ITP001 Axioms: ITP076_5.ax
%------------------------------------------------------------------------------
% File : ITP076_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 : ternaryComparisons_2.ax [Gau20]
% : HL4076_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 104 ( 36 unt; 34 typ; 0 def)
% Number of atoms : 1089 ( 105 equ)
% Maximal formula atoms : 23 ( 10 avg)
% Number of connectives : 213 ( 4 ~; 4 |; 59 &)
% ( 17 <=>; 129 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 810 ( 810 fml; 0 var)
% Number of types : 2 ( 1 usr)
% Number of type conns : 28 ( 16 >; 12 *; 0 +; 0 <<)
% Number of predicates : 41 ( 38 usr; 19 prp; 0-2 aty)
% Number of functors : 33 ( 33 usr; 17 con; 0-4 aty)
% Number of variables : 191 ( 187 !; 4 ?; 191 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_ty_2EternaryComparisons_2Eordering,type,
ty_2EternaryComparisons_2Eordering: del ).
tff(stp_ty_2EternaryComparisons_2Eordering,type,
tp__ty_2EternaryComparisons_2Eordering: $tType ).
tff(stp_inj_ty_2EternaryComparisons_2Eordering,type,
inj__ty_2EternaryComparisons_2Eordering: tp__ty_2EternaryComparisons_2Eordering > $i ).
tff(stp_surj_ty_2EternaryComparisons_2Eordering,type,
surj__ty_2EternaryComparisons_2Eordering: $i > tp__ty_2EternaryComparisons_2Eordering ).
tff(stp_inj_surj_ty_2EternaryComparisons_2Eordering,axiom,
! [X: tp__ty_2EternaryComparisons_2Eordering] : ( surj__ty_2EternaryComparisons_2Eordering(inj__ty_2EternaryComparisons_2Eordering(X)) = X ) ).
tff(stp_inj_mem_ty_2EternaryComparisons_2Eordering,axiom,
! [X: tp__ty_2EternaryComparisons_2Eordering] : mem(inj__ty_2EternaryComparisons_2Eordering(X),ty_2EternaryComparisons_2Eordering) ).
tff(stp_iso_mem_ty_2EternaryComparisons_2Eordering,axiom,
! [X: $i] :
( mem(X,ty_2EternaryComparisons_2Eordering)
=> ( X = inj__ty_2EternaryComparisons_2Eordering(surj__ty_2EternaryComparisons_2Eordering(X)) ) ) ).
tff(tp_c_2EternaryComparisons_2EEQUAL,type,
c_2EternaryComparisons_2EEQUAL: $i ).
tff(mem_c_2EternaryComparisons_2EEQUAL,axiom,
mem(c_2EternaryComparisons_2EEQUAL,ty_2EternaryComparisons_2Eordering) ).
tff(stp_fo_c_2EternaryComparisons_2EEQUAL,type,
fo__c_2EternaryComparisons_2EEQUAL: tp__ty_2EternaryComparisons_2Eordering ).
tff(stp_eq_fo_c_2EternaryComparisons_2EEQUAL,axiom,
inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2EEQUAL) = c_2EternaryComparisons_2EEQUAL ).
tff(tp_c_2EternaryComparisons_2EGREATER,type,
c_2EternaryComparisons_2EGREATER: $i ).
tff(mem_c_2EternaryComparisons_2EGREATER,axiom,
mem(c_2EternaryComparisons_2EGREATER,ty_2EternaryComparisons_2Eordering) ).
tff(stp_fo_c_2EternaryComparisons_2EGREATER,type,
fo__c_2EternaryComparisons_2EGREATER: tp__ty_2EternaryComparisons_2Eordering ).
tff(stp_eq_fo_c_2EternaryComparisons_2EGREATER,axiom,
inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2EGREATER) = c_2EternaryComparisons_2EGREATER ).
tff(tp_c_2EternaryComparisons_2ELESS,type,
c_2EternaryComparisons_2ELESS: $i ).
tff(mem_c_2EternaryComparisons_2ELESS,axiom,
mem(c_2EternaryComparisons_2ELESS,ty_2EternaryComparisons_2Eordering) ).
tff(stp_fo_c_2EternaryComparisons_2ELESS,type,
fo__c_2EternaryComparisons_2ELESS: tp__ty_2EternaryComparisons_2Eordering ).
tff(stp_eq_fo_c_2EternaryComparisons_2ELESS,axiom,
inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2ELESS) = c_2EternaryComparisons_2ELESS ).
tff(tp_c_2EternaryComparisons_2Ebool__compare,type,
c_2EternaryComparisons_2Ebool__compare: $i ).
tff(mem_c_2EternaryComparisons_2Ebool__compare,axiom,
mem(c_2EternaryComparisons_2Ebool__compare,arr(bool,arr(bool,ty_2EternaryComparisons_2Eordering))) ).
tff(stp_fo_c_2EternaryComparisons_2Ebool__compare,type,
fo__c_2EternaryComparisons_2Ebool__compare: ( tp__o * tp__o ) > tp__ty_2EternaryComparisons_2Eordering ).
tff(stp_eq_fo_c_2EternaryComparisons_2Ebool__compare,axiom,
! [X0: tp__o,X1: tp__o] : ( inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2Ebool__compare(X0,X1)) = ap(ap(c_2EternaryComparisons_2Ebool__compare,inj__o(X0)),inj__o(X1)) ) ).
tff(tp_c_2EternaryComparisons_2Echar__compare,type,
c_2EternaryComparisons_2Echar__compare: $i ).
tff(mem_c_2EternaryComparisons_2Echar__compare,axiom,
mem(c_2EternaryComparisons_2Echar__compare,arr(ty_2Estring_2Echar,arr(ty_2Estring_2Echar,ty_2EternaryComparisons_2Eordering))) ).
tff(stp_fo_c_2EternaryComparisons_2Echar__compare,type,
fo__c_2EternaryComparisons_2Echar__compare: ( tp__ty_2Estring_2Echar * tp__ty_2Estring_2Echar ) > tp__ty_2EternaryComparisons_2Eordering ).
tff(stp_eq_fo_c_2EternaryComparisons_2Echar__compare,axiom,
! [X0: tp__ty_2Estring_2Echar,X1: tp__ty_2Estring_2Echar] : ( inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2Echar__compare(X0,X1)) = ap(ap(c_2EternaryComparisons_2Echar__compare,inj__ty_2Estring_2Echar(X0)),inj__ty_2Estring_2Echar(X1)) ) ).
