__dt > $i ).
thf(stp_surj_ty_2Etoto_2Enum__dt,type,
surj__ty_2Etoto_2Enum__dt: $i > tp__ty_2Etoto_2Enum__dt ).
thf(stp_inj_surj_ty_2Etoto_2Enum__dt,axiom,
! [X: tp__ty_2Etoto_2Enum__dt] :
( ( surj__ty_2Etoto_2Enum__dt @ ( inj__ty_2Etoto_2Enum__dt @ X ) )
= X ) ).
thf(stp_inj_mem_ty_2Etoto_2Enum__dt,axiom,
! [X: tp__ty_2Etoto_2Enum__dt] : ( mem @ ( inj__ty_2Etoto_2Enum__dt @ X ) @ ty_2Etoto_2Enum__dt ) ).
thf(stp_iso_mem_ty_2Etoto_2Enum__dt,axiom,
! [X: $i] :
( ( mem @ X @ ty_2Etoto_2Enum__dt )
=> ( X
= ( inj__ty_2Etoto_2Enum__dt @ ( surj__ty_2Etoto_2Enum__dt @ X ) ) ) ) ).
thf(tp_c_2Etoto_2Ebit1,type,
c_2Etoto_2Ebit1: $i ).
thf(mem_c_2Etoto_2Ebit1,axiom,
mem @ c_2Etoto_2Ebit1 @ ( arr @ ty_2Etoto_2Enum__dt @ ty_2Etoto_2Enum__dt ) ).
thf(stp_fo_c_2Etoto_2Ebit1,type,
fo__c_2Etoto_2Ebit1: tp__ty_2Etoto_2Enum__dt > tp__ty_2Etoto_2Enum__dt ).
thf(stp_eq_fo_c_2Etoto_2Ebit1,axiom,
! [X0: tp__ty_2Etoto_2Enum__dt] :
( ( inj__ty_2Etoto_2Enum__dt @ ( fo__c_2Etoto_2Ebit1 @ X0 ) )
= ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ X0 ) ) ) ).
thf(tp_c_2Etoto_2Ebit2,type,
c_2Etoto_2Ebit2: $i ).
thf(mem_c_2Etoto_2Ebit2,axiom,
mem @ c_2Etoto_2Ebit2 @ ( arr @ ty_2Etoto_2Enum__dt @ ty_2Etoto_2Enum__dt ) ).
thf(stp_fo_c_2Etoto_2Ebit2,type,
fo__c_2Etoto_2Ebit2: tp__ty_2Etoto_2Enum__dt > tp__ty_2Etoto_2Enum__dt ).
thf(stp_eq_fo_c_2Etoto_2Ebit2,axiom,
! [X0: tp__ty_2Etoto_2Enum__dt] :
( ( inj__ty_2Etoto_2Enum__dt @ ( fo__c_2Etoto_2Ebit2 @ X0 ) )
= ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ X0 ) ) ) ).
thf(tp_c_2Etoto_2EcharOrd,type,
c_2Etoto_2EcharOrd: $i ).
thf(mem_c_2Etoto_2EcharOrd,axiom,
mem @ c_2Etoto_2EcharOrd @ ( arr @ ty_2Estring_2Echar @ ( arr @ ty_2Estring_2Echar @ ty_2EternaryComparisons_2Eordering ) ) ).
thf(stp_fo_c_2Etoto_2EcharOrd,type,
fo__c_2Etoto_2EcharOrd: tp__ty_2Estring_2Echar > tp__ty_2Estring_2Echar > tp__ty_2EternaryComparisons_2Eordering ).
thf(stp_eq_fo_c_2Etoto_2EcharOrd,axiom,
! [X0: tp__ty_2Estring_2Echar,X1: tp__ty_2Estring_2Echar] :
( ( inj__ty_2EternaryComparisons_2Eordering @ ( fo__c_2Etoto_2EcharOrd @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Etoto_2EcharOrd @ ( inj__ty_2Estring_2Echar @ X0 ) ) @ ( inj__ty_2Estring_2Echar @ X1 ) ) ) ).
thf(stp_c_ty_2Etoto_2Etoto_ty_2Estring_2Echar,type,
tp__c_ty_2Etoto_2Etoto_ty_2Estring_2Echar: $tType ).
thf(stp_inj_c_ty_2Etoto_2Etoto_ty_2Estring_2Echar,type,
inj__c_ty_2Etoto_2Etoto_ty_2Estring_2Echar: tp__c_ty_2Etoto_2Etoto_ty_2Estring_2Echar > $i ).
thf(stp_surj_c_ty_2Etoto_2Etoto_ty_2Estring_2Echar,type,
surj__c_ty_2Etoto_2Etoto_ty_2Estring_2Echar: $i > tp__c_ty_2Etoto_2Etoto_ty_2Estring_2Echar ).
thf(stp_inj_surj_c_ty_2Etoto_2Etoto_ty_2Estring_2Echar,axiom,
! [X: tp__c_ty_2Etoto_2Etoto_ty_2Estring_2Echar] :
( ( surj__c_ty_2Etoto_2Etoto_ty_2Estring_2Echar @ ( inj__c_ty_2Etoto_2Etoto_ty_2Estring_2Echar @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Etoto_2Etoto_ty_2Estring_2Echar,axiom,
! [X: tp__c_ty_2Etoto_2Etoto_ty_2Estring_2Echar] : ( mem @ ( inj__c_ty_2Etoto_2Etoto_ty_2Estring_2Echar @ X ) @ ( ty_2Etoto_2Etoto @ ty_2Estring_2Echar ) ) ).
thf(stp_iso_mem_c_ty_2Etoto_2Etoto_ty_2Estring_2Echar,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Etoto_2Etoto @ ty_2Estring_2Echar ) )
=> ( X
= ( inj__c_ty_2Etoto_2Etoto_ty_2Estring_2Echar @ ( surj__c_ty_2Etoto_2Etoto_ty_2Estring_2Echar @ X ) ) ) ) ).
thf(tp_c_2Etoto_2Echarto,type,
c_2Etoto_2Echarto: $i ).
thf(mem_c_2Etoto_2Echarto,axiom,
mem @ c_2Etoto_2Echarto @ ( ty_2Etoto_2Etoto @ ty_2Estring_2Echar ) ).
thf(tp_c_2Etoto_2EimageOrd,type,
c_2Etoto_2EimageOrd: del > del > $i ).
thf(mem_c_2Etoto_2EimageOrd,axiom,
! [A_27a: del,A_27c: del] : ( mem @ ( c_2Etoto_2EimageOrd @ A_27a @ A_27c ) @ ( arr @ ( arr @ A_27a @ A_27c ) @ ( arr @ ( arr @ A_27c @ ( arr @ A_27c @ ty_2EternaryComparisons_2Eordering ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) ) ) ) ).
thf(tp_c_2Etoto_2ElexTO,type,
c_2Etoto_2ElexTO: del > del > $i ).
thf(mem_c_2Etoto_2ElexTO,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Etoto_2ElexTO @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ty_2EternaryComparisons_2Eordering ) ) ) ) ) ).
thf(tp_c_2Etoto_2Elextoto,type,
c_2Etoto_2Elextoto: del > del > $i ).
thf(mem_c_2Etoto_2Elextoto,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Etoto_2Elextoto @ A_27a @ A_27b ) @ ( arr @ ( ty_2Etoto_2Etoto @ A_27a ) @ ( arr @ ( ty_2Etoto_2Etoto @ A_27b ) @ ( ty_2Etoto_2Etoto @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ) ) ).
thf(tp_c_2Etoto_2Elistorder,type,
c_2Etoto_2Elistorder: del > $i ).
thf(mem_c_2Etoto_2Elistorder,axiom,
! [A_27a: del] : ( mem @ ( c_2Etoto_2Elistorder @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) ) ) ).
thf(tp_c_2Etoto_2Elistoto,type,
c_2Etoto_2Elistoto: del > $i ).
thf(mem_c_2Etoto_2Elistoto,axiom,
! [A_27a: del] : ( mem @ ( c_2Etoto_2Elistoto @ A_27a ) @ ( arr @ ( ty_2Etoto_2Etoto @ A_27a ) @ ( ty_2Etoto_2Etoto @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(tp_c_2Etoto_2EnumOrd,type,
c_2Etoto_2EnumOrd: $i ).
thf(mem_c_2Etoto_2EnumOrd,axiom,
mem @ c_2Etoto_2EnumOrd @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ty_2EternaryComparisons_2Eordering ) ) ).
thf(stp_fo_c_2Etoto_2EnumOrd,type,
fo__c_2Etoto_2EnumOrd: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum > tp__ty_2EternaryComparisons_2Eordering ).
thf(stp_eq_fo_c_2Etoto_2EnumOrd,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] :
( ( inj__ty_2EternaryComparisons_2Eordering @ ( fo__c_2Etoto_2EnumOrd @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Etoto_2EnumOrd @ ( inj__ty_2Enum_2Enum @ X0 ) ) @ ( inj__ty_2Enum_2Enum @ X1 ) ) ) ).
thf(tp_c_2Etoto_2Enum__dtOrd,type,
c_2Etoto_2Enum__dtOrd: $i ).
thf(mem_c_2Etoto_2Enum__dtOrd,axiom,
mem @ c_2Etoto_2Enum__dtOrd @ ( arr @ ty_2Etoto_2Enum__dt @ ( arr @ ty_2Etoto_2Enum__dt @ ty_2EternaryComparisons_2Eordering ) ) ).
thf(stp_fo_c_2Etoto_2Enum__dtOrd,type,
fo__c_2Etoto_2Enum__dtOrd: tp__ty_2Etoto_2Enum__dt > tp__ty_2Etoto_2Enum__dt > tp__ty_2EternaryComparisons_2Eordering ).
thf(stp_eq_fo_c_2Etoto_2Enum__dtOrd,axiom,
! [X0: tp__ty_2Etoto_2Enum__dt,X1: tp__ty_2Etoto_2Enum__dt] :
( ( inj__ty_2EternaryComparisons_2Eordering @ ( fo__c_2Etoto_2Enum__dtOrd @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Etoto_2Enum__dtOrd @ ( inj__ty_2Etoto_2Enum__dt @ X0 ) ) @ ( inj__ty_2Etoto_2Enum__dt @ X1 ) ) ) ).
thf(tp_c_2Etoto_2Enum__dt__CASE,type,
c_2Etoto_2Enum__dt__CASE: del > $i ).
