ITP001 Axioms: ITP057+5.ax
%------------------------------------------------------------------------------
% File : ITP057+5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : wellorder+2.ax [Gau20]
% : HL4057+5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 170 ( 1 unt; 0 def)
% Number of atoms : 1079 ( 117 equ)
% Maximal formula atoms : 24 ( 6 avg)
% Number of connectives : 920 ( 11 ~; 13 |; 125 &)
% ( 43 <=>; 728 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 9 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of predicates : 5 ( 3 usr; 1 prp; 0-2 aty)
% Number of functors : 97 ( 97 usr; 10 con; 0-5 aty)
% Number of variables : 687 ( 654 !; 33 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(ne_ty_2Ewellorder_2Ewellorder,axiom,
! [A0] :
( ne(A0)
=> ne(ty_2Ewellorder_2Ewellorder(A0)) ) ).
fof(mem_c_2Ewellorder_2EADD1,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ewellorder_2EADD1(A_27a),arr(A_27a,arr(ty_2Ewellorder_2Ewellorder(A_27a),ty_2Ewellorder_2Ewellorder(A_27a)))) ) ).
fof(mem_c_2Ewellorder_2EChain,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ewellorder_2EChain(A_27a),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Ewellorder_2EelsOf,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ewellorder_2EelsOf(A_27a),arr(ty_2Ewellorder_2Ewellorder(A_27a),arr(A_27a,bool))) ) ).
fof(mem_c_2Ewellorder_2Efinite,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ewellorder_2Efinite(A_27a),arr(ty_2Ewellorder_2Ewellorder(A_27a),bool)) ) ).
fof(mem_c_2Ewellorder_2Efl,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ewellorder_2Efl(A_27a),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),arr(A_27a,bool))) ) ).
fof(mem_c_2Ewellorder_2EfromNatWO,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ewellorder_2EfromNatWO(A_27a),arr(ty_2Enum_2Enum,ty_2Ewellorder_2Ewellorder(ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a)))) ) ).
fof(mem_c_2Ewellorder_2Eiseg,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ewellorder_2Eiseg(A_27a),arr(ty_2Ewellorder_2Ewellorder(A_27a),arr(A_27a,arr(A_27a,bool)))) ) ).
fof(mem_c_2Ewellorder_2Eorderiso,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Ewellorder_2Eorderiso(A_27a,A_27b),arr(ty_2Ewellorder_2Ewellorder(A_27a),arr(ty_2Ewellorder_2Ewellorder(A_27b),bool))) ) ) ).
fof(mem_c_2Ewellorder_2Eorderlt,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Ewellorder_2Eorderlt(A_27a,A_27b),arr(ty_2Ewellorder_2Ewellorder(A_27a),arr(ty_2Ewellorder_2Ewellorder(A_27b),bool))) ) ) ).
fof(mem_c_2Ewellorder_2Eposet,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ewellorder_2Eposet(A_27a),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),bool)) ) ).
fof(mem_c_2Ewellorder_2Eremove,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ewellorder_2Eremove(A_27a),arr(A_27a,arr(ty_2Ewellorder_2Ewellorder(A_27a),ty_2Ewellorder_2Ewellorder(A_27a)))) ) ).
fof(mem_c_2Ewellorder_2EwZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ewellorder_2EwZERO(A_27a),ty_2Ewellorder_2Ewellorder(A_27a)) ) ).
fof(mem_c_2Ewellorder_2Ewellfounded,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ewellorder_2Ewellfounded(A_27a),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),bool)) ) ).
fof(mem_c_2Ewellorder_2Ewellorder,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ewellorder_2Ewellorder(A_27a),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),bool)) ) ).
fof(mem_c_2Ewellorder_2Ewellorder__ABS,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ewellorder_2Ewellorder__ABS(A_27a),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),ty_2Ewellorder_2Ewellorder(A_27a))) ) ).
fof(mem_c_2Ewellorder_2Ewellorder__REP,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ewellorder_2Ewellorder__REP(A_27a),arr(ty_2Ewellorder_2Ewellorder(A_27a),arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))) ) ).
fof(mem_c_2Ewellorder_2Ewleast,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ewellorder_2Ewleast(A_27a),arr(ty_2Ewellorder_2Ewellorder(A_27a),arr(arr(A_27a,bool),ty_2Eoption_2Eoption(A_27a)))) ) ).
fof(mem_c_2Ewellorder_2Ewo2wo,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Ewellorder_2Ewo2wo(A_27a,A_27b),arr(ty_2Ewellorder_2Ewellorder(A_27a),arr(ty_2Ewellorder_2Ewellorder(A_27b),arr(A_27a,ty_2Eoption_2Eoption(A_27b))))) ) ) ).
fof(mem_c_2Ewellorder_2Ewobound,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ewellorder_2Ewobound(A_27a),arr(A_27a,arr(ty_2Ewellorder_2Ewellorder(A_27a),ty_2Ewellorder_2Ewellorder(A_27a)))) ) ).
fof(ax_thm_2Ewellorder_2Ewellfounded__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ( p(ap(c_2Ewellorder_2Ewellfounded(A_27a),V0R))
<=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ? [V2w] :
( mem(V2w,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V2w),V1s)) )
=> ? [V3min] :
( mem(V3min,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V3min),V1s))
& ! [V4w] :
( mem(V4w,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V4w),V3min)),V0R))
=> ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V4w),V1s)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Ewellfounded__WF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ( p(ap(c_2Ewellorder_2Ewellfounded(A_27a),V0R))
<=> p(ap(c_2Erelation_2EWF(A_27a),ap(c_2Epair_2ECURRY(A_27a,A_27a,bool),V0R))) ) ) ) ).
fof(ax_thm_2Ewellorder_2Ewellorder__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ( p(ap(c_2Ewellorder_2Ewellorder(A_27a),V0R))
<=> ( p(ap(c_2Ewellorder_2Ewellfounded(A_27a),ap(c_2Eset__relation_2Estrict(A_27a),V0R)))
& p(ap(ap(c_2Eset__relation_2Elinear__order(A_27a),V0R),ap(ap(c_2Epred__set_2EUNION(A_27a),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),V0R)),ap(c_2Eset__relation_2Erange(A_27a,A_27a),V0R))))
& p(ap(ap(c_2Eset__relation_2Ereflexive(A_27a),V0R),ap(ap(c_2Epred__set_2EUNION(A_27a),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),V0R)),ap(c_2Eset__relation_2Erange(A_27a,A_27a),V0R)))) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Ewellorder__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> p(ap(c_2Ewellorder_2Ewellorder(A_27a),c_2Epred__set_2EEMPTY(ty_2Epair_2Eprod(A_27a,A_27a)))) ) ).
fof(conj_thm_2Ewellorder_2Ewellorder__SING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ( p(ap(c_2Ewellorder_2Ewellorder(A_27a),ap(ap(c_2Epred__set_2EINSERT(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V1y)),c_2Epred__set_2EEMPTY(ty_2Epair_2Eprod(A_27a,A_27a)))))
<=> V0x = V1y ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Errestrict__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Eset__relation_2Errestrict(A_27a),V0r),V1s)),V0r)) ) ) ) ).
fof(conj_thm_2Ewellorder_2Ewellfounded__subset,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r0] :
( mem(V0r0,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1r] :
( mem(V1r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ( ( p(ap(c_2Ewellorder_2Ewellfounded(A_27a),V1r))
& p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Epair_2Eprod(A_27a,A_27a)),V0r0),V1r)) )
=> p(ap(c_2Ewellorder_2Ewellfounded(A_27a),V0r0)) ) ) ) ) ).
fof(ax_thm_2Ewellorder_2Ewellorder__TY__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ? [V0rep] :
( mem(V0rep,arr(ty_2Ewellorder_2Ewellorder(A_27a),arr(ty_2Epair_2Eprod(A_27a,A_27a),bool)))
& p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),ty_2Ewellorder_2Ewellorder(A_27a)),c_2Ewellorder_2Ewellorder(A_27a)),V0rep)) ) ) ).
fof(ax_thm_2Ewellorder_2Ewellorder__ABSREP,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0a] :
( mem(V0a,ty_2Ewellorder_2Ewellorder(A_27a))
=> ap(c_2Ewellorder_2Ewellorder__ABS(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V0a)) = V0a )
& ! [V1r] :
( mem(V1r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ( p(ap(c_2Ewellorder_2Ewellorder(A_27a),V1r))
<=> ap(c_2Ewellorder_2Ewellorder__REP(A_27a),ap(c_2Ewellorder_2Ewellorder__ABS(A_27a),V1r)) = V1r ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EmkWO__destWO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Ewellorder_2Ewellorder(A_27a))
=> ap(c_2Ewellorder_2Ewellorder__ABS(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V0a)) = V0a ) ) ).
fof(conj_thm_2Ewellorder_2EdestWO__mkWO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ( p(ap(c_2Ewellorder_2Ewellorder(A_27a),V0r))
=> ap(c_2Ewellorder_2Ewellorder__REP(A_27a),ap(c_2Ewellorder_2Ewellorder__ABS(A_27a),V0r)) = V0r ) ) ) ).
fof(ax_thm_2Ewellorder_2EelsOf__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ap(c_2Ewellorder_2EelsOf(A_27a),V0w) = ap(ap(c_2Epred__set_2EUNION(A_27a),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V0w))),ap(c_2Eset__relation_2Erange(A_27a,A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V0w))) ) ) ).