tff(tp_c_2EternaryComparisons_2Einvert__comparison,type,
c_2EternaryComparisons_2Einvert__comparison: $i ).
tff(mem_c_2EternaryComparisons_2Einvert__comparison,axiom,
mem(c_2EternaryComparisons_2Einvert__comparison,arr(ty_2EternaryComparisons_2Eordering,ty_2EternaryComparisons_2Eordering)) ).
tff(stp_fo_c_2EternaryComparisons_2Einvert__comparison,type,
fo__c_2EternaryComparisons_2Einvert__comparison: tp__ty_2EternaryComparisons_2Eordering > tp__ty_2EternaryComparisons_2Eordering ).
tff(stp_eq_fo_c_2EternaryComparisons_2Einvert__comparison,axiom,
! [X0: tp__ty_2EternaryComparisons_2Eordering] : ( inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2Einvert__comparison(X0)) = ap(c_2EternaryComparisons_2Einvert__comparison,inj__ty_2EternaryComparisons_2Eordering(X0)) ) ).
tff(tp_c_2EternaryComparisons_2Elist__compare,type,
c_2EternaryComparisons_2Elist__compare: ( del * del ) > $i ).
tff(mem_c_2EternaryComparisons_2Elist__compare,axiom,
! [A_27a: del,A_27b: del] : mem(c_2EternaryComparisons_2Elist__compare(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,ty_2EternaryComparisons_2Eordering)),arr(ty_2Elist_2Elist(A_27a),arr(ty_2Elist_2Elist(A_27b),ty_2EternaryComparisons_2Eordering)))) ).
tff(tp_c_2EternaryComparisons_2Elist__merge,type,
c_2EternaryComparisons_2Elist__merge: del > $i ).
tff(mem_c_2EternaryComparisons_2Elist__merge,axiom,
! [A_27a: del] : mem(c_2EternaryComparisons_2Elist__merge(A_27a),arr(arr(A_27a,arr(A_27a,bool)),arr(ty_2Elist_2Elist(A_27a),arr(ty_2Elist_2Elist(A_27a),ty_2Elist_2Elist(A_27a))))) ).
tff(tp_c_2EternaryComparisons_2Enum2ordering,type,
c_2EternaryComparisons_2Enum2ordering: $i ).
tff(mem_c_2EternaryComparisons_2Enum2ordering,axiom,
mem(c_2EternaryComparisons_2Enum2ordering,arr(ty_2Enum_2Enum,ty_2EternaryComparisons_2Eordering)) ).
tff(stp_fo_c_2EternaryComparisons_2Enum2ordering,type,
fo__c_2EternaryComparisons_2Enum2ordering: tp__ty_2Enum_2Enum > tp__ty_2EternaryComparisons_2Eordering ).
tff(stp_eq_fo_c_2EternaryComparisons_2Enum2ordering,axiom,
! [X0: tp__ty_2Enum_2Enum] : ( inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2Enum2ordering(X0)) = ap(c_2EternaryComparisons_2Enum2ordering,inj__ty_2Enum_2Enum(X0)) ) ).
tff(tp_c_2EternaryComparisons_2Enum__compare,type,
c_2EternaryComparisons_2Enum__compare: $i ).
tff(mem_c_2EternaryComparisons_2Enum__compare,axiom,
mem(c_2EternaryComparisons_2Enum__compare,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2EternaryComparisons_2Eordering))) ).
tff(stp_fo_c_2EternaryComparisons_2Enum__compare,type,
fo__c_2EternaryComparisons_2Enum__compare: ( tp__ty_2Enum_2Enum * tp__ty_2Enum_2Enum ) > tp__ty_2EternaryComparisons_2Eordering ).
tff(stp_eq_fo_c_2EternaryComparisons_2Enum__compare,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] : ( inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2Enum__compare(X0,X1)) = ap(ap(c_2EternaryComparisons_2Enum__compare,inj__ty_2Enum_2Enum(X0)),inj__ty_2Enum_2Enum(X1)) ) ).
tff(tp_c_2EternaryComparisons_2Eoption__compare,type,
c_2EternaryComparisons_2Eoption__compare: ( del * del ) > $i ).
tff(mem_c_2EternaryComparisons_2Eoption__compare,axiom,
! [A_27a: del,A_27b: del] : mem(c_2EternaryComparisons_2Eoption__compare(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_2EternaryComparisons_2Eordering2num,type,
c_2EternaryComparisons_2Eordering2num: $i ).
tff(mem_c_2EternaryComparisons_2Eordering2num,axiom,
mem(c_2EternaryComparisons_2Eordering2num,arr(ty_2EternaryComparisons_2Eordering,ty_2Enum_2Enum)) ).
tff(stp_fo_c_2EternaryComparisons_2Eordering2num,type,
fo__c_2EternaryComparisons_2Eordering2num: tp__ty_2EternaryComparisons_2Eordering > tp__ty_2Enum_2Enum ).
tff(stp_eq_fo_c_2EternaryComparisons_2Eordering2num,axiom,
! [X0: tp__ty_2EternaryComparisons_2Eordering] : ( inj__ty_2Enum_2Enum(fo__c_2EternaryComparisons_2Eordering2num(X0)) = ap(c_2EternaryComparisons_2Eordering2num,inj__ty_2EternaryComparisons_2Eordering(X0)) ) ).
tff(tp_c_2EternaryComparisons_2Eordering__CASE,type,
c_2EternaryComparisons_2Eordering__CASE: del > $i ).
tff(mem_c_2EternaryComparisons_2Eordering__CASE,axiom,
! [A_27a: del] : mem(c_2EternaryComparisons_2Eordering__CASE(A_27a),arr(ty_2EternaryComparisons_2Eordering,arr(A_27a,arr(A_27a,arr(A_27a,A_27a))))) ).
tff(tp_c_2EternaryComparisons_2Eordering__size,type,
c_2EternaryComparisons_2Eordering__size: $i ).
tff(mem_c_2EternaryComparisons_2Eordering__size,axiom,
mem(c_2EternaryComparisons_2Eordering__size,arr(ty_2EternaryComparisons_2Eordering,ty_2Enum_2Enum)) ).
