ITP001 Axioms: ITP097_5.ax
%------------------------------------------------------------------------------
% File : ITP097_5 : TPTP v9.0.0. 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 : 75 ( 15 unt; 13 typ; 0 def)
% Number of atoms : 1480 ( 143 equ)
% Maximal formula atoms : 54 ( 19 avg)
% Number of connectives : 326 ( 3 ~; 2 |; 63 &)
% ( 24 <=>; 234 =>; 0 <=; 0 <~>)
% Maximal formula depth : 35 ( 9 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 1095 (1095 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 34 ( 13 >; 21 *; 0 +; 0 <<)
% Number of predicates : 67 ( 66 usr; 25 prp; 0-4 aty)
% Number of functors : 13 ( 13 usr; 0 con; 1-4 aty)
% Number of variables : 289 ( 289 !; 0 ?; 289 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2Ecomparison_2Eequiv__inj,type,
c_2Ecomparison_2Eequiv__inj: ( del * del ) > $i ).
tff(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)))) ).
tff(tp_c_2Ecomparison_2Egood__cmp,type,
c_2Ecomparison_2Egood__cmp: del > $i ).
tff(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)) ).
tff(tp_c_2Ecomparison_2Eoption__cmp2,type,
c_2Ecomparison_2Eoption__cmp2: ( del * del ) > $i ).
tff(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)))) ).
tff(tp_c_2Ecomparison_2Eresp__equiv,type,
c_2Ecomparison_2Eresp__equiv: ( del * del * del ) > $i ).
tff(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))) ).
tff(tp_c_2Ecomparison_2Eresp__equiv2,type,
c_2Ecomparison_2Eresp__equiv2: ( del * del ) > $i ).
tff(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)))) ).
tff(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 ) ) ) ) ) ) ) ) ).
tff(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 ) ) ) ) ) ) ) ) ) ).
tff(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 ) ) ) ) ) ) ) ) ) ).
tff(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)) ) ) ) ) ) ) ) ).
tff(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)) ) ) ) ) ) ).
tff(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)) ) ) ) ) ) ) ).
tff(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))) ) ) ) ) ) ).
tff(conj_thm_2Ecomparison_2Ebool__cmp__def,axiom,
( ( surj__ty_2EternaryComparisons_2Eordering(ap(ap(c_2EternaryComparisons_2Ebool__compare,inj__o(fo__c_2Ebool_2ET)),inj__o(fo__c_2Ebool_2ET))) = fo__c_2EternaryComparisons_2EEQUAL )
& ( surj__ty_2EternaryComparisons_2Eordering(ap(ap(c_2EternaryComparisons_2Ebool__compare,inj__o(fo__c_2Ebool_2EF)),inj__o(fo__c_2Ebool_2EF))) = fo__c_2EternaryComparisons_2EEQUAL )
& ( surj__ty_2EternaryComparisons_2Eordering(ap(ap(c_2EternaryComparisons_2Ebool__compare,inj__o(fo__c_2Ebool_2ET)),inj__o(fo__c_2Ebool_2EF))) = fo__c_2EternaryComparisons_2EGREATER )
& ( surj__ty_2EternaryComparisons_2Eordering(ap(ap(c_2EternaryComparisons_2Ebool__compare,inj__o(fo__c_2Ebool_2EF)),inj__o(fo__c_2Ebool_2ET))) = fo__c_2EternaryComparisons_2ELESS ) ) ).
tff(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)))) ) ).
tff(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)))) ) ).
tff(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) ).
tff(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)) ) ) ).
tff(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))) ) ) ) ).
tff(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))) ) ) ).
tff(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)) ) ) ) ).
tff(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))) ) ) ).
tff(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) ) ) ) ) ).
tff(conj_thm_2Ecomparison_2Enum__cmp__numOrd,axiom,
c_2EternaryComparisons_2Enum__compare = c_2Etoto_2EnumOrd ).
tff(conj_thm_2Ecomparison_2Echar__cmp__charOrd,axiom,
c_2EternaryComparisons_2Echar__compare = c_2Etoto_2EcharOrd ).
