ITP001 Axioms: ITP097^5.ax
%------------------------------------------------------------------------------
% File : ITP097^5 : 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 : comparison^2.ax [Gau20]
% : HL4097^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 59 ( 6 unt; 5 typ; 0 def)
% Number of atoms : 1438 ( 135 equ; 0 cnn)
% Maximal formula atoms : 70 ( 24 avg)
% Number of connectives : 2285 ( 3 ~; 2 |; 63 &;1969 @)
% ( 24 <=>; 224 =>; 0 <=; 0 <~>)
% Maximal formula depth : 35 ( 13 avg;1969 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 10 ( 10 >; 0 *; 0 +; 0 <<)
% Number of symbols : 70 ( 69 usr; 64 con; 0-3 aty)
% Number of variables : 269 ( 12 ^ 257 !; 0 ?; 269 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Ecomparison_2Eequiv__inj,type,
c_2Ecomparison_2Eequiv__inj: del > del > $i ).
thf(mem_c_2Ecomparison_2Eequiv__inj,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ecomparison_2Eequiv__inj @ 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 @ ( arr @ A_27a @ A_27b ) @ bool ) ) ) ) ).
thf(tp_c_2Ecomparison_2Egood__cmp,type,
c_2Ecomparison_2Egood__cmp: del > $i ).
thf(mem_c_2Ecomparison_2Egood__cmp,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecomparison_2Egood__cmp @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) @ bool ) ) ).
thf(tp_c_2Ecomparison_2Eoption__cmp2,type,
c_2Ecomparison_2Eoption__cmp2: del > del > $i ).
thf(mem_c_2Ecomparison_2Eoption__cmp2,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ecomparison_2Eoption__cmp2 @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) @ ( arr @ ( ty_2Eoption_2Eoption @ A_27a ) @ ( arr @ ( ty_2Eoption_2Eoption @ A_27b ) @ ty_2EternaryComparisons_2Eordering ) ) ) ) ).
thf(tp_c_2Ecomparison_2Eresp__equiv,type,
c_2Ecomparison_2Eresp__equiv: del > del > del > $i ).
thf(mem_c_2Ecomparison_2Eresp__equiv,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Ecomparison_2Eresp__equiv @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) @ bool ) ) ) ).
thf(tp_c_2Ecomparison_2Eresp__equiv2,type,
c_2Ecomparison_2Eresp__equiv2: del > del > $i ).
thf(mem_c_2Ecomparison_2Eresp__equiv2,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ecomparison_2Eresp__equiv2 @ 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 @ ( arr @ A_27a @ A_27b ) @ bool ) ) ) ) ).
thf(ax_thm_2Ecomparison_2Egood__cmp__def,axiom,
! [A_27a: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Ecomparison_2Egood__cmp @ A_27a ) @ V0cmp ) )
<=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V1x ) @ V1x ) )
= fo__c_2EternaryComparisons_2EEQUAL ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V2x ) @ V3y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V3y ) @ V2x ) )
= fo__c_2EternaryComparisons_2EEQUAL ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V4x ) @ V5y ) )
= fo__c_2EternaryComparisons_2EGREATER )
<=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V5y ) @ V4x ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7y: $i] :
( ( mem @ V7y @ A_27a )
=> ! [V8z: $i] :
( ( mem @ V8z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V6x ) @ V7y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V7y ) @ V8z ) )
= fo__c_2EternaryComparisons_2ELESS ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V6x ) @ V8z ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) )
& ! [V9x: $i] :
( ( mem @ V9x @ A_27a )
=> ! [V10y: $i] :
( ( mem @ V10y @ A_27a )
=> ! [V11z: $i] :
( ( mem @ V11z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V9x ) @ V10y ) )
= fo__c_2EternaryComparisons_2ELESS )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V10y ) @ V11z ) )
= fo__c_2EternaryComparisons_2EEQUAL ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V9x ) @ V11z ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) )
& ! [V12x: $i] :
( ( mem @ V12x @ A_27a )
=> ! [V13y: $i] :
( ( mem @ V13y @ A_27a )
=> ! [V14z: $i] :
( ( mem @ V14z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V12x ) @ V13y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V13y ) @ V14z ) )
= fo__c_2EternaryComparisons_2EEQUAL ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V12x ) @ V14z ) )
= fo__c_2EternaryComparisons_2EEQUAL ) ) ) ) )
& ! [V15x: $i] :
( ( mem @ V15x @ A_27a )
=> ! [V16y: $i] :
( ( mem @ V16y @ A_27a )
=> ! [V17z: $i] :
( ( mem @ V17z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V15x ) @ V16y ) )
= fo__c_2EternaryComparisons_2ELESS )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V16y ) @ V17z ) )
= fo__c_2EternaryComparisons_2ELESS ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V15x ) @ V17z ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Egood__cmp__thm,axiom,
! [A_27a: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Ecomparison_2Egood__cmp @ A_27a ) @ V0cmp ) )
<=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V1x ) @ V1x ) )
= fo__c_2EternaryComparisons_2EEQUAL ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ! [V4z: $i] :
( ( mem @ V4z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V2x ) @ V3y ) )
= fo__c_2EternaryComparisons_2EGREATER )
<=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V3y ) @ V2x ) )