tff(stp_fo_c_2EternaryComparisons_2Eordering__size,type,
fo__c_2EternaryComparisons_2Eordering__size: tp__ty_2EternaryComparisons_2Eordering > tp__ty_2Enum_2Enum ).
tff(stp_eq_fo_c_2EternaryComparisons_2Eordering__size,axiom,
! [X0: tp__ty_2EternaryComparisons_2Eordering] : ( inj__ty_2Enum_2Enum(fo__c_2EternaryComparisons_2Eordering__size(X0)) = ap(c_2EternaryComparisons_2Eordering__size,inj__ty_2EternaryComparisons_2Eordering(X0)) ) ).
tff(tp_c_2EternaryComparisons_2Epair__compare,type,
c_2EternaryComparisons_2Epair__compare: ( del * del * del * del ) > $i ).
tff(mem_c_2EternaryComparisons_2Epair__compare,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : mem(c_2EternaryComparisons_2Epair__compare(A_27a,A_27b,A_27c,A_27d),arr(arr(A_27a,arr(A_27b,ty_2EternaryComparisons_2Eordering)),arr(arr(A_27c,arr(A_27d,ty_2EternaryComparisons_2Eordering)),arr(ty_2Epair_2Eprod(A_27a,A_27c),arr(ty_2Epair_2Eprod(A_27b,A_27d),ty_2EternaryComparisons_2Eordering))))) ).
tff(tp_c_2EternaryComparisons_2Estring__compare,type,
c_2EternaryComparisons_2Estring__compare: $i ).
tff(mem_c_2EternaryComparisons_2Estring__compare,axiom,
mem(c_2EternaryComparisons_2Estring__compare,arr(ty_2Elist_2Elist(ty_2Estring_2Echar),arr(ty_2Elist_2Elist(ty_2Estring_2Echar),ty_2EternaryComparisons_2Eordering))) ).
tff(stp_fo_c_2EternaryComparisons_2Estring__compare,type,
fo__c_2EternaryComparisons_2Estring__compare: ( tp__c_ty_2Elist_2Elist_ty_2Estring_2Echar * tp__c_ty_2Elist_2Elist_ty_2Estring_2Echar ) > tp__ty_2EternaryComparisons_2Eordering ).
tff(stp_eq_fo_c_2EternaryComparisons_2Estring__compare,axiom,
! [X0: tp__c_ty_2Elist_2Elist_ty_2Estring_2Echar,X1: tp__c_ty_2Elist_2Elist_ty_2Estring_2Echar] : ( inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2Estring__compare(X0,X1)) = ap(ap(c_2EternaryComparisons_2Estring__compare,inj__c_ty_2Elist_2Elist_ty_2Estring_2Echar(X0)),inj__c_ty_2Elist_2Elist_ty_2Estring_2Echar(X1)) ) ).
tff(lamtp_f1949,type,
f1949: $i ).
tff(lameq_f1949,axiom,
! [V1n: tp__ty_2Enum_2Enum] : ( ap(f1949,inj__ty_2Enum_2Enum(V1n)) = ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V1n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) ).
tff(ax_thm_2EternaryComparisons_2Eordering__TY__DEF,axiom,
? [V0rep: $i] :
( mem(V0rep,arr(ty_2EternaryComparisons_2Eordering,ty_2Enum_2Enum))
& p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(ty_2Enum_2Enum,ty_2EternaryComparisons_2Eordering),f1949),V0rep)) ) ).
tff(lamtp_f1950,type,
f1950: $i ).
tff(lameq_f1950,axiom,
! [V2n: tp__ty_2Enum_2Enum] : ( ap(f1950,inj__ty_2Enum_2Enum(V2n)) = ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V2n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) ).
tff(ax_thm_2EternaryComparisons_2Eordering__BIJ,axiom,
( ! [V0a: tp__ty_2EternaryComparisons_2Eordering] : ( surj__ty_2EternaryComparisons_2Eordering(ap(c_2EternaryComparisons_2Enum2ordering,ap(c_2EternaryComparisons_2Eordering2num,inj__ty_2EternaryComparisons_2Eordering(V0a)))) = V0a )
& ! [V1r: tp__ty_2Enum_2Enum] :
( p(ap(f1950,inj__ty_2Enum_2Enum(V1r)))
<=> ( surj__ty_2Enum_2Enum(ap(c_2EternaryComparisons_2Eordering2num,ap(c_2EternaryComparisons_2Enum2ordering,inj__ty_2Enum_2Enum(V1r)))) = V1r ) ) ) ).
tff(conj_thm_2EternaryComparisons_2Enum2ordering__ordering2num,axiom,
! [V0a: tp__ty_2EternaryComparisons_2Eordering] : ( surj__ty_2EternaryComparisons_2Eordering(ap(c_2EternaryComparisons_2Enum2ordering,ap(c_2EternaryComparisons_2Eordering2num,inj__ty_2EternaryComparisons_2Eordering(V0a)))) = V0a ) ).
tff(conj_thm_2EternaryComparisons_2Eordering2num__num2ordering,axiom,
! [V0r: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V0r)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))
<=> ( surj__ty_2Enum_2Enum(ap(c_2EternaryComparisons_2Eordering2num,ap(c_2EternaryComparisons_2Enum2ordering,inj__ty_2Enum_2Enum(V0r)))) = V0r ) ) ).
tff(conj_thm_2EternaryComparisons_2Enum2ordering__11,axiom,
! [V0r: tp__ty_2Enum_2Enum,V1r_27: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V0r)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V1r_27)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))
=> ( ( surj__ty_2EternaryComparisons_2Eordering(ap(c_2EternaryComparisons_2Enum2ordering,inj__ty_2Enum_2Enum(V0r))) = surj__ty_2EternaryComparisons_2Eordering(ap(c_2EternaryComparisons_2Enum2ordering,inj__ty_2Enum_2Enum(V1r_27))) )
<=> ( V0r = V1r_27 ) ) ) ) ).
tff(conj_thm_2EternaryComparisons_2Eordering2num__11,axiom,
! [V0a: tp__ty_2EternaryComparisons_2Eordering,V1a_27: tp__ty_2EternaryComparisons_2Eordering] :
( ( surj__ty_2Enum_2Enum(ap(c_2EternaryComparisons_2Eordering2num,inj__ty_2EternaryComparisons_2Eordering(V0a))) = surj__ty_2Enum_2Enum(ap(c_2EternaryComparisons_2Eordering2num,inj__ty_2EternaryComparisons_2Eordering(V1a_27))) )
<=> ( V0a = V1a_27 ) ) ).