tff(conj_thm_2Ecomparison_2Estring__cmp__stringto,axiom,
c_2EternaryComparisons_2Estring__compare = ap(c_2Etoto_2Eapto(ty_2Elist_2Elist(ty_2Estring_2Echar)),c_2Etoto_2Estringto) ).
tff(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))) ) ) ).
tff(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))) ) ) ).
tff(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))) ) ) ).
tff(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))) ) ) ) ).
tff(conj_thm_2Ecomparison_2Ebool__cmp__good,axiom,
p(ap(c_2Ecomparison_2Egood__cmp(bool),c_2EternaryComparisons_2Ebool__compare)) ).
tff(conj_thm_2Ecomparison_2Enum__cmp__good,axiom,
p(ap(c_2Ecomparison_2Egood__cmp(ty_2Enum_2Enum),c_2EternaryComparisons_2Enum__compare)) ).
tff(conj_thm_2Ecomparison_2Echar__cmp__good,axiom,
p(ap(c_2Ecomparison_2Egood__cmp(ty_2Estring_2Echar),c_2EternaryComparisons_2Echar__compare)) ).
tff(conj_thm_2Ecomparison_2Estring__cmp__good,axiom,
p(ap(c_2Ecomparison_2Egood__cmp(ty_2Elist_2Elist(ty_2Estring_2Echar)),c_2EternaryComparisons_2Estring__compare)) ).
tff(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)) ) ) ) ) ) ) ) ) ).
tff(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)) ) ) ) ) ) ) ) ) ).
tff(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)) ) ) ) ) ) ) ) ) ).
tff(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)) ) ) ) ) ) ) ) ) ) ) ).
tff(lamtp_f2280,type,
f2280: ( del * del * $i * $i ) > $i ).
tff(lameq_f2280,axiom,
! [A_27b: del,A_27a: del,V1k: $i] :
( mem(V1k,A_27a)
=> ! [V0cmp: $i] :
( mem(V0cmp,arr(A_27a,arr(A_27a,ty_2EternaryComparisons_2Eordering)))
=> ! [V3k_27: $i] : ( ap(f2280(A_27b,A_27a,V1k,V0cmp),V3k_27) = k(A_27b,ap(ap(c_2Emin_2E_3D(ty_2EternaryComparisons_2Eordering),ap(ap(V0cmp,V1k),V3k_27)),inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2ELESS))) ) ) ) ).
tff(lamtp_f2281,type,
f2281: ( del * del * $i ) > $i ).
tff(lameq_f2281,axiom,
! [A_27b: del,A_27a: del,V0cmp: $i] :
( mem(V0cmp,arr(A_27a,arr(A_27a,ty_2EternaryComparisons_2Eordering)))
=> ! [V1k: $i] : ( ap(f2281(A_27b,A_27a,V0cmp),V1k) = k(A_27b,ap(c_2Epair_2EUNCURRY(A_27a,A_27b,bool),f2280(A_27b,A_27a,V1k,V0cmp))) ) ) ).
tff(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)),f2281(A_27b,A_27a,V0cmp)))) ) ) ).
tff(lamtp_f2282,type,
f2282: ( del * $i * $i ) > $i ).
tff(lameq_f2282,axiom,
! [A_27a: del,V1k: $i] :
( mem(V1k,A_27a)
=> ! [V0cmp: $i] :
( mem(V0cmp,arr(A_27a,arr(A_27a,ty_2EternaryComparisons_2Eordering)))
=> ! [V2k_27: $i] : ( ap(f2282(A_27a,V1k,V0cmp),V2k_27) = ap(ap(c_2Emin_2E_3D(ty_2EternaryComparisons_2Eordering),ap(ap(V0cmp,V1k),V2k_27)),inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2ELESS)) ) ) ) ).
tff(lamtp_f2283,type,
f2283: ( del * $i ) > $i ).