= fo__c_2EternaryComparisons_2ELESS ) )
& ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V2x ) @ V3y ) )
= fo__c_2EternaryComparisons_2ELESS )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V3y ) @ V4z ) )
= fo__c_2EternaryComparisons_2EEQUAL ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V2x ) @ V4z ) )
= fo__c_2EternaryComparisons_2ELESS ) )
& ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V2x ) @ V3y ) )
= fo__c_2EternaryComparisons_2ELESS )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V3y ) @ V4z ) )
= fo__c_2EternaryComparisons_2ELESS ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V2x ) @ V4z ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Ecmp__thms,axiom,
! [A_27a: del] :
( ( fo__c_2EternaryComparisons_2ELESS != fo__c_2EternaryComparisons_2EEQUAL )
& ( fo__c_2EternaryComparisons_2ELESS != fo__c_2EternaryComparisons_2EGREATER )
& ( fo__c_2EternaryComparisons_2EEQUAL != fo__c_2EternaryComparisons_2EGREATER )
& ! [V0v0: $i] :
( ( mem @ V0v0 @ A_27a )
=> ! [V1v1: $i] :
( ( mem @ V1v1 @ A_27a )
=> ! [V2v2: $i] :
( ( mem @ V2v2 @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ A_27a ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2ELESS ) ) @ V0v0 ) @ V1v1 ) @ V2v2 )
= V0v0 ) ) ) )
& ! [V3v0: $i] :
( ( mem @ V3v0 @ A_27a )
=> ! [V4v1: $i] :
( ( mem @ V4v1 @ A_27a )
=> ! [V5v2: $i] :
( ( mem @ V5v2 @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ A_27a ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2EEQUAL ) ) @ V3v0 ) @ V4v1 ) @ V5v2 )
= V4v1 ) ) ) )
& ! [V6v0: $i] :
( ( mem @ V6v0 @ A_27a )
=> ! [V7v1: $i] :
( ( mem @ V7v1 @ A_27a )
=> ! [V8v2: $i] :
( ( mem @ V8v2 @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ A_27a ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2EGREATER ) ) @ V6v0 ) @ V7v1 ) @ V8v2 )
= V8v2 ) ) ) )
& ! [V9a: tp__ty_2EternaryComparisons_2Eordering] :
( ( V9a = fo__c_2EternaryComparisons_2ELESS )
| ( V9a = fo__c_2EternaryComparisons_2EEQUAL )
| ( V9a = fo__c_2EternaryComparisons_2EGREATER ) )
& ! [V10cmp: $i] :
( ( mem @ V10cmp @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Ecomparison_2Egood__cmp @ A_27a ) @ V10cmp ) )
<=> ( ! [V11x: $i] :
( ( mem @ V11x @ A_27a )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V10cmp @ V11x ) @ V11x ) )
= fo__c_2EternaryComparisons_2EEQUAL ) )
& ! [V12x: $i] :
( ( mem @ V12x @ A_27a )
=> ! [V13y: $i] :
( ( mem @ V13y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V10cmp @ V12x ) @ V13y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V10cmp @ V13y ) @ V12x ) )
= fo__c_2EternaryComparisons_2EEQUAL ) ) ) )
& ! [V14x: $i] :
( ( mem @ V14x @ A_27a )
=> ! [V15y: $i] :
( ( mem @ V15y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V10cmp @ V14x ) @ V15y ) )
= fo__c_2EternaryComparisons_2EGREATER )
<=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V10cmp @ V15y ) @ V14x ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) )
& ! [V16x: $i] :
( ( mem @ V16x @ A_27a )
=> ! [V17y: $i] :
( ( mem @ V17y @ A_27a )
=> ! [V18z: $i] :
( ( mem @ V18z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V10cmp @ V16x ) @ V17y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V10cmp @ V17y ) @ V18z ) )
= fo__c_2EternaryComparisons_2ELESS ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V10cmp @ V16x ) @ V18z ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) )
& ! [V19x: $i] :
( ( mem @ V19x @ A_27a )
=> ! [V20y: $i] :
( ( mem @ V20y @ A_27a )
=> ! [V21z: $i] :
( ( mem @ V21z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V10cmp @ V19x ) @ V20y ) )
= fo__c_2EternaryComparisons_2ELESS )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V10cmp @ V20y ) @ V21z ) )
= fo__c_2EternaryComparisons_2EEQUAL ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V10cmp @ V19x ) @ V21z ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) )
& ! [V22x: $i] :
( ( mem @ V22x @ A_27a )
=> ! [V23y: $i] :
( ( mem @ V23y @ A_27a )
=> ! [V24z: $i] :
( ( mem @ V24z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V10cmp @ V22x ) @ V23y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V10cmp @ V23y ) @ V24z ) )
= fo__c_2EternaryComparisons_2EEQUAL ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V10cmp @ V22x ) @ V24z ) )
= fo__c_2EternaryComparisons_2EEQUAL ) ) ) ) )
& ! [V25x: $i] :
( ( mem @ V25x @ A_27a )
=> ! [V26y: $i] :
( ( mem @ V26y @ A_27a )
=> ! [V27z: $i] :
( ( mem @ V27z @ A_27a )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V10cmp @ V25x ) @ V26y ) )
= fo__c_2EternaryComparisons_2ELESS )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V10cmp @ V26y ) @ V27z ) )
= fo__c_2EternaryComparisons_2ELESS ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V10cmp @ V25x ) @ V27z ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Eoption__cmp__def,axiom,
! [A_27a: del,A_27b: del,V0c: $i] :
( ( mem @ V0c @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1v0: $i] :
( ( mem @ V1v0 @ A_27b )
=> ! [V2v3: $i] :
( ( mem @ V2v3 @ A_27a )
=> ! [V3v1: $i] :
( ( mem @ V3v1 @ A_27a )
=> ! [V4v2: $i] :
( ( mem @ V4v2 @ A_27b )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eoption__compare @ A_27a @ A_27b ) @ V0c ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( c_2Eoption_2ENONE @ A_27b ) ) )