tff(conj_thm_2EternaryComparisons_2Enum2ordering__ONTO,axiom,
! [V0a: tp__ty_2EternaryComparisons_2Eordering] :
? [V1r: tp__ty_2Enum_2Enum] :
( ( V0a = surj__ty_2EternaryComparisons_2Eordering(ap(c_2EternaryComparisons_2Enum2ordering,inj__ty_2Enum_2Enum(V1r))) )
& p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V1r)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) ) ).
tff(conj_thm_2EternaryComparisons_2Eordering2num__ONTO,axiom,
! [V0r: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V0r)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))
<=> ? [V1a: tp__ty_2EternaryComparisons_2Eordering] : ( V0r = surj__ty_2Enum_2Enum(ap(c_2EternaryComparisons_2Eordering2num,inj__ty_2EternaryComparisons_2Eordering(V1a))) ) ) ).
tff(conj_thm_2EternaryComparisons_2Enum2ordering__thm,axiom,
( ( surj__ty_2EternaryComparisons_2Eordering(ap(c_2EternaryComparisons_2Enum2ordering,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) = fo__c_2EternaryComparisons_2ELESS )
& ( surj__ty_2EternaryComparisons_2Eordering(ap(c_2EternaryComparisons_2Enum2ordering,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) = fo__c_2EternaryComparisons_2EEQUAL )
& ( surj__ty_2EternaryComparisons_2Eordering(ap(c_2EternaryComparisons_2Enum2ordering,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) = fo__c_2EternaryComparisons_2EGREATER ) ) ).
tff(conj_thm_2EternaryComparisons_2Eordering2num__thm,axiom,
( ( surj__ty_2Enum_2Enum(ap(c_2EternaryComparisons_2Eordering2num,inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2ELESS))) = fo__c_2Enum_2E0 )
& ( surj__ty_2Enum_2Enum(ap(c_2EternaryComparisons_2Eordering2num,inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2EEQUAL))) = surj__ty_2Enum_2Enum(ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))) )
& ( surj__ty_2Enum_2Enum(ap(c_2EternaryComparisons_2Eordering2num,inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2EGREATER))) = surj__ty_2Enum_2Enum(ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))) ) ) ).
tff(conj_thm_2EternaryComparisons_2Eordering__EQ__ordering,axiom,
! [V0a: tp__ty_2EternaryComparisons_2Eordering,V1a_27: tp__ty_2EternaryComparisons_2Eordering] :
( ( V0a = V1a_27 )
<=> ( surj__ty_2Enum_2Enum(ap(c_2EternaryComparisons_2Eordering2num,inj__ty_2EternaryComparisons_2Eordering(V0a))) = surj__ty_2Enum_2Enum(ap(c_2EternaryComparisons_2Eordering2num,inj__ty_2EternaryComparisons_2Eordering(V1a_27))) ) ) ).
tff(ax_thm_2EternaryComparisons_2Eordering__size__def,axiom,
! [V0x: tp__ty_2EternaryComparisons_2Eordering] : ( surj__ty_2Enum_2Enum(ap(c_2EternaryComparisons_2Eordering__size,inj__ty_2EternaryComparisons_2Eordering(V0x))) = fo__c_2Enum_2E0 ) ).
tff(lamtp_f1951,type,
f1951: ( del * $i * $i * $i ) > $i ).
tff(lameq_f1951,axiom,
! [A_27a: del,V3v2: $i] :
( mem(V3v2,A_27a)
=> ! [V2v1: $i] :
( mem(V2v1,A_27a)
=> ! [V1v0: $i] :
( mem(V1v0,A_27a)
=> ! [V4m: tp__ty_2Enum_2Enum] : ( ap(f1951(A_27a,V3v2,V2v1,V1v0),inj__ty_2Enum_2Enum(V4m)) = ap(ap(ap(c_2Ebool_2ECOND(A_27a),ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V4m)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),V1v0),ap(ap(ap(c_2Ebool_2ECOND(A_27a),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V4m)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),V2v1),V3v2)) ) ) ) ) ).
tff(ax_thm_2EternaryComparisons_2Eordering__CASE,axiom,
! [A_27a: del,V0x: tp__ty_2EternaryComparisons_2Eordering,V1v0: $i] :
( mem(V1v0,A_27a)
=> ! [V2v1: $i] :
( mem(V2v1,A_27a)
=> ! [V3v2: $i] :
( mem(V3v2,A_27a)
=> ( ap(ap(ap(ap(c_2EternaryComparisons_2Eordering__CASE(A_27a),inj__ty_2EternaryComparisons_2Eordering(V0x)),V1v0),V2v1),V3v2) = ap(f1951(A_27a,V3v2,V2v1,V1v0),ap(c_2EternaryComparisons_2Eordering2num,inj__ty_2EternaryComparisons_2Eordering(V0x))) ) ) ) ) ).
tff(conj_thm_2EternaryComparisons_2Eordering__case__def,axiom,
! [A_27a: del] :
( ! [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 ) ) ) ) ) ).
tff(conj_thm_2EternaryComparisons_2Edatatype__ordering,axiom,
! [V0ordering: $i] :
( mem(V0ordering,arr(ty_2EternaryComparisons_2Eordering,arr(ty_2EternaryComparisons_2Eordering,arr(ty_2EternaryComparisons_2Eordering,bool))))
=> p(ap(c_2Ebool_2EDATATYPE(bool),ap(ap(ap(V0ordering,inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2ELESS)),inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2EEQUAL)),inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2EGREATER)))) ) ).
tff(conj_thm_2EternaryComparisons_2Eordering__distinct,axiom,
( ( fo__c_2EternaryComparisons_2ELESS != fo__c_2EternaryComparisons_2EEQUAL )
& ( fo__c_2EternaryComparisons_2ELESS != fo__c_2EternaryComparisons_2EGREATER )
& ( fo__c_2EternaryComparisons_2EEQUAL != fo__c_2EternaryComparisons_2EGREATER ) ) ).