thf(mem_c_2Etoto_2Enum__dt__CASE,axiom,
! [A_27a: del] : ( mem @ ( c_2Etoto_2Enum__dt__CASE @ A_27a ) @ ( arr @ ty_2Etoto_2Enum__dt @ ( arr @ A_27a @ ( arr @ ( arr @ ty_2Etoto_2Enum__dt @ A_27a ) @ ( arr @ ( arr @ ty_2Etoto_2Enum__dt @ A_27a ) @ A_27a ) ) ) ) ) ).
thf(tp_c_2Etoto_2Enum__dt__size,type,
c_2Etoto_2Enum__dt__size: $i ).
thf(mem_c_2Etoto_2Enum__dt__size,axiom,
mem @ c_2Etoto_2Enum__dt__size @ ( arr @ ty_2Etoto_2Enum__dt @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Etoto_2Enum__dt__size,type,
fo__c_2Etoto_2Enum__dt__size: tp__ty_2Etoto_2Enum__dt > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Etoto_2Enum__dt__size,axiom,
! [X0: tp__ty_2Etoto_2Enum__dt] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Etoto_2Enum__dt__size @ X0 ) )
= ( ap @ c_2Etoto_2Enum__dt__size @ ( inj__ty_2Etoto_2Enum__dt @ X0 ) ) ) ).
thf(tp_c_2Etoto_2Enum__to__dt,type,
c_2Etoto_2Enum__to__dt: $i ).
thf(mem_c_2Etoto_2Enum__to__dt,axiom,
mem @ c_2Etoto_2Enum__to__dt @ ( arr @ ty_2Enum_2Enum @ ty_2Etoto_2Enum__dt ) ).
thf(stp_fo_c_2Etoto_2Enum__to__dt,type,
fo__c_2Etoto_2Enum__to__dt: tp__ty_2Enum_2Enum > tp__ty_2Etoto_2Enum__dt ).
thf(stp_eq_fo_c_2Etoto_2Enum__to__dt,axiom,
! [X0: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Etoto_2Enum__dt @ ( fo__c_2Etoto_2Enum__to__dt @ X0 ) )
= ( ap @ c_2Etoto_2Enum__to__dt @ ( inj__ty_2Enum_2Enum @ X0 ) ) ) ).
thf(stp_c_ty_2Etoto_2Etoto_ty_2Enum_2Enum,type,
tp__c_ty_2Etoto_2Etoto_ty_2Enum_2Enum: $tType ).
thf(stp_inj_c_ty_2Etoto_2Etoto_ty_2Enum_2Enum,type,
inj__c_ty_2Etoto_2Etoto_ty_2Enum_2Enum: tp__c_ty_2Etoto_2Etoto_ty_2Enum_2Enum > $i ).
thf(stp_surj_c_ty_2Etoto_2Etoto_ty_2Enum_2Enum,type,
surj__c_ty_2Etoto_2Etoto_ty_2Enum_2Enum: $i > tp__c_ty_2Etoto_2Etoto_ty_2Enum_2Enum ).
thf(stp_inj_surj_c_ty_2Etoto_2Etoto_ty_2Enum_2Enum,axiom,
! [X: tp__c_ty_2Etoto_2Etoto_ty_2Enum_2Enum] :
( ( surj__c_ty_2Etoto_2Etoto_ty_2Enum_2Enum @ ( inj__c_ty_2Etoto_2Etoto_ty_2Enum_2Enum @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Etoto_2Etoto_ty_2Enum_2Enum,axiom,
! [X: tp__c_ty_2Etoto_2Etoto_ty_2Enum_2Enum] : ( mem @ ( inj__c_ty_2Etoto_2Etoto_ty_2Enum_2Enum @ X ) @ ( ty_2Etoto_2Etoto @ ty_2Enum_2Enum ) ) ).
thf(stp_iso_mem_c_ty_2Etoto_2Etoto_ty_2Enum_2Enum,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Etoto_2Etoto @ ty_2Enum_2Enum ) )
=> ( X
= ( inj__c_ty_2Etoto_2Etoto_ty_2Enum_2Enum @ ( surj__c_ty_2Etoto_2Etoto_ty_2Enum_2Enum @ X ) ) ) ) ).
thf(tp_c_2Etoto_2Enumto,type,
c_2Etoto_2Enumto: $i ).
thf(mem_c_2Etoto_2Enumto,axiom,
mem @ c_2Etoto_2Enumto @ ( ty_2Etoto_2Etoto @ ty_2Enum_2Enum ) ).
thf(tp_c_2Etoto_2Eqk__numOrd,type,
c_2Etoto_2Eqk__numOrd: $i ).
thf(mem_c_2Etoto_2Eqk__numOrd,axiom,
mem @ c_2Etoto_2Eqk__numOrd @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ty_2EternaryComparisons_2Eordering ) ) ).
thf(stp_fo_c_2Etoto_2Eqk__numOrd,type,
fo__c_2Etoto_2Eqk__numOrd: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum > tp__ty_2EternaryComparisons_2Eordering ).
thf(stp_eq_fo_c_2Etoto_2Eqk__numOrd,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] :
( ( inj__ty_2EternaryComparisons_2Eordering @ ( fo__c_2Etoto_2Eqk__numOrd @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Etoto_2Eqk__numOrd @ ( inj__ty_2Enum_2Enum @ X0 ) ) @ ( inj__ty_2Enum_2Enum @ X1 ) ) ) ).
thf(tp_c_2Etoto_2Eqk__numto,type,
c_2Etoto_2Eqk__numto: $i ).
thf(mem_c_2Etoto_2Eqk__numto,axiom,
mem @ c_2Etoto_2Eqk__numto @ ( ty_2Etoto_2Etoto @ ty_2Enum_2Enum ) ).
thf(stp_c_ty_2Etoto_2Etoto_c_ty_2Elist_2Elist_ty_2Estring_2Echar,type,
tp__c_ty_2Etoto_2Etoto_c_ty_2Elist_2Elist_ty_2Estring_2Echar: $tType ).
thf(stp_inj_c_ty_2Etoto_2Etoto_c_ty_2Elist_2Elist_ty_2Estring_2Echar,type,
inj__c_ty_2Etoto_2Etoto_c_ty_2Elist_2Elist_ty_2Estring_2Echar: tp__c_ty_2Etoto_2Etoto_c_ty_2Elist_2Elist_ty_2Estring_2Echar > $i ).
thf(stp_surj_c_ty_2Etoto_2Etoto_c_ty_2Elist_2Elist_ty_2Estring_2Echar,type,
surj__c_ty_2Etoto_2Etoto_c_ty_2Elist_2Elist_ty_2Estring_2Echar: $i > tp__c_ty_2Etoto_2Etoto_c_ty_2Elist_2Elist_ty_2Estring_2Echar ).
thf(stp_inj_surj_c_ty_2Etoto_2Etoto_c_ty_2Elist_2Elist_ty_2Estring_2Echar,axiom,
! [X: tp__c_ty_2Etoto_2Etoto_c_ty_2Elist_2Elist_ty_2Estring_2Echar] :
( ( surj__c_ty_2Etoto_2Etoto_c_ty_2Elist_2Elist_ty_2Estring_2Echar @ ( inj__c_ty_2Etoto_2Etoto_c_ty_2Elist_2Elist_ty_2Estring_2Echar @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Etoto_2Etoto_c_ty_2Elist_2Elist_ty_2Estring_2Echar,axiom,
! [X: tp__c_ty_2Etoto_2Etoto_c_ty_2Elist_2Elist_ty_2Estring_2Echar] : ( mem @ ( inj__c_ty_2Etoto_2Etoto_c_ty_2Elist_2Elist_ty_2Estring_2Echar @ X ) @ ( ty_2Etoto_2Etoto @ ( ty_2Elist_2Elist @ ty_2Estring_2Echar ) ) ) ).
thf(stp_iso_mem_c_ty_2Etoto_2Etoto_c_ty_2Elist_2Elist_ty_2Estring_2Echar,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Etoto_2Etoto @ ( ty_2Elist_2Elist @ ty_2Estring_2Echar ) ) )
=> ( X
= ( inj__c_ty_2Etoto_2Etoto_c_ty_2Elist_2Elist_ty_2Estring_2Echar @ ( surj__c_ty_2Etoto_2Etoto_c_ty_2Elist_2Elist_ty_2Estring_2Echar @ X ) ) ) ) ).
thf(tp_c_2Etoto_2Estringto,type,
c_2Etoto_2Estringto: $i ).
thf(mem_c_2Etoto_2Estringto,axiom,
mem @ c_2Etoto_2Estringto @ ( ty_2Etoto_2Etoto @ ( ty_2Elist_2Elist @ ty_2Estring_2Echar ) ) ).
thf(tp_c_2Etoto_2Etoto__inv,type,
c_2Etoto_2Etoto__inv: del > $i ).
thf(mem_c_2Etoto_2Etoto__inv,axiom,
! [A_27a: del] : ( mem @ ( c_2Etoto_2Etoto__inv @ A_27a ) @ ( arr @ ( ty_2Etoto_2Etoto @ A_27a ) @ ( ty_2Etoto_2Etoto @ A_27a ) ) ) ).
thf(tp_c_2Etoto_2Etoto__of__LinearOrder,type,
c_2Etoto_2Etoto__of__LinearOrder: del > $i ).
thf(mem_c_2Etoto_2Etoto__of__LinearOrder,axiom,
! [A_27a: del] : ( mem @ ( c_2Etoto_2Etoto__of__LinearOrder @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( ty_2Etoto_2Etoto @ A_27a ) ) ) ).
thf(tp_c_2Etoto_2Ezer,type,
c_2Etoto_2Ezer: $i ).
thf(mem_c_2Etoto_2Ezer,axiom,
mem @ c_2Etoto_2Ezer @ ty_2Etoto_2Enum__dt ).
thf(stp_fo_c_2Etoto_2Ezer,type,
fo__c_2Etoto_2Ezer: tp__ty_2Etoto_2Enum__dt ).
thf(stp_eq_fo_c_2Etoto_2Ezer,axiom,
( ( inj__ty_2Etoto_2Enum__dt @ fo__c_2Etoto_2Ezer )
= c_2Etoto_2Ezer ) ).
thf(conj_thm_2Etoto_2EStrongLinearOrderExists,axiom,
! [A_27a: del] :
? [V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
& ( p @ ( ap @ ( c_2Erelation_2EStrongLinearOrder @ A_27a ) @ V0R ) ) ) ).