fof(conj_thm_2Ewellorder_2EWIN__elsOf,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ! [V2w] :
( mem(V2w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V1y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V2w))))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Ewellorder_2EelsOf(A_27a),V2w)))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V1y),ap(c_2Ewellorder_2EelsOf(A_27a),V2w))) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EWLE__elsOf,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ! [V2w] :
( mem(V2w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V1y)),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V2w)))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Ewellorder_2EelsOf(A_27a),V2w)))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V1y),ap(c_2Ewellorder_2EelsOf(A_27a),V2w))) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EWIN__trichotomy,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),ap(c_2Ewellorder_2EelsOf(A_27a),V0w)))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V2y),ap(c_2Ewellorder_2EelsOf(A_27a),V0w))) )
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1x),V2y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V0w))))
| V1x = V2y
| p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2y),V1x)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V0w)))) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EWIN__REFL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1w] :
( mem(V1w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V0x)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V1w))))
<=> $false ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EWLE__TRANS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ! [V2w] :
( mem(V2w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V3z] :
( mem(V3z,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V1y)),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V2w)))
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1y),V3z)),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V2w))) )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V3z)),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V2w))) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EWLE__ANTISYM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ! [V2w] :
( mem(V2w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V1y)),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V2w)))
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1y),V0x)),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V2w))) )
=> V0x = V1y ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EWIN__WLE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ! [V2w] :
( mem(V2w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V1y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V2w))))
=> p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V1y)),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V2w))) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EelsOf__WLE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1w] :
( mem(V1w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Ewellorder_2EelsOf(A_27a),V1w)))
<=> p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V0x)),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V1w))) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Etransitive__strict,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ( ( p(ap(c_2Eset__relation_2Etransitive(A_27a),V0r))
& p(ap(c_2Eset__relation_2Eantisym(A_27a),V0r)) )
=> p(ap(c_2Eset__relation_2Etransitive(A_27a),ap(c_2Eset__relation_2Estrict(A_27a),V0r))) ) ) ) ).
fof(conj_thm_2Ewellorder_2EWIN__TRANS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ! [V2w] :
( mem(V2w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V3z] :
( mem(V3z,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V1y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V2w))))
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1y),V3z)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V2w)))) )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V3z)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V2w)))) ) ) ) ) ) ) ).
fof(lameq_f1018,axiom,
! [A_27a,V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1p] : ap(f1018(A_27a,V0w),V1p) = ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),V1p),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V0w))) ) ).
fof(conj_thm_2Ewellorder_2EWIN__WF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> p(ap(c_2Ewellorder_2Ewellfounded(A_27a),f1018(A_27a,V0w))) ) ) ).
fof(lameq_f1019,axiom,
! [A_27a,V1x] :
( mem(V1x,A_27a)
=> ! [V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V2y] : ap(f1019(A_27a,V1x,V0w),V2y) = ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1x),V2y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V0w))) ) ) ).
fof(lameq_f1020,axiom,
! [A_27a,V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1x] : ap(f1020(A_27a,V0w),V1x) = f1019(A_27a,V1x,V0w) ) ).
fof(conj_thm_2Ewellorder_2EWIN__WF2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> p(ap(c_2Erelation_2EWF(A_27a),f1020(A_27a,V0w))) ) ) ).
fof(lameq_f1021,axiom,
! [A_27a,V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] : ap(f1021(A_27a,V0w,V1x),V2y) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V2y),ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2y),V1x)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V0w)))) ) ) ).
fof(ax_thm_2Ewellorder_2Eiseg__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ap(ap(c_2Ewellorder_2Eiseg(A_27a),V0w),V1x) = ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f1021(A_27a,V0w,V1x)) ) ) ) ).
fof(conj_thm_2Ewellorder_2Estrict__subset,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r1] :
( mem(V0r1,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1r2] :
( mem(V1r2,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Epair_2Eprod(A_27a,A_27a)),V0r1),V1r2))
=> p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Epair_2Eprod(A_27a,A_27a)),ap(c_2Eset__relation_2Estrict(A_27a),V0r1)),ap(c_2Eset__relation_2Estrict(A_27a),V1r2))) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Etransitive__rrestrict,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Eset__relation_2Etransitive(A_27a),V0r))
=> p(ap(c_2Eset__relation_2Etransitive(A_27a),ap(ap(c_2Eset__relation_2Errestrict(A_27a),V0r),V1s))) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Eantisym__rrestrict,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Eset__relation_2Eantisym(A_27a),V0r))
=> p(ap(c_2Eset__relation_2Eantisym(A_27a),ap(ap(c_2Eset__relation_2Errestrict(A_27a),V0r),V1s))) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Elinear__order__rrestrict,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Eset__relation_2Elinear__order(A_27a),V0r),ap(ap(c_2Epred__set_2EUNION(A_27a),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),V0r)),ap(c_2Eset__relation_2Erange(A_27a,A_27a),V0r))))
=> p(ap(ap(c_2Eset__relation_2Elinear__order(A_27a),ap(ap(c_2Eset__relation_2Errestrict(A_27a),V0r),V1s)),ap(ap(c_2Epred__set_2EUNION(A_27a),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),ap(ap(c_2Eset__relation_2Errestrict(A_27a),V0r),V1s))),ap(c_2Eset__relation_2Erange(A_27a,A_27a),ap(ap(c_2Eset__relation_2Errestrict(A_27a),V0r),V1s))))) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Ereflexive__rrestrict,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Eset__relation_2Ereflexive(A_27a),V0r),ap(ap(c_2Epred__set_2EUNION(A_27a),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),V0r)),ap(c_2Eset__relation_2Erange(A_27a,A_27a),V0r))))
=> p(ap(ap(c_2Eset__relation_2Ereflexive(A_27a),ap(ap(c_2Eset__relation_2Errestrict(A_27a),V0r),V1s)),ap(ap(c_2Epred__set_2EUNION(A_27a),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),ap(ap(c_2Eset__relation_2Errestrict(A_27a),V0r),V1s))),ap(c_2Eset__relation_2Erange(A_27a,A_27a),ap(ap(c_2Eset__relation_2Errestrict(A_27a),V0r),V1s))))) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Ewellorder__rrestrict,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> p(ap(c_2Ewellorder_2Ewellorder(A_27a),ap(ap(c_2Eset__relation_2Errestrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V0w)),V1s))) ) ) ) ).
fof(ax_thm_2Ewellorder_2Ewobound__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1w] :
( mem(V1w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ap(ap(c_2Ewellorder_2Ewobound(A_27a),V0x),V1w) = ap(c_2Ewellorder_2Ewellorder__ABS(A_27a),ap(ap(c_2Eset__relation_2Errestrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V1w)),ap(ap(c_2Ewellorder_2Eiseg(A_27a),V1w),V0x))) ) ) ) ).
fof(conj_thm_2Ewellorder_2EWIN__wobound,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ! [V2z] :
( mem(V2z,A_27a)
=> ! [V3w] :
( mem(V3w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V1y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),ap(ap(c_2Ewellorder_2Ewobound(A_27a),V2z),V3w)))))
<=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V2z)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V3w))))
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1y),V2z)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V3w))))
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V1y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V3w)))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EWLE__wobound,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ! [V2z] :
( mem(V2z,A_27a)
=> ! [V3w] :
( mem(V3w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V1y)),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),ap(ap(c_2Ewellorder_2Ewobound(A_27a),V2z),V3w))))
<=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V2z)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V3w))))
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1y),V2z)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V3w))))
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V1y)),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V3w))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Ewellorder__cases,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ? [V1s] :
( mem(V1s,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
& p(ap(c_2Ewellorder_2Ewellorder(A_27a),V1s))
& V0w = ap(c_2Ewellorder_2Ewellorder__ABS(A_27a),V1s) ) ) ) ).
fof(conj_thm_2Ewellorder_2EWEXTENSION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0w1] :
( mem(V0w1,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27a))
=> ( V0w1 = V1w2
<=> ! [V2a] :
( mem(V2a,A_27a)
=> ! [V3b] :
( mem(V3b,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2a),V3b)),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V0w1)))
<=> p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2a),V3b)),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V1w2))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Ewobound2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,A_27a)
=> ! [V1b] :
( mem(V1b,A_27a)
=> ! [V2w] :
( mem(V2w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0a),V1b)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V2w))))
=> ap(ap(c_2Ewellorder_2Ewobound(A_27a),V0a),ap(ap(c_2Ewellorder_2Ewobound(A_27a),V1b),V2w)) = ap(ap(c_2Ewellorder_2Ewobound(A_27a),V0a),V2w) ) ) ) ) ) ).
fof(lameq_f1022,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1i] :
( mem(V1i,ty_2Enum_2Enum)
=> ! [V2j] : ap(f1022(V0n,V1i),V2j) = ap(ap(c_2Epair_2E_2C(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum),bool),ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),V1i),V2j)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Earithmetic_2E_3C_3D,V1i),V2j)),ap(ap(c_2Eprim__rec_2E_3C,V2j),V0n))) ) ) ).
fof(lameq_f1023,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1i] : ap(f1023(V0n),V1i) = f1022(V0n,V1i) ) ).
fof(conj_thm_2Ewellorder_2Ewellorder__fromNat,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> p(ap(c_2Ewellorder_2Ewellorder(ty_2Enum_2Enum),ap(c_2Epred__set_2EGSPEC(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum),ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum)),ap(c_2Epair_2EUNCURRY(ty_2Enum_2Enum,ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum),bool)),f1023(V0n))))) ) ).