tff(conj_thm_2EternaryComparisons_2Eordering__nchotomy,axiom,
! [V0a: tp__ty_2EternaryComparisons_2Eordering] :
( ( V0a = fo__c_2EternaryComparisons_2ELESS )
| ( V0a = fo__c_2EternaryComparisons_2EEQUAL )
| ( V0a = fo__c_2EternaryComparisons_2EGREATER ) ) ).
tff(conj_thm_2EternaryComparisons_2Eordering__Axiom,axiom,
! [A_27a: del,V0x0: $i] :
( mem(V0x0,A_27a)
=> ! [V1x1: $i] :
( mem(V1x1,A_27a)
=> ! [V2x2: $i] :
( mem(V2x2,A_27a)
=> ? [V3f: $i] :
( mem(V3f,arr(ty_2EternaryComparisons_2Eordering,A_27a))
& ( ap(V3f,inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2ELESS)) = V0x0 )
& ( ap(V3f,inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2EEQUAL)) = V1x1 )
& ( ap(V3f,inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2EGREATER)) = V2x2 ) ) ) ) ) ).
tff(conj_thm_2EternaryComparisons_2Eordering__induction,axiom,
! [V0P: $i] :
( mem(V0P,arr(ty_2EternaryComparisons_2Eordering,bool))
=> ( ( p(ap(V0P,inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2EEQUAL)))
& p(ap(V0P,inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2EGREATER)))
& p(ap(V0P,inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2ELESS))) )
=> ! [V1a: tp__ty_2EternaryComparisons_2Eordering] : p(ap(V0P,inj__ty_2EternaryComparisons_2Eordering(V1a))) ) ) ).
tff(conj_thm_2EternaryComparisons_2Eordering__case__cong,axiom,
! [A_27a: del,V0v0_27: $i] :
( mem(V0v0_27,A_27a)
=> ! [V1v1_27: $i] :
( mem(V1v1_27,A_27a)
=> ! [V2v2_27: $i] :
( mem(V2v2_27,A_27a)
=> ! [V3M: tp__ty_2EternaryComparisons_2Eordering,V4M_27: tp__ty_2EternaryComparisons_2Eordering,V5v0: $i] :
( mem(V5v0,A_27a)
=> ! [V6v1: $i] :
( mem(V6v1,A_27a)
=> ! [V7v2: $i] :
( mem(V7v2,A_27a)
=> ( ( ( V3M = V4M_27 )
& ( ( V4M_27 = fo__c_2EternaryComparisons_2ELESS )
=> ( V5v0 = V0v0_27 ) )
& ( ( V4M_27 = fo__c_2EternaryComparisons_2EEQUAL )
=> ( V6v1 = V1v1_27 ) )
& ( ( V4M_27 = fo__c_2EternaryComparisons_2EGREATER )
=> ( V7v2 = V2v2_27 ) ) )
=> ( ap(ap(ap(ap(c_2EternaryComparisons_2Eordering__CASE(A_27a),inj__ty_2EternaryComparisons_2Eordering(V3M)),V5v0),V6v1),V7v2) = ap(ap(ap(ap(c_2EternaryComparisons_2Eordering__CASE(A_27a),inj__ty_2EternaryComparisons_2Eordering(V4M_27)),V0v0_27),V1v1_27),V2v2_27) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2EternaryComparisons_2Eordering__case__eq,axiom,
! [A_27a: del,V0x: tp__ty_2EternaryComparisons_2Eordering,V1v0: $i] :
( mem(V1v0,A_27a)
=> ! [V2v1: $i] :
( mem(V2v1,A_27a)
=> ! [V3v2: $i] :
( mem(V3v2,A_27a)
=> ! [V4v: $i] :
( mem(V4v,A_27a)
=> ( ( ap(ap(ap(ap(c_2EternaryComparisons_2Eordering__CASE(A_27a),inj__ty_2EternaryComparisons_2Eordering(V0x)),V1v0),V2v1),V3v2) = V4v )
<=> ( ( ( V0x = fo__c_2EternaryComparisons_2ELESS )
& ( V1v0 = V4v ) )
| ( ( V0x = fo__c_2EternaryComparisons_2EEQUAL )
& ( V2v1 = V4v ) )
| ( ( V0x = fo__c_2EternaryComparisons_2EGREATER )
& ( V3v2 = V4v ) ) ) ) ) ) ) ) ).
tff(conj_thm_2EternaryComparisons_2Eordering__eq__dec,axiom,
( ! [V0x: tp__ty_2EternaryComparisons_2Eordering] :
( ( V0x = V0x )
<=> $true )
& ( ( fo__c_2EternaryComparisons_2ELESS = fo__c_2EternaryComparisons_2EEQUAL )
<=> $false )
& ( ( fo__c_2EternaryComparisons_2ELESS = fo__c_2EternaryComparisons_2EGREATER )
<=> $false )
& ( ( fo__c_2EternaryComparisons_2EEQUAL = fo__c_2EternaryComparisons_2EGREATER )
<=> $false )
& ( ( fo__c_2EternaryComparisons_2EEQUAL = fo__c_2EternaryComparisons_2ELESS )
<=> $false )
& ( ( fo__c_2EternaryComparisons_2EGREATER = fo__c_2EternaryComparisons_2ELESS )
<=> $false )
& ( ( fo__c_2EternaryComparisons_2EGREATER = fo__c_2EternaryComparisons_2EEQUAL )
<=> $false )
& ( surj__ty_2Enum_2Enum(ap(c_2EternaryComparisons_2Eordering2num,inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2ELESS))) = fo__c_2Enum_2E0 )
& ( surj__ty_2Enum_2Enum(ap(c_2EternaryComparisons_2Eordering2num,inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2EEQUAL))) = surj__ty_2Enum_2Enum(ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))) )
& ( surj__ty_2Enum_2Enum(ap(c_2EternaryComparisons_2Eordering2num,inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2EGREATER))) = surj__ty_2Enum_2Enum(ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))) )
& ( surj__ty_2EternaryComparisons_2Eordering(ap(c_2EternaryComparisons_2Enum2ordering,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) = fo__c_2EternaryComparisons_2ELESS )
& ( surj__ty_2EternaryComparisons_2Eordering(ap(c_2EternaryComparisons_2Enum2ordering,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) = fo__c_2EternaryComparisons_2EEQUAL )
& ( surj__ty_2EternaryComparisons_2Eordering(ap(c_2EternaryComparisons_2Enum2ordering,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) = fo__c_2EternaryComparisons_2EGREATER ) ) ).
