ITP001 Axioms: ITP097^7.ax
%------------------------------------------------------------------------------
% File : ITP097^7 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : comparison.ax [Gau19]
% : HL4097^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 118 ( 17 unt; 62 typ; 0 def)
% Number of atoms : 207 ( 137 equ; 4 cnn)
% Maximal formula atoms : 30 ( 1 avg)
% Number of connectives : 888 ( 4 ~; 3 |; 64 &; 741 @)
% ( 30 <=>; 46 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 9 avg; 741 nst)
% Number of types : 5 ( 4 usr)
% Number of type conns : 280 ( 280 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 58 usr; 7 con; 0-8 aty)
% Number of variables : 330 ( 12 ^ 262 !; 1 ?; 330 :)
% ( 55 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Elist_2Elist,type,
tyop_2Elist_2Elist: $tType > $tType ).
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Eoption_2Eoption,type,
tyop_2Eoption_2Eoption: $tType > $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(tyop_2Estring_2Echar,type,
tyop_2Estring_2Echar: $tType ).
thf(tyop_2EternaryComparisons_2Eordering,type,
tyop_2EternaryComparisons_2Eordering: $tType ).
thf(tyop_2Etoto_2Etoto,type,
tyop_2Etoto_2Etoto: $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Eprim__rec_2E_3C,type,
c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Epair_2ECURRY,type,
c_2Epair_2ECURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) > A_27a > A_27b > A_27c ) ).
thf(c_2EternaryComparisons_2EEQUAL,type,
c_2EternaryComparisons_2EEQUAL: tyop_2EternaryComparisons_2Eordering ).
thf(c_2Ebool_2EF,type,
c_2Ebool_2EF: $o ).
thf(c_2Epair_2EFST,type,
c_2Epair_2EFST:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27a ) ).
thf(c_2EternaryComparisons_2EGREATER,type,
c_2EternaryComparisons_2EGREATER: tyop_2EternaryComparisons_2Eordering ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2EternaryComparisons_2ELESS,type,
c_2EternaryComparisons_2ELESS: tyop_2EternaryComparisons_2Eordering ).
thf(c_2Elist_2ELIST__REL,type,
c_2Elist_2ELIST__REL:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27b ) > $o ) ).
thf(c_2Elist_2ELIST__TO__SET,type,
c_2Elist_2ELIST__TO__SET:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > A_27a > $o ) ).
thf(c_2Elist_2ELLEX,type,
c_2Elist_2ELLEX:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
thf(c_2Etoto_2EListOrd,type,
c_2Etoto_2EListOrd:
!>[A_27a: $tType] : ( ( tyop_2Etoto_2Etoto @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > tyop_2EternaryComparisons_2Eordering ) ).
thf(c_2Eoption_2ENONE,type,
c_2Eoption_2ENONE:
!>[A_27a: $tType] : ( tyop_2Eoption_2Eoption @ A_27a ) ).
thf(c_2Estring_2EORD,type,
c_2Estring_2EORD: tyop_2Estring_2Echar > tyop_2Enum_2Enum ).
thf(c_2Epair_2ESND,type,
c_2Epair_2ESND:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27b ) ).
thf(c_2Eoption_2ESOME,type,
c_2Eoption_2ESOME:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Ebool_2ET,type,
c_2Ebool_2ET: $o ).
thf(c_2Etoto_2ETO,type,
c_2Etoto_2ETO:
!>[A_27a: $tType] : ( ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) > ( tyop_2Etoto_2Etoto @ A_27a ) ) ).
thf(c_2Etoto_2ETO__inv,type,
c_2Etoto_2ETO__inv:
!>[A_27a: $tType] : ( ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) > A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) ).
thf(c_2Etoto_2ETO__of__LinearOrder,type,
c_2Etoto_2ETO__of__LinearOrder:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) ).
thf(c_2Etoto_2ETotOrd,type,
c_2Etoto_2ETotOrd:
!>[A_27a: $tType] : ( ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) > $o ) ).
thf(c_2Epair_2EUNCURRY,type,
c_2Epair_2EUNCURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Etoto_2Eapto,type,
c_2Etoto_2Eapto:
!>[A_27a: $tType] : ( ( tyop_2Etoto_2Etoto @ A_27a ) > A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) ).
thf(c_2EternaryComparisons_2Ebool__compare,type,
c_2EternaryComparisons_2Ebool__compare: $o > $o > tyop_2EternaryComparisons_2Eordering ).
thf(c_2Etoto_2EcharOrd,type,
c_2Etoto_2EcharOrd: tyop_2Estring_2Echar > tyop_2Estring_2Echar > tyop_2EternaryComparisons_2Eordering ).
thf(c_2EternaryComparisons_2Echar__compare,type,
c_2EternaryComparisons_2Echar__compare: tyop_2Estring_2Echar > tyop_2Estring_2Echar > tyop_2EternaryComparisons_2Eordering ).
thf(c_2Ecomparison_2Eequiv__inj,type,
c_2Ecomparison_2Eequiv__inj:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) > ( A_27b > A_27b > tyop_2EternaryComparisons_2Eordering ) > ( A_27a > A_27b ) > $o ) ).
thf(c_2Ecomparison_2Egood__cmp,type,
c_2Ecomparison_2Egood__cmp:
!>[A_27a: $tType] : ( ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) > $o ) ).
thf(c_2EternaryComparisons_2Einvert__comparison,type,
c_2EternaryComparisons_2Einvert__comparison: tyop_2EternaryComparisons_2Eordering > tyop_2EternaryComparisons_2Eordering ).
thf(c_2Erelation_2Eirreflexive,type,
c_2Erelation_2Eirreflexive:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Etoto_2ElexTO,type,
c_2Etoto_2ElexTO:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) > ( A_27b > A_27b > tyop_2EternaryComparisons_2Eordering ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > tyop_2EternaryComparisons_2Eordering ) ).
thf(c_2EternaryComparisons_2Elist__compare,type,
c_2EternaryComparisons_2Elist__compare:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > tyop_2EternaryComparisons_2Eordering ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27b ) > tyop_2EternaryComparisons_2Eordering ) ).
thf(c_2Etoto_2EnumOrd,type,
c_2Etoto_2EnumOrd: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2EternaryComparisons_2Eordering ).