fof(conj_thm_2Ewellorder_2EINJ__preserves__transitive,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ( ( p(ap(c_2Eset__relation_2Etransitive(A_27a),V0r))
& p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V1f),ap(ap(c_2Epred__set_2EUNION(A_27a),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),V0r)),ap(c_2Eset__relation_2Erange(A_27a,A_27a),V0r))),V2t)) )
=> p(ap(c_2Eset__relation_2Etransitive(A_27b),ap(ap(c_2Epred__set_2EIMAGE(ty_2Epair_2Eprod(A_27a,A_27a),ty_2Epair_2Eprod(A_27b,A_27b)),ap(ap(c_2Epair_2E_23_23(A_27a,A_27a,A_27b,A_27b),V1f),V1f)),V0r))) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EINJ__preserves__antisym,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ( ( p(ap(c_2Eset__relation_2Eantisym(A_27a),V0r))
& p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V1f),ap(ap(c_2Epred__set_2EUNION(A_27a),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),V0r)),ap(c_2Eset__relation_2Erange(A_27a,A_27a),V0r))),V2t)) )
=> p(ap(c_2Eset__relation_2Eantisym(A_27b),ap(ap(c_2Epred__set_2EIMAGE(ty_2Epair_2Eprod(A_27a,A_27a),ty_2Epair_2Eprod(A_27b,A_27b)),ap(ap(c_2Epair_2E_23_23(A_27a,A_27a,A_27b,A_27b),V1f),V1f)),V0r))) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EINJ__preserves__linear__order,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ( ( p(ap(ap(c_2Eset__relation_2Elinear__order(A_27a),V0r),ap(ap(c_2Epred__set_2EUNION(A_27a),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),V0r)),ap(c_2Eset__relation_2Erange(A_27a,A_27a),V0r))))
& p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V1f),ap(ap(c_2Epred__set_2EUNION(A_27a),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),V0r)),ap(c_2Eset__relation_2Erange(A_27a,A_27a),V0r))),V2t)) )
=> p(ap(ap(c_2Eset__relation_2Elinear__order(A_27b),ap(ap(c_2Epred__set_2EIMAGE(ty_2Epair_2Eprod(A_27a,A_27a),ty_2Epair_2Eprod(A_27b,A_27b)),ap(ap(c_2Epair_2E_23_23(A_27a,A_27a,A_27b,A_27b),V1f),V1f)),V0r)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f),ap(ap(c_2Epred__set_2EUNION(A_27a),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),V0r)),ap(c_2Eset__relation_2Erange(A_27a,A_27a),V0r))))) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Edomain__IMAGE__ff,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0f] :
( mem(V0f,arr(A_27c,A_27a))
=> ! [V1g] :
( mem(V1g,arr(A_27d,A_27b))
=> ! [V2r] :
( mem(V2r,arr(ty_2Epair_2Eprod(A_27c,A_27d),bool))
=> ap(c_2Eset__relation_2Edomain(A_27a,A_27b),ap(ap(c_2Epred__set_2EIMAGE(ty_2Epair_2Eprod(A_27c,A_27d),ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2E_23_23(A_27c,A_27d,A_27a,A_27b),V0f),V1g)),V2r)) = ap(ap(c_2Epred__set_2EIMAGE(A_27c,A_27a),V0f),ap(c_2Eset__relation_2Edomain(A_27c,A_27d),V2r)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Erange__IMAGE__ff,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0f] :
( mem(V0f,arr(A_27c,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27d,A_27a))
=> ! [V2r] :
( mem(V2r,arr(ty_2Epair_2Eprod(A_27c,A_27d),bool))
=> ap(c_2Eset__relation_2Erange(A_27a,A_27b),ap(ap(c_2Epred__set_2EIMAGE(ty_2Epair_2Eprod(A_27c,A_27d),ty_2Epair_2Eprod(A_27b,A_27a)),ap(ap(c_2Epair_2E_23_23(A_27c,A_27d,A_27b,A_27a),V0f),V1g)),V2r)) = ap(ap(c_2Epred__set_2EIMAGE(A_27d,A_27a),V1g),ap(c_2Eset__relation_2Erange(A_27d,A_27c),V2r)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EINJ__preserves__wellorder,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ( ( p(ap(c_2Ewellorder_2Ewellorder(A_27a),V0r))
& p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V1f),ap(ap(c_2Epred__set_2EUNION(A_27a),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),V0r)),ap(c_2Eset__relation_2Erange(A_27a,A_27a),V0r))),V2t)) )
=> p(ap(c_2Ewellorder_2Ewellorder(A_27b),ap(ap(c_2Epred__set_2EIMAGE(ty_2Epair_2Eprod(A_27a,A_27a),ty_2Epair_2Eprod(A_27b,A_27b)),ap(ap(c_2Epair_2E_23_23(A_27a,A_27a,A_27b,A_27b),V1f),V1f)),V0r))) ) ) ) ) ) ) ).
fof(lameq_f1024,axiom,
! [A_27a,V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1i] :
( mem(V1i,ty_2Enum_2Enum)
=> ! [V2j] : ap(f1024(A_27a,V0n,V1i),V2j) = ap(ap(c_2Epair_2E_2C(ty_2Epair_2Eprod(ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a),ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a)),bool),ap(ap(c_2Epair_2E_2C(ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a),ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a)),ap(c_2Esum_2EINL(ty_2Enum_2Enum,A_27a),V1i)),ap(c_2Esum_2EINL(ty_2Enum_2Enum,A_27a),V2j))),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Earithmetic_2E_3C_3D,V1i),V2j)),ap(ap(c_2Eprim__rec_2E_3C,V2j),V0n))) ) ) ).
fof(lameq_f1025,axiom,
! [A_27a,V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1i] : ap(f1025(A_27a,V0n),V1i) = f1024(A_27a,V0n,V1i) ) ).
fof(conj_thm_2Ewellorder_2Ewellorder__fromNat__SUM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> p(ap(c_2Ewellorder_2Ewellorder(ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a)),ap(c_2Epred__set_2EGSPEC(ty_2Epair_2Eprod(ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a),ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a)),ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum)),ap(c_2Epair_2EUNCURRY(ty_2Enum_2Enum,ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epair_2Eprod(ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a),ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a)),bool)),f1025(A_27a,V0n))))) ) ) ).
fof(ax_thm_2Ewellorder_2EfromNatWO__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(c_2Ewellorder_2EfromNatWO(A_27a),V0n) = ap(c_2Ewellorder_2Ewellorder__ABS(ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a)),ap(c_2Epred__set_2EGSPEC(ty_2Epair_2Eprod(ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a),ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a)),ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum)),ap(c_2Epair_2EUNCURRY(ty_2Enum_2Enum,ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Epair_2Eprod(ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a),ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a)),bool)),f1025(A_27a,V0n)))) ) ) ).
fof(conj_thm_2Ewellorder_2EfromNatWO__11,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0i] :
( mem(V0i,ty_2Enum_2Enum)
=> ! [V1j] :
( mem(V1j,ty_2Enum_2Enum)
=> ( ap(c_2Ewellorder_2EfromNatWO(A_27a),V0i) = ap(c_2Ewellorder_2EfromNatWO(A_27a),V1j)
<=> V0i = V1j ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EelsOf__fromNatWO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(c_2Ewellorder_2EelsOf(ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a)),ap(c_2Ewellorder_2EfromNatWO(A_27a),V0n)) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a)),c_2Esum_2EINL(ty_2Enum_2Enum,A_27a)),ap(c_2Epred__set_2Ecount,V0n)) ) ) ).
fof(conj_thm_2Ewellorder_2EWLE__WIN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ! [V2w] :
( mem(V2w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V1y)),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V2w)))
=> ( V0x = V1y
| p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V1y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V2w)))) ) ) ) ) ) ) ).
fof(lameq_f1026,axiom,
! [A_27a,V1w] :
( mem(V1w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V2y] : ap(f1026(A_27a,V1w,V0x),V2y) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V2y),ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2y),V0x)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V1w)))) ) ) ).
fof(conj_thm_2Ewellorder_2EelsOf__wobound,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1w] :
( mem(V1w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ap(c_2Ewellorder_2EelsOf(A_27a),ap(ap(c_2Ewellorder_2Ewobound(A_27a),V0x),V1w)) = ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f1026(A_27a,V1w,V0x)) ) ) ) ).
fof(ax_thm_2Ewellorder_2Eorderiso__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0w1] :
( mem(V0w1,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ( p(ap(ap(c_2Ewellorder_2Eorderiso(A_27a,A_27b),V0w1),V1w2))
<=> ? [V2f] :
( mem(V2f,arr(A_27a,A_27b))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(c_2Ewellorder_2EelsOf(A_27a),V0w1)))
=> p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V2f,V3x)),ap(c_2Ewellorder_2EelsOf(A_27b),V1w2))) ) )
& ! [V4x1] :
( mem(V4x1,A_27a)
=> ! [V5x2] :
( mem(V5x2,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x1),ap(c_2Ewellorder_2EelsOf(A_27a),V0w1)))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V5x2),ap(c_2Ewellorder_2EelsOf(A_27a),V0w1))) )
=> ( ap(V2f,V4x1) = ap(V2f,V5x2)
<=> V4x1 = V5x2 ) ) ) )
& ! [V6y] :
( mem(V6y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V6y),ap(c_2Ewellorder_2EelsOf(A_27b),V1w2)))
=> ? [V7x] :
( mem(V7x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V7x),ap(c_2Ewellorder_2EelsOf(A_27a),V0w1)))
& ap(V2f,V7x) = V6y ) ) )
& ! [V8x] :
( mem(V8x,A_27a)
=> ! [V9y] :
( mem(V9y,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V8x),V9y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V0w1))))
=> p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27b,A_27b)),ap(ap(c_2Epair_2E_2C(A_27b,A_27b),ap(V2f,V8x)),ap(V2f,V9y))),ap(c_2Eset__relation_2Estrict(A_27b),ap(c_2Ewellorder_2Ewellorder__REP(A_27b),V1w2)))) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Eorderiso__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0w1] :
( mem(V0w1,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ( p(ap(ap(c_2Ewellorder_2Eorderiso(A_27a,A_27b),V0w1),V1w2))
<=> ? [V2f] :
( mem(V2f,arr(A_27a,A_27b))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V2f),ap(c_2Ewellorder_2EelsOf(A_27a),V0w1)),ap(c_2Ewellorder_2EelsOf(A_27b),V1w2)))
& ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( mem(V4y,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V3x),V4y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V0w1))))
=> p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27b,A_27b)),ap(ap(c_2Epair_2E_2C(A_27b,A_27b),ap(V2f,V3x)),ap(V2f,V4y))),ap(c_2Eset__relation_2Estrict(A_27b),ap(c_2Ewellorder_2Ewellorder__REP(A_27b),V1w2)))) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Eorderiso__REFL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> p(ap(ap(c_2Ewellorder_2Eorderiso(A_27a,A_27a),V0w),V0w)) ) ) ).
fof(conj_thm_2Ewellorder_2Eorderiso__SYM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0w1] :
( mem(V0w1,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ( p(ap(ap(c_2Ewellorder_2Eorderiso(A_27a,A_27b),V0w1),V1w2))
=> p(ap(ap(c_2Ewellorder_2Eorderiso(A_27b,A_27a),V1w2),V0w1)) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Eorderiso__TRANS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0w1] :
( mem(V0w1,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ! [V2w3] :
( mem(V2w3,ty_2Ewellorder_2Ewellorder(A_27c))
=> ( ( p(ap(ap(c_2Ewellorder_2Eorderiso(A_27a,A_27b),V0w1),V1w2))
& p(ap(ap(c_2Ewellorder_2Eorderiso(A_27b,A_27c),V1w2),V2w3)) )
=> p(ap(ap(c_2Ewellorder_2Eorderiso(A_27a,A_27c),V0w1),V2w3)) ) ) ) ) ) ) ) ).