tff(conj_thm_2EternaryComparisons_2Ebool__compare__ind,axiom,
! [V0P: $i] :
( mem(V0P,arr(bool,arr(bool,bool)))
=> ( ( p(ap(ap(V0P,inj__o(fo__c_2Ebool_2ET)),inj__o(fo__c_2Ebool_2ET)))
& p(ap(ap(V0P,inj__o(fo__c_2Ebool_2EF)),inj__o(fo__c_2Ebool_2EF)))
& p(ap(ap(V0P,inj__o(fo__c_2Ebool_2ET)),inj__o(fo__c_2Ebool_2EF)))
& p(ap(ap(V0P,inj__o(fo__c_2Ebool_2EF)),inj__o(fo__c_2Ebool_2ET))) )
=> ! [V1v: tp__o,V2v1: tp__o] : p(ap(ap(V0P,inj__o(V1v)),inj__o(V2v1))) ) ) ).
tff(conj_thm_2EternaryComparisons_2Ebool__compare__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(ax_thm_2EternaryComparisons_2Epair__compare__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)))
=> ! [V2a: $i] :
( mem(V2a,A_27a)
=> ! [V3b: $i] :
( mem(V3b,A_27c)
=> ! [V4x: $i] :
( mem(V4x,A_27b)
=> ! [V5y: $i] :
( mem(V5y,A_27d)
=> ( surj__ty_2EternaryComparisons_2Eordering(ap(ap(ap(ap(c_2EternaryComparisons_2Epair__compare(A_27a,A_27b,A_27c,A_27d),V0c1),V1c2),ap(ap(c_2Epair_2E_2C(A_27a,A_27c),V2a),V3b)),ap(ap(c_2Epair_2E_2C(A_27b,A_27d),V4x),V5y))) = surj__ty_2EternaryComparisons_2Eordering(ap(ap(ap(ap(c_2EternaryComparisons_2Eordering__CASE(ty_2EternaryComparisons_2Eordering),ap(ap(V0c1,V2a),V4x)),inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2ELESS)),ap(ap(V1c2,V3b),V5y)),inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2EGREATER))) ) ) ) ) ) ) ) ).
tff(conj_thm_2EternaryComparisons_2Eoption__compare__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))))
=> ( ( ! [V1c: $i] :
( mem(V1c,arr(A_27a,arr(A_27b,ty_2EternaryComparisons_2Eordering)))
=> p(ap(ap(ap(V0P,V1c),c_2Eoption_2ENONE(A_27a)),c_2Eoption_2ENONE(A_27b))) )
& ! [V2c: $i] :
( mem(V2c,arr(A_27a,arr(A_27b,ty_2EternaryComparisons_2Eordering)))
=> ! [V3v0: $i] :
( mem(V3v0,A_27b)
=> p(ap(ap(ap(V0P,V2c),c_2Eoption_2ENONE(A_27a)),ap(c_2Eoption_2ESOME(A_27b),V3v0))) ) )
& ! [V4c: $i] :
( mem(V4c,arr(A_27a,arr(A_27b,ty_2EternaryComparisons_2Eordering)))
=> ! [V5v3: $i] :
( mem(V5v3,A_27a)
=> p(ap(ap(ap(V0P,V4c),ap(c_2Eoption_2ESOME(A_27a),V5v3)),c_2Eoption_2ENONE(A_27b))) ) )
& ! [V6c: $i] :
( mem(V6c,arr(A_27a,arr(A_27b,ty_2EternaryComparisons_2Eordering)))
=> ! [V7v1: $i] :
( mem(V7v1,A_27a)
=> ! [V8v2: $i] :
( mem(V8v2,A_27b)
=> p(ap(ap(ap(V0P,V6c),ap(c_2Eoption_2ESOME(A_27a),V7v1)),ap(c_2Eoption_2ESOME(A_27b),V8v2))) ) ) ) )
=> ! [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_2EternaryComparisons_2Eoption__compare__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(ax_thm_2EternaryComparisons_2Enum__compare__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(ax_thm_2EternaryComparisons_2Echar__compare__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_2EternaryComparisons_2Elist__compare__ind,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,arr(A_27b,ty_2EternaryComparisons_2Eordering)),arr(ty_2Elist_2Elist(A_27a),arr(ty_2Elist_2Elist(A_27b),bool))))
=> ( ( ! [V1cmp: $i] :
( mem(V1cmp,arr(A_27a,arr(A_27b,ty_2EternaryComparisons_2Eordering)))
=> p(ap(ap(ap(V0P,V1cmp),c_2Elist_2ENIL(A_27a)),c_2Elist_2ENIL(A_27b))) )
& ! [V2cmp: $i] :
( mem(V2cmp,arr(A_27a,arr(A_27b,ty_2EternaryComparisons_2Eordering)))
=> ! [V3v8: $i] :
( mem(V3v8,A_27b)
=> ! [V4v9: $i] :
( mem(V4v9,ty_2Elist_2Elist(A_27b))
=> p(ap(ap(ap(V0P,V2cmp),c_2Elist_2ENIL(A_27a)),ap(ap(c_2Elist_2ECONS(A_27b),V3v8),V4v9))) ) ) )
& ! [V5cmp: $i] :
( mem(V5cmp,arr(A_27a,arr(A_27b,ty_2EternaryComparisons_2Eordering)))
=> ! [V6v4: $i] :
( mem(V6v4,A_27a)
=> ! [V7v5: $i] :
( mem(V7v5,ty_2Elist_2Elist(A_27a))
=> p(ap(ap(ap(V0P,V5cmp),ap(ap(c_2Elist_2ECONS(A_27a),V6v4),V7v5)),c_2Elist_2ENIL(A_27b))) ) ) )
& ! [V8cmp: $i] :
( mem(V8cmp,arr(A_27a,arr(A_27b,ty_2EternaryComparisons_2Eordering)))
=> ! [V9x: $i] :
( mem(V9x,A_27a)
=> ! [V10l1: $i] :
( mem(V10l1,ty_2Elist_2Elist(A_27a))
=> ! [V11y: $i] :
( mem(V11y,A_27b)
=> ! [V12l2: $i] :
( mem(V12l2,ty_2Elist_2Elist(A_27b))
=> ( ( ( surj__ty_2EternaryComparisons_2Eordering(ap(ap(V8cmp,V9x),V11y)) = fo__c_2EternaryComparisons_2EEQUAL )
=> p(ap(ap(ap(V0P,V8cmp),V10l1),V12l2)) )
=> p(ap(ap(ap(V0P,V8cmp),ap(ap(c_2Elist_2ECONS(A_27a),V9x),V10l1)),ap(ap(c_2Elist_2ECONS(A_27b),V11y),V12l2))) ) ) ) ) ) ) )
=> ! [V13v: $i] :
( mem(V13v,arr(A_27a,arr(A_27b,ty_2EternaryComparisons_2Eordering)))
=> ! [V14v1: $i] :
( mem(V14v1,ty_2Elist_2Elist(A_27a))
=> ! [V15v2: $i] :
( mem(V15v2,ty_2Elist_2Elist(A_27b))
=> p(ap(ap(ap(V0P,V13v),V14v1),V15v2)) ) ) ) ) ) ).