thf(ax_thm_2Etoto_2ETotOrd,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ V0c ) )
<=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0c @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
<=> ( V1x = V2y ) ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0c @ V3x ) @ V4y ) )
= fo__c_2EternaryComparisons_2EGREATER )
<=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0c @ V4y ) @ V3x ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6y: $i] :
( ( mem @ V6y @ A_27a )
=> ! [V7z: $i] :
( ( mem @ V7z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0c @ V5x ) @ V6y ) )
= fo__c_2EternaryComparisons_2ELESS )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0c @ V6y ) @ V7z ) )
= fo__c_2EternaryComparisons_2ELESS ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0c @ V5x ) @ V7z ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Etoto_2ETO__of__LinearOrder,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a ) @ V0r ) @ V1x ) @ V2y ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1x ) @ V2y ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2EEQUAL ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ V0r @ V1x ) @ V2y ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2ELESS ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2EGREATER ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Etrichotomous__ALT,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etrichotomous @ A_27a ) @ V0R ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ~ ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
& ~ ( p @ ( ap @ ( ap @ V0R @ V2y ) @ V1x ) ) )
=> ( V1x = V2y ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ETotOrd__TO__of__LO,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2ELinearOrder @ A_27a ) @ V0r ) )
=> ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ ( ap @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a ) @ V0r ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ESPLIT__PAIRS,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( V0x = V1y )
<=> ( ( ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0x )
= ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V1y ) )
& ( ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V0x )
= ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V1y ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Eall__cpn__distinct,axiom,
( ( fo__c_2EternaryComparisons_2ELESS != fo__c_2EternaryComparisons_2EEQUAL )
& ( fo__c_2EternaryComparisons_2ELESS != fo__c_2EternaryComparisons_2EGREATER )
& ( fo__c_2EternaryComparisons_2EEQUAL != fo__c_2EternaryComparisons_2EGREATER )
& ( fo__c_2EternaryComparisons_2EEQUAL != fo__c_2EternaryComparisons_2ELESS )
& ( fo__c_2EternaryComparisons_2EGREATER != fo__c_2EternaryComparisons_2ELESS )
& ( fo__c_2EternaryComparisons_2EGREATER != fo__c_2EternaryComparisons_2EEQUAL ) ) ).
thf(conj_thm_2Etoto_2ETO__exists,axiom,
! [A_27a: del] :
? [V0x: $i] :
( ( mem @ V0x @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
& ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ V0x ) ) ) ).
thf(ax_thm_2Etoto_2Etoto__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2Etoto_2Etoto @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) @ ( ty_2Etoto_2Etoto @ A_27a ) ) @ ( c_2Etoto_2ETotOrd @ A_27a ) ) @ V0rep ) ) ) ).
thf(ax_thm_2Etoto_2Eto__bij,axiom,
! [A_27a: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ( ( ap @ ( c_2Etoto_2ETO @ A_27a ) @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0a ) )
= V0a ) )
& ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ V1r ) )
<=> ( ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ ( ap @ ( c_2Etoto_2ETO @ A_27a ) @ V1r ) )
= V1r ) ) ) ) ).
thf(conj_thm_2Etoto_2ETO__apto__ID,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ( ( ap @ ( c_2Etoto_2ETO @ A_27a ) @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0a ) )
= V0a ) ) ).
thf(conj_thm_2Etoto_2ETO__apto__TO__ID,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ V0r ) )
<=> ( ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ ( ap @ ( c_2Etoto_2ETO @ A_27a ) @ V0r ) )
= V0r ) ) ) ).
thf(conj_thm_2Etoto_2ETO__11,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1r_27: $i] :
( ( mem @ V1r_27 @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ V0r ) )
=> ( ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ V1r_27 ) )
=> ( ( ( ap @ ( c_2Etoto_2ETO @ A_27a ) @ V0r )
= ( ap @ ( c_2Etoto_2ETO @ A_27a ) @ V1r_27 ) )
<=> ( V0r = V1r_27 ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Eonto__apto,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ V0r ) )
<=> ? [V1a: $i] :
( ( mem @ V1a @ ( ty_2Etoto_2Etoto @ A_27a ) )
& ( V0r
= ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V1a ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ETO__onto,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ? [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
& ( V0a
= ( ap @ ( c_2Etoto_2ETO @ A_27a ) @ V1r ) )
& ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ V1r ) ) ) ) ).
thf(conj_thm_2Etoto_2ETotOrd__apto,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) ) ) ) ).
thf(conj_thm_2Etoto_2ETO__apto__TO__IMP,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ V0r ) )
=> ( ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ ( ap @ ( c_2Etoto_2ETO @ A_27a ) @ V0r ) )
= V0r ) ) ) ).
thf(conj_thm_2Etoto_2Etoto__thm,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
<=> ( V1x = V2y ) ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V3x ) @ V4y ) )
= fo__c_2EternaryComparisons_2EGREATER )
<=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V4y ) @ V3x ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6y: $i] :
( ( mem @ V6y @ A_27a )
=> ! [V7z: $i] :
( ( mem @ V7z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V5x ) @ V6y ) )
= fo__c_2EternaryComparisons_2ELESS )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V6y ) @ V7z ) )
= fo__c_2EternaryComparisons_2ELESS ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V5x ) @ V7z ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ETO__equal__eq,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ V0c ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0c @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
<=> ( V1x = V2y ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Etoto__equal__eq,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
<=> ( V1x = V2y ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Etoto__equal__imp__eq,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
=> ( V1x = V2y ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ETO__refl,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ V0c ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0c @ V1x ) @ V1x ) )
= fo__c_2EternaryComparisons_2EEQUAL ) ) ) ) ).
thf(conj_thm_2Etoto_2Etoto__refl,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V1x ) )
= fo__c_2EternaryComparisons_2EEQUAL ) ) ) ).
thf(conj_thm_2Etoto_2Etoto__equal__sym,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
<=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V2y ) @ V1x ) )
= fo__c_2EternaryComparisons_2EEQUAL ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ETO__antisym,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ V0c ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0c @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2EGREATER )
<=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0c @ V2y ) @ V1x ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Etoto__antisym,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2EGREATER )
<=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V2y ) @ V1x ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Etoto__not__less__refl,axiom,
! [A_27a: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1h: $i] :
( ( mem @ V1h @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0cmp ) @ V1h ) @ V1h ) )
= fo__c_2EternaryComparisons_2ELESS )
<=> $false ) ) ) ).
thf(conj_thm_2Etoto_2Etoto__swap__cases,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V2y ) @ V1x ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V2y ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2EGREATER ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2EEQUAL ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Etoto__glneq,axiom,
! [A_27a: del] :
( ! [V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2ELESS )
=> ( V1x != V2y ) ) ) ) )
& ! [V3c: $i] :
( ( mem @ V3c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V3c ) @ V4x ) @ V5y ) )
= fo__c_2EternaryComparisons_2EGREATER )
=> ( V4x != V5y ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Etoto__cpn__eqn,axiom,
! [A_27a: del] :
( ! [V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
=> ( V1x = V2y ) ) ) ) )
& ! [V3c: $i] :
( ( mem @ V3c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V3c ) @ V4x ) @ V5y ) )
= fo__c_2EternaryComparisons_2ELESS )
=> ( V4x != V5y ) ) ) ) )
& ! [V6c: $i] :
( ( mem @ V6c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8y: $i] :
( ( mem @ V8y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V6c ) @ V7x ) @ V8y ) )
= fo__c_2EternaryComparisons_2EGREATER )
=> ( V7x != V8y ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ETO__cpn__eqn,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ V0c ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0c @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2ELESS )
=> ( V1x != V2y ) ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0c @ V3x ) @ V4y ) )
= fo__c_2EternaryComparisons_2EGREATER )
=> ( V3x != V4y ) ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6y: $i] :
( ( mem @ V6y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0c @ V5x ) @ V6y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
=> ( V5x = V6y ) ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ENOT__EQ__LESS__IMP,axiom,
! [A_27a: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0cmp ) @ V1x ) @ V2y ) )
!= fo__c_2EternaryComparisons_2ELESS )
=> ( ( V1x = V2y )
| ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0cmp ) @ V2y ) @ V1x ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2EtotoEEtrans,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V2y ) @ V3z ) )
= fo__c_2EternaryComparisons_2EEQUAL ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V3z ) )
= fo__c_2EternaryComparisons_2EEQUAL ) )
& ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V3z ) @ V2y ) )
= fo__c_2EternaryComparisons_2EEQUAL ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V3z ) )
= fo__c_2EternaryComparisons_2EEQUAL ) ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2EtotoLLtrans,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2ELESS )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V2y ) @ V3z ) )
= fo__c_2EternaryComparisons_2ELESS ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V3z ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2EtotoLGtrans,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2ELESS )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V3z ) @ V2y ) )
= fo__c_2EternaryComparisons_2EGREATER ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V3z ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2EtotoGGtrans,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V2y ) @ V1x ) )
= fo__c_2EternaryComparisons_2EGREATER )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V3z ) @ V2y ) )
= fo__c_2EternaryComparisons_2EGREATER ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V3z ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2EtotoGLtrans,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V2y ) @ V1x ) )
= fo__c_2EternaryComparisons_2EGREATER )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V2y ) @ V3z ) )
= fo__c_2EternaryComparisons_2ELESS ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V3z ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2EtotoLEtrans,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2ELESS )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V2y ) @ V3z ) )
= fo__c_2EternaryComparisons_2EEQUAL ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V3z ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2EtotoELtrans,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V2y ) @ V3z ) )
= fo__c_2EternaryComparisons_2ELESS ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V3z ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Etoto__trans__less,axiom,
! [A_27a: del] :
( ! [V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2ELESS )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V2y ) @ V3z ) )
= fo__c_2EternaryComparisons_2ELESS ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V1x ) @ V3z ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) )
& ! [V4c: $i] :
( ( mem @ V4c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6y: $i] :
( ( mem @ V6y @ A_27a )
=> ! [V7z: $i] :
( ( mem @ V7z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V4c ) @ V5x ) @ V6y ) )
= fo__c_2EternaryComparisons_2ELESS )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V4c ) @ V7z ) @ V6y ) )
= fo__c_2EternaryComparisons_2EGREATER ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V4c ) @ V5x ) @ V7z ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) )
& ! [V8c: $i] :
( ( mem @ V8c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V9x: $i] :
( ( mem @ V9x @ A_27a )
=> ! [V10y: $i] :
( ( mem @ V10y @ A_27a )
=> ! [V11z: $i] :
( ( mem @ V11z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V8c ) @ V10y ) @ V9x ) )
= fo__c_2EternaryComparisons_2EGREATER )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V8c ) @ V11z ) @ V10y ) )
= fo__c_2EternaryComparisons_2EGREATER ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V8c ) @ V9x ) @ V11z ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) )
& ! [V12c: $i] :
( ( mem @ V12c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V13x: $i] :
( ( mem @ V13x @ A_27a )
=> ! [V14y: $i] :
( ( mem @ V14y @ A_27a )
=> ! [V15z: $i] :
( ( mem @ V15z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V12c ) @ V14y ) @ V13x ) )
= fo__c_2EternaryComparisons_2EGREATER )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V12c ) @ V14y ) @ V15z ) )
= fo__c_2EternaryComparisons_2ELESS ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V12c ) @ V13x ) @ V15z ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) )
& ! [V16c: $i] :
( ( mem @ V16c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V17x: $i] :
( ( mem @ V17x @ A_27a )
=> ! [V18y: $i] :
( ( mem @ V18y @ A_27a )
=> ! [V19z: $i] :
( ( mem @ V19z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V16c ) @ V17x ) @ V18y ) )
= fo__c_2EternaryComparisons_2ELESS )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V16c ) @ V18y ) @ V19z ) )
= fo__c_2EternaryComparisons_2EEQUAL ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V16c ) @ V17x ) @ V19z ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) )
& ! [V20c: $i] :
( ( mem @ V20c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V21x: $i] :
( ( mem @ V21x @ A_27a )
=> ! [V22y: $i] :
( ( mem @ V22y @ A_27a )
=> ! [V23z: $i] :
( ( mem @ V23z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V20c ) @ V21x ) @ V22y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V20c ) @ V22y ) @ V23z ) )
= fo__c_2EternaryComparisons_2ELESS ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V20c ) @ V21x ) @ V23z ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) ) ) ).
