TPTP Problem File: ITP005+5.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP005+5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Eset__relation_2Erel__to__reln__inv.p, 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 : thm_2Eset__relation_2Erel__to__reln__inv.p [Gau20]
% : HL402001+5.p [TPAP]
% Status : Theorem
% Rating : 0.78 v8.1.0, 0.75 v7.5.0
% Syntax : Number of formulae : 3163 ( 220 unt; 0 def)
% Number of atoms : 16581 (2359 equ)
% Maximal formula atoms : 94 ( 5 avg)
% Number of connectives : 13787 ( 369 ~; 262 |;1754 &)
% ( 969 <=>;10433 =>; 0 <=; 0 <~>)
% Maximal formula depth : 38 ( 8 avg)
% Maximal term depth : 16 ( 2 avg)
% Number of predicates : 6 ( 3 usr; 2 prp; 0-2 aty)
% Number of functors : 753 ( 753 usr; 147 con; 0-7 aty)
% Number of variables : 10130 (9813 !; 317 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001+2.ax').
include('Axioms/ITP001/ITP002+5.ax').
include('Axioms/ITP001/ITP003+5.ax').
include('Axioms/ITP001/ITP004+5.ax').
include('Axioms/ITP001/ITP007+5.ax').
include('Axioms/ITP001/ITP006+5.ax').
include('Axioms/ITP001/ITP005+5.ax').
include('Axioms/ITP001/ITP008+5.ax').
include('Axioms/ITP001/ITP009+5.ax').
include('Axioms/ITP001/ITP010+5.ax').
include('Axioms/ITP001/ITP012+5.ax').
include('Axioms/ITP001/ITP011+5.ax').
include('Axioms/ITP001/ITP013+5.ax').
include('Axioms/ITP001/ITP014+5.ax').
include('Axioms/ITP001/ITP015+5.ax').
include('Axioms/ITP001/ITP017+5.ax').
include('Axioms/ITP001/ITP016+5.ax').
include('Axioms/ITP001/ITP019+5.ax').
include('Axioms/ITP001/ITP018+5.ax').
include('Axioms/ITP001/ITP021+5.ax').
include('Axioms/ITP001/ITP022+5.ax').
include('Axioms/ITP001/ITP020+5.ax').
include('Axioms/ITP001/ITP024+5.ax').
include('Axioms/ITP001/ITP023+5.ax').
include('Axioms/ITP001/ITP025+5.ax').
include('Axioms/ITP001/ITP026+5.ax').
include('Axioms/ITP001/ITP027+5.ax').
include('Axioms/ITP001/ITP028+5.ax').
include('Axioms/ITP001/ITP031+5.ax').
include('Axioms/ITP001/ITP029+5.ax').
%------------------------------------------------------------------------------
fof(mem_c_2Eset__relation_2ERREFL__EXP,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2ERREFL__EXP(A_27a),arr(arr(A_27a,arr(A_27a,bool)),arr(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool))))) ) ).
fof(mem_c_2Eset__relation_2ERRUNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2ERRUNIV(A_27a),arr(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool)))) ) ).
fof(mem_c_2Eset__relation_2Eacyclic,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2Eacyclic(A_27a),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),bool)) ) ).
fof(mem_c_2Eset__relation_2Eall__choices,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2Eall__choices(A_27a),arr(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Eset__relation_2Eantisym,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2Eantisym(A_27a),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),bool)) ) ).
fof(mem_c_2Eset__relation_2Echain,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2Echain(A_27a),arr(arr(A_27a,bool),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),bool))) ) ).
fof(mem_c_2Eset__relation_2Edomain,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Eset__relation_2Edomain(A_27a,A_27b),arr(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),arr(A_27a,bool))) ) ) ).
fof(mem_c_2Eset__relation_2Efchains,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2Efchains(A_27a),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Eset__relation_2Efinite__prefixes,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Eset__relation_2Efinite__prefixes(A_27a,A_27b),arr(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),arr(arr(A_27b,bool),bool))) ) ) ).
fof(mem_c_2Eset__relation_2Eget__min,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2Eget__min(A_27a),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(ty_2Epair_2Eprod(A_27a,A_27a),bool)),ty_2Eoption_2Eoption(A_27a)))) ) ).