tff(conj_thm_2EternaryComparisons_2Elist__compare__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0cmp: $i] :
( mem(V0cmp,arr(A_27a,arr(A_27b,ty_2EternaryComparisons_2Eordering)))
=> ( surj__ty_2EternaryComparisons_2Eordering(ap(ap(ap(c_2EternaryComparisons_2Elist__compare(A_27a,A_27b),V0cmp),c_2Elist_2ENIL(A_27a)),c_2Elist_2ENIL(A_27b))) = fo__c_2EternaryComparisons_2EEQUAL ) )
& ! [V1v9: $i] :
( mem(V1v9,ty_2Elist_2Elist(A_27b))
=> ! [V2v8: $i] :
( mem(V2v8,A_27b)
=> ! [V3cmp: $i] :
( mem(V3cmp,arr(A_27a,arr(A_27b,ty_2EternaryComparisons_2Eordering)))
=> ( surj__ty_2EternaryComparisons_2Eordering(ap(ap(ap(c_2EternaryComparisons_2Elist__compare(A_27a,A_27b),V3cmp),c_2Elist_2ENIL(A_27a)),ap(ap(c_2Elist_2ECONS(A_27b),V2v8),V1v9))) = fo__c_2EternaryComparisons_2ELESS ) ) ) )
& ! [V4v5: $i] :
( mem(V4v5,ty_2Elist_2Elist(A_27a))
=> ! [V5v4: $i] :
( mem(V5v4,A_27a)
=> ! [V6cmp: $i] :
( mem(V6cmp,arr(A_27a,arr(A_27b,ty_2EternaryComparisons_2Eordering)))
=> ( surj__ty_2EternaryComparisons_2Eordering(ap(ap(ap(c_2EternaryComparisons_2Elist__compare(A_27a,A_27b),V6cmp),ap(ap(c_2Elist_2ECONS(A_27a),V5v4),V4v5)),c_2Elist_2ENIL(A_27b))) = fo__c_2EternaryComparisons_2EGREATER ) ) ) )
& ! [V7y: $i] :
( mem(V7y,A_27b)
=> ! [V8x: $i] :
( mem(V8x,A_27a)
=> ! [V9l2: $i] :
( mem(V9l2,ty_2Elist_2Elist(A_27b))
=> ! [V10l1: $i] :
( mem(V10l1,ty_2Elist_2Elist(A_27a))
=> ! [V11cmp: $i] :
( mem(V11cmp,arr(A_27a,arr(A_27b,ty_2EternaryComparisons_2Eordering)))
=> ( surj__ty_2EternaryComparisons_2Eordering(ap(ap(ap(c_2EternaryComparisons_2Elist__compare(A_27a,A_27b),V11cmp),ap(ap(c_2Elist_2ECONS(A_27a),V8x),V10l1)),ap(ap(c_2Elist_2ECONS(A_27b),V7y),V9l2))) = surj__ty_2EternaryComparisons_2Eordering(ap(ap(ap(ap(c_2EternaryComparisons_2Eordering__CASE(ty_2EternaryComparisons_2Eordering),ap(ap(V11cmp,V8x),V7y)),inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2ELESS)),ap(ap(ap(c_2EternaryComparisons_2Elist__compare(A_27a,A_27b),V11cmp),V10l1),V9l2)),inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2EGREATER))) ) ) ) ) ) ) ) ).
tff(ax_thm_2EternaryComparisons_2Estring__compare__def,axiom,
c_2EternaryComparisons_2Estring__compare = ap(c_2EternaryComparisons_2Elist__compare(ty_2Estring_2Echar,ty_2Estring_2Echar),c_2EternaryComparisons_2Echar__compare) ).
tff(conj_thm_2EternaryComparisons_2Ecompare__equal,axiom,
! [A_27a: del,V0cmp: $i] :
( mem(V0cmp,arr(A_27a,arr(A_27a,ty_2EternaryComparisons_2Eordering)))
=> ( ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( ( surj__ty_2EternaryComparisons_2Eordering(ap(ap(V0cmp,V1x),V2y)) = fo__c_2EternaryComparisons_2EEQUAL )
<=> ( V1x = V2y ) ) ) )
=> ! [V3l1: $i] :
( mem(V3l1,ty_2Elist_2Elist(A_27a))
=> ! [V4l2: $i] :
( mem(V4l2,ty_2Elist_2Elist(A_27a))
=> ( ( surj__ty_2EternaryComparisons_2Eordering(ap(ap(ap(c_2EternaryComparisons_2Elist__compare(A_27a,A_27a),V0cmp),V3l1),V4l2)) = fo__c_2EternaryComparisons_2EEQUAL )
<=> ( V3l1 = V4l2 ) ) ) ) ) ) ).