thf(c_2EternaryComparisons_2Enum__compare,type,
c_2EternaryComparisons_2Enum__compare: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2EternaryComparisons_2Eordering ).
thf(c_2Ecombin_2Eo,type,
c_2Ecombin_2Eo:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > A_27c ) > A_27a > A_27b ) ).
thf(c_2Ecomparison_2Eoption__cmp2,type,
c_2Ecomparison_2Eoption__cmp2:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > tyop_2EternaryComparisons_2Eordering ) > ( tyop_2Eoption_2Eoption @ A_27a ) > ( tyop_2Eoption_2Eoption @ A_27b ) > tyop_2EternaryComparisons_2Eordering ) ).
thf(c_2EternaryComparisons_2Eoption__compare,type,
c_2EternaryComparisons_2Eoption__compare:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > tyop_2EternaryComparisons_2Eordering ) > ( tyop_2Eoption_2Eoption @ A_27a ) > ( tyop_2Eoption_2Eoption @ A_27b ) > tyop_2EternaryComparisons_2Eordering ) ).
thf(c_2EternaryComparisons_2Eordering__CASE,type,
c_2EternaryComparisons_2Eordering__CASE:
!>[A_27a: $tType] : ( tyop_2EternaryComparisons_2Eordering > A_27a > A_27a > A_27a > A_27a ) ).
thf(c_2EternaryComparisons_2Epair__compare,type,
c_2EternaryComparisons_2Epair__compare:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( A_27a > A_27b > tyop_2EternaryComparisons_2Eordering ) > ( A_27c > A_27d > tyop_2EternaryComparisons_2Eordering ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) > ( tyop_2Epair_2Eprod @ A_27b @ A_27d ) > tyop_2EternaryComparisons_2Eordering ) ).
thf(c_2Ecomparison_2Eresp__equiv,type,
c_2Ecomparison_2Eresp__equiv:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) > ( A_27a > A_27b > A_27c ) > $o ) ).
thf(c_2Ecomparison_2Eresp__equiv2,type,
c_2Ecomparison_2Eresp__equiv2:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) > ( A_27b > A_27b > tyop_2EternaryComparisons_2Eordering ) > ( A_27a > A_27b ) > $o ) ).
thf(c_2EternaryComparisons_2Estring__compare,type,
c_2EternaryComparisons_2Estring__compare: ( tyop_2Elist_2Elist @ tyop_2Estring_2Echar ) > ( tyop_2Elist_2Elist @ tyop_2Estring_2Echar ) > tyop_2EternaryComparisons_2Eordering ).
thf(c_2Etoto_2Estringto,type,
c_2Etoto_2Estringto: tyop_2Etoto_2Etoto @ ( tyop_2Elist_2Elist @ tyop_2Estring_2Echar ) ).
thf(c_2Erelation_2Etransitive,type,
c_2Erelation_2Etransitive:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Ecomparison_2Egood__cmp__def,axiom,
! [A_27a: $tType,V0cmp: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Ecomparison_2Egood__cmp @ A_27a @ V0cmp )
<=> ( ! [V1x: A_27a] :
( ( V0cmp @ V1x @ V1x )
= c_2EternaryComparisons_2EEQUAL )
& ! [V2x: A_27a,V3y: A_27a] :
( ( ( V0cmp @ V2x @ V3y )
= c_2EternaryComparisons_2EEQUAL )
=> ( ( V0cmp @ V3y @ V2x )
= c_2EternaryComparisons_2EEQUAL ) )
& ! [V4x: A_27a,V5y: A_27a] :
( ( ( V0cmp @ V4x @ V5y )
= c_2EternaryComparisons_2EGREATER )
<=> ( ( V0cmp @ V5y @ V4x )
= c_2EternaryComparisons_2ELESS ) )
& ! [V6x: A_27a,V7y: A_27a,V8z: A_27a] :
( ( ( ( V0cmp @ V6x @ V7y )
= c_2EternaryComparisons_2EEQUAL )
& ( ( V0cmp @ V7y @ V8z )
= c_2EternaryComparisons_2ELESS ) )
=> ( ( V0cmp @ V6x @ V8z )
= c_2EternaryComparisons_2ELESS ) )
& ! [V9x: A_27a,V10y: A_27a,V11z: A_27a] :
( ( ( ( V0cmp @ V9x @ V10y )
= c_2EternaryComparisons_2ELESS )
& ( ( V0cmp @ V10y @ V11z )
= c_2EternaryComparisons_2EEQUAL ) )
=> ( ( V0cmp @ V9x @ V11z )
= c_2EternaryComparisons_2ELESS ) )
& ! [V12x: A_27a,V13y: A_27a,V14z: A_27a] :
( ( ( ( V0cmp @ V12x @ V13y )
= c_2EternaryComparisons_2EEQUAL )
& ( ( V0cmp @ V13y @ V14z )
= c_2EternaryComparisons_2EEQUAL ) )
=> ( ( V0cmp @ V12x @ V14z )
= c_2EternaryComparisons_2EEQUAL ) )
& ! [V15x: A_27a,V16y: A_27a,V17z: A_27a] :
( ( ( ( V0cmp @ V15x @ V16y )
= c_2EternaryComparisons_2ELESS )
& ( ( V0cmp @ V16y @ V17z )
= c_2EternaryComparisons_2ELESS ) )
=> ( ( V0cmp @ V15x @ V17z )
= c_2EternaryComparisons_2ELESS ) ) ) ) ).
thf(thm_2Ecomparison_2Eresp__equiv__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0cmp: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1f: A_27a > A_27b > A_27c] :
( ( c_2Ecomparison_2Eresp__equiv @ A_27a @ A_27b @ A_27c @ V0cmp @ V1f )
<=> ! [V2k1: A_27a,V3k2: A_27a,V4v: A_27b] :
( ( ( V0cmp @ V2k1 @ V3k2 )
= c_2EternaryComparisons_2EEQUAL )
=> ( ( V1f @ V2k1 @ V4v )
= ( V1f @ V3k2 @ V4v ) ) ) ) ).