= fo__c_2EternaryComparisons_2EEQUAL )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eoption__compare @ A_27a @ A_27b ) @ V0c ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V1v0 ) ) )
= fo__c_2EternaryComparisons_2ELESS )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eoption__compare @ A_27a @ A_27b ) @ V0c ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2v3 ) ) @ ( c_2Eoption_2ENONE @ A_27b ) ) )
= fo__c_2EternaryComparisons_2EGREATER )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eoption__compare @ A_27a @ A_27b ) @ V0c ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3v1 ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V4v2 ) ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0c @ V3v1 ) @ V4v2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Eoption__cmp2__ind,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) @ ( arr @ ( ty_2Eoption_2Eoption @ A_27a ) @ ( arr @ ( ty_2Eoption_2Eoption @ A_27b ) @ bool ) ) ) )
=> ( ( ! [V1cmp: $i] :
( ( mem @ V1cmp @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V1cmp ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( c_2Eoption_2ENONE @ A_27b ) ) ) )
& ! [V2cmp: $i] :
( ( mem @ V2cmp @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27b )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V2cmp ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V3x ) ) ) ) )
& ! [V4cmp: $i] :
( ( mem @ V4cmp @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V4cmp ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V5x ) ) @ ( c_2Eoption_2ENONE @ A_27b ) ) ) ) )
& ! [V6cmp: $i] :
( ( mem @ V6cmp @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8y: $i] :
( ( mem @ V8y @ A_27b )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V6cmp ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V7x ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V8y ) ) ) ) ) ) )
=> ! [V9v: $i] :
( ( mem @ V9v @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V10v1: $i] :
( ( mem @ V10v1 @ ( ty_2Eoption_2Eoption @ A_27a ) )
=> ! [V11v2: $i] :
( ( mem @ V11v2 @ ( ty_2Eoption_2Eoption @ A_27b ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V9v ) @ V10v1 ) @ V11v2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Eoption__cmp2__def,axiom,
! [A_27a: del,A_27b: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1x_27: $i] :
( ( mem @ V1x_27 @ A_27b )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Ecomparison_2Eoption__cmp2 @ A_27a @ A_27b ) @ V0cmp ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( c_2Eoption_2ENONE @ A_27b ) ) )
= fo__c_2EternaryComparisons_2EEQUAL )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Ecomparison_2Eoption__cmp2 @ A_27a @ A_27b ) @ V0cmp ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V1x_27 ) ) )
= fo__c_2EternaryComparisons_2EGREATER )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Ecomparison_2Eoption__cmp2 @ A_27a @ A_27b ) @ V0cmp ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2x ) ) @ ( c_2Eoption_2ENONE @ A_27b ) ) )
= fo__c_2EternaryComparisons_2ELESS )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Ecomparison_2Eoption__cmp2 @ A_27a @ A_27b ) @ V0cmp ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2x ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V3y ) ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V2x ) @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Epair__cmp__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0c1: $i] :
( ( mem @ V0c1 @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1c2: $i] :
( ( mem @ V1c2 @ ( arr @ A_27c @ ( arr @ A_27d @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) )
=> ! [V3y: $i] :
( ( mem @ V3y @ ( ty_2Epair_2Eprod @ A_27b @ A_27d ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Epair__compare @ A_27a @ A_27b @ A_27c @ A_27d ) @ V0c1 ) @ V1c2 ) @ V2x ) @ V3y ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ V0c1 @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27c ) @ V2x ) ) @ ( ap @ ( c_2Epair_2EFST @ A_27b @ A_27d ) @ V3y ) ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2ELESS ) ) @ ( ap @ ( ap @ V1c2 @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27c ) @ V2x ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27b @ A_27d ) @ V3y ) ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2EGREATER ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Ebool__cmp__def,axiom,
( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2EternaryComparisons_2Ebool__compare @ c_2Ebool_2ET ) @ c_2Ebool_2ET ) )
= fo__c_2EternaryComparisons_2EEQUAL )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2EternaryComparisons_2Ebool__compare @ c_2Ebool_2EF ) @ c_2Ebool_2EF ) )
= fo__c_2EternaryComparisons_2EEQUAL )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2EternaryComparisons_2Ebool__compare @ c_2Ebool_2ET ) @ c_2Ebool_2EF ) )
= fo__c_2EternaryComparisons_2EGREATER )
& ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2EternaryComparisons_2Ebool__compare @ c_2Ebool_2EF ) @ c_2Ebool_2ET ) )
= fo__c_2EternaryComparisons_2ELESS ) ) ).