thf(ax_thm_2Etoto_2EWeakLinearOrder__of__TO,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2EWeakLinearOrder__of__TO @ A_27a ) @ V0c ) @ V1x ) @ V2y ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ bool ) @ ( ap @ ( ap @ V0c @ V1x ) @ V2y ) ) @ c_2Ebool_2ET ) @ c_2Ebool_2ET ) @ c_2Ebool_2EF ) ) ) ) ) ) ).
thf(ax_thm_2Etoto_2EStrongLinearOrder__of__TO,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a ) @ V0c ) @ V1x ) @ V2y ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ bool ) @ ( ap @ ( ap @ V0c @ V1x ) @ V2y ) ) @ c_2Ebool_2ET ) @ c_2Ebool_2EF ) @ c_2Ebool_2EF ) ) ) ) ) ) ).
thf(ax_thm_2Etoto_2Etoto__of__LinearOrder,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( c_2Etoto_2Etoto__of__LinearOrder @ A_27a ) @ V0r )
= ( ap @ ( c_2Etoto_2ETO @ A_27a ) @ ( ap @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a ) @ V0r ) ) ) ) ).
thf(conj_thm_2Etoto_2EWeak__Weak__of,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ( p @ ( ap @ ( c_2Erelation_2EWeakLinearOrder @ A_27a ) @ ( ap @ ( c_2Etoto_2EWeakLinearOrder__of__TO @ A_27a ) @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ESTRORD__SLO,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWeakLinearOrder @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2EStrongLinearOrder @ A_27a ) @ ( ap @ ( c_2Erelation_2ESTRORD @ A_27a ) @ V0R ) ) ) ) ) ).
thf(conj_thm_2Etoto_2EStrongof__toto__STRORD,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ( ( ap @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a ) @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) )
= ( ap @ ( c_2Erelation_2ESTRORD @ A_27a ) @ ( ap @ ( c_2Etoto_2EWeakLinearOrder__of__TO @ A_27a ) @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) ) ) ) ) ).
thf(conj_thm_2Etoto_2EStrong__Strong__of,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ( p @ ( ap @ ( c_2Erelation_2EStrongLinearOrder @ A_27a ) @ ( ap @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a ) @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) ) ) ) ) ).
thf(conj_thm_2Etoto_2EStrong__Strong__of__TO,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ V0c ) )
=> ( p @ ( ap @ ( c_2Erelation_2EStrongLinearOrder @ A_27a ) @ ( ap @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a ) @ V0c ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ETotOrd__TO__of__Weak,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWeakLinearOrder @ A_27a ) @ V0r ) )
=> ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ ( ap @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a ) @ V0r ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ETotOrd__TO__of__Strong,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EStrongLinearOrder @ A_27a ) @ V0r ) )
=> ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ ( ap @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a ) @ V0r ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Etoto__Weak__thm,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ( ( ap @ ( c_2Etoto_2Etoto__of__LinearOrder @ A_27a ) @ ( ap @ ( c_2Etoto_2EWeakLinearOrder__of__TO @ A_27a ) @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) ) )
= V0c ) ) ).
thf(conj_thm_2Etoto_2Etoto__Strong__thm,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ( ( ap @ ( c_2Etoto_2Etoto__of__LinearOrder @ A_27a ) @ ( ap @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a ) @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) ) )
= V0c ) ) ).
thf(conj_thm_2Etoto_2EWeak__toto__thm,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWeakLinearOrder @ A_27a ) @ V0r ) )
=> ( ( ap @ ( c_2Etoto_2EWeakLinearOrder__of__TO @ A_27a ) @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ ( ap @ ( c_2Etoto_2Etoto__of__LinearOrder @ A_27a ) @ V0r ) ) )
= V0r ) ) ) ).
thf(conj_thm_2Etoto_2EStrong__toto__thm,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EStrongLinearOrder @ A_27a ) @ V0r ) )
=> ( ( ap @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a ) @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ ( ap @ ( c_2Etoto_2Etoto__of__LinearOrder @ A_27a ) @ V0r ) ) )
= V0r ) ) ) ).
thf(ax_thm_2Etoto_2ETO__inv,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2ETO__inv @ A_27a ) @ V0c ) @ V1x ) @ V2y ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0c @ V2y ) @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ETotOrd__inv,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ V0c ) )
=> ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ ( ap @ ( c_2Etoto_2ETO__inv @ A_27a ) @ V0c ) ) ) ) ) ).
thf(ax_thm_2Etoto_2Etoto__inv,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ( ( ap @ ( c_2Etoto_2Etoto__inv @ A_27a ) @ V0c )
= ( ap @ ( c_2Etoto_2ETO @ A_27a ) @ ( ap @ ( c_2Etoto_2ETO__inv @ A_27a ) @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Einv__TO,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ V0r ) )
=> ( ( ap @ ( c_2Etoto_2Etoto__inv @ A_27a ) @ ( ap @ ( c_2Etoto_2ETO @ A_27a ) @ V0r ) )
= ( ap @ ( c_2Etoto_2ETO @ A_27a ) @ ( ap @ ( c_2Etoto_2ETO__inv @ A_27a ) @ V0r ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Eapto__inv,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ( ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ ( ap @ ( c_2Etoto_2Etoto__inv @ A_27a ) @ V0c ) )
= ( ap @ ( c_2Etoto_2ETO__inv @ A_27a ) @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) ) ) ) ).
thf(conj_thm_2Etoto_2EWeak__toto__inv,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ( ( ap @ ( c_2Etoto_2EWeakLinearOrder__of__TO @ A_27a ) @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ ( ap @ ( c_2Etoto_2Etoto__inv @ A_27a ) @ V0c ) ) )
= ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ ( ap @ ( c_2Etoto_2EWeakLinearOrder__of__TO @ A_27a ) @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) ) ) ) ) ).
thf(conj_thm_2Etoto_2EStrong__toto__inv,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ( ( ap @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a ) @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ ( ap @ ( c_2Etoto_2Etoto__inv @ A_27a ) @ V0c ) ) )
= ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ ( ap @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a ) @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ETO__inv__TO__inv,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( ap @ ( c_2Etoto_2ETO__inv @ A_27a ) @ ( ap @ ( c_2Etoto_2ETO__inv @ A_27a ) @ V0c ) )
= V0c ) ) ).
thf(conj_thm_2Etoto_2Etoto__inv__toto__inv,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ( ( ap @ ( c_2Etoto_2Etoto__inv @ A_27a ) @ ( ap @ ( c_2Etoto_2Etoto__inv @ A_27a ) @ V0c ) )
= V0c ) ) ).
thf(conj_thm_2Etoto_2ETO__inv__Ord,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a ) @ ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ V0r ) )
= ( ap @ ( c_2Etoto_2ETO__inv @ A_27a ) @ ( ap @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a ) @ V0r ) ) ) ) ).
thf(conj_thm_2Etoto_2ETO__of__less__rel,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EStrongLinearOrder @ A_27a ) @ V0r ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a ) @ V0r ) @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2ELESS )
<=> ( p @ ( ap @ ( ap @ V0r @ V1x ) @ V2y ) ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ETO__of__greater__ler,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EStrongLinearOrder @ A_27a ) @ V0r ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a ) @ V0r ) @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2EGREATER )
<=> ( p @ ( ap @ ( ap @ V0r @ V2y ) @ V1x ) ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Etoto__equal__imp,axiom,
! [A_27a: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1phi: $i] :
( ( mem @ V1phi @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2ELinearOrder @ A_27a ) @ V1phi ) )
& ( V0cmp
= ( ap @ ( c_2Etoto_2Etoto__of__LinearOrder @ A_27a ) @ V1phi ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( V2x = V3y )
<=> $true )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0cmp ) @ V2x ) @ V3y ) )
= fo__c_2EternaryComparisons_2EEQUAL ) ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Etoto__unequal__imp,axiom,
! [A_27a: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1phi: $i] :
( ( mem @ V1phi @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2ELinearOrder @ A_27a ) @ V1phi ) )
& ( V0cmp
= ( ap @ ( c_2Etoto_2Etoto__of__LinearOrder @ A_27a ) @ V1phi ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( V2x = V3y )
<=> $false )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ ( ap @ V1phi @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0cmp ) @ V2x ) @ V3y ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2ELESS ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0cmp ) @ V2x ) @ V3y ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2EGREATER ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2EStrongOrder__ALT,axiom,
! [A_27a: del,V0Z: $i] :
( ( mem @ V0Z @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EStrongOrder @ A_27a ) @ V0Z ) )
<=> ( ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ V0Z ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0Z ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ELEX__ALT,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1U: $i] :
( ( mem @ V1U @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V3d: $i] :
( ( mem @ V3d @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epair_2ELEX @ A_27a @ A_27b ) @ V0R ) @ V1U ) @ V2c ) @ V3d ) )
<=> ( ( p @ ( ap @ ( ap @ V0R @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V2c ) ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V3d ) ) )
| ( ( ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V2c )
= ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V3d ) )
& ( p @ ( ap @ ( ap @ V1U @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V2c ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V3d ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ESLO__LEX,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1V: $i] :
( ( mem @ V1V @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2EStrongLinearOrder @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2EStrongLinearOrder @ A_27b ) @ V1V ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2EStrongLinearOrder @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2ELEX @ A_27a @ A_27b ) @ V0R ) @ V1V ) ) ) ) ) ) ).