thf(thm_2Ecomparison_2Eresp__equiv2__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0cmp: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1cmp2: A_27b > A_27b > tyop_2EternaryComparisons_2Eordering,V2f: A_27a > A_27b] :
( ( c_2Ecomparison_2Eresp__equiv2 @ A_27a @ A_27b @ V0cmp @ V1cmp2 @ V2f )
<=> ! [V3k1: A_27a,V4k2: A_27a] :
( ( ( V0cmp @ V3k1 @ V4k2 )
= c_2EternaryComparisons_2EEQUAL )
=> ( ( V1cmp2 @ ( V2f @ V3k1 ) @ ( V2f @ V4k2 ) )
= c_2EternaryComparisons_2EEQUAL ) ) ) ).
thf(thm_2Ecomparison_2Eequiv__inj__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0cmp: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1cmp2: A_27b > A_27b > tyop_2EternaryComparisons_2Eordering,V2f: A_27a > A_27b] :
( ( c_2Ecomparison_2Eequiv__inj @ A_27a @ A_27b @ V0cmp @ V1cmp2 @ V2f )
<=> ! [V3k1: A_27a,V4k2: A_27a] :
( ( ( V1cmp2 @ ( V2f @ V3k1 ) @ ( V2f @ V4k2 ) )
= c_2EternaryComparisons_2EEQUAL )
=> ( ( V0cmp @ V3k1 @ V4k2 )
= c_2EternaryComparisons_2EEQUAL ) ) ) ).
thf(thm_2Ecomparison_2Egood__cmp__thm,axiom,
! [A_27a: $tType,V0cmp: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Ecomparison_2Egood__cmp @ A_27a @ V0cmp )
<=> ( ! [V1x: A_27a] :
( ( V0cmp @ V1x @ V1x )
= c_2EternaryComparisons_2EEQUAL )
& ! [V2x: A_27a,V3y: A_27a,V4z: A_27a] :
( ( ( ( V0cmp @ V2x @ V3y )
= c_2EternaryComparisons_2EGREATER )
<=> ( ( V0cmp @ V3y @ V2x )
= c_2EternaryComparisons_2ELESS ) )
& ( ( ( ( V0cmp @ V2x @ V3y )
= c_2EternaryComparisons_2ELESS )
& ( ( V0cmp @ V3y @ V4z )
= c_2EternaryComparisons_2EEQUAL ) )
=> ( ( V0cmp @ V2x @ V4z )
= c_2EternaryComparisons_2ELESS ) )
& ( ( ( ( V0cmp @ V2x @ V3y )
= c_2EternaryComparisons_2ELESS )
& ( ( V0cmp @ V3y @ V4z )
= c_2EternaryComparisons_2ELESS ) )
=> ( ( V0cmp @ V2x @ V4z )
= c_2EternaryComparisons_2ELESS ) ) ) ) ) ).
thf(thm_2Ecomparison_2Ecmp__thms,axiom,
! [A_27a: $tType] :
( ( (~) @ ( c_2EternaryComparisons_2ELESS = c_2EternaryComparisons_2EEQUAL ) )
& ( (~) @ ( c_2EternaryComparisons_2ELESS = c_2EternaryComparisons_2EGREATER ) )
& ( (~) @ ( c_2EternaryComparisons_2EEQUAL = c_2EternaryComparisons_2EGREATER ) )
& ! [V0v0: A_27a,V1v1: A_27a,V2v2: A_27a] :
( ( c_2EternaryComparisons_2Eordering__CASE @ A_27a @ c_2EternaryComparisons_2ELESS @ V0v0 @ V1v1 @ V2v2 )
= V0v0 )
& ! [V3v0: A_27a,V4v1: A_27a,V5v2: A_27a] :
( ( c_2EternaryComparisons_2Eordering__CASE @ A_27a @ c_2EternaryComparisons_2EEQUAL @ V3v0 @ V4v1 @ V5v2 )
= V4v1 )
& ! [V6v0: A_27a,V7v1: A_27a,V8v2: A_27a] :
( ( c_2EternaryComparisons_2Eordering__CASE @ A_27a @ c_2EternaryComparisons_2EGREATER @ V6v0 @ V7v1 @ V8v2 )
= V8v2 )
& ! [V9a: tyop_2EternaryComparisons_2Eordering] :
( ( V9a = c_2EternaryComparisons_2ELESS )
| ( V9a = c_2EternaryComparisons_2EEQUAL )
| ( V9a = c_2EternaryComparisons_2EGREATER ) )
& ! [V10cmp: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Ecomparison_2Egood__cmp @ A_27a @ V10cmp )
<=> ( ! [V11x: A_27a] :
( ( V10cmp @ V11x @ V11x )
= c_2EternaryComparisons_2EEQUAL )
& ! [V12x: A_27a,V13y: A_27a] :
( ( ( V10cmp @ V12x @ V13y )
= c_2EternaryComparisons_2EEQUAL )
=> ( ( V10cmp @ V13y @ V12x )
= c_2EternaryComparisons_2EEQUAL ) )
& ! [V14x: A_27a,V15y: A_27a] :
( ( ( V10cmp @ V14x @ V15y )
= c_2EternaryComparisons_2EGREATER )
<=> ( ( V10cmp @ V15y @ V14x )
= c_2EternaryComparisons_2ELESS ) )
& ! [V16x: A_27a,V17y: A_27a,V18z: A_27a] :
( ( ( ( V10cmp @ V16x @ V17y )
= c_2EternaryComparisons_2EEQUAL )
& ( ( V10cmp @ V17y @ V18z )
= c_2EternaryComparisons_2ELESS ) )
=> ( ( V10cmp @ V16x @ V18z )
= c_2EternaryComparisons_2ELESS ) )
& ! [V19x: A_27a,V20y: A_27a,V21z: A_27a] :
( ( ( ( V10cmp @ V19x @ V20y )
= c_2EternaryComparisons_2ELESS )
& ( ( V10cmp @ V20y @ V21z )
= c_2EternaryComparisons_2EEQUAL ) )
=> ( ( V10cmp @ V19x @ V21z )
= c_2EternaryComparisons_2ELESS ) )
& ! [V22x: A_27a,V23y: A_27a,V24z: A_27a] :
( ( ( ( V10cmp @ V22x @ V23y )
= c_2EternaryComparisons_2EEQUAL )
& ( ( V10cmp @ V23y @ V24z )
= c_2EternaryComparisons_2EEQUAL ) )
=> ( ( V10cmp @ V22x @ V24z )
= c_2EternaryComparisons_2EEQUAL ) )
& ! [V25x: A_27a,V26y: A_27a,V27z: A_27a] :
( ( ( ( V10cmp @ V25x @ V26y )
= c_2EternaryComparisons_2ELESS )
& ( ( V10cmp @ V26y @ V27z )
= c_2EternaryComparisons_2ELESS ) )
=> ( ( V10cmp @ V25x @ V27z )
= c_2EternaryComparisons_2ELESS ) ) ) ) ) ).