thf(conj_thm_2Ecomparison_2Enum__cmp__def,axiom,
! [V0n1: tp__ty_2Enum_2Enum,V1n2: tp__ty_2Enum_2Enum] :
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2EternaryComparisons_2Enum__compare @ ( inj__ty_2Enum_2Enum @ V0n1 ) ) @ ( inj__ty_2Enum_2Enum @ V1n2 ) ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0n1 ) ) @ ( inj__ty_2Enum_2Enum @ V1n2 ) ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2EEQUAL ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n1 ) ) @ ( inj__ty_2Enum_2Enum @ V1n2 ) ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2ELESS ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2EGREATER ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Echar__cmp__def,axiom,
! [V0c1: tp__ty_2Estring_2Echar,V1c2: tp__ty_2Estring_2Echar] :
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2EternaryComparisons_2Echar__compare @ ( inj__ty_2Estring_2Echar @ V0c1 ) ) @ ( inj__ty_2Estring_2Echar @ V1c2 ) ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2EternaryComparisons_2Enum__compare @ ( ap @ c_2Estring_2EORD @ ( inj__ty_2Estring_2Echar @ V0c1 ) ) ) @ ( ap @ c_2Estring_2EORD @ ( inj__ty_2Estring_2Echar @ V1c2 ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Estring__cmp__def,axiom,
( c_2EternaryComparisons_2Estring__compare
= ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Estring_2Echar @ ty_2Estring_2Echar ) @ c_2EternaryComparisons_2Echar__compare ) ) ).
thf(conj_thm_2Ecomparison_2ETotOrder__imp__good__cmp,axiom,
! [A_27a: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Etoto_2ETotOrd @ A_27a ) @ V0cmp ) )
=> ( p @ ( ap @ ( c_2Ecomparison_2Egood__cmp @ A_27a ) @ V0cmp ) ) ) ) ).
thf(conj_thm_2Ecomparison_2ETO__inv__invert,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 ) )
=> ( ( ap @ ( c_2Etoto_2ETO__inv @ A_27a ) @ V0c )
= ( ap @ ( c_2Epair_2ECURRY @ A_27a @ A_27a @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ty_2EternaryComparisons_2Eordering @ ty_2EternaryComparisons_2Eordering ) @ c_2EternaryComparisons_2Einvert__comparison ) @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ty_2EternaryComparisons_2Eordering ) @ V0c ) ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Eoption__cmp2__TO__inv,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( ap @ ( c_2Ecomparison_2Eoption__cmp2 @ A_27a @ A_27a ) @ V0c )
= ( ap @ ( c_2Etoto_2ETO__inv @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( c_2EternaryComparisons_2Eoption__compare @ A_27a @ A_27a ) @ ( ap @ ( c_2Etoto_2ETO__inv @ A_27a ) @ V0c ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Elist__cmp__ListOrd,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 ) )
=> ( ( ap @ ( c_2EternaryComparisons_2Elist__compare @ A_27a @ A_27a ) @ V0c )
= ( ap @ ( c_2Etoto_2EListOrd @ A_27a ) @ ( ap @ ( c_2Etoto_2ETO @ A_27a ) @ V0c ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2ETotOrd__list__cmp,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 @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ A_27a @ A_27a ) @ V0c ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Epair__cmp__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 ) ) )
=> ( ( ap @ ( ap @ ( c_2EternaryComparisons_2Epair__compare @ A_27a @ A_27a @ A_27b @ A_27b ) @ V0R ) @ V1V )
= ( ap @ ( ap @ ( c_2Etoto_2ElexTO @ A_27a @ A_27b ) @ V0R ) @ V1V ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Enum__cmp__numOrd,axiom,
c_2EternaryComparisons_2Enum__compare = c_2Etoto_2EnumOrd ).