tff(lameq_f2283,axiom,
! [A_27a: del,V0cmp: $i] :
( mem(V0cmp,arr(A_27a,arr(A_27a,ty_2EternaryComparisons_2Eordering)))
=> ! [V1k: $i] : ( ap(f2283(A_27a,V0cmp),V1k) = f2282(A_27a,V1k,V0cmp) ) ) ).
tff(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),f2283(A_27a,V0cmp))) ) ) ).
tff(lamtp_f2284,type,
f2284: ( del * $i * $i ) > $i ).
tff(lameq_f2284,axiom,
! [A_27a: del,V3k: $i] :
( mem(V3k,A_27a)
=> ! [V0cmp: $i] :
( mem(V0cmp,arr(A_27a,arr(A_27a,ty_2EternaryComparisons_2Eordering)))
=> ! [V4k_27: $i] : ( ap(f2284(A_27a,V3k,V0cmp),V4k_27) = ap(ap(c_2Emin_2E_3D(ty_2EternaryComparisons_2Eordering),ap(ap(V0cmp,V3k),V4k_27)),inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2ELESS)) ) ) ) ).
tff(lamtp_f2285,type,
f2285: ( del * $i ) > $i ).
tff(lameq_f2285,axiom,
! [A_27a: del,V0cmp: $i] :
( mem(V0cmp,arr(A_27a,arr(A_27a,ty_2EternaryComparisons_2Eordering)))
=> ! [V3k: $i] : ( ap(f2285(A_27a,V0cmp),V3k) = f2284(A_27a,V3k,V0cmp) ) ) ).
tff(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),f2283(A_27a,V0cmp)))
& p(ap(c_2Erelation_2Etransitive(A_27a),f2285(A_27a,V0cmp))) ) ) ) ).
tff(conj_thm_2Ecomparison_2Ebool__cmp__antisym,axiom,
! [V0x: tp__o,V1y: tp__o] :
( ( surj__ty_2EternaryComparisons_2Eordering(ap(ap(c_2EternaryComparisons_2Ebool__compare,inj__o(V0x)),inj__o(V1y))) = fo__c_2EternaryComparisons_2EEQUAL )
<=> ( p(inj__o(V0x))
<=> p(inj__o(V1y)) ) ) ).
tff(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 ) ) ).
tff(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 ) ) ).
tff(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 ) ) ) ) ) ) ).
tff(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 ) ) ).
tff(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 ) ) ) ) ) ) ) ).
tff(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 ) ) ) ) ) ) ).
tff(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 ) ) ) ) ) ) ).
tff(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) ) ) ) ) ) ) ) ) ).
tff(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 ) ) ) ) ) ) ) ) ).
tff(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 ) ) ) ) ) ) ) ) ).
tff(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)) ) ) ) ) ) ) ).
tff(lamtp_f2286,type,
f2286: ( del * del * $i * $i ) > $i ).
tff(lameq_f2286,axiom,
! [A_27a: del,A_27b: del,V3x: $i] :
( mem(V3x,A_27a)
=> ! [V0cmp: $i] :
( mem(V0cmp,arr(A_27a,arr(A_27b,ty_2EternaryComparisons_2Eordering)))
=> ! [V4y: $i] : ( ap(f2286(A_27a,A_27b,V3x,V0cmp),V4y) = ap(ap(c_2Emin_2E_3D(ty_2EternaryComparisons_2Eordering),ap(ap(V0cmp,V3x),V4y)),inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2EEQUAL)) ) ) ) ).
tff(lamtp_f2287,type,
f2287: ( del * del * $i ) > $i ).
tff(lameq_f2287,axiom,
! [A_27b: del,A_27a: del,V0cmp: $i] :
( mem(V0cmp,arr(A_27a,arr(A_27b,ty_2EternaryComparisons_2Eordering)))
=> ! [V3x: $i] : ( ap(f2287(A_27b,A_27a,V0cmp),V3x) = f2286(A_27a,A_27b,V3x,V0cmp) ) ) ).
tff(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),f2287(A_27b,A_27a,V0cmp)),V1l1),V2l2)) ) ) ) ) ).
tff(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)) ) ) ) ).
%------------------------------------------------------------------------------