thf(thm_2Ecomparison_2Eoption__cmp__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0v3: A_27a,V1v2: A_27b,V2v1: A_27a,V3v0: A_27b,V4c: A_27a > A_27b > tyop_2EternaryComparisons_2Eordering] :
( ( ( c_2EternaryComparisons_2Eoption__compare @ A_27a @ A_27b @ V4c @ ( c_2Eoption_2ENONE @ A_27a ) @ ( c_2Eoption_2ENONE @ A_27b ) )
= c_2EternaryComparisons_2EEQUAL )
& ( ( c_2EternaryComparisons_2Eoption__compare @ A_27a @ A_27b @ V4c @ ( c_2Eoption_2ENONE @ A_27a ) @ ( c_2Eoption_2ESOME @ A_27b @ V3v0 ) )
= c_2EternaryComparisons_2ELESS )
& ( ( c_2EternaryComparisons_2Eoption__compare @ A_27a @ A_27b @ V4c @ ( c_2Eoption_2ESOME @ A_27a @ V0v3 ) @ ( c_2Eoption_2ENONE @ A_27b ) )
= c_2EternaryComparisons_2EGREATER )
& ( ( c_2EternaryComparisons_2Eoption__compare @ A_27a @ A_27b @ V4c @ ( c_2Eoption_2ESOME @ A_27a @ V2v1 ) @ ( c_2Eoption_2ESOME @ A_27b @ V1v2 ) )
= ( V4c @ V2v1 @ V1v2 ) ) ) ).
thf(thm_2Ecomparison_2Eoption__cmp2__ind,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: ( A_27a > A_27b > tyop_2EternaryComparisons_2Eordering ) > ( tyop_2Eoption_2Eoption @ A_27a ) > ( tyop_2Eoption_2Eoption @ A_27b ) > $o] :
( ( ! [V1cmp: A_27a > A_27b > tyop_2EternaryComparisons_2Eordering] : ( V0P @ V1cmp @ ( c_2Eoption_2ENONE @ A_27a ) @ ( c_2Eoption_2ENONE @ A_27b ) )
& ! [V2cmp: A_27a > A_27b > tyop_2EternaryComparisons_2Eordering,V3x: A_27b] : ( V0P @ V2cmp @ ( c_2Eoption_2ENONE @ A_27a ) @ ( c_2Eoption_2ESOME @ A_27b @ V3x ) )
& ! [V4cmp: A_27a > A_27b > tyop_2EternaryComparisons_2Eordering,V5x: A_27a] : ( V0P @ V4cmp @ ( c_2Eoption_2ESOME @ A_27a @ V5x ) @ ( c_2Eoption_2ENONE @ A_27b ) )
& ! [V6cmp: A_27a > A_27b > tyop_2EternaryComparisons_2Eordering,V7x: A_27a,V8y: A_27b] : ( V0P @ V6cmp @ ( c_2Eoption_2ESOME @ A_27a @ V7x ) @ ( c_2Eoption_2ESOME @ A_27b @ V8y ) ) )
=> ! [V9v: A_27a > A_27b > tyop_2EternaryComparisons_2Eordering,V10v1: tyop_2Eoption_2Eoption @ A_27a,V11v2: tyop_2Eoption_2Eoption @ A_27b] : ( V0P @ V9v @ V10v1 @ V11v2 ) ) ).
thf(thm_2Ecomparison_2Eoption__cmp2__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27b,V1x_27: A_27b,V2x: A_27a,V3cmp: A_27a > A_27b > tyop_2EternaryComparisons_2Eordering] :
( ( ( c_2Ecomparison_2Eoption__cmp2 @ A_27a @ A_27b @ V3cmp @ ( c_2Eoption_2ENONE @ A_27a ) @ ( c_2Eoption_2ENONE @ A_27b ) )
= c_2EternaryComparisons_2EEQUAL )
& ( ( c_2Ecomparison_2Eoption__cmp2 @ A_27a @ A_27b @ V3cmp @ ( c_2Eoption_2ENONE @ A_27a ) @ ( c_2Eoption_2ESOME @ A_27b @ V1x_27 ) )
= c_2EternaryComparisons_2EGREATER )
& ( ( c_2Ecomparison_2Eoption__cmp2 @ A_27a @ A_27b @ V3cmp @ ( c_2Eoption_2ESOME @ A_27a @ V2x ) @ ( c_2Eoption_2ENONE @ A_27b ) )
= c_2EternaryComparisons_2ELESS )
& ( ( c_2Ecomparison_2Eoption__cmp2 @ A_27a @ A_27b @ V3cmp @ ( c_2Eoption_2ESOME @ A_27a @ V2x ) @ ( c_2Eoption_2ESOME @ A_27b @ V0y ) )
= ( V3cmp @ V2x @ V0y ) ) ) ).
thf(thm_2Ecomparison_2Epair__cmp__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0y: tyop_2Epair_2Eprod @ A_27b @ A_27d,V1x: tyop_2Epair_2Eprod @ A_27a @ A_27c,V2c2: A_27c > A_27d > tyop_2EternaryComparisons_2Eordering,V3c1: A_27a > A_27b > tyop_2EternaryComparisons_2Eordering] :
( ( c_2EternaryComparisons_2Epair__compare @ A_27a @ A_27b @ A_27c @ A_27d @ V3c1 @ V2c2 @ V1x @ V0y )
= ( c_2EternaryComparisons_2Eordering__CASE @ tyop_2EternaryComparisons_2Eordering @ ( V3c1 @ ( c_2Epair_2EFST @ A_27a @ A_27c @ V1x ) @ ( c_2Epair_2EFST @ A_27b @ A_27d @ V0y ) ) @ c_2EternaryComparisons_2ELESS @ ( V2c2 @ ( c_2Epair_2ESND @ A_27a @ A_27c @ V1x ) @ ( c_2Epair_2ESND @ A_27b @ A_27d @ V0y ) ) @ c_2EternaryComparisons_2EGREATER ) ) ).