fof(ax_thm_2Ewellorder_2Eorderlt__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0w1] :
( mem(V0w1,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ( p(ap(ap(c_2Ewellorder_2Eorderlt(A_27a,A_27b),V0w1),V1w2))
<=> ? [V2x] :
( mem(V2x,A_27b)
& p(ap(ap(c_2Ebool_2EIN(A_27b),V2x),ap(c_2Ewellorder_2EelsOf(A_27b),V1w2)))
& p(ap(ap(c_2Ewellorder_2Eorderiso(A_27a,A_27b),V0w1),ap(ap(c_2Ewellorder_2Ewobound(A_27b),V2x),V1w2))) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Eorderlt__REFL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ( p(ap(ap(c_2Ewellorder_2Eorderlt(A_27a,A_27a),V0w),V0w))
<=> $false ) ) ) ).
fof(conj_thm_2Ewellorder_2Ewobounds__preserve__bijections,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1w1] :
( mem(V1w1,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V2w2] :
( mem(V2w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ! [V3x] :
( mem(V3x,A_27a)
=> ( ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),ap(c_2Ewellorder_2EelsOf(A_27a),V1w1)),ap(c_2Ewellorder_2EelsOf(A_27b),V2w2)))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(c_2Ewellorder_2EelsOf(A_27a),V1w1)))
& ! [V4x] :
( mem(V4x,A_27a)
=> ! [V5y] :
( mem(V5y,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V4x),V5y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V1w1))))
=> p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27b,A_27b)),ap(ap(c_2Epair_2E_2C(A_27b,A_27b),ap(V0f,V4x)),ap(V0f,V5y))),ap(c_2Eset__relation_2Estrict(A_27b),ap(c_2Ewellorder_2Ewellorder__REP(A_27b),V2w2)))) ) ) ) )
=> p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),ap(c_2Ewellorder_2EelsOf(A_27a),ap(ap(c_2Ewellorder_2Ewobound(A_27a),V3x),V1w1))),ap(c_2Ewellorder_2EelsOf(A_27b),ap(ap(c_2Ewellorder_2Ewobound(A_27b),ap(V0f,V3x)),V2w2)))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Eorderlt__TRANS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0w1] :
( mem(V0w1,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ! [V2w3] :
( mem(V2w3,ty_2Ewellorder_2Ewellorder(A_27c))
=> ( ( p(ap(ap(c_2Ewellorder_2Eorderlt(A_27a,A_27b),V0w1),V1w2))
& p(ap(ap(c_2Ewellorder_2Eorderlt(A_27b,A_27c),V1w2),V2w3)) )
=> p(ap(ap(c_2Ewellorder_2Eorderlt(A_27a,A_27c),V0w1),V2w3)) ) ) ) ) ) ) ) ).
fof(lameq_f1027,axiom,
! [A_27a,V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ! [V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V3y] : ap(f1027(A_27a,V1s,V2x,V0w),V3y) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V3y),ap(c_2Ewellorder_2EelsOf(A_27a),V0w))),ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Ebool_2E_7E,ap(ap(c_2Ebool_2EIN(A_27a),V3y),V1s))),ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(A_27a),V2x),V3y))))),ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2x),V3y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V0w)))) ) ) ) ).
fof(lameq_f1028,axiom,
! [A_27a,V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V2x] : ap(f1028(A_27a,V1s,V0w),V2x) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Ewellorder_2EelsOf(A_27a),V0w))),ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Ebool_2E_7E,ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))),ap(c_2Ebool_2E_21(A_27a),f1027(A_27a,V1s,V2x,V0w)))) ) ) ).
fof(ax_thm_2Ewellorder_2Ewleast__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ap(ap(c_2Ewellorder_2Ewleast(A_27a),V0w),V1s) = ap(c_2Eoption_2Esome(A_27a),f1028(A_27a,V1s,V0w)) ) ) ) ).
fof(lameq_f1029,axiom,
! [A_27a,V2x] :
( mem(V2x,A_27a)
=> ! [V0w1] :
( mem(V0w1,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V3y] : ap(f1029(A_27a,V2x,V0w1),V3y) = ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2x),V3y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V0w1))) ) ) ).
fof(lameq_f1030,axiom,
! [A_27a,V0w1] :
( mem(V0w1,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V2x] : ap(f1030(A_27a,V0w1),V2x) = f1029(A_27a,V2x,V0w1) ) ).
fof(lameq_f1031,axiom,
! [A_27b,V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ! [V7s1] : ap(f1031(A_27b,V1w2),V7s1) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eoption_2Eoption(A_27b)),ap(ap(c_2Emin_2E_3D(arr(A_27b,bool)),V7s1),ap(c_2Ewellorder_2EelsOf(A_27b),V1w2))),c_2Eoption_2ENONE(A_27b)),ap(ap(c_2Ewellorder_2Ewleast(A_27b),V1w2),V7s1)) ) ).
fof(lameq_f1032,axiom,
! [A_27b,V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ! [V6s0] : ap(f1032(A_27b,V1w2),V6s0) = ap(ap(c_2Ebool_2ELET(arr(A_27b,bool),ty_2Eoption_2Eoption(A_27b)),f1031(A_27b,V1w2)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Eoption_2Eoption(A_27b),A_27b),c_2Eoption_2ETHE(A_27b)),ap(ap(c_2Epred__set_2EDELETE(ty_2Eoption_2Eoption(A_27b)),V6s0),c_2Eoption_2ENONE(A_27b)))) ) ).
fof(lameq_f1033,axiom,
! [A_27b,A_27a,V4f] :
( mem(V4f,arr(A_27a,ty_2Eoption_2Eoption(A_27b)))
=> ! [V0w1] :
( mem(V0w1,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ! [V5x] : ap(f1033(A_27b,A_27a,V4f,V0w1,V1w2),V5x) = ap(ap(c_2Ebool_2ELET(arr(ty_2Eoption_2Eoption(A_27b),bool),ty_2Eoption_2Eoption(A_27b)),f1032(A_27b,V1w2)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eoption_2Eoption(A_27b)),V4f),ap(ap(c_2Ewellorder_2Eiseg(A_27a),V0w1),V5x))) ) ) ) ).
fof(lameq_f1034,axiom,
! [A_27b,A_27a,V0w1] :
( mem(V0w1,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ! [V4f] : ap(f1034(A_27b,A_27a,V0w1,V1w2),V4f) = f1033(A_27b,A_27a,V4f,V0w1,V1w2) ) ) ).
fof(ax_thm_2Ewellorder_2Ewo2wo__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0w1] :
( mem(V0w1,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ap(ap(c_2Ewellorder_2Ewo2wo(A_27a,A_27b),V0w1),V1w2) = ap(ap(c_2Erelation_2EWFREC(A_27a,ty_2Eoption_2Eoption(A_27b)),f1030(A_27a,V0w1)),f1034(A_27b,A_27a,V0w1,V1w2)) ) ) ) ) ).
fof(lameq_f1035,axiom,
! [A_27b,V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ! [V4s1] : ap(f1035(A_27b,V1w2),V4s1) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eoption_2Eoption(A_27b)),ap(ap(c_2Emin_2E_3D(arr(A_27b,bool)),V4s1),ap(c_2Ewellorder_2EelsOf(A_27b),V1w2))),c_2Eoption_2ENONE(A_27b)),ap(ap(c_2Ewellorder_2Ewleast(A_27b),V1w2),V4s1)) ) ).
fof(lameq_f1036,axiom,
! [A_27b,V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ! [V3s0] : ap(f1036(A_27b,V1w2),V3s0) = ap(ap(c_2Ebool_2ELET(arr(A_27b,bool),ty_2Eoption_2Eoption(A_27b)),f1035(A_27b,V1w2)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Eoption_2Eoption(A_27b),A_27b),c_2Eoption_2ETHE(A_27b)),ap(ap(c_2Epred__set_2EDELETE(ty_2Eoption_2Eoption(A_27b)),V3s0),c_2Eoption_2ENONE(A_27b)))) ) ).