thf(ax_thm_2Etoto_2ElexTO,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1V: $i] :
( ( mem @ V1V @ ( arr @ A_27b @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( ap @ ( ap @ ( c_2Etoto_2ElexTO @ A_27a @ A_27b ) @ V0R ) @ V1V )
= ( ap @ ( c_2Etoto_2ETO__of__LinearOrder @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2ELEX @ A_27a @ A_27b ) @ ( ap @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a ) @ V0R ) ) @ ( ap @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27b ) @ V1V ) ) ) ) ) ) ).
thf(ax_thm_2Etoto_2Elextoto,axiom,
! [A_27a: del,A_27b: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( ty_2Etoto_2Etoto @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Etoto_2Elextoto @ A_27a @ A_27b ) @ V0c ) @ V1v )
= ( ap @ ( c_2Etoto_2ETO @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Etoto_2ElexTO @ A_27a @ A_27b ) @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) ) @ ( ap @ ( c_2Etoto_2Eapto @ A_27b ) @ V1v ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ElexTO__thm,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1V: $i] :
( ( mem @ V1V @ ( arr @ A_27b @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27b ) @ V1V ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V3y: $i] :
( ( mem @ V3y @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2ElexTO @ A_27a @ A_27b ) @ V0R ) @ V1V ) @ V2x ) @ V3y ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ V0R @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V2x ) ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V3y ) ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2ELESS ) ) @ ( ap @ ( ap @ V1V @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V2x ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V3y ) ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2EGREATER ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ElexTO__ALT,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1V: $i] :
( ( mem @ V1V @ ( arr @ A_27b @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27b ) @ V1V ) ) )
=> ( p
@ ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ bool )
@ ( lam @ A_27a
@ ^ [V2r: $i] :
( lam @ A_27b
@ ^ [V3u: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ bool )
@ ( lam @ A_27a
@ ^ [V4r_27: $i] :
( lam @ A_27b
@ ^ [V5u_27: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2ElexTO @ A_27a @ A_27b ) @ V0R ) @ V1V ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2r ) @ V3u ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V4r_27 ) @ V5u_27 ) ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ V0R @ V2r ) @ V4r_27 ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2ELESS ) ) @ ( ap @ ( ap @ V1V @ V3u ) @ V5u_27 ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2EGREATER ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ETO__lexTO,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1V: $i] :
( ( mem @ V1V @ ( arr @ A_27b @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27b ) @ V1V ) ) )
=> ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Etoto_2ElexTO @ A_27a @ A_27b ) @ V0R ) @ V1V ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Epre__aplextoto,axiom,
! [A_27a: del,A_27b: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( ty_2Etoto_2Etoto @ A_27b ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V3y: $i] :
( ( mem @ V3y @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Etoto_2Elextoto @ A_27a @ A_27b ) @ V0c ) @ V1v ) ) @ V2x ) @ V3y ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V2x ) ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V3y ) ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2ELESS ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27b ) @ V1v ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V2x ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V3y ) ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2EGREATER ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Eaplextoto,axiom,
! [A_27a: del,A_27b: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( ty_2Etoto_2Etoto @ A_27b ) )
=> ! [V2x1: $i] :
( ( mem @ V2x1 @ A_27a )
=> ! [V3x2: $i] :
( ( mem @ V3x2 @ A_27b )
=> ! [V4y1: $i] :
( ( mem @ V4y1 @ A_27a )
=> ! [V5y2: $i] :
( ( mem @ V5y2 @ A_27b )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Etoto_2Elextoto @ A_27a @ A_27b ) @ V0c ) @ V1v ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x1 ) @ V3x2 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V4y1 ) @ V5y2 ) ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V2x1 ) @ V4y1 ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2ELESS ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27b ) @ V1v ) @ V3x2 ) @ V5y2 ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2EGREATER ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2EStrongLinearOrder__LESS,axiom,
p @ ( ap @ ( c_2Erelation_2EStrongLinearOrder @ ty_2Enum_2Enum ) @ c_2Eprim__rec_2E_3C ) ).
thf(ax_thm_2Etoto_2EnumOrd,axiom,
( c_2Etoto_2EnumOrd
= ( ap @ ( c_2Etoto_2ETO__of__LinearOrder @ ty_2Enum_2Enum ) @ c_2Eprim__rec_2E_3C ) ) ).
thf(conj_thm_2Etoto_2ETO__numOrd,axiom,
p @ ( ap @ ( c_2Etoto_2ETotOrd @ ty_2Enum_2Enum ) @ c_2Etoto_2EnumOrd ) ).
thf(ax_thm_2Etoto_2Enumto,axiom,
( ( surj__c_ty_2Etoto_2Etoto_ty_2Enum_2Enum @ c_2Etoto_2Enumto )
= ( surj__c_ty_2Etoto_2Etoto_ty_2Enum_2Enum @ ( ap @ ( c_2Etoto_2ETO @ ty_2Enum_2Enum ) @ c_2Etoto_2EnumOrd ) ) ) ).
thf(conj_thm_2Etoto_2Eapnumto__thm,axiom,
( ( ap @ ( c_2Etoto_2Eapto @ ty_2Enum_2Enum ) @ c_2Etoto_2Enumto )
= c_2Etoto_2EnumOrd ) ).
thf(conj_thm_2Etoto_2EnumeralOrd,axiom,
! [V0x: tp__ty_2Enum_2Enum,V1y: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2EnumOrd @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
= fo__c_2EternaryComparisons_2EEQUAL )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2EnumOrd @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
= fo__c_2EternaryComparisons_2ELESS )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2EnumOrd @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
= fo__c_2EternaryComparisons_2ELESS )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2EnumOrd @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
= fo__c_2EternaryComparisons_2EGREATER )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2EnumOrd @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
= fo__c_2EternaryComparisons_2EGREATER )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2EnumOrd @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2EnumOrd @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2EnumOrd @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2EnumOrd @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2EnumOrd @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ c_2Etoto_2EnumOrd @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2ELESS ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2ELESS ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2EGREATER ) ) ) )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2EnumOrd @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ c_2Etoto_2EnumOrd @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2ELESS ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2EGREATER ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2EGREATER ) ) ) ) ) ).
thf(ax_thm_2Etoto_2Enum__dt__TY__DEF,axiom,
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ty_2Etoto_2Enum__dt @ ( ty_2Eind__type_2Erecspace @ bool ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( ty_2Eind__type_2Erecspace @ bool ) @ ty_2Etoto_2Enum__dt )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ bool )
@ ^ [V1a0: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( ty_2Eind__type_2Erecspace @ bool ) @ bool ) )
@ ( lam @ ( arr @ ( ty_2Eind__type_2Erecspace @ bool ) @ bool )
@ ^ [V2_27num__dt_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Eind__type_2Erecspace @ bool ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ bool )
@ ^ [V3a0: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap
@ ( ap @ c_2Ebool_2E_5C_2F
@ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ bool ) ) @ V3a0 )
@ ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ bool ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( c_2Ebool_2EARB @ bool ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( c_2Eind__type_2EBOTTOM @ bool ) ) ) ) )
@ ( ap
@ ( ap @ c_2Ebool_2E_5C_2F
@ ( ap @ ( c_2Ebool_2E_3F @ ( ty_2Eind__type_2Erecspace @ bool ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ bool )
@ ^ [V5a: $i] :
( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ bool ) ) @ V3a0 )
@ ( ap
@ ( lam @ ( ty_2Eind__type_2Erecspace @ bool )
@ ^ [V6a: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ bool ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( c_2Ebool_2EARB @ bool ) )
@ ( ap @ ( ap @ ( c_2Eind__type_2EFCONS @ ( ty_2Eind__type_2Erecspace @ bool ) ) @ V6a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7n: $i] : ( c_2Eind__type_2EBOTTOM @ bool ) ) ) ) )
@ V5a ) ) )
@ ( ap @ V2_27num__dt_27 @ V5a ) ) ) ) )
@ ( ap @ ( c_2Ebool_2E_3F @ ( ty_2Eind__type_2Erecspace @ bool ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ bool )
@ ^ [V8a: $i] :
( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ bool ) ) @ V3a0 )
@ ( ap
@ ( lam @ ( ty_2Eind__type_2Erecspace @ bool )
@ ^ [V9a: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ bool ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( c_2Ebool_2EARB @ bool ) )
@ ( ap @ ( ap @ ( c_2Eind__type_2EFCONS @ ( ty_2Eind__type_2Erecspace @ bool ) ) @ V9a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V10n: $i] : ( c_2Eind__type_2EBOTTOM @ bool ) ) ) ) )
@ V8a ) ) )
@ ( ap @ V2_27num__dt_27 @ V8a ) ) ) ) ) ) )
@ ( ap @ V2_27num__dt_27 @ V3a0 ) ) ) ) )
@ ( ap @ V2_27num__dt_27 @ V1a0 ) ) ) ) ) )
@ V0rep ) ) ) ).