thf(thm_2Ecomparison_2Ebool__cmp__def,axiom,
( ( ( c_2EternaryComparisons_2Ebool__compare @ c_2Ebool_2ET @ c_2Ebool_2ET )
= c_2EternaryComparisons_2EEQUAL )
& ( ( c_2EternaryComparisons_2Ebool__compare @ c_2Ebool_2EF @ c_2Ebool_2EF )
= c_2EternaryComparisons_2EEQUAL )
& ( ( c_2EternaryComparisons_2Ebool__compare @ c_2Ebool_2ET @ c_2Ebool_2EF )
= c_2EternaryComparisons_2EGREATER )
& ( ( c_2EternaryComparisons_2Ebool__compare @ c_2Ebool_2EF @ c_2Ebool_2ET )
= c_2EternaryComparisons_2ELESS ) ) ).
thf(thm_2Ecomparison_2Enum__cmp__def,axiom,
! [V0n1: tyop_2Enum_2Enum,V1n2: tyop_2Enum_2Enum] :
( ( c_2EternaryComparisons_2Enum__compare @ V0n1 @ V1n2 )
= ( c_2Ebool_2ECOND @ tyop_2EternaryComparisons_2Eordering @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V0n1 @ V1n2 ) @ c_2EternaryComparisons_2EEQUAL @ ( c_2Ebool_2ECOND @ tyop_2EternaryComparisons_2Eordering @ ( c_2Eprim__rec_2E_3C @ V0n1 @ V1n2 ) @ c_2EternaryComparisons_2ELESS @ c_2EternaryComparisons_2EGREATER ) ) ) ).
thf(thm_2Ecomparison_2Echar__cmp__def,axiom,
! [V0c1: tyop_2Estring_2Echar,V1c2: tyop_2Estring_2Echar] :
( ( c_2EternaryComparisons_2Echar__compare @ V0c1 @ V1c2 )
= ( c_2EternaryComparisons_2Enum__compare @ ( c_2Estring_2EORD @ V0c1 ) @ ( c_2Estring_2EORD @ V1c2 ) ) ) ).
thf(thm_2Ecomparison_2Estring__cmp__def,axiom,
( c_2EternaryComparisons_2Estring__compare
= ( c_2EternaryComparisons_2Elist__compare @ tyop_2Estring_2Echar @ tyop_2Estring_2Echar @ c_2EternaryComparisons_2Echar__compare ) ) ).
thf(thm_2Ecomparison_2ETotOrder__imp__good__cmp,axiom,
! [A_27a: $tType,V0cmp: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Etoto_2ETotOrd @ A_27a @ V0cmp )
=> ( c_2Ecomparison_2Egood__cmp @ A_27a @ V0cmp ) ) ).
thf(thm_2Ecomparison_2ETO__inv__invert,axiom,
! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Etoto_2ETotOrd @ A_27a @ V0c )
=> ( ( c_2Etoto_2ETO__inv @ A_27a @ V0c )
= ( c_2Epair_2ECURRY @ A_27a @ A_27a @ tyop_2EternaryComparisons_2Eordering @ ( c_2Ecombin_2Eo @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ tyop_2EternaryComparisons_2Eordering @ tyop_2EternaryComparisons_2Eordering @ c_2EternaryComparisons_2Einvert__comparison @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ tyop_2EternaryComparisons_2Eordering @ V0c ) ) ) ) ) ).
thf(thm_2Ecomparison_2Eoption__cmp2__TO__inv,axiom,
! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Ecomparison_2Eoption__cmp2 @ A_27a @ A_27a @ V0c )
= ( c_2Etoto_2ETO__inv @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2EternaryComparisons_2Eoption__compare @ A_27a @ A_27a @ ( c_2Etoto_2ETO__inv @ A_27a @ V0c ) ) ) ) ).
thf(thm_2Ecomparison_2Elist__cmp__ListOrd,axiom,
! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Etoto_2ETotOrd @ A_27a @ V0c )
=> ( ( c_2EternaryComparisons_2Elist__compare @ A_27a @ A_27a @ V0c )
= ( c_2Etoto_2EListOrd @ A_27a @ ( c_2Etoto_2ETO @ A_27a @ V0c ) ) ) ) ).
thf(thm_2Ecomparison_2ETotOrd__list__cmp,axiom,
! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Etoto_2ETotOrd @ A_27a @ V0c )
=> ( c_2Etoto_2ETotOrd @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2EternaryComparisons_2Elist__compare @ A_27a @ A_27a @ V0c ) ) ) ).
thf(thm_2Ecomparison_2Epair__cmp__lexTO,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1V: A_27b > A_27b > tyop_2EternaryComparisons_2Eordering] :
( ( ( c_2Etoto_2ETotOrd @ A_27a @ V0R )
& ( c_2Etoto_2ETotOrd @ A_27b @ V1V ) )
=> ( ( c_2EternaryComparisons_2Epair__compare @ A_27a @ A_27a @ A_27b @ A_27b @ V0R @ V1V )
= ( c_2Etoto_2ElexTO @ A_27a @ A_27b @ V0R @ V1V ) ) ) ).
thf(thm_2Ecomparison_2Enum__cmp__numOrd,axiom,
c_2EternaryComparisons_2Enum__compare = c_2Etoto_2EnumOrd ).
thf(thm_2Ecomparison_2Echar__cmp__charOrd,axiom,
c_2EternaryComparisons_2Echar__compare = c_2Etoto_2EcharOrd ).
thf(thm_2Ecomparison_2Estring__cmp__stringto,axiom,
( c_2EternaryComparisons_2Estring__compare
= ( c_2Etoto_2Eapto @ ( tyop_2Elist_2Elist @ tyop_2Estring_2Echar ) @ c_2Etoto_2Estringto ) ) ).