fof(conj_thm_2Ewellorder_2Ewo2wo__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ap(ap(ap(c_2Ewellorder_2Ewo2wo(A_27a,A_27b),V0w),V1w2),V2x) = ap(ap(c_2Ebool_2ELET(arr(ty_2Eoption_2Eoption(A_27b),bool),ty_2Eoption_2Eoption(A_27b)),f1036(A_27b,V1w2)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eoption_2Eoption(A_27b)),ap(ap(c_2Ewellorder_2Ewo2wo(A_27a,A_27b),V0w),V1w2)),ap(ap(c_2Ewellorder_2Eiseg(A_27a),V0w),V2x))) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Ewleast__IN__wo,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ( ap(ap(c_2Ewellorder_2Ewleast(A_27a),V0w),V1s) = ap(c_2Eoption_2ESOME(A_27a),V2x)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Ewellorder_2EelsOf(A_27a),V0w)))
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
& ! [V3y] :
( mem(V3y,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3y),ap(c_2Ewellorder_2EelsOf(A_27a),V0w)))
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3y),V1s))
& V2x != V3y )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2x),V3y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V0w)))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Ewleast__EQ__NONE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ap(ap(c_2Ewellorder_2Ewleast(A_27a),V0w),V1s) = c_2Eoption_2ENONE(A_27a)
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Ewellorder_2EelsOf(A_27a),V0w)),V1s)) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Ewo2wo__IN__w2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0w1] :
( mem(V0w1,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( mem(V3y,A_27b)
=> ( ap(ap(ap(c_2Ewellorder_2Ewo2wo(A_27a,A_27b),V0w1),V1w2),V2x) = ap(c_2Eoption_2ESOME(A_27b),V3y)
=> p(ap(ap(c_2Ebool_2EIN(A_27b),V3y),ap(c_2Ewellorder_2EelsOf(A_27b),V1w2))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EIMAGE__wo2wo__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0w1] :
( mem(V0w1,ty_2Ewellorder_2Ewellorder(A_27b))
=> ! [V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V2x] :
( mem(V2x,A_27b)
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Eoption_2Eoption(A_27a),A_27a),c_2Eoption_2ETHE(A_27a)),ap(ap(c_2Epred__set_2EDELETE(ty_2Eoption_2Eoption(A_27a)),ap(ap(c_2Epred__set_2EIMAGE(A_27b,ty_2Eoption_2Eoption(A_27a)),ap(ap(c_2Ewellorder_2Ewo2wo(A_27b,A_27a),V0w1),V1w2)),ap(ap(c_2Ewellorder_2Eiseg(A_27b),V0w1),V2x))),c_2Eoption_2ENONE(A_27a)))),ap(c_2Ewellorder_2EelsOf(A_27a),V1w2))) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Ewo2wo__EQ__NONE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0w1] :
( mem(V0w1,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ( ap(ap(ap(c_2Ewellorder_2Ewo2wo(A_27a,A_27b),V0w1),V1w2),V2x) = c_2Eoption_2ENONE(A_27b)
=> ! [V3y] :
( mem(V3y,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2x),V3y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V0w1))))
=> ap(ap(ap(c_2Ewellorder_2Ewo2wo(A_27a,A_27b),V0w1),V1w2),V3y) = c_2Eoption_2ENONE(A_27b) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Ewo2wo__EQ__SOME__downwards,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0w1] :
( mem(V0w1,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( mem(V3y,A_27b)
=> ( ap(ap(ap(c_2Ewellorder_2Ewo2wo(A_27a,A_27b),V0w1),V1w2),V2x) = ap(c_2Eoption_2ESOME(A_27b),V3y)
=> ! [V4x0] :
( mem(V4x0,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V4x0),V2x)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V0w1))))
=> ? [V5y0] :
( mem(V5y0,A_27b)
& ap(ap(ap(c_2Ewellorder_2Ewo2wo(A_27a,A_27b),V0w1),V1w2),V4x0) = ap(c_2Eoption_2ESOME(A_27b),V5y0) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Emono__woseg,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x1] :
( mem(V0x1,A_27a)
=> ! [V1x2] :
( mem(V1x2,A_27a)
=> ! [V2w1] :
( mem(V2w1,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V3w2] :
( mem(V3w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x1),V1x2)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V2w1))))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),ap(ap(c_2Epred__set_2EIMAGE(ty_2Eoption_2Eoption(A_27b),A_27b),c_2Eoption_2ETHE(A_27b)),ap(ap(c_2Epred__set_2EDELETE(ty_2Eoption_2Eoption(A_27b)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eoption_2Eoption(A_27b)),ap(ap(c_2Ewellorder_2Ewo2wo(A_27a,A_27b),V2w1),V3w2)),ap(ap(c_2Ewellorder_2Eiseg(A_27a),V2w1),V0x1))),c_2Eoption_2ENONE(A_27b)))),ap(ap(c_2Epred__set_2EIMAGE(ty_2Eoption_2Eoption(A_27b),A_27b),c_2Eoption_2ETHE(A_27b)),ap(ap(c_2Epred__set_2EDELETE(ty_2Eoption_2Eoption(A_27b)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eoption_2Eoption(A_27b)),ap(ap(c_2Ewellorder_2Ewo2wo(A_27a,A_27b),V2w1),V3w2)),ap(ap(c_2Ewellorder_2Eiseg(A_27a),V2w1),V1x2))),c_2Eoption_2ENONE(A_27b))))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Ewo2wo__11,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x1] :
( mem(V0x1,A_27a)
=> ! [V1w1] :
( mem(V1w1,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V2x2] :
( mem(V2x2,A_27a)
=> ! [V3w2] :
( mem(V3w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ! [V4y] :
( mem(V4y,A_27b)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x1),ap(c_2Ewellorder_2EelsOf(A_27a),V1w1)))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V2x2),ap(c_2Ewellorder_2EelsOf(A_27a),V1w1)))
& ap(ap(ap(c_2Ewellorder_2Ewo2wo(A_27a,A_27b),V1w1),V3w2),V0x1) = ap(c_2Eoption_2ESOME(A_27b),V4y)
& ap(ap(ap(c_2Ewellorder_2Ewo2wo(A_27a,A_27b),V1w1),V3w2),V2x2) = ap(c_2Eoption_2ESOME(A_27b),V4y) )
=> V0x1 = V2x2 ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Ewleast__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1s1] :
( mem(V1s1,arr(A_27a,bool))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3s2] :
( mem(V3s2,arr(A_27a,bool))
=> ! [V4y] :
( mem(V4y,A_27a)
=> ( ( ap(ap(c_2Ewellorder_2Ewleast(A_27a),V0w),V1s1) = ap(c_2Eoption_2ESOME(A_27a),V2x)
& ap(ap(c_2Ewellorder_2Ewleast(A_27a),V0w),V3s2) = ap(c_2Eoption_2ESOME(A_27a),V4y)
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s1),V3s2)) )
=> ( V2x = V4y
| p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2x),V4y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V0w)))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Ewo2wo__mono,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0w1] :
( mem(V0w1,ty_2Ewellorder_2Ewellorder(A_27b))
=> ! [V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V2x0] :
( mem(V2x0,A_27b)
=> ! [V3y0] :
( mem(V3y0,A_27a)
=> ! [V4x] :
( mem(V4x,A_27b)
=> ! [V5y] :
( mem(V5y,A_27a)
=> ( ( ap(ap(ap(c_2Ewellorder_2Ewo2wo(A_27b,A_27a),V0w1),V1w2),V2x0) = ap(c_2Eoption_2ESOME(A_27a),V3y0)
& ap(ap(ap(c_2Ewellorder_2Ewo2wo(A_27b,A_27a),V0w1),V1w2),V4x) = ap(c_2Eoption_2ESOME(A_27a),V5y)
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27b,A_27b)),ap(ap(c_2Epair_2E_2C(A_27b,A_27b),V2x0),V4x)),ap(c_2Eset__relation_2Estrict(A_27b),ap(c_2Ewellorder_2Ewellorder__REP(A_27b),V0w1)))) )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V3y0),V5y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V1w2)))) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Ewo2wo__ONTO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1w1] :
( mem(V1w1,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V2w2] :
( mem(V2w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ! [V3y] :
( mem(V3y,A_27b)
=> ! [V4y0] :
( mem(V4y0,A_27b)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Ewellorder_2EelsOf(A_27a),V1w1)))
& ap(ap(ap(c_2Ewellorder_2Ewo2wo(A_27a,A_27b),V1w1),V2w2),V0x) = ap(c_2Eoption_2ESOME(A_27b),V3y)
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27b,A_27b)),ap(ap(c_2Epair_2E_2C(A_27b,A_27b),V4y0),V3y)),ap(c_2Eset__relation_2Estrict(A_27b),ap(c_2Ewellorder_2Ewellorder__REP(A_27b),V2w2)))) )
=> ? [V5x0] :
( mem(V5x0,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V5x0),ap(c_2Ewellorder_2EelsOf(A_27a),V1w1)))
& ap(ap(ap(c_2Ewellorder_2Ewo2wo(A_27a,A_27b),V1w1),V2w2),V5x0) = ap(c_2Eoption_2ESOME(A_27b),V4y0) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Ewo2wo__EQ__NONE__woseg,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0w1] :
( mem(V0w1,ty_2Ewellorder_2Ewellorder(A_27b))
=> ! [V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V2x] :
( mem(V2x,A_27b)
=> ( ap(ap(ap(c_2Ewellorder_2Ewo2wo(A_27b,A_27a),V0w1),V1w2),V2x) = c_2Eoption_2ENONE(A_27a)
=> ap(c_2Ewellorder_2EelsOf(A_27a),V1w2) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Eoption_2Eoption(A_27a),A_27a),c_2Eoption_2ETHE(A_27a)),ap(ap(c_2Epred__set_2EDELETE(ty_2Eoption_2Eoption(A_27a)),ap(ap(c_2Epred__set_2EIMAGE(A_27b,ty_2Eoption_2Eoption(A_27a)),ap(ap(c_2Ewellorder_2Ewo2wo(A_27b,A_27a),V0w1),V1w2)),ap(ap(c_2Ewellorder_2Eiseg(A_27b),V0w1),V2x))),c_2Eoption_2ENONE(A_27a))) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Eorderlt__trichotomy,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0w1] :
( mem(V0w1,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ( p(ap(ap(c_2Ewellorder_2Eorderlt(A_27a,A_27b),V0w1),V1w2))
| p(ap(ap(c_2Ewellorder_2Eorderiso(A_27a,A_27b),V0w1),V1w2))
| p(ap(ap(c_2Ewellorder_2Eorderlt(A_27b,A_27a),V1w2),V0w1)) ) ) ) ) ) ).
fof(ax_thm_2Ewellorder_2EwZERO__def,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Ewellorder_2EwZERO(A_27a) = ap(c_2Ewellorder_2Ewellorder__ABS(A_27a),c_2Epred__set_2EEMPTY(ty_2Epair_2Eprod(A_27a,A_27a))) ) ).
fof(conj_thm_2Ewellorder_2EelsOf__wZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> ap(c_2Ewellorder_2EelsOf(A_27a),c_2Ewellorder_2EwZERO(A_27a)) = c_2Epred__set_2EEMPTY(A_27a) ) ).
fof(conj_thm_2Ewellorder_2EWIN__wZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V1y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),c_2Ewellorder_2EwZERO(A_27a)))))
<=> $false ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EWLE__wZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V1y)),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),c_2Ewellorder_2EwZERO(A_27a))))
<=> $false ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Eorderiso__wZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27b))
=> ( p(ap(ap(c_2Ewellorder_2Eorderiso(A_27a,A_27b),c_2Ewellorder_2EwZERO(A_27a)),V0w))
<=> V0w = c_2Ewellorder_2EwZERO(A_27b) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EelsOf__EQ__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ( ap(c_2Ewellorder_2EelsOf(A_27a),V0w) = c_2Epred__set_2EEMPTY(A_27a)
<=> V0w = c_2Ewellorder_2EwZERO(A_27a) ) ) ) ).
fof(conj_thm_2Ewellorder_2ELT__wZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ( p(ap(ap(c_2Ewellorder_2Eorderlt(A_27a,A_27b),V0w),c_2Ewellorder_2EwZERO(A_27b)))
<=> $false ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Eorderlt__WF,axiom,
! [A_27a] :
( ne(A_27a)
=> p(ap(c_2Erelation_2EWF(ty_2Ewellorder_2Ewellorder(A_27a)),c_2Ewellorder_2Eorderlt(A_27a,A_27a))) ) ).