thf(conj_thm_2Ecomparison_2Echar__cmp__charOrd,axiom,
c_2EternaryComparisons_2Echar__compare = c_2Etoto_2EcharOrd ).
thf(conj_thm_2Ecomparison_2Estring__cmp__stringto,axiom,
( c_2EternaryComparisons_2Estring__compare
= ( ap @ ( c_2Etoto_2Eapto @ ( ty_2Elist_2Elist @ ty_2Estring_2Echar ) ) @ c_2Etoto_2Estringto ) ) ).
thf(conj_thm_2Ecomparison_2Eoption__cmp__good,axiom,
! [A_27a: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Ecomparison_2Egood__cmp @ A_27a ) @ V0cmp ) )
=> ( p @ ( ap @ ( c_2Ecomparison_2Egood__cmp @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( c_2EternaryComparisons_2Eoption__compare @ A_27a @ A_27a ) @ V0cmp ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Eoption__cmp2__good,axiom,
! [A_27a: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Ecomparison_2Egood__cmp @ A_27a ) @ V0cmp ) )
=> ( p @ ( ap @ ( c_2Ecomparison_2Egood__cmp @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( c_2Ecomparison_2Eoption__cmp2 @ A_27a @ A_27a ) @ V0cmp ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Elist__cmp__good,axiom,
! [A_27a: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Ecomparison_2Egood__cmp @ A_27a ) @ V0cmp ) )
=> ( p @ ( ap @ ( c_2Ecomparison_2Egood__cmp @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ A_27a @ A_27a ) @ V0cmp ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Epair__cmp__good,axiom,
! [A_27a: del,A_27b: del,V0cmp1: $i] :
( ( mem @ V0cmp1 @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1cmp2: $i] :
( ( mem @ V1cmp2 @ ( arr @ A_27b @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( ( p @ ( ap @ ( c_2Ecomparison_2Egood__cmp @ A_27a ) @ V0cmp1 ) )
& ( p @ ( ap @ ( c_2Ecomparison_2Egood__cmp @ A_27b ) @ V1cmp2 ) ) )
=> ( p @ ( ap @ ( c_2Ecomparison_2Egood__cmp @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Epair__compare @ A_27a @ A_27a @ A_27b @ A_27b ) @ V0cmp1 ) @ V1cmp2 ) ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Ebool__cmp__good,axiom,
p @ ( ap @ ( c_2Ecomparison_2Egood__cmp @ bool ) @ c_2EternaryComparisons_2Ebool__compare ) ).
thf(conj_thm_2Ecomparison_2Enum__cmp__good,axiom,
p @ ( ap @ ( c_2Ecomparison_2Egood__cmp @ ty_2Enum_2Enum ) @ c_2EternaryComparisons_2Enum__compare ) ).
thf(conj_thm_2Ecomparison_2Echar__cmp__good,axiom,
p @ ( ap @ ( c_2Ecomparison_2Egood__cmp @ ty_2Estring_2Echar ) @ c_2EternaryComparisons_2Echar__compare ) ).
thf(conj_thm_2Ecomparison_2Estring__cmp__good,axiom,
p @ ( ap @ ( c_2Ecomparison_2Egood__cmp @ ( ty_2Elist_2Elist @ ty_2Estring_2Echar ) ) @ c_2EternaryComparisons_2Estring__compare ) ).
thf(conj_thm_2Ecomparison_2Elist__cmp__cong,axiom,
! [A_27a: del,A_27b: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V3cmp_27: $i] :
( ( mem @ V3cmp_27 @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V4l1_27: $i] :
( ( mem @ V4l1_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5l2_27: $i] :
( ( mem @ V5l2_27 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( V1l1 = V4l1_27 )
& ( V2l2 = V5l2_27 )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7x_27: $i] :
( ( mem @ V7x_27 @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V4l1_27 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V7x_27 ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27b ) @ V5l2_27 ) ) ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V6x ) @ V7x_27 ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V3cmp_27 @ V6x ) @ V7x_27 ) ) ) ) ) ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ A_27a @ A_27b ) @ V0cmp ) @ V1l1 ) @ V2l2 ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ A_27a @ A_27b ) @ V3cmp_27 ) @ V4l1_27 ) @ V5l2_27 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Eoption__cmp__cong,axiom,
! [A_27a: del,A_27b: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1v1: $i] :
( ( mem @ V1v1 @ ( ty_2Eoption_2Eoption @ A_27a ) )
=> ! [V2v2: $i] :
( ( mem @ V2v2 @ ( ty_2Eoption_2Eoption @ A_27b ) )
=> ! [V3cmp_27: $i] :
( ( mem @ V3cmp_27 @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V4v1_27: $i] :
( ( mem @ V4v1_27 @ ( ty_2Eoption_2Eoption @ A_27a ) )
=> ! [V5v2_27: $i] :
( ( mem @ V5v2_27 @ ( ty_2Eoption_2Eoption @ A_27b ) )
=> ( ( ( V1v1 = V4v1_27 )
& ( V2v2 = V5v2_27 )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7x_27: $i] :
( ( mem @ V7x_27 @ A_27b )
=> ( ( ( V4v1_27
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V6x ) )
& ( V5v2_27
= ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V7x_27 ) ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V6x ) @ V7x_27 ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V3cmp_27 @ V6x ) @ V7x_27 ) ) ) ) ) ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eoption__compare @ A_27a @ A_27b ) @ V0cmp ) @ V1v1 ) @ V2v2 ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eoption__compare @ A_27a @ A_27b ) @ V3cmp_27 ) @ V4v1_27 ) @ V5v2_27 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Eoption__cmp2__cong,axiom,
! [A_27a: del,A_27b: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1v1: $i] :
( ( mem @ V1v1 @ ( ty_2Eoption_2Eoption @ A_27a ) )
=> ! [V2v2: $i] :
( ( mem @ V2v2 @ ( ty_2Eoption_2Eoption @ A_27b ) )
=> ! [V3cmp_27: $i] :
( ( mem @ V3cmp_27 @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V4v1_27: $i] :
( ( mem @ V4v1_27 @ ( ty_2Eoption_2Eoption @ A_27a ) )
=> ! [V5v2_27: $i] :
( ( mem @ V5v2_27 @ ( ty_2Eoption_2Eoption @ A_27b ) )
=> ( ( ( V1v1 = V4v1_27 )
& ( V2v2 = V5v2_27 )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7x_27: $i] :
( ( mem @ V7x_27 @ A_27b )
=> ( ( ( V4v1_27
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V6x ) )
& ( V5v2_27
= ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V7x_27 ) ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V6x ) @ V7x_27 ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V3cmp_27 @ V6x ) @ V7x_27 ) ) ) ) ) ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Ecomparison_2Eoption__cmp2 @ A_27a @ A_27b ) @ V0cmp ) @ V1v1 ) @ V2v2 ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Ecomparison_2Eoption__cmp2 @ A_27a @ A_27b ) @ V3cmp_27 ) @ V4v1_27 ) @ V5v2_27 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Epair__cmp__cong,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0cmp1: $i] :
( ( mem @ V0cmp1 @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1cmp2: $i] :
( ( mem @ V1cmp2 @ ( arr @ A_27c @ ( arr @ A_27d @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V2v1: $i] :
( ( mem @ V2v1 @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) )
=> ! [V3v2: $i] :
( ( mem @ V3v2 @ ( ty_2Epair_2Eprod @ A_27b @ A_27d ) )
=> ! [V4cmp1_27: $i] :
( ( mem @ V4cmp1_27 @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V5cmp2_27: $i] :
( ( mem @ V5cmp2_27 @ ( arr @ A_27c @ ( arr @ A_27d @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V6v1_27: $i] :
( ( mem @ V6v1_27 @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) )
=> ! [V7v2_27: $i] :
( ( mem @ V7v2_27 @ ( ty_2Epair_2Eprod @ A_27b @ A_27d ) )
=> ( ( ( V2v1 = V6v1_27 )
& ( V3v2 = V7v2_27 )
& ! [V8a: $i] :
( ( mem @ V8a @ A_27a )
=> ! [V9b: $i] :
( ( mem @ V9b @ A_27c )
=> ! [V10c: $i] :
( ( mem @ V10c @ A_27b )
=> ! [V11d: $i] :
( ( mem @ V11d @ A_27d )
=> ( ( ( V6v1_27
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27c ) @ V8a ) @ V9b ) )
& ( V7v2_27
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27d ) @ V10c ) @ V11d ) ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp1 @ V8a ) @ V10c ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V4cmp1_27 @ V8a ) @ V10c ) ) ) ) ) ) ) )
& ! [V12a: $i] :
( ( mem @ V12a @ A_27a )
=> ! [V13b: $i] :
( ( mem @ V13b @ A_27c )
=> ! [V14c: $i] :
( ( mem @ V14c @ A_27b )
=> ! [V15d: $i] :
( ( mem @ V15d @ A_27d )
=> ( ( ( V6v1_27
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27c ) @ V12a ) @ V13b ) )
& ( V7v2_27
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27d ) @ V14c ) @ V15d ) ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V1cmp2 @ V13b ) @ V15d ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V5cmp2_27 @ V13b ) @ V15d ) ) ) ) ) ) ) ) )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Epair__compare @ A_27a @ A_27b @ A_27c @ A_27d ) @ V0cmp1 ) @ V1cmp2 ) @ V2v1 ) @ V3v2 ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Epair__compare @ A_27a @ A_27b @ A_27c @ A_27d ) @ V4cmp1_27 ) @ V5cmp2_27 ) @ V6v1_27 ) @ V7v2_27 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Egood__cmp__trans,axiom,
! [A_27a: del,A_27b: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Ecomparison_2Egood__cmp @ A_27a ) @ V0cmp ) )
=> ( p
@ ( ap @ ( c_2Erelation_2Etransitive @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
@ ( lam @ A_27a
@ ^ [V1k: $i] :
( lam @ A_27b
@ ^ [V2v: $i] :
( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ bool )
@ ( lam @ A_27a
@ ^ [V3k_27: $i] :
( lam @ A_27b
@ ^ [V4v_27: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ V0cmp @ V1k ) @ V3k_27 ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Egood__cmp__Less__trans,axiom,
! [A_27a: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Ecomparison_2Egood__cmp @ A_27a ) @ V0cmp ) )
=> ( p
@ ( ap @ ( c_2Erelation_2Etransitive @ A_27a )
@ ( lam @ A_27a
@ ^ [V1k: $i] :
( lam @ A_27a