thf(thm_2Ecomparison_2Eoption__cmp__good,axiom,
! [A_27a: $tType,V0cmp: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Ecomparison_2Egood__cmp @ A_27a @ V0cmp )
=> ( c_2Ecomparison_2Egood__cmp @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2EternaryComparisons_2Eoption__compare @ A_27a @ A_27a @ V0cmp ) ) ) ).
thf(thm_2Ecomparison_2Eoption__cmp2__good,axiom,
! [A_27a: $tType,V0cmp: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Ecomparison_2Egood__cmp @ A_27a @ V0cmp )
=> ( c_2Ecomparison_2Egood__cmp @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Ecomparison_2Eoption__cmp2 @ A_27a @ A_27a @ V0cmp ) ) ) ).
thf(thm_2Ecomparison_2Elist__cmp__good,axiom,
! [A_27a: $tType,V0cmp: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Ecomparison_2Egood__cmp @ A_27a @ V0cmp )
=> ( c_2Ecomparison_2Egood__cmp @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2EternaryComparisons_2Elist__compare @ A_27a @ A_27a @ V0cmp ) ) ) ).
thf(thm_2Ecomparison_2Epair__cmp__good,axiom,
! [A_27a: $tType,A_27b: $tType,V0cmp1: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1cmp2: A_27b > A_27b > tyop_2EternaryComparisons_2Eordering] :
( ( ( c_2Ecomparison_2Egood__cmp @ A_27a @ V0cmp1 )
& ( c_2Ecomparison_2Egood__cmp @ A_27b @ V1cmp2 ) )
=> ( c_2Ecomparison_2Egood__cmp @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2EternaryComparisons_2Epair__compare @ A_27a @ A_27a @ A_27b @ A_27b @ V0cmp1 @ V1cmp2 ) ) ) ).
thf(thm_2Ecomparison_2Ebool__cmp__good,axiom,
c_2Ecomparison_2Egood__cmp @ $o @ c_2EternaryComparisons_2Ebool__compare ).
thf(thm_2Ecomparison_2Enum__cmp__good,axiom,
c_2Ecomparison_2Egood__cmp @ tyop_2Enum_2Enum @ c_2EternaryComparisons_2Enum__compare ).
thf(thm_2Ecomparison_2Echar__cmp__good,axiom,
c_2Ecomparison_2Egood__cmp @ tyop_2Estring_2Echar @ c_2EternaryComparisons_2Echar__compare ).
thf(thm_2Ecomparison_2Estring__cmp__good,axiom,
c_2Ecomparison_2Egood__cmp @ ( tyop_2Elist_2Elist @ tyop_2Estring_2Echar ) @ c_2EternaryComparisons_2Estring__compare ).
thf(thm_2Ecomparison_2Elist__cmp__cong,axiom,
! [A_27a: $tType,A_27b: $tType,V0cmp: A_27a > A_27b > tyop_2EternaryComparisons_2Eordering,V1l1: tyop_2Elist_2Elist @ A_27a,V2l2: tyop_2Elist_2Elist @ A_27b,V3cmp_27: A_27a > A_27b > tyop_2EternaryComparisons_2Eordering,V4l1_27: tyop_2Elist_2Elist @ A_27a,V5l2_27: tyop_2Elist_2Elist @ A_27b] :
( ( ( V1l1 = V4l1_27 )
& ( V2l2 = V5l2_27 )
& ! [V6x: A_27a,V7x_27: A_27b] :
( ( ( c_2Ebool_2EIN @ A_27a @ V6x @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V4l1_27 ) )
& ( c_2Ebool_2EIN @ A_27b @ V7x_27 @ ( c_2Elist_2ELIST__TO__SET @ A_27b @ V5l2_27 ) ) )
=> ( ( V0cmp @ V6x @ V7x_27 )
= ( V3cmp_27 @ V6x @ V7x_27 ) ) ) )
=> ( ( c_2EternaryComparisons_2Elist__compare @ A_27a @ A_27b @ V0cmp @ V1l1 @ V2l2 )
= ( c_2EternaryComparisons_2Elist__compare @ A_27a @ A_27b @ V3cmp_27 @ V4l1_27 @ V5l2_27 ) ) ) ).
thf(thm_2Ecomparison_2Eoption__cmp__cong,axiom,
! [A_27a: $tType,A_27b: $tType,V0cmp: A_27a > A_27b > tyop_2EternaryComparisons_2Eordering,V1v1: tyop_2Eoption_2Eoption @ A_27a,V2v2: tyop_2Eoption_2Eoption @ A_27b,V3cmp_27: A_27a > A_27b > tyop_2EternaryComparisons_2Eordering,V4v1_27: tyop_2Eoption_2Eoption @ A_27a,V5v2_27: tyop_2Eoption_2Eoption @ A_27b] :
( ( ( V1v1 = V4v1_27 )
& ( V2v2 = V5v2_27 )
& ! [V6x: A_27a,V7x_27: A_27b] :
( ( ( V4v1_27
= ( c_2Eoption_2ESOME @ A_27a @ V6x ) )
& ( V5v2_27
= ( c_2Eoption_2ESOME @ A_27b @ V7x_27 ) ) )
=> ( ( V0cmp @ V6x @ V7x_27 )
= ( V3cmp_27 @ V6x @ V7x_27 ) ) ) )
=> ( ( c_2EternaryComparisons_2Eoption__compare @ A_27a @ A_27b @ V0cmp @ V1v1 @ V2v2 )
= ( c_2EternaryComparisons_2Eoption__compare @ A_27a @ A_27b @ V3cmp_27 @ V4v1_27 @ V5v2_27 ) ) ) ).