thf(ax_thm_2Etoto_2Enum__dt__case__def,axiom,
! [A_27a: del] :
( ! [V0v: $i] :
( ( mem @ V0v @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Etoto_2Enum__dt @ A_27a ) )
=> ! [V2f1: $i] :
( ( mem @ V2f1 @ ( arr @ ty_2Etoto_2Enum__dt @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Enum__dt__CASE @ A_27a ) @ ( inj__ty_2Etoto_2Enum__dt @ fo__c_2Etoto_2Ezer ) ) @ V0v ) @ V1f ) @ V2f1 )
= V0v ) ) ) )
& ! [V3a: tp__ty_2Etoto_2Enum__dt,V4v: $i] :
( ( mem @ V4v @ A_27a )
=> ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ ty_2Etoto_2Enum__dt @ A_27a ) )
=> ! [V6f1: $i] :
( ( mem @ V6f1 @ ( arr @ ty_2Etoto_2Enum__dt @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Enum__dt__CASE @ A_27a ) @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V3a ) ) ) @ V4v ) @ V5f ) @ V6f1 )
= ( ap @ V5f @ ( inj__ty_2Etoto_2Enum__dt @ V3a ) ) ) ) ) )
& ! [V7a: tp__ty_2Etoto_2Enum__dt,V8v: $i] :
( ( mem @ V8v @ A_27a )
=> ! [V9f: $i] :
( ( mem @ V9f @ ( arr @ ty_2Etoto_2Enum__dt @ A_27a ) )
=> ! [V10f1: $i] :
( ( mem @ V10f1 @ ( arr @ ty_2Etoto_2Enum__dt @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Enum__dt__CASE @ A_27a ) @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V7a ) ) ) @ V8v ) @ V9f ) @ V10f1 )
= ( ap @ V10f1 @ ( inj__ty_2Etoto_2Enum__dt @ V7a ) ) ) ) ) ) ) ).
thf(ax_thm_2Etoto_2Enum__dt__size__def,axiom,
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Etoto_2Enum__dt__size @ ( inj__ty_2Etoto_2Enum__dt @ fo__c_2Etoto_2Ezer ) ) )
= fo__c_2Enum_2E0 )
& ! [V0a: tp__ty_2Etoto_2Enum__dt] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Etoto_2Enum__dt__size @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V0a ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ c_2Etoto_2Enum__dt__size @ ( inj__ty_2Etoto_2Enum__dt @ V0a ) ) ) ) )
& ! [V1a: tp__ty_2Etoto_2Enum__dt] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Etoto_2Enum__dt__size @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V1a ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ c_2Etoto_2Enum__dt__size @ ( inj__ty_2Etoto_2Enum__dt @ V1a ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Edatatype__num__dt,axiom,
! [V0num__dt: $i] :
( ( mem @ V0num__dt @ ( arr @ ty_2Etoto_2Enum__dt @ ( arr @ ( arr @ ty_2Etoto_2Enum__dt @ ty_2Etoto_2Enum__dt ) @ ( arr @ ( arr @ ty_2Etoto_2Enum__dt @ ty_2Etoto_2Enum__dt ) @ bool ) ) ) )
=> ( p @ ( ap @ ( c_2Ebool_2EDATATYPE @ bool ) @ ( ap @ ( ap @ ( ap @ V0num__dt @ ( inj__ty_2Etoto_2Enum__dt @ fo__c_2Etoto_2Ezer ) ) @ c_2Etoto_2Ebit1 ) @ c_2Etoto_2Ebit2 ) ) ) ) ).
thf(conj_thm_2Etoto_2Enum__dt__11,axiom,
( ! [V0a: tp__ty_2Etoto_2Enum__dt,V1a_27: tp__ty_2Etoto_2Enum__dt] :
( ( ( surj__ty_2Etoto_2Enum__dt @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V0a ) ) )
= ( surj__ty_2Etoto_2Enum__dt @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V1a_27 ) ) ) )
<=> ( V0a = V1a_27 ) )
& ! [V2a: tp__ty_2Etoto_2Enum__dt,V3a_27: tp__ty_2Etoto_2Enum__dt] :
( ( ( surj__ty_2Etoto_2Enum__dt @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V2a ) ) )
= ( surj__ty_2Etoto_2Enum__dt @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V3a_27 ) ) ) )
<=> ( V2a = V3a_27 ) ) ) ).
thf(conj_thm_2Etoto_2Enum__dt__distinct,axiom,
( ! [V0a: tp__ty_2Etoto_2Enum__dt] :
( fo__c_2Etoto_2Ezer
!= ( surj__ty_2Etoto_2Enum__dt @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V0a ) ) ) )
& ! [V1a: tp__ty_2Etoto_2Enum__dt] :
( fo__c_2Etoto_2Ezer
!= ( surj__ty_2Etoto_2Enum__dt @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V1a ) ) ) )
& ! [V2a_27: tp__ty_2Etoto_2Enum__dt,V3a: tp__ty_2Etoto_2Enum__dt] :
( ( surj__ty_2Etoto_2Enum__dt @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V3a ) ) )
!= ( surj__ty_2Etoto_2Enum__dt @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V2a_27 ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Enum__dt__nchotomy,axiom,
! [V0nn: tp__ty_2Etoto_2Enum__dt] :
( ( V0nn = fo__c_2Etoto_2Ezer )
| ? [V1n: tp__ty_2Etoto_2Enum__dt] :
( V0nn
= ( surj__ty_2Etoto_2Enum__dt @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V1n ) ) ) )
| ? [V2n: tp__ty_2Etoto_2Enum__dt] :
( V0nn
= ( surj__ty_2Etoto_2Enum__dt @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V2n ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Enum__dt__Axiom,axiom,
! [A_27a: del,V0f0: $i] :
( ( mem @ V0f0 @ A_27a )
=> ! [V1f1: $i] :
( ( mem @ V1f1 @ ( arr @ ty_2Etoto_2Enum__dt @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ ( arr @ ty_2Etoto_2Enum__dt @ ( arr @ A_27a @ A_27a ) ) )
=> ? [V3fn: $i] :
( ( mem @ V3fn @ ( arr @ ty_2Etoto_2Enum__dt @ A_27a ) )
& ( ( ap @ V3fn @ ( inj__ty_2Etoto_2Enum__dt @ fo__c_2Etoto_2Ezer ) )
= V0f0 )
& ! [V4a: tp__ty_2Etoto_2Enum__dt] :
( ( ap @ V3fn @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V4a ) ) )
= ( ap @ ( ap @ V1f1 @ ( inj__ty_2Etoto_2Enum__dt @ V4a ) ) @ ( ap @ V3fn @ ( inj__ty_2Etoto_2Enum__dt @ V4a ) ) ) )
& ! [V5a: tp__ty_2Etoto_2Enum__dt] :
( ( ap @ V3fn @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V5a ) ) )
= ( ap @ ( ap @ V2f2 @ ( inj__ty_2Etoto_2Enum__dt @ V5a ) ) @ ( ap @ V3fn @ ( inj__ty_2Etoto_2Enum__dt @ V5a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Enum__dt__induction,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Etoto_2Enum__dt @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( inj__ty_2Etoto_2Enum__dt @ fo__c_2Etoto_2Ezer ) ) )
& ! [V1n: tp__ty_2Etoto_2Enum__dt] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Etoto_2Enum__dt @ V1n ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V1n ) ) ) ) )
& ! [V2n: tp__ty_2Etoto_2Enum__dt] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Etoto_2Enum__dt @ V2n ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V2n ) ) ) ) ) )
=> ! [V3n: tp__ty_2Etoto_2Enum__dt] : ( p @ ( ap @ V0P @ ( inj__ty_2Etoto_2Enum__dt @ V3n ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Enum__dt__case__cong,axiom,
! [A_27a: del,V0v_27: $i] :
( ( mem @ V0v_27 @ A_27a )
=> ! [V1f_27: $i] :
( ( mem @ V1f_27 @ ( arr @ ty_2Etoto_2Enum__dt @ A_27a ) )
=> ! [V2f1_27: $i] :
( ( mem @ V2f1_27 @ ( arr @ ty_2Etoto_2Enum__dt @ A_27a ) )
=> ! [V3M: tp__ty_2Etoto_2Enum__dt,V4M_27: tp__ty_2Etoto_2Enum__dt,V5v: $i] :
( ( mem @ V5v @ A_27a )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ ty_2Etoto_2Enum__dt @ A_27a ) )
=> ! [V7f1: $i] :
( ( mem @ V7f1 @ ( arr @ ty_2Etoto_2Enum__dt @ A_27a ) )
=> ( ( ( V3M = V4M_27 )
& ( ( V4M_27 = fo__c_2Etoto_2Ezer )
=> ( V5v = V0v_27 ) )
& ! [V8a: tp__ty_2Etoto_2Enum__dt] :
( ( V4M_27
= ( surj__ty_2Etoto_2Enum__dt @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V8a ) ) ) )
=> ( ( ap @ V6f @ ( inj__ty_2Etoto_2Enum__dt @ V8a ) )
= ( ap @ V1f_27 @ ( inj__ty_2Etoto_2Enum__dt @ V8a ) ) ) )
& ! [V9a: tp__ty_2Etoto_2Enum__dt] :
( ( V4M_27
= ( surj__ty_2Etoto_2Enum__dt @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V9a ) ) ) )
=> ( ( ap @ V7f1 @ ( inj__ty_2Etoto_2Enum__dt @ V9a ) )
= ( ap @ V2f1_27 @ ( inj__ty_2Etoto_2Enum__dt @ V9a ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Enum__dt__CASE @ A_27a ) @ ( inj__ty_2Etoto_2Enum__dt @ V3M ) ) @ V5v ) @ V6f ) @ V7f1 )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Enum__dt__CASE @ A_27a ) @ ( inj__ty_2Etoto_2Enum__dt @ V4M_27 ) ) @ V0v_27 ) @ V1f_27 ) @ V2f1_27 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Enum__dt__case__eq,axiom,
! [A_27a: del,V0x: tp__ty_2Etoto_2Enum__dt,V1v: $i] :
( ( mem @ V1v @ A_27a )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Etoto_2Enum__dt @ A_27a ) )
=> ! [V3f1: $i] :
( ( mem @ V3f1 @ ( arr @ ty_2Etoto_2Enum__dt @ A_27a ) )
=> ! [V4v_27: $i] :
( ( mem @ V4v_27 @ A_27a )
=> ( ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Enum__dt__CASE @ A_27a ) @ ( inj__ty_2Etoto_2Enum__dt @ V0x ) ) @ V1v ) @ V2f ) @ V3f1 )
= V4v_27 )
<=> ( ( ( V0x = fo__c_2Etoto_2Ezer )
& ( V1v = V4v_27 ) )
| ? [V5n: tp__ty_2Etoto_2Enum__dt] :
( ( V0x
= ( surj__ty_2Etoto_2Enum__dt @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V5n ) ) ) )
& ( ( ap @ V2f @ ( inj__ty_2Etoto_2Enum__dt @ V5n ) )
= V4v_27 ) )
| ? [V6n: tp__ty_2Etoto_2Enum__dt] :
( ( V0x
= ( surj__ty_2Etoto_2Enum__dt @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V6n ) ) ) )
& ( ( ap @ V3f1 @ ( inj__ty_2Etoto_2Enum__dt @ V6n ) )
= V4v_27 ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Etoto_2Enum__to__dt__primitive,axiom,
( c_2Etoto_2Enum__to__dt
= ( ap
@ ( ap @ ( c_2Erelation_2EWFREC @ ty_2Enum_2Enum @ ty_2Etoto_2Enum__dt )
@ ( ap @ ( c_2Emin_2E_40 @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) )
@ ^ [V0R: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( c_2Erelation_2EWF @ ty_2Enum_2Enum ) @ V0R ) )
@ ( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( c_2Ebool_2E_21 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V1n ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( ap @ c_2Earithmetic_2EODD @ V1n ) ) ) @ ( ap @ ( ap @ V0R @ ( ap @ c_2Earithmetic_2EDIV2 @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V1n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ V1n ) ) ) ) )