fof(conj_thm_2Ewellorder_2Eorderlt__orderiso,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0x0] :
( mem(V0x0,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1y0] :
( mem(V1y0,ty_2Ewellorder_2Ewellorder(A_27b))
=> ! [V2a0] :
( mem(V2a0,ty_2Ewellorder_2Ewellorder(A_27c))
=> ! [V3b0] :
( mem(V3b0,ty_2Ewellorder_2Ewellorder(A_27d))
=> ( ( p(ap(ap(c_2Ewellorder_2Eorderiso(A_27a,A_27b),V0x0),V1y0))
& p(ap(ap(c_2Ewellorder_2Eorderiso(A_27c,A_27d),V2a0),V3b0)) )
=> ( p(ap(ap(c_2Ewellorder_2Eorderlt(A_27a,A_27c),V0x0),V2a0))
<=> p(ap(ap(c_2Ewellorder_2Eorderlt(A_27b,A_27d),V1y0),V3b0)) ) ) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Ewellorder_2Efinite__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0w] :
( mem(V0w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ( p(ap(c_2Ewellorder_2Efinite(A_27a),V0w))
<=> p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Ewellorder_2EelsOf(A_27a),V0w))) ) ) ) ).
fof(conj_thm_2Ewellorder_2Efinite__iso,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0w1] :
( mem(V0w1,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V1w2] :
( mem(V1w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ( p(ap(ap(c_2Ewellorder_2Eorderiso(A_27a,A_27b),V0w1),V1w2))
=> ( p(ap(c_2Ewellorder_2Efinite(A_27a),V0w1))
<=> p(ap(c_2Ewellorder_2Efinite(A_27b),V1w2)) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Efinite__wZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> p(ap(c_2Ewellorder_2Efinite(A_27a),c_2Ewellorder_2EwZERO(A_27a))) ) ).
fof(conj_thm_2Ewellorder_2Eorderiso__unique,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f1] :
( mem(V0f1,arr(A_27a,A_27b))
=> ! [V1w1] :
( mem(V1w1,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V2w2] :
( mem(V2w2,ty_2Ewellorder_2Ewellorder(A_27b))
=> ! [V3f2] :
( mem(V3f2,arr(A_27a,A_27b))
=> ( ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f1),ap(c_2Ewellorder_2EelsOf(A_27a),V1w1)),ap(c_2Ewellorder_2EelsOf(A_27b),V2w2)))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V3f2),ap(c_2Ewellorder_2EelsOf(A_27a),V1w1)),ap(c_2Ewellorder_2EelsOf(A_27b),V2w2)))
& ! [V4x] :
( mem(V4x,A_27a)
=> ! [V5y] :
( mem(V5y,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V4x),V5y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V1w1))))
=> p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27b,A_27b)),ap(ap(c_2Epair_2E_2C(A_27b,A_27b),ap(V0f1,V4x)),ap(V0f1,V5y))),ap(c_2Eset__relation_2Estrict(A_27b),ap(c_2Ewellorder_2Ewellorder__REP(A_27b),V2w2)))) ) ) )
& ! [V6x] :
( mem(V6x,A_27a)
=> ! [V7y] :
( mem(V7y,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V6x),V7y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V1w1))))
=> p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27b,A_27b)),ap(ap(c_2Epair_2E_2C(A_27b,A_27b),ap(V3f2,V6x)),ap(V3f2,V7y))),ap(c_2Eset__relation_2Estrict(A_27b),ap(c_2Ewellorder_2Ewellorder__REP(A_27b),V2w2)))) ) ) ) )
=> ! [V8x] :
( mem(V8x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V8x),ap(c_2Ewellorder_2EelsOf(A_27a),V1w1)))
=> ap(V0f1,V8x) = ap(V3f2,V8x) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Eseteq__wlog,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,arr(A_27a,bool)))
=> ! [V1f] :
( mem(V1f,arr(A_27a,arr(A_27b,bool)))
=> ( ( ! [V2a] :
( mem(V2a,A_27a)
=> ! [V3b] :
( mem(V3b,A_27a)
=> ( p(ap(ap(V0P,V2a),V3b))
=> p(ap(ap(V0P,V3b),V2a)) ) ) )
& ! [V4x] :
( mem(V4x,A_27b)
=> ! [V5a] :
( mem(V5a,A_27a)
=> ! [V6b] :
( mem(V6b,A_27a)
=> ( ( p(ap(ap(V0P,V5a),V6b))
& p(ap(ap(c_2Ebool_2EIN(A_27b),V4x),ap(V1f,V5a))) )
=> p(ap(ap(c_2Ebool_2EIN(A_27b),V4x),ap(V1f,V6b))) ) ) ) ) )
=> ! [V7a] :
( mem(V7a,A_27a)
=> ! [V8b] :
( mem(V8b,A_27a)
=> ( p(ap(ap(V0P,V7a),V8b))
=> ap(V1f,V7a) = ap(V1f,V8b) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Ewo__INDUCTION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1w] :
( mem(V1w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ( ! [V3y] :
( mem(V3y,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V3y),V2x)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V1w))))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3y),ap(c_2Ewellorder_2EelsOf(A_27a),V1w)))
=> p(ap(V0P,V3y)) ) ) )
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Ewellorder_2EelsOf(A_27a),V1w)))
=> p(ap(V0P,V2x)) ) ) )
=> ! [V4x] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),ap(c_2Ewellorder_2EelsOf(A_27a),V1w)))
=> p(ap(V0P,V4x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EFORALL__NUM,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Enum_2Enum,bool))
=> ( ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> p(ap(V0P,V1n)) )
<=> ( p(ap(V0P,c_2Enum_2E0))
& ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(V0P,ap(c_2Enum_2ESUC,V2n))) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Estrict__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r1] :
( mem(V0r1,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1r2] :
( mem(V1r2,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ap(c_2Eset__relation_2Estrict(A_27a),ap(ap(c_2Epred__set_2EUNION(ty_2Epair_2Eprod(A_27a,A_27a)),V0r1),V1r2)) = ap(ap(c_2Epred__set_2EUNION(ty_2Epair_2Eprod(A_27a,A_27a)),ap(c_2Eset__relation_2Estrict(A_27a),V0r1)),ap(c_2Eset__relation_2Estrict(A_27a),V1r2)) ) ) ) ).
fof(conj_thm_2Ewellorder_2EWLE__WIN__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ! [V2w] :
( mem(V2w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V1y)),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V2w)))
<=> ( ( V0x = V1y
& p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Ewellorder_2EelsOf(A_27a),V2w))) )
| p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V1y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V2w)))) ) ) ) ) ) ) ).
fof(lameq_f1037,axiom,
! [A_27a,V0e] :
( mem(V0e,A_27a)
=> ! [V1w] :
( mem(V1w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] : ap(f1037(A_27a,V0e,V1w,V2x),V3y) = ap(ap(c_2Epair_2E_2C(ty_2Epair_2Eprod(A_27a,A_27a),bool),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2x),V3y)),ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(A_27a),V2x),V0e))),ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(A_27a),V3y),V0e))),ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2x),V3y)),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V1w))))) ) ) ) ).
fof(lameq_f1038,axiom,
! [A_27a,V0e] :
( mem(V0e,A_27a)
=> ! [V1w] :
( mem(V1w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V2x] : ap(f1038(A_27a,V0e,V1w),V2x) = f1037(A_27a,V0e,V1w,V2x) ) ) ).
fof(ax_thm_2Ewellorder_2Eremove__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0e] :
( mem(V0e,A_27a)
=> ! [V1w] :
( mem(V1w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ap(ap(c_2Ewellorder_2Eremove(A_27a),V0e),V1w) = ap(c_2Ewellorder_2Ewellorder__ABS(A_27a),ap(c_2Epred__set_2EGSPEC(ty_2Epair_2Eprod(A_27a,A_27a),ty_2Epair_2Eprod(A_27a,A_27a)),ap(c_2Epair_2EUNCURRY(A_27a,A_27a,ty_2Epair_2Eprod(ty_2Epair_2Eprod(A_27a,A_27a),bool)),f1038(A_27a,V0e,V1w)))) ) ) ) ).
fof(conj_thm_2Ewellorder_2Ewellorder__remove,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0e] :
( mem(V0e,A_27a)
=> ! [V1w] :
( mem(V1w,ty_2Ewellorder_2Ewellorder(A_27a))
=> p(ap(c_2Ewellorder_2Ewellorder(A_27a),ap(c_2Epred__set_2EGSPEC(ty_2Epair_2Eprod(A_27a,A_27a),ty_2Epair_2Eprod(A_27a,A_27a)),ap(c_2Epair_2EUNCURRY(A_27a,A_27a,ty_2Epair_2Eprod(ty_2Epair_2Eprod(A_27a,A_27a),bool)),f1038(A_27a,V0e,V1w))))) ) ) ) ).
fof(conj_thm_2Ewellorder_2EelsOf__remove,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0e] :
( mem(V0e,A_27a)
=> ! [V1w] :
( mem(V1w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ap(c_2Ewellorder_2EelsOf(A_27a),ap(ap(c_2Ewellorder_2Eremove(A_27a),V0e),V1w)) = ap(ap(c_2Epred__set_2EDELETE(A_27a),ap(c_2Ewellorder_2EelsOf(A_27a),V1w)),V0e) ) ) ) ).
fof(conj_thm_2Ewellorder_2EWIN__remove,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ! [V2e] :
( mem(V2e,A_27a)
=> ! [V3w] :
( mem(V3w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V1y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),ap(ap(c_2Ewellorder_2Eremove(A_27a),V2e),V3w)))))
<=> ( V0x != V2e
& V1y != V2e
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V1y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V3w)))) ) ) ) ) ) ) ) ).
fof(lameq_f1039,axiom,
! [A_27a,V0e] :
( mem(V0e,A_27a)
=> ! [V1w] :
( mem(V1w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ! [V2x] : ap(f1039(A_27a,V0e,V1w),V2x) = ap(ap(c_2Epair_2E_2C(ty_2Epair_2Eprod(A_27a,A_27a),bool),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2x),V0e)),ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Ewellorder_2EelsOf(A_27a),V1w))) ) ) ).