thf(thm_2Ecomparison_2Eoption__cmp2__cong,axiom,
! [A_27a: $tType,A_27b: $tType,V0cmp: A_27a > A_27b > tyop_2EternaryComparisons_2Eordering,V1v1: tyop_2Eoption_2Eoption @ A_27a,V2v2: tyop_2Eoption_2Eoption @ A_27b,V3cmp_27: A_27a > A_27b > tyop_2EternaryComparisons_2Eordering,V4v1_27: tyop_2Eoption_2Eoption @ A_27a,V5v2_27: tyop_2Eoption_2Eoption @ A_27b] :
( ( ( V1v1 = V4v1_27 )
& ( V2v2 = V5v2_27 )
& ! [V6x: A_27a,V7x_27: A_27b] :
( ( ( V4v1_27
= ( c_2Eoption_2ESOME @ A_27a @ V6x ) )
& ( V5v2_27
= ( c_2Eoption_2ESOME @ A_27b @ V7x_27 ) ) )
=> ( ( V0cmp @ V6x @ V7x_27 )
= ( V3cmp_27 @ V6x @ V7x_27 ) ) ) )
=> ( ( c_2Ecomparison_2Eoption__cmp2 @ A_27a @ A_27b @ V0cmp @ V1v1 @ V2v2 )
= ( c_2Ecomparison_2Eoption__cmp2 @ A_27a @ A_27b @ V3cmp_27 @ V4v1_27 @ V5v2_27 ) ) ) ).
thf(thm_2Ecomparison_2Epair__cmp__cong,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0cmp1: A_27a > A_27b > tyop_2EternaryComparisons_2Eordering,V1cmp2: A_27c > A_27d > tyop_2EternaryComparisons_2Eordering,V2v1: tyop_2Epair_2Eprod @ A_27a @ A_27c,V3v2: tyop_2Epair_2Eprod @ A_27b @ A_27d,V4cmp1_27: A_27a > A_27b > tyop_2EternaryComparisons_2Eordering,V5cmp2_27: A_27c > A_27d > tyop_2EternaryComparisons_2Eordering,V6v1_27: tyop_2Epair_2Eprod @ A_27a @ A_27c,V7v2_27: tyop_2Epair_2Eprod @ A_27b @ A_27d] :
( ( ( V2v1 = V6v1_27 )
& ( V3v2 = V7v2_27 )
& ! [V8a: A_27a,V9b: A_27c,V10c: A_27b,V11d: A_27d] :
( ( ( V6v1_27
= ( c_2Epair_2E_2C @ A_27a @ A_27c @ V8a @ V9b ) )
& ( V7v2_27
= ( c_2Epair_2E_2C @ A_27b @ A_27d @ V10c @ V11d ) ) )
=> ( ( V0cmp1 @ V8a @ V10c )
= ( V4cmp1_27 @ V8a @ V10c ) ) )
& ! [V12a: A_27a,V13b: A_27c,V14c: A_27b,V15d: A_27d] :
( ( ( V6v1_27
= ( c_2Epair_2E_2C @ A_27a @ A_27c @ V12a @ V13b ) )
& ( V7v2_27
= ( c_2Epair_2E_2C @ A_27b @ A_27d @ V14c @ V15d ) ) )
=> ( ( V1cmp2 @ V13b @ V15d )
= ( V5cmp2_27 @ V13b @ V15d ) ) ) )
=> ( ( c_2EternaryComparisons_2Epair__compare @ A_27a @ A_27b @ A_27c @ A_27d @ V0cmp1 @ V1cmp2 @ V2v1 @ V3v2 )
= ( c_2EternaryComparisons_2Epair__compare @ A_27a @ A_27b @ A_27c @ A_27d @ V4cmp1_27 @ V5cmp2_27 @ V6v1_27 @ V7v2_27 ) ) ) ).
thf(thm_2Ecomparison_2Egood__cmp__trans,axiom,
! [A_27a: $tType,A_27b: $tType,V0cmp: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Ecomparison_2Egood__cmp @ A_27a @ V0cmp )
=> ( c_2Erelation_2Etransitive @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o )
@ ^ [V1k: A_27a,V2v: A_27b] :
( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ $o
@ ^ [V3k_27: A_27a,V4v_27: A_27b] : ( c_2Emin_2E_3D @ tyop_2EternaryComparisons_2Eordering @ ( V0cmp @ V1k @ V3k_27 ) @ c_2EternaryComparisons_2ELESS ) ) ) ) ) ).
thf(thm_2Ecomparison_2Egood__cmp__Less__trans,axiom,
! [A_27a: $tType,V0cmp: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Ecomparison_2Egood__cmp @ A_27a @ V0cmp )
=> ( c_2Erelation_2Etransitive @ A_27a
@ ^ [V1k: A_27a,V2k_27: A_27a] : ( c_2Emin_2E_3D @ tyop_2EternaryComparisons_2Eordering @ ( V0cmp @ V1k @ V2k_27 ) @ c_2EternaryComparisons_2ELESS ) ) ) ).
thf(thm_2Ecomparison_2Egood__cmp__Less__irrefl__trans,axiom,
! [A_27a: $tType,V0cmp: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Ecomparison_2Egood__cmp @ A_27a @ V0cmp )
=> ( ( c_2Erelation_2Eirreflexive @ A_27a
@ ^ [V1k: A_27a,V2k_27: A_27a] : ( c_2Emin_2E_3D @ tyop_2EternaryComparisons_2Eordering @ ( V0cmp @ V1k @ V2k_27 ) @ c_2EternaryComparisons_2ELESS ) )
& ( c_2Erelation_2Etransitive @ A_27a
@ ^ [V3k: A_27a,V4k_27: A_27a] : ( c_2Emin_2E_3D @ tyop_2EternaryComparisons_2Eordering @ ( V0cmp @ V3k @ V4k_27 ) @ c_2EternaryComparisons_2ELESS ) ) ) ) ).
thf(thm_2Ecomparison_2Ebool__cmp__antisym,axiom,
! [V0x: $o,V1y: $o] :
( ( ( c_2EternaryComparisons_2Ebool__compare @ V0x @ V1y )
= c_2EternaryComparisons_2EEQUAL )
<=> ( V0x = V1y ) ) ).
thf(thm_2Ecomparison_2Enum__cmp__antisym,axiom,
! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum] :
( ( ( c_2EternaryComparisons_2Enum__compare @ V0x @ V1y )
= c_2EternaryComparisons_2EEQUAL )
<=> ( V0x = V1y ) ) ).
thf(thm_2Ecomparison_2Echar__cmp__antisym,axiom,
! [V0x: tyop_2Estring_2Echar,V1y: tyop_2Estring_2Echar] :
( ( ( c_2EternaryComparisons_2Echar__compare @ V0x @ V1y )
= c_2EternaryComparisons_2EEQUAL )
<=> ( V0x = V1y ) ) ).