@ ( ap @ ( c_2Ebool_2E_21 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V2n ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ c_2Earithmetic_2EODD @ V2n ) ) ) ) @ ( ap @ ( ap @ V0R @ ( ap @ c_2Earithmetic_2EDIV2 @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V2n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ V2n ) ) ) ) ) ) ) ) )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ ty_2Etoto_2Enum__dt )
@ ^ [V3num__to__dt: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V4a: $i] : ( ap @ ( c_2Ecombin_2EI @ ty_2Etoto_2Enum__dt ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Etoto_2Enum__dt ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V4a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Etoto_2Enum__dt @ fo__c_2Etoto_2Ezer ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Etoto_2Enum__dt ) @ ( ap @ c_2Earithmetic_2EODD @ V4a ) ) @ ( ap @ c_2Etoto_2Ebit1 @ ( ap @ V3num__to__dt @ ( ap @ c_2Earithmetic_2EDIV2 @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V4a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) @ ( ap @ c_2Etoto_2Ebit2 @ ( ap @ V3num__to__dt @ ( ap @ c_2Earithmetic_2EDIV2 @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V4a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Enum__dtOrd__ind,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Etoto_2Enum__dt @ ( arr @ ty_2Etoto_2Enum__dt @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Etoto_2Enum__dt @ fo__c_2Etoto_2Ezer ) ) @ ( inj__ty_2Etoto_2Enum__dt @ fo__c_2Etoto_2Ezer ) ) )
& ! [V1x: tp__ty_2Etoto_2Enum__dt] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Etoto_2Enum__dt @ fo__c_2Etoto_2Ezer ) ) @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V1x ) ) ) )
& ! [V2x: tp__ty_2Etoto_2Enum__dt] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Etoto_2Enum__dt @ fo__c_2Etoto_2Ezer ) ) @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V2x ) ) ) )
& ! [V3x: tp__ty_2Etoto_2Enum__dt] : ( p @ ( ap @ ( ap @ V0P @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V3x ) ) ) @ ( inj__ty_2Etoto_2Enum__dt @ fo__c_2Etoto_2Ezer ) ) )
& ! [V4x: tp__ty_2Etoto_2Enum__dt] : ( p @ ( ap @ ( ap @ V0P @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V4x ) ) ) @ ( inj__ty_2Etoto_2Enum__dt @ fo__c_2Etoto_2Ezer ) ) )
& ! [V5x: tp__ty_2Etoto_2Enum__dt,V6y: tp__ty_2Etoto_2Enum__dt] : ( p @ ( ap @ ( ap @ V0P @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V5x ) ) ) @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V6y ) ) ) )
& ! [V7x: tp__ty_2Etoto_2Enum__dt,V8y: tp__ty_2Etoto_2Enum__dt] : ( p @ ( ap @ ( ap @ V0P @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V7x ) ) ) @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V8y ) ) ) )
& ! [V9x: tp__ty_2Etoto_2Enum__dt,V10y: tp__ty_2Etoto_2Enum__dt] :
( ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Etoto_2Enum__dt @ V9x ) ) @ ( inj__ty_2Etoto_2Enum__dt @ V10y ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V9x ) ) ) @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V10y ) ) ) ) )
& ! [V11x: tp__ty_2Etoto_2Enum__dt,V12y: tp__ty_2Etoto_2Enum__dt] :
( ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Etoto_2Enum__dt @ V11x ) ) @ ( inj__ty_2Etoto_2Enum__dt @ V12y ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V11x ) ) ) @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V12y ) ) ) ) ) )
=> ! [V13v: tp__ty_2Etoto_2Enum__dt,V14v1: tp__ty_2Etoto_2Enum__dt] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Etoto_2Enum__dt @ V13v ) ) @ ( inj__ty_2Etoto_2Enum__dt @ V14v1 ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Enum__dtOrd,axiom,
( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Enum__dtOrd @ ( inj__ty_2Etoto_2Enum__dt @ fo__c_2Etoto_2Ezer ) ) @ ( inj__ty_2Etoto_2Enum__dt @ fo__c_2Etoto_2Ezer ) ) )
= fo__c_2EternaryComparisons_2EEQUAL )
& ! [V0x: tp__ty_2Etoto_2Enum__dt] :
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Enum__dtOrd @ ( inj__ty_2Etoto_2Enum__dt @ fo__c_2Etoto_2Ezer ) ) @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V0x ) ) ) )
= fo__c_2EternaryComparisons_2ELESS )
& ! [V1x: tp__ty_2Etoto_2Enum__dt] :
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Enum__dtOrd @ ( inj__ty_2Etoto_2Enum__dt @ fo__c_2Etoto_2Ezer ) ) @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V1x ) ) ) )
= fo__c_2EternaryComparisons_2ELESS )
& ! [V2x: tp__ty_2Etoto_2Enum__dt] :
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Enum__dtOrd @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V2x ) ) ) @ ( inj__ty_2Etoto_2Enum__dt @ fo__c_2Etoto_2Ezer ) ) )
= fo__c_2EternaryComparisons_2EGREATER )
& ! [V3x: tp__ty_2Etoto_2Enum__dt] :
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Enum__dtOrd @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V3x ) ) ) @ ( inj__ty_2Etoto_2Enum__dt @ fo__c_2Etoto_2Ezer ) ) )
= fo__c_2EternaryComparisons_2EGREATER )
& ! [V4y: tp__ty_2Etoto_2Enum__dt,V5x: tp__ty_2Etoto_2Enum__dt] :
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Enum__dtOrd @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V5x ) ) ) @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V4y ) ) ) )
= fo__c_2EternaryComparisons_2ELESS )
& ! [V6y: tp__ty_2Etoto_2Enum__dt,V7x: tp__ty_2Etoto_2Enum__dt] :
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Enum__dtOrd @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V7x ) ) ) @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V6y ) ) ) )
= fo__c_2EternaryComparisons_2EGREATER )
& ! [V8y: tp__ty_2Etoto_2Enum__dt,V9x: tp__ty_2Etoto_2Enum__dt] :
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Enum__dtOrd @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V9x ) ) ) @ ( ap @ c_2Etoto_2Ebit1 @ ( inj__ty_2Etoto_2Enum__dt @ V8y ) ) ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Enum__dtOrd @ ( inj__ty_2Etoto_2Enum__dt @ V9x ) ) @ ( inj__ty_2Etoto_2Enum__dt @ V8y ) ) ) )
& ! [V10y: tp__ty_2Etoto_2Enum__dt,V11x: tp__ty_2Etoto_2Enum__dt] :
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Enum__dtOrd @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V11x ) ) ) @ ( ap @ c_2Etoto_2Ebit2 @ ( inj__ty_2Etoto_2Enum__dt @ V10y ) ) ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Enum__dtOrd @ ( inj__ty_2Etoto_2Enum__dt @ V11x ) ) @ ( inj__ty_2Etoto_2Enum__dt @ V10y ) ) ) ) ) ).
thf(ax_thm_2Etoto_2Eqk__numOrd__def,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Eqk__numOrd @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Enum__dtOrd @ ( ap @ c_2Etoto_2Enum__to__dt @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) @ ( ap @ c_2Etoto_2Enum__to__dt @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ETO__qk__numOrd,axiom,
p @ ( ap @ ( c_2Etoto_2ETotOrd @ ty_2Enum_2Enum ) @ c_2Etoto_2Eqk__numOrd ) ).
thf(conj_thm_2Etoto_2Eqk__numeralOrd,axiom,
! [V0x: tp__ty_2Enum_2Enum,V1y: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Eqk__numOrd @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
= fo__c_2EternaryComparisons_2EEQUAL )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Eqk__numOrd @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
= fo__c_2EternaryComparisons_2ELESS )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Eqk__numOrd @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
= fo__c_2EternaryComparisons_2ELESS )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Eqk__numOrd @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
= fo__c_2EternaryComparisons_2EGREATER )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Eqk__numOrd @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
= fo__c_2EternaryComparisons_2EGREATER )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Eqk__numOrd @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Eqk__numOrd @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Eqk__numOrd @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Eqk__numOrd @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Eqk__numOrd @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
= fo__c_2EternaryComparisons_2ELESS )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2Eqk__numOrd @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
= fo__c_2EternaryComparisons_2EGREATER ) ) ).
thf(ax_thm_2Etoto_2Eqk__numto,axiom,
( ( surj__c_ty_2Etoto_2Etoto_ty_2Enum_2Enum @ c_2Etoto_2Eqk__numto )
= ( surj__c_ty_2Etoto_2Etoto_ty_2Enum_2Enum @ ( ap @ ( c_2Etoto_2ETO @ ty_2Enum_2Enum ) @ c_2Etoto_2Eqk__numOrd ) ) ) ).
thf(conj_thm_2Etoto_2Eap__qk__numto__thm,axiom,
( ( ap @ ( c_2Etoto_2Eapto @ ty_2Enum_2Enum ) @ c_2Etoto_2Eqk__numto )
= c_2Etoto_2Eqk__numOrd ) ).
thf(ax_thm_2Etoto_2EcharOrd,axiom,
! [V0a: tp__ty_2Estring_2Echar,V1b: tp__ty_2Estring_2Echar] :
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2EcharOrd @ ( inj__ty_2Estring_2Echar @ V0a ) ) @ ( inj__ty_2Estring_2Echar @ V1b ) ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2EnumOrd @ ( ap @ c_2Estring_2EORD @ ( inj__ty_2Estring_2Echar @ V0a ) ) ) @ ( ap @ c_2Estring_2EORD @ ( inj__ty_2Estring_2Echar @ V1b ) ) ) ) ) ).