fof(ax_thm_2Ewellorder_2EADD1__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0e] :
( mem(V0e,A_27a)
=> ! [V1w] :
( mem(V1w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ap(ap(c_2Ewellorder_2EADD1(A_27a),V0e),V1w) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Ewellorder_2Ewellorder(A_27a)),ap(ap(c_2Ebool_2EIN(A_27a),V0e),ap(c_2Ewellorder_2EelsOf(A_27a),V1w))),V1w),ap(c_2Ewellorder_2Ewellorder__ABS(A_27a),ap(ap(c_2Epred__set_2EUNION(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epred__set_2EUNION(ty_2Epair_2Eprod(A_27a,A_27a)),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V1w)),ap(c_2Epred__set_2EGSPEC(ty_2Epair_2Eprod(A_27a,A_27a),A_27a),f1039(A_27a,V0e,V1w)))),ap(ap(c_2Epred__set_2EINSERT(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0e),V0e)),c_2Epred__set_2EEMPTY(ty_2Epair_2Eprod(A_27a,A_27a)))))) ) ) ) ).
fof(conj_thm_2Ewellorder_2Ewellorder__ADD1,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0e] :
( mem(V0e,A_27a)
=> ! [V1w] :
( mem(V1w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ( ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V0e),ap(c_2Ewellorder_2EelsOf(A_27a),V1w)))
=> p(ap(c_2Ewellorder_2Ewellorder(A_27a),ap(ap(c_2Epred__set_2EUNION(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epred__set_2EUNION(ty_2Epair_2Eprod(A_27a,A_27a)),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V1w)),ap(c_2Epred__set_2EGSPEC(ty_2Epair_2Eprod(A_27a,A_27a),A_27a),f1039(A_27a,V0e,V1w)))),ap(ap(c_2Epred__set_2EINSERT(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0e),V0e)),c_2Epred__set_2EEMPTY(ty_2Epair_2Eprod(A_27a,A_27a)))))) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EelsOf__ADD1,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0e] :
( mem(V0e,A_27a)
=> ! [V1w] :
( mem(V1w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ap(c_2Ewellorder_2EelsOf(A_27a),ap(ap(c_2Ewellorder_2EADD1(A_27a),V0e),V1w)) = ap(ap(c_2Epred__set_2EINSERT(A_27a),V0e),ap(c_2Ewellorder_2EelsOf(A_27a),V1w)) ) ) ) ).
fof(conj_thm_2Ewellorder_2EWIN__ADD1,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ! [V2e] :
( mem(V2e,A_27a)
=> ! [V3w] :
( mem(V3w,ty_2Ewellorder_2Ewellorder(A_27a))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V1y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),ap(ap(c_2Ewellorder_2EADD1(A_27a),V2e),V3w)))))
<=> ( ( ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V2e),ap(c_2Ewellorder_2EelsOf(A_27a),V3w)))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Ewellorder_2EelsOf(A_27a),V3w)))
& V1y = V2e )
| p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V0x),V1y)),ap(c_2Eset__relation_2Estrict(A_27a),ap(c_2Ewellorder_2Ewellorder__REP(A_27a),V3w)))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EelsOf__cardeq__iso,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27b,A_27a))
=> ! [V1wo] :
( mem(V1wo,ty_2Ewellorder_2Ewellorder(A_27b))
=> ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27b,A_27a),V0f),ap(c_2Ewellorder_2EelsOf(A_27b),V1wo)),c_2Epred__set_2EUNIV(A_27a)))
=> ? [V2wo_27] :
( mem(V2wo_27,ty_2Ewellorder_2Ewellorder(A_27a))
& p(ap(ap(c_2Ewellorder_2Eorderiso(A_27b,A_27a),V1wo),V2wo_27)) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2Eallsets__wellorderable,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ? [V1wo] :
( mem(V1wo,ty_2Ewellorder_2Ewellorder(A_27a))
& ap(c_2Ewellorder_2EelsOf(A_27a),V1wo) = V0s ) ) ) ).
fof(conj_thm_2Ewellorder_2EWF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0_3C_3C] :
( mem(V0_3C_3C,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EWF(A_27a),V0_3C_3C))
<=> ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> ( ? [V2x] :
( mem(V2x,A_27a)
& p(ap(V1P,V2x)) )
=> ? [V3x] :
( mem(V3x,A_27a)
& p(ap(V1P,V3x))
& ! [V4y] :
( mem(V4y,A_27a)
=> ( p(ap(ap(V0_3C_3C,V4y),V3x))
=> ~ p(ap(V1P,V4y)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EWF__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0_3C_3C] :
( mem(V0_3C_3C,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EWF(A_27a),V0_3C_3C))
<=> ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> ( ? [V2x] :
( mem(V2x,A_27a)
& p(ap(V1P,V2x)) )
<=> ? [V3x] :
( mem(V3x,A_27a)
& p(ap(V1P,V3x))
& ! [V4y] :
( mem(V4y,A_27a)
=> ( p(ap(ap(V0_3C_3C,V4y),V3x))
=> ~ p(ap(V1P,V4y)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EWF__IND,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0_3C_3C] :
( mem(V0_3C_3C,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EWF(A_27a),V0_3C_3C))
<=> ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ( ! [V3y] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V0_3C_3C,V3y),V2x))
=> p(ap(V1P,V3y)) ) )
=> p(ap(V1P,V2x)) ) )
=> ! [V4x] :
( mem(V4x,A_27a)
=> p(ap(V1P,V4x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EWF__DCHAIN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0_3C_3C] :
( mem(V0_3C_3C,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EWF(A_27a),V0_3C_3C))
<=> ~ ? [V1s] :
( mem(V1s,arr(ty_2Enum_2Enum,A_27a))
& ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(V0_3C_3C,ap(V1s,ap(c_2Enum_2ESUC,V2n))),ap(V1s,V2n))) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EWF__UREC,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0_3C_3C] :
( mem(V0_3C_3C,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EWF(A_27a),V0_3C_3C))
=> ! [V1H] :
( mem(V1H,arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))
=> ( ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V3g] :
( mem(V3g,arr(A_27a,A_27b))
=> ! [V4x] :
( mem(V4x,A_27a)
=> ( ! [V5z] :
( mem(V5z,A_27a)
=> ( p(ap(ap(V0_3C_3C,V5z),V4x))
=> ap(V2f,V5z) = ap(V3g,V5z) ) )
=> ap(ap(V1H,V2f),V4x) = ap(ap(V1H,V3g),V4x) ) ) ) )
=> ! [V6f] :
( mem(V6f,arr(A_27a,A_27b))
=> ! [V7g] :
( mem(V7g,arr(A_27a,A_27b))
=> ( ( ! [V8x] :
( mem(V8x,A_27a)
=> ap(V6f,V8x) = ap(ap(V1H,V6f),V8x) )
& ! [V9x] :
( mem(V9x,A_27a)
=> ap(V7g,V9x) = ap(ap(V1H,V7g),V9x) ) )
=> V6f = V7g ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EWF__UREC__WF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0_3C_3C] :
( mem(V0_3C_3C,arr(A_27a,arr(A_27a,bool)))
=> ( ! [V1H] :
( mem(V1H,arr(arr(A_27a,bool),arr(A_27a,bool)))
=> ( ! [V2f] :
( mem(V2f,arr(A_27a,bool))
=> ! [V3g] :
( mem(V3g,arr(A_27a,bool))
=> ! [V4x] :
( mem(V4x,A_27a)
=> ( ! [V5z] :
( mem(V5z,A_27a)
=> ( p(ap(ap(V0_3C_3C,V5z),V4x))
=> ( p(ap(V2f,V5z))
<=> p(ap(V3g,V5z)) ) ) )
=> ( p(ap(ap(V1H,V2f),V4x))
<=> p(ap(ap(V1H,V3g),V4x)) ) ) ) ) )
=> ! [V6f] :
( mem(V6f,arr(A_27a,bool))
=> ! [V7g] :
( mem(V7g,arr(A_27a,bool))
=> ( ( ! [V8x] :
( mem(V8x,A_27a)
=> ( p(ap(V6f,V8x))
<=> p(ap(ap(V1H,V6f),V8x)) ) )
& ! [V9x] :
( mem(V9x,A_27a)
=> ( p(ap(V7g,V9x))
<=> p(ap(ap(V1H,V7g),V9x)) ) ) )
=> V6f = V7g ) ) ) ) )
=> p(ap(c_2Erelation_2EWF(A_27a),V0_3C_3C)) ) ) ) ).
fof(conj_thm_2Ewellorder_2EWF__REC__INVARIANT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0_3C_3C] :
( mem(V0_3C_3C,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EWF(A_27a),V0_3C_3C))
=> ! [V1H] :
( mem(V1H,arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))
=> ! [V2S] :
( mem(V2S,arr(A_27a,arr(A_27b,bool)))
=> ( ! [V3f] :
( mem(V3f,arr(A_27a,A_27b))
=> ! [V4g] :
( mem(V4g,arr(A_27a,A_27b))
=> ! [V5x] :
( mem(V5x,A_27a)
=> ( ! [V6z] :
( mem(V6z,A_27a)
=> ( p(ap(ap(V0_3C_3C,V6z),V5x))
=> ( ap(V3f,V6z) = ap(V4g,V6z)
& p(ap(ap(V2S,V6z),ap(V3f,V6z))) ) ) )
=> ( ap(ap(V1H,V3f),V5x) = ap(ap(V1H,V4g),V5x)
& p(ap(ap(V2S,V5x),ap(ap(V1H,V3f),V5x))) ) ) ) ) )
=> ? [V7f] :
( mem(V7f,arr(A_27a,A_27b))
& ! [V8x] :
( mem(V8x,A_27a)
=> ap(V7f,V8x) = ap(ap(V1H,V7f),V8x) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EWF__REC,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0_3C_3C] :
( mem(V0_3C_3C,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Erelation_2EWF(A_27a),V0_3C_3C))
=> ! [V1H] :
( mem(V1H,arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))
=> ( ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V3g] :
( mem(V3g,arr(A_27a,A_27b))
=> ! [V4x] :
( mem(V4x,A_27a)
=> ( ! [V5z] :
( mem(V5z,A_27a)
=> ( p(ap(ap(V0_3C_3C,V5z),V4x))
=> ap(V2f,V5z) = ap(V3g,V5z) ) )
=> ap(ap(V1H,V2f),V4x) = ap(ap(V1H,V3g),V4x) ) ) ) )
=> ? [V6f] :
( mem(V6f,arr(A_27a,A_27b))
& ! [V7x] :
( mem(V7x,A_27a)
=> ap(V6f,V7x) = ap(ap(V1H,V6f),V7x) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EWF__num,axiom,
p(ap(c_2Erelation_2EWF(ty_2Enum_2Enum),c_2Eprim__rec_2E_3C)) ).