fof(mem_c_2Eset__relation_2Eirreflexive,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2Eirreflexive(A_27a),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Eset__relation_2Elinear__order,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2Elinear__order(A_27a),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Eset__relation_2Emaximal__elements,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2Emaximal__elements(A_27a),arr(arr(A_27a,bool),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),arr(A_27a,bool)))) ) ).
fof(mem_c_2Eset__relation_2Eminimal__elements,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2Eminimal__elements(A_27a),arr(arr(A_27a,bool),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),arr(A_27a,bool)))) ) ).
fof(mem_c_2Eset__relation_2Enth__min,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2Enth__min(A_27a),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(ty_2Epair_2Eprod(A_27a,A_27a),bool)),arr(ty_2Enum_2Enum,ty_2Eoption_2Eoption(A_27a))))) ) ).
fof(mem_c_2Eset__relation_2Enum__order,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2Enum__order(A_27a),arr(arr(A_27a,ty_2Enum_2Enum),arr(arr(A_27a,bool),arr(ty_2Epair_2Eprod(A_27a,A_27a),bool)))) ) ).
fof(mem_c_2Eset__relation_2Epartial__order,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2Epartial__order(A_27a),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Eset__relation_2Eper,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2Eper(A_27a),arr(arr(A_27a,bool),arr(arr(arr(A_27a,bool),bool),bool))) ) ).
fof(mem_c_2Eset__relation_2Eper__restrict,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2Eper__restrict(A_27a),arr(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))) ) ).
fof(mem_c_2Eset__relation_2Erange,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Eset__relation_2Erange(A_27a,A_27b),arr(arr(ty_2Epair_2Eprod(A_27b,A_27a),bool),arr(A_27a,bool))) ) ) ).
fof(mem_c_2Eset__relation_2Ercomp,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> mem(c_2Eset__relation_2Ercomp(A_27a,A_27b,A_27c),arr(arr(ty_2Epair_2Eprod(A_27a,A_27c),bool),arr(arr(ty_2Epair_2Eprod(A_27c,A_27b),bool),arr(ty_2Epair_2Eprod(A_27a,A_27b),bool)))) ) ) ) ).
fof(mem_c_2Eset__relation_2Ereflexive,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2Ereflexive(A_27a),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Eset__relation_2Erel__to__reln,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Eset__relation_2Erel__to__reln(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))) ) ) ).
fof(mem_c_2Eset__relation_2Ereln__to__rel,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Eset__relation_2Ereln__to__rel(A_27a,A_27b),arr(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),arr(A_27a,arr(A_27b,bool)))) ) ) ).
fof(mem_c_2Eset__relation_2Errestrict,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2Errestrict(A_27a),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),arr(arr(A_27a,bool),arr(ty_2Epair_2Eprod(A_27a,A_27a),bool)))) ) ).
fof(mem_c_2Eset__relation_2Estrict,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2Estrict(A_27a),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))) ) ).
fof(mem_c_2Eset__relation_2Estrict__linear__order,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2Estrict__linear__order(A_27a),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Eset__relation_2Etc,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2Etc(A_27a),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))) ) ).
fof(mem_c_2Eset__relation_2Etransitive,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2Etransitive(A_27a),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),bool)) ) ).
fof(mem_c_2Eset__relation_2Euniv__reln,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eset__relation_2Euniv__reln(A_27a),arr(arr(A_27a,bool),arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))) ) ).
fof(mem_c_2Eset__relation_2Eupper__bounds,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Eset__relation_2Eupper__bounds(A_27a,A_27b),arr(arr(A_27b,bool),arr(arr(ty_2Epair_2Eprod(A_27b,A_27a),bool),arr(A_27a,bool)))) ) ) ).
fof(conj_thm_2Eset__relation_2Erextension,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ! [V1t] :
( mem(V1t,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ( V0s = V1t
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( mem(V3y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V2x),V3y)),V0s))
<=> p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V2x),V3y)),V1t)) ) ) ) ) ) ) ) ) ).
fof(lameq_f317,axiom,
! [A_27a,A_27b,V1x] :
( mem(V1x,A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ! [V2y] : ap(f317(A_27a,A_27b,V1x,V0r),V2y) = ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V1x),V2y)),V0r) ) ) ).