@ ^ [V2k_27: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ V0cmp @ V1k ) @ V2k_27 ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Egood__cmp__Less__irrefl__trans,axiom,
! [A_27a: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Ecomparison_2Egood__cmp @ A_27a ) @ V0cmp ) )
=> ( ( p
@ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a )
@ ( lam @ A_27a
@ ^ [V1k: $i] :
( lam @ A_27a
@ ^ [V2k_27: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ V0cmp @ V1k ) @ V2k_27 ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) )
& ( p
@ ( ap @ ( c_2Erelation_2Etransitive @ A_27a )
@ ( lam @ A_27a
@ ^ [V3k: $i] :
( lam @ A_27a
@ ^ [V4k_27: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ V0cmp @ V3k ) @ V4k_27 ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2ELESS ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Ebool__cmp__antisym,axiom,
! [V0x: $i] :
( ( mem @ V0x @ bool )
=> ! [V1y: $i] :
( ( mem @ V1y @ bool )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2EternaryComparisons_2Ebool__compare @ V0x ) @ V1y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
<=> ( ( p @ V0x )
<=> ( p @ V1y ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Enum__cmp__antisym,axiom,
! [V0x: tp__ty_2Enum_2Enum,V1y: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2EternaryComparisons_2Enum__compare @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) )
= fo__c_2EternaryComparisons_2EEQUAL )
<=> ( V0x = V1y ) ) ).
thf(conj_thm_2Ecomparison_2Echar__cmp__antisym,axiom,
! [V0x: tp__ty_2Estring_2Echar,V1y: tp__ty_2Estring_2Echar] :
( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2EternaryComparisons_2Echar__compare @ ( inj__ty_2Estring_2Echar @ V0x ) ) @ ( inj__ty_2Estring_2Echar @ V1y ) ) )
= fo__c_2EternaryComparisons_2EEQUAL )
<=> ( V0x = V1y ) ) ).
thf(conj_thm_2Ecomparison_2Elist__cmp__antisym,axiom,
! [A_27a: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V3x ) @ V4y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
<=> ( V3x = V4y ) ) ) )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ A_27a @ A_27a ) @ V0cmp ) @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
<=> ( V1x = V2y ) ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Estring__cmp__antisym,axiom,
! [V0x: tp__c_ty_2Elist_2Elist_ty_2Estring_2Echar,V1y: tp__c_ty_2Elist_2Elist_ty_2Estring_2Echar] :
( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2EternaryComparisons_2Estring__compare @ ( inj__c_ty_2Elist_2Elist_ty_2Estring_2Echar @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Estring_2Echar @ V1y ) ) )
= fo__c_2EternaryComparisons_2EEQUAL )
<=> ( V0x = V1y ) ) ).
thf(conj_thm_2Ecomparison_2Epair__cmp__antisym,axiom,
! [A_27a: del,A_27b: del,V0cmp1: $i] :
( ( mem @ V0cmp1 @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1cmp2: $i] :
( ( mem @ V1cmp2 @ ( arr @ A_27b @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V3y: $i] :
( ( mem @ V3y @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp1 @ V4x ) @ V5y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
<=> ( V4x = V5y ) ) ) )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27b )
=> ! [V7y: $i] :
( ( mem @ V7y @ A_27b )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V1cmp2 @ V6x ) @ V7y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
<=> ( V6x = V7y ) ) ) ) )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Epair__compare @ A_27a @ A_27a @ A_27b @ A_27b ) @ V0cmp1 ) @ V1cmp2 ) @ V2x ) @ V3y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
<=> ( V2x = V3y ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Eoption__cmp__antisym,axiom,
! [A_27a: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Eoption_2Eoption @ A_27a ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ ( ty_2Eoption_2Eoption @ A_27a ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V3x ) @ V4y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
<=> ( V3x = V4y ) ) ) )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eoption__compare @ A_27a @ A_27a ) @ V0cmp ) @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
<=> ( V1x = V2y ) ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Eoption__cmp2__antisym,axiom,
! [A_27a: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Eoption_2Eoption @ A_27a ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ ( ty_2Eoption_2Eoption @ A_27a ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V3x ) @ V4y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
<=> ( V3x = V4y ) ) ) )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2Ecomparison_2Eoption__cmp2 @ A_27a @ A_27a ) @ V0cmp ) @ V1x ) @ V2y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
<=> ( V1x = V2y ) ) ) ) ) ) ).