tff(conj_thm_2EternaryComparisons_2Elist__merge__ind,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,arr(A_27a,bool)),arr(ty_2Elist_2Elist(A_27a),arr(ty_2Elist_2Elist(A_27a),bool))))
=> ( ( ! [V1a__lt: $i] :
( mem(V1a__lt,arr(A_27a,arr(A_27a,bool)))
=> ! [V2l1: $i] :
( mem(V2l1,ty_2Elist_2Elist(A_27a))
=> p(ap(ap(ap(V0P,V1a__lt),V2l1),c_2Elist_2ENIL(A_27a))) ) )
& ! [V3a__lt: $i] :
( mem(V3a__lt,arr(A_27a,arr(A_27a,bool)))
=> ! [V4v4: $i] :
( mem(V4v4,A_27a)
=> ! [V5v5: $i] :
( mem(V5v5,ty_2Elist_2Elist(A_27a))
=> p(ap(ap(ap(V0P,V3a__lt),c_2Elist_2ENIL(A_27a)),ap(ap(c_2Elist_2ECONS(A_27a),V4v4),V5v5))) ) ) )
& ! [V6a__lt: $i] :
( mem(V6a__lt,arr(A_27a,arr(A_27a,bool)))
=> ! [V7x: $i] :
( mem(V7x,A_27a)
=> ! [V8l1: $i] :
( mem(V8l1,ty_2Elist_2Elist(A_27a))
=> ! [V9y: $i] :
( mem(V9y,A_27a)
=> ! [V10l2: $i] :
( mem(V10l2,ty_2Elist_2Elist(A_27a))
=> ( ( ( ~ p(ap(ap(V6a__lt,V7x),V9y))
=> p(ap(ap(ap(V0P,V6a__lt),ap(ap(c_2Elist_2ECONS(A_27a),V7x),V8l1)),V10l2)) )
& ( p(ap(ap(V6a__lt,V7x),V9y))
=> p(ap(ap(ap(V0P,V6a__lt),V8l1),ap(ap(c_2Elist_2ECONS(A_27a),V9y),V10l2))) ) )
=> p(ap(ap(ap(V0P,V6a__lt),ap(ap(c_2Elist_2ECONS(A_27a),V7x),V8l1)),ap(ap(c_2Elist_2ECONS(A_27a),V9y),V10l2))) ) ) ) ) ) ) )
=> ! [V11v: $i] :
( mem(V11v,arr(A_27a,arr(A_27a,bool)))
=> ! [V12v1: $i] :
( mem(V12v1,ty_2Elist_2Elist(A_27a))
=> ! [V13v2: $i] :
( mem(V13v2,ty_2Elist_2Elist(A_27a))
=> p(ap(ap(ap(V0P,V11v),V12v1),V13v2)) ) ) ) ) ) ).
tff(conj_thm_2EternaryComparisons_2Elist__merge__def,axiom,
! [A_27a: del] :
( ! [V0l1: $i] :
( mem(V0l1,ty_2Elist_2Elist(A_27a))
=> ! [V1a__lt: $i] :
( mem(V1a__lt,arr(A_27a,arr(A_27a,bool)))
=> ( ap(ap(ap(c_2EternaryComparisons_2Elist__merge(A_27a),V1a__lt),V0l1),c_2Elist_2ENIL(A_27a)) = V0l1 ) ) )
& ! [V2v5: $i] :
( mem(V2v5,ty_2Elist_2Elist(A_27a))
=> ! [V3v4: $i] :
( mem(V3v4,A_27a)
=> ! [V4a__lt: $i] :
( mem(V4a__lt,arr(A_27a,arr(A_27a,bool)))
=> ( ap(ap(ap(c_2EternaryComparisons_2Elist__merge(A_27a),V4a__lt),c_2Elist_2ENIL(A_27a)),ap(ap(c_2Elist_2ECONS(A_27a),V3v4),V2v5)) = ap(ap(c_2Elist_2ECONS(A_27a),V3v4),V2v5) ) ) ) )
& ! [V5y: $i] :
( mem(V5y,A_27a)
=> ! [V6x: $i] :
( mem(V6x,A_27a)
=> ! [V7l2: $i] :
( mem(V7l2,ty_2Elist_2Elist(A_27a))
=> ! [V8l1: $i] :
( mem(V8l1,ty_2Elist_2Elist(A_27a))
=> ! [V9a__lt: $i] :
( mem(V9a__lt,arr(A_27a,arr(A_27a,bool)))
=> ( ap(ap(ap(c_2EternaryComparisons_2Elist__merge(A_27a),V9a__lt),ap(ap(c_2Elist_2ECONS(A_27a),V6x),V8l1)),ap(ap(c_2Elist_2ECONS(A_27a),V5y),V7l2)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Elist_2Elist(A_27a)),ap(ap(V9a__lt,V6x),V5y)),ap(ap(c_2Elist_2ECONS(A_27a),V6x),ap(ap(ap(c_2EternaryComparisons_2Elist__merge(A_27a),V9a__lt),V8l1),ap(ap(c_2Elist_2ECONS(A_27a),V5y),V7l2)))),ap(ap(c_2Elist_2ECONS(A_27a),V5y),ap(ap(ap(c_2EternaryComparisons_2Elist__merge(A_27a),V9a__lt),ap(ap(c_2Elist_2ECONS(A_27a),V6x),V8l1)),V7l2))) ) ) ) ) ) ) ) ).
tff(ax_thm_2EternaryComparisons_2Einvert__comparison__def,axiom,
( ( surj__ty_2EternaryComparisons_2Eordering(ap(c_2EternaryComparisons_2Einvert__comparison,inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2EGREATER))) = fo__c_2EternaryComparisons_2ELESS )
& ( surj__ty_2EternaryComparisons_2Eordering(ap(c_2EternaryComparisons_2Einvert__comparison,inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2ELESS))) = fo__c_2EternaryComparisons_2EGREATER )
& ( surj__ty_2EternaryComparisons_2Eordering(ap(c_2EternaryComparisons_2Einvert__comparison,inj__ty_2EternaryComparisons_2Eordering(fo__c_2EternaryComparisons_2EEQUAL))) = fo__c_2EternaryComparisons_2EEQUAL ) ) ).
tff(conj_thm_2EternaryComparisons_2Einvert__eq__EQUAL,axiom,
! [V0x: tp__ty_2EternaryComparisons_2Eordering] :
( ( surj__ty_2EternaryComparisons_2Eordering(ap(c_2EternaryComparisons_2Einvert__comparison,inj__ty_2EternaryComparisons_2Eordering(V0x))) = fo__c_2EternaryComparisons_2EEQUAL )
<=> ( V0x = fo__c_2EternaryComparisons_2EEQUAL ) ) ).
%------------------------------------------------------------------------------