fof(conj_thm_2Ewellorder_2EWF__REC__num,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0H] :
( mem(V0H,arr(arr(ty_2Enum_2Enum,A_27a),arr(ty_2Enum_2Enum,A_27a)))
=> ( ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,A_27a))
=> ! [V2g] :
( mem(V2g,arr(ty_2Enum_2Enum,A_27a))
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( ! [V4m] :
( mem(V4m,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V4m),V3n))
=> ap(V1f,V4m) = ap(V2g,V4m) ) )
=> ap(ap(V0H,V1f),V3n) = ap(ap(V0H,V2g),V3n) ) ) ) )
=> ? [V5f] :
( mem(V5f,arr(ty_2Enum_2Enum,A_27a))
& ! [V6n] :
( mem(V6n,ty_2Enum_2Enum)
=> ap(V5f,V6n) = ap(ap(V0H,V5f),V6n) ) ) ) ) ) ).
fof(ax_thm_2Ewellorder_2Efl,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0l] :
( mem(V0l,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ( p(ap(ap(c_2Ewellorder_2Efl(A_27a),V0l),V1x))
<=> ? [V2y] :
( mem(V2y,A_27a)
& ( p(ap(V0l,ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1x),V2y)))
| p(ap(V0l,ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2y),V1x))) ) ) ) ) ) ) ).
fof(ax_thm_2Ewellorder_2Eposet,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0l] :
( mem(V0l,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ( p(ap(c_2Ewellorder_2Eposet(A_27a),V0l))
<=> ( ! [V1x] :
( mem(V1x,A_27a)
=> ( p(ap(ap(c_2Ewellorder_2Efl(A_27a),V0l),V1x))
=> p(ap(V0l,ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1x),V1x))) ) )
& ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( mem(V3y,A_27a)
=> ! [V4z] :
( mem(V4z,A_27a)
=> ( ( p(ap(V0l,ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2x),V3y)))
& p(ap(V0l,ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V3y),V4z))) )
=> p(ap(V0l,ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2x),V4z))) ) ) ) )
& ! [V5x] :
( mem(V5x,A_27a)
=> ! [V6y] :
( mem(V6y,A_27a)
=> ( ( p(ap(V0l,ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V5x),V6y)))
& p(ap(V0l,ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V6y),V5x))) )
=> V5x = V6y ) ) ) ) ) ) ) ).
fof(ax_thm_2Ewellorder_2Echain,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0l] :
( mem(V0l,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ewellorder_2EChain(A_27a),V0l),V1P))
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( mem(V3y,A_27a)
=> ( ( p(ap(V1P,V2x))
& p(ap(V1P,V3y)) )
=> ( p(ap(V0l,ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2x),V3y)))
| p(ap(V0l,ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V3y),V2x))) ) ) ) ) ) ) ) ) ).
fof(lameq_f1040,axiom,
! [A_27a,V0l] :
( mem(V0l,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V2x] : ap(f1040(A_27a,V0l),V2x) = ap(ap(c_2Ewellorder_2Efl(A_27a),V0l),V2x) ) ).
fof(conj_thm_2Ewellorder_2EZL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0l] :
( mem(V0l,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ( ( p(ap(c_2Ewellorder_2Eposet(A_27a),V0l))
& ? [V1x] :
( mem(V1x,A_27a)
& p(ap(f1040(A_27a,V0l),V1x)) )
& ! [V3P] :
( mem(V3P,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ewellorder_2EChain(A_27a),V0l),V3P))
=> ? [V4y] :
( mem(V4y,A_27a)
& p(ap(ap(c_2Ewellorder_2Efl(A_27a),V0l),V4y))
& ! [V5x] :
( mem(V5x,A_27a)
=> ( p(ap(V3P,V5x))
=> p(ap(V0l,ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V5x),V4y))) ) ) ) ) ) )
=> ? [V6y] :
( mem(V6y,A_27a)
& p(ap(ap(c_2Ewellorder_2Efl(A_27a),V0l),V6y))
& ! [V7x] :
( mem(V7x,A_27a)
=> ( p(ap(V0l,ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V6y),V7x)))
=> V6y = V7x ) ) ) ) ) ) ).
fof(lameq_f1041,axiom,
! [A_27a,V1x] :
( mem(V1x,arr(A_27a,bool))
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V2y] : ap(f1041(A_27a,V1x,V0P),V2y) = ap(ap(c_2Ebool_2E_2F_5C,ap(V0P,V1x)),ap(ap(c_2Ebool_2E_2F_5C,ap(V0P,V2y)),ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1x),V2y))) ) ) ).
fof(lameq_f1042,axiom,
! [A_27a,V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1x] : ap(f1042(A_27a,V0P),V1x) = f1041(A_27a,V1x,V0P) ) ).
fof(conj_thm_2Ewellorder_2EPOSET__RESTRICTED__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> p(ap(c_2Ewellorder_2Eposet(arr(A_27a,bool)),ap(c_2Epair_2EUNCURRY(arr(A_27a,bool),arr(A_27a,bool),bool),f1042(A_27a,V0P)))) ) ) ).
fof(conj_thm_2Ewellorder_2EFL__RESTRICTED__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ap(c_2Ewellorder_2Efl(arr(A_27a,bool)),ap(c_2Epair_2EUNCURRY(arr(A_27a,bool),arr(A_27a,bool),bool),f1042(A_27a,V0P))) = V0P ) ) ).
fof(conj_thm_2Ewellorder_2EZL__SUBSETS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ( ! [V1c] :
( mem(V1c,arr(arr(A_27a,bool),bool))
=> ( ( ! [V2x] :
( mem(V2x,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2x),V1c))
=> p(ap(V0P,V2x)) ) )
& ! [V3x] :
( mem(V3x,arr(A_27a,bool))
=> ! [V4y] :
( mem(V4y,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3x),V1c))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V4y),V1c)) )
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V3x),V4y))
| p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V4y),V3x)) ) ) ) ) )
=> ? [V5z] :
( mem(V5z,arr(A_27a,bool))
& p(ap(V0P,V5z))
& ! [V6x] :
( mem(V6x,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V6x),V1c))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V6x),V5z)) ) ) ) ) )
=> ? [V7a] :
( mem(V7a,arr(A_27a,bool))
& p(ap(V0P,V7a))
& ! [V8x] :
( mem(V8x,arr(A_27a,bool))
=> ( ( p(ap(V0P,V8x))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V7a),V8x)) )
=> V7a = V8x ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EZL__SUBSETS__BIGUNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ( ! [V1c] :
( mem(V1c,arr(arr(A_27a,bool),bool))
=> ( ( ! [V2x] :
( mem(V2x,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2x),V1c))
=> p(ap(V0P,V2x)) ) )
& ! [V3x] :
( mem(V3x,arr(A_27a,bool))
=> ! [V4y] :
( mem(V4y,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3x),V1c))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V4y),V1c)) )
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V3x),V4y))
| p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V4y),V3x)) ) ) ) ) )
=> p(ap(V0P,ap(c_2Epred__set_2EBIGUNION(A_27a),V1c))) ) )
=> ? [V5a] :
( mem(V5a,arr(A_27a,bool))
& p(ap(V0P,V5a))
& ! [V6x] :
( mem(V6x,arr(A_27a,bool))
=> ( ( p(ap(V0P,V6x))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V5a),V6x)) )
=> V5a = V6x ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EZL__SUBSETS__BIGUNION__NONEMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ( ( ? [V1x] :
( mem(V1x,arr(A_27a,bool))
& p(ap(V0P,V1x)) )
& ! [V2c] :
( mem(V2c,arr(arr(A_27a,bool),bool))
=> ( ( ? [V3x] :
( mem(V3x,arr(A_27a,bool))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3x),V2c)) )
& ! [V4x] :
( mem(V4x,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V4x),V2c))
=> p(ap(V0P,V4x)) ) )
& ! [V5x] :
( mem(V5x,arr(A_27a,bool))
=> ! [V6y] :
( mem(V6y,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V5x),V2c))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V6y),V2c)) )
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V5x),V6y))
| p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V6y),V5x)) ) ) ) ) )
=> p(ap(V0P,ap(c_2Epred__set_2EBIGUNION(A_27a),V2c))) ) ) )
=> ? [V7a] :
( mem(V7a,arr(A_27a,bool))
& p(ap(V0P,V7a))
& ! [V8x] :
( mem(V8x,arr(A_27a,bool))
=> ( ( p(ap(V0P,V8x))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V7a),V8x)) )
=> V7a = V8x ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EFLATTEN__LEMMA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0g] :
( mem(V0g,arr(A_27b,A_27a))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> ap(V0g,ap(V1f,V3x)) = V3x ) )
<=> ! [V4y] :
( mem(V4y,A_27b)
=> ! [V5x] :
( mem(V5x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V2s))
& V4y = ap(V1f,V5x) )
=> ap(V0g,V4y) = V5x ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ewellorder_2ETARSKI__SET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(arr(A_27a,bool),arr(A_27a,bool)))
=> ( ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V0f,V1s)),ap(V0f,V2t))) ) ) )
=> ? [V3s] :
( mem(V3s,arr(A_27a,bool))
& ap(V0f,V3s) = V3s ) ) ) ) ).
fof(conj_thm_2Ewellorder_2EINJECTIVE__LEFT__INVERSE__NONEMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ( ? [V3x] :
( mem(V3x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) )
=> ( ! [V4x] :
( mem(V4x,A_27a)
=> ! [V5y] :
( mem(V5y,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V1s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V5y),V1s))
& ap(V0f,V4x) = ap(V0f,V5y) )
=> V4x = V5y ) ) )
<=> ? [V6g] :
( mem(V6g,arr(A_27b,A_27a))
& ! [V7y] :
( mem(V7y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V7y),V2t))
=> p(ap(ap(c_2Ebool_2EIN(A_27a),ap(V6g,V7y)),V1s)) ) )
& ! [V8x] :
( mem(V8x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V8x),V1s))
=> ap(V6g,ap(V0f,V8x)) = V8x ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------