thf(ax_thm_2Etoto_2Echarto,axiom,
( ( surj__c_ty_2Etoto_2Etoto_ty_2Estring_2Echar @ c_2Etoto_2Echarto )
= ( surj__c_ty_2Etoto_2Etoto_ty_2Estring_2Echar @ ( ap @ ( c_2Etoto_2ETO @ ty_2Estring_2Echar ) @ c_2Etoto_2EcharOrd ) ) ) ).
thf(conj_thm_2Etoto_2ETO__charOrd,axiom,
p @ ( ap @ ( c_2Etoto_2ETotOrd @ ty_2Estring_2Echar ) @ c_2Etoto_2EcharOrd ) ).
thf(conj_thm_2Etoto_2Eapcharto__thm,axiom,
( ( ap @ ( c_2Etoto_2Eapto @ ty_2Estring_2Echar ) @ c_2Etoto_2Echarto )
= c_2Etoto_2EcharOrd ) ).
thf(conj_thm_2Etoto_2EcharOrd__lt__lem,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2EnumOrd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
= fo__c_2EternaryComparisons_2ELESS )
=> ( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) )
<=> $true )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2EcharOrd @ ( ap @ c_2Estring_2ECHR @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) @ ( ap @ c_2Estring_2ECHR @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ).
thf(conj_thm_2Etoto_2EcharOrd__gt__lem,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2EnumOrd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
= fo__c_2EternaryComparisons_2EGREATER )
=> ( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) )
<=> $true )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2EcharOrd @ ( ap @ c_2Estring_2ECHR @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) @ ( ap @ c_2Estring_2ECHR @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) )
= fo__c_2EternaryComparisons_2EGREATER ) ) ) ).
thf(conj_thm_2Etoto_2EcharOrd__eq__lem,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2EnumOrd @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
= fo__c_2EternaryComparisons_2EEQUAL )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2EcharOrd @ ( ap @ c_2Estring_2ECHR @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) @ ( ap @ c_2Estring_2ECHR @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) )
= fo__c_2EternaryComparisons_2EEQUAL ) ) ).
thf(conj_thm_2Etoto_2EcharOrd__thm,axiom,
( c_2Etoto_2EcharOrd
= ( ap @ ( c_2Etoto_2ETO__of__LinearOrder @ ty_2Estring_2Echar ) @ c_2Estring_2Echar__lt ) ) ).
thf(conj_thm_2Etoto_2Elistorder__ind,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) ) )
=> ( ( ! [V1V: $i] :
( ( mem @ V1V @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V1V ) @ V2l ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) )
& ! [V3V: $i] :
( ( mem @ V3V @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V4s: $i] :
( ( mem @ V4s @ A_27a )
=> ! [V5m: $i] :
( ( mem @ V5m @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V3V ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4s ) @ V5m ) ) ) ) ) )
& ! [V6V: $i] :
( ( mem @ V6V @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V7r: $i] :
( ( mem @ V7r @ A_27a )
=> ! [V8l: $i] :
( ( mem @ V8l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V9s: $i] :
( ( mem @ V9s @ A_27a )
=> ! [V10m: $i] :
( ( mem @ V10m @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ V0P @ V6V ) @ V8l ) @ V10m ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V6V ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V7r ) @ V8l ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V9s ) @ V10m ) ) ) ) ) ) ) ) ) )
=> ! [V11v: $i] :
( ( mem @ V11v @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V12v1: $i] :
( ( mem @ V12v1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V13v2: $i] :
( ( mem @ V13v2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V11v ) @ V12v1 ) @ V13v2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2Elistorder,axiom,
! [A_27a: del] :
( ! [V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1V: $i] :
( ( mem @ V1V @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Elistorder @ A_27a ) @ V1V ) @ V0l ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
<=> $false ) ) )
& ! [V2s: $i] :
( ( mem @ V2s @ A_27a )
=> ! [V3m: $i] :
( ( mem @ V3m @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4V: $i] :
( ( mem @ V4V @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Elistorder @ A_27a ) @ V4V ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2s ) @ V3m ) ) )
<=> $true ) ) ) )
& ! [V5s: $i] :
( ( mem @ V5s @ A_27a )
=> ! [V6r: $i] :
( ( mem @ V6r @ A_27a )
=> ! [V7m: $i] :
( ( mem @ V7m @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V8l: $i] :
( ( mem @ V8l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V9V: $i] :
( ( mem @ V9V @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Elistorder @ A_27a ) @ V9V ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V6r ) @ V8l ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V5s ) @ V7m ) ) )
<=> ( ( p @ ( ap @ ( ap @ V9V @ V6r ) @ V5s ) )
| ( ( V6r = V5s )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Elistorder @ A_27a ) @ V9V ) @ V8l ) @ V7m ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ESLO__listorder,axiom,
! [A_27a: del,V0V: $i] :
( ( mem @ V0V @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EStrongLinearOrder @ A_27a ) @ V0V ) )
=> ( p @ ( ap @ ( c_2Erelation_2EStrongLinearOrder @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Etoto_2Elistorder @ A_27a ) @ V0V ) ) ) ) ) ).
thf(ax_thm_2Etoto_2EListOrd,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ( ( ap @ ( c_2Etoto_2EListOrd @ A_27a ) @ V0c )
= ( ap @ ( c_2Etoto_2ETO__of__LinearOrder @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Etoto_2Elistorder @ A_27a ) @ ( ap @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a ) @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ETO__ListOrd,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Etoto_2EListOrd @ A_27a ) @ V0c ) ) ) ) ).
thf(conj_thm_2Etoto_2EListOrd__THM,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2EListOrd @ A_27a ) @ V0c ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
= fo__c_2EternaryComparisons_2EEQUAL )
& ! [V1b: $i] :
( ( mem @ V1b @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2EListOrd @ A_27a ) @ V0c ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1b ) @ V2y ) ) )
= fo__c_2EternaryComparisons_2ELESS ) ) )
& ! [V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2EListOrd @ A_27a ) @ V0c ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3a ) @ V4x ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
= fo__c_2EternaryComparisons_2EGREATER ) ) )
& ! [V5a: $i] :
( ( mem @ V5a @ A_27a )
=> ! [V6x: $i] :
( ( mem @ V6x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V7b: $i] :
( ( mem @ V7b @ A_27a )
=> ! [V8y: $i] :
( ( mem @ V8y @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2EListOrd @ A_27a ) @ V0c ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V5a ) @ V6x ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V7b ) @ V8y ) ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V5a ) @ V7b ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2ELESS ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2EListOrd @ A_27a ) @ V0c ) @ V6x ) @ V8y ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2EGREATER ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Etoto_2Elistoto,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ( ( ap @ ( c_2Etoto_2Elistoto @ A_27a ) @ V0c )
= ( ap @ ( c_2Etoto_2ETO @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Etoto_2EListOrd @ A_27a ) @ V0c ) ) ) ) ).
thf(conj_thm_2Etoto_2Eaplistoto,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Etoto_2Etoto @ A_27a ) )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Etoto_2Elistoto @ A_27a ) @ V0c ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
= fo__c_2EternaryComparisons_2EEQUAL )
& ! [V1b: $i] :
( ( mem @ V1b @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Etoto_2Elistoto @ A_27a ) @ V0c ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1b ) @ V2y ) ) )
= fo__c_2EternaryComparisons_2ELESS ) ) )
& ! [V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Etoto_2Elistoto @ A_27a ) @ V0c ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3a ) @ V4x ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
= fo__c_2EternaryComparisons_2EGREATER ) ) )
& ! [V5a: $i] :
( ( mem @ V5a @ A_27a )
=> ! [V6x: $i] :
( ( mem @ V6x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V7b: $i] :
( ( mem @ V7b @ A_27a )
=> ! [V8y: $i] :
( ( mem @ V8y @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Etoto_2Elistoto @ A_27a ) @ V0c ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V5a ) @ V6x ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V7b ) @ V8y ) ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ A_27a ) @ V0c ) @ V5a ) @ V7b ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2ELESS ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2Eapto @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Etoto_2Elistoto @ A_27a ) @ V0c ) ) @ V6x ) @ V8y ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2EGREATER ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Etoto_2Estringto,axiom,
( ( surj__c_ty_2Etoto_2Etoto_c_ty_2Elist_2Elist_ty_2Estring_2Echar @ c_2Etoto_2Estringto )
= ( surj__c_ty_2Etoto_2Etoto_c_ty_2Elist_2Elist_ty_2Estring_2Echar @ ( ap @ ( c_2Etoto_2Elistoto @ ty_2Estring_2Echar ) @ c_2Etoto_2Echarto ) ) ) ).
thf(ax_thm_2Etoto_2EimageOrd,axiom,
! [A_27a: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27c ) )
=> ! [V1cp: $i] :
( ( mem @ V1cp @ ( arr @ A_27c @ ( arr @ A_27c @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3b: $i] :
( ( mem @ V3b @ A_27a )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Etoto_2EimageOrd @ A_27a @ A_27c ) @ V0f ) @ V1cp ) @ V2a ) @ V3b ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V1cp @ ( ap @ V0f @ V2a ) ) @ ( ap @ V0f @ V3b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2ETO__injection,axiom,
! [A_27c: del,A_27d: del,V0cp: $i] :
( ( mem @ V0cp @ ( arr @ A_27c @ ( arr @ A_27c @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27c ) @ V0cp ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27d @ A_27c ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2EONE__ONE @ A_27d @ A_27c ) @ V1f ) )
=> ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27d ) @ ( ap @ ( ap @ ( c_2Etoto_2EimageOrd @ A_27d @ A_27c ) @ V1f ) @ V0cp ) ) ) ) ) ) ) ).
thf(conj_thm_2Etoto_2EStrongLinearOrder__of__TO__TO__of__LinearOrder,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ V0R ) )
=> ( ( ap @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a ) @ ( ap @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a ) @ V0R ) )
= V0R ) ) ) ).
thf(conj_thm_2Etoto_2ETO__of__LinearOrder__LEX,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1V: $i] :
( ( mem @ V1V @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27b ) @ V1V ) ) )
=> ( ( ap @ ( c_2Etoto_2ETO__of__LinearOrder @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2ELEX @ A_27a @ A_27b ) @ V0R ) @ V1V ) )
= ( ap @ ( ap @ ( c_2Etoto_2ElexTO @ A_27a @ A_27b ) @ ( ap @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a ) @ V0R ) ) @ ( ap @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27b ) @ V1V ) ) ) ) ) ) ).
%------------------------------------------------------------------------------