thf(thm_2Ecomparison_2Elist__cmp__antisym,axiom,
! [A_27a: $tType,V0cmp: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1x: tyop_2Elist_2Elist @ A_27a,V2y: tyop_2Elist_2Elist @ A_27a] :
( ! [V3x: A_27a,V4y: A_27a] :
( ( ( V0cmp @ V3x @ V4y )
= c_2EternaryComparisons_2EEQUAL )
<=> ( V3x = V4y ) )
=> ( ( ( c_2EternaryComparisons_2Elist__compare @ A_27a @ A_27a @ V0cmp @ V1x @ V2y )
= c_2EternaryComparisons_2EEQUAL )
<=> ( V1x = V2y ) ) ) ).
thf(thm_2Ecomparison_2Estring__cmp__antisym,axiom,
! [V0x: tyop_2Elist_2Elist @ tyop_2Estring_2Echar,V1y: tyop_2Elist_2Elist @ tyop_2Estring_2Echar] :
( ( ( c_2EternaryComparisons_2Estring__compare @ V0x @ V1y )
= c_2EternaryComparisons_2EEQUAL )
<=> ( V0x = V1y ) ) ).
thf(thm_2Ecomparison_2Epair__cmp__antisym,axiom,
! [A_27a: $tType,A_27b: $tType,V0cmp1: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1cmp2: A_27b > A_27b > tyop_2EternaryComparisons_2Eordering,V2x: tyop_2Epair_2Eprod @ A_27a @ A_27b,V3y: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( ! [V4x: A_27a,V5y: A_27a] :
( ( ( V0cmp1 @ V4x @ V5y )
= c_2EternaryComparisons_2EEQUAL )
<=> ( V4x = V5y ) )
& ! [V6x: A_27b,V7y: A_27b] :
( ( ( V1cmp2 @ V6x @ V7y )
= c_2EternaryComparisons_2EEQUAL )
<=> ( V6x = V7y ) ) )
=> ( ( ( c_2EternaryComparisons_2Epair__compare @ A_27a @ A_27a @ A_27b @ A_27b @ V0cmp1 @ V1cmp2 @ V2x @ V3y )
= c_2EternaryComparisons_2EEQUAL )
<=> ( V2x = V3y ) ) ) ).
thf(thm_2Ecomparison_2Eoption__cmp__antisym,axiom,
! [A_27a: $tType,V0cmp: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1x: tyop_2Eoption_2Eoption @ A_27a,V2y: tyop_2Eoption_2Eoption @ A_27a] :
( ! [V3x: A_27a,V4y: A_27a] :
( ( ( V0cmp @ V3x @ V4y )
= c_2EternaryComparisons_2EEQUAL )
<=> ( V3x = V4y ) )
=> ( ( ( c_2EternaryComparisons_2Eoption__compare @ A_27a @ A_27a @ V0cmp @ V1x @ V2y )
= c_2EternaryComparisons_2EEQUAL )
<=> ( V1x = V2y ) ) ) ).
thf(thm_2Ecomparison_2Eoption__cmp2__antisym,axiom,
! [A_27a: $tType,V0cmp: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1x: tyop_2Eoption_2Eoption @ A_27a,V2y: tyop_2Eoption_2Eoption @ A_27a] :
( ! [V3x: A_27a,V4y: A_27a] :
( ( ( V0cmp @ V3x @ V4y )
= c_2EternaryComparisons_2EEQUAL )
<=> ( V3x = V4y ) )
=> ( ( ( c_2Ecomparison_2Eoption__cmp2 @ A_27a @ A_27a @ V0cmp @ V1x @ V2y )
= c_2EternaryComparisons_2EEQUAL )
<=> ( V1x = V2y ) ) ) ).
thf(thm_2Ecomparison_2Eantisym__resp__equiv,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0cmp: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1f: A_27a > A_27b > A_27c] :
( ! [V2x: A_27a,V3y: A_27a] :
( ( ( V0cmp @ V2x @ V3y )
= c_2EternaryComparisons_2EEQUAL )
=> ( V2x = V3y ) )
=> ( ( c_2Ecomparison_2Eresp__equiv @ A_27a @ A_27b @ A_27c @ V0cmp @ V1f )
& ! [V4cmp2: ( A_27b > A_27c ) > ( A_27b > A_27c ) > tyop_2EternaryComparisons_2Eordering] :
( ( c_2Ecomparison_2Egood__cmp @ ( A_27b > A_27c ) @ V4cmp2 )
=> ( c_2Ecomparison_2Eresp__equiv2 @ A_27a @ ( A_27b > A_27c ) @ V0cmp @ V4cmp2 @ V1f ) ) ) ) ).
thf(thm_2Ecomparison_2Elist__cmp__equal__list__rel,axiom,
! [A_27a: $tType,A_27b: $tType,V0cmp: A_27a > A_27b > tyop_2EternaryComparisons_2Eordering,V1l1: tyop_2Elist_2Elist @ A_27a,V2l2: tyop_2Elist_2Elist @ A_27b] :
( ( ( c_2EternaryComparisons_2Elist__compare @ A_27a @ A_27b @ V0cmp @ V1l1 @ V2l2 )
= c_2EternaryComparisons_2EEQUAL )
<=> ( c_2Elist_2ELIST__REL @ A_27a @ A_27b
@ ^ [V3x: A_27a,V4y: A_27b] : ( c_2Emin_2E_3D @ tyop_2EternaryComparisons_2Eordering @ ( V0cmp @ V3x @ V4y ) @ c_2EternaryComparisons_2EEQUAL )
@ V1l1
@ V2l2 ) ) ).
thf(thm_2Ecomparison_2ETO__of__LinearOrder__LLEX,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2Eirreflexive @ A_27a @ V0R )
=> ( ( c_2Etoto_2ETO__of__LinearOrder @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Elist_2ELLEX @ A_27a @ V0R ) )
= ( c_2EternaryComparisons_2Elist__compare @ A_27a @ A_27a @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ V0R ) ) ) ) ).
%------------------------------------------------------------------------------