thf(ax_thm_2Ecomparison_2Eresp__equiv__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecomparison_2Eresp__equiv @ A_27a @ A_27b @ A_27c ) @ V0cmp ) @ V1f ) )
<=> ! [V2k1: $i] :
( ( mem @ V2k1 @ A_27a )
=> ! [V3k2: $i] :
( ( mem @ V3k2 @ A_27a )
=> ! [V4v: $i] :
( ( mem @ V4v @ A_27b )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V2k1 ) @ V3k2 ) )
= fo__c_2EternaryComparisons_2EEQUAL )
=> ( ( ap @ ( ap @ V1f @ V2k1 ) @ V4v )
= ( ap @ ( ap @ V1f @ V3k2 ) @ V4v ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ecomparison_2Eresp__equiv2__def,axiom,
! [A_27a: del,A_27b: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1cmp2: $i] :
( ( mem @ V1cmp2 @ ( arr @ A_27b @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ecomparison_2Eresp__equiv2 @ A_27a @ A_27b ) @ V0cmp ) @ V1cmp2 ) @ V2f ) )
<=> ! [V3k1: $i] :
( ( mem @ V3k1 @ A_27a )
=> ! [V4k2: $i] :
( ( mem @ V4k2 @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V3k1 ) @ V4k2 ) )
= fo__c_2EternaryComparisons_2EEQUAL )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V1cmp2 @ ( ap @ V2f @ V3k1 ) ) @ ( ap @ V2f @ V4k2 ) ) )
= fo__c_2EternaryComparisons_2EEQUAL ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ecomparison_2Eequiv__inj__def,axiom,
! [A_27a: del,A_27b: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1cmp2: $i] :
( ( mem @ V1cmp2 @ ( arr @ A_27b @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ecomparison_2Eequiv__inj @ A_27a @ A_27b ) @ V0cmp ) @ V1cmp2 ) @ V2f ) )
<=> ! [V3k1: $i] :
( ( mem @ V3k1 @ A_27a )
=> ! [V4k2: $i] :
( ( mem @ V4k2 @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V1cmp2 @ ( ap @ V2f @ V3k1 ) ) @ ( ap @ V2f @ V4k2 ) ) )
= fo__c_2EternaryComparisons_2EEQUAL )
=> ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V3k1 ) @ V4k2 ) )
= fo__c_2EternaryComparisons_2EEQUAL ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Eantisym__resp__equiv,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ V0cmp @ V2x ) @ V3y ) )
= fo__c_2EternaryComparisons_2EEQUAL )
=> ( V2x = V3y ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecomparison_2Eresp__equiv @ A_27a @ A_27b @ A_27c ) @ V0cmp ) @ V1f ) )
& ! [V4cmp2: $i] :
( ( mem @ V4cmp2 @ ( arr @ ( arr @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27b @ A_27c ) @ ty_2EternaryComparisons_2Eordering ) ) )
=> ( ( p @ ( ap @ ( c_2Ecomparison_2Egood__cmp @ ( arr @ A_27b @ A_27c ) ) @ V4cmp2 ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ecomparison_2Eresp__equiv2 @ A_27a @ ( arr @ A_27b @ A_27c ) ) @ V0cmp ) @ V4cmp2 ) @ V1f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2Elist__cmp__equal__list__rel,axiom,
! [A_27a: del,A_27b: del,V0cmp: $i] :
( ( mem @ V0cmp @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2EternaryComparisons_2Eordering ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ A_27a @ A_27b ) @ V0cmp ) @ V1l1 ) @ V2l2 ) )
= fo__c_2EternaryComparisons_2EEQUAL )
<=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2ELIST__REL @ A_27a @ A_27b )
@ ( lam @ A_27a
@ ^ [V3x: $i] :
( lam @ A_27b
@ ^ [V4y: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2EternaryComparisons_2Eordering ) @ ( ap @ ( ap @ V0cmp @ V3x ) @ V4y ) ) @ ( inj__ty_2EternaryComparisons_2Eordering @ fo__c_2EternaryComparisons_2EEQUAL ) ) ) ) )
@ V1l1 )
@ V2l2 ) ) ) ) ) ) ).
thf(conj_thm_2Ecomparison_2ETO__of__LinearOrder__LLEX,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_2ETO__of__LinearOrder @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Elist_2ELLEX @ A_27a ) @ V0R ) )
= ( ap @ ( c_2EternaryComparisons_2Elist__compare @ A_27a @ A_27a ) @ ( ap @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a ) @ V0R ) ) ) ) ) ).
%------------------------------------------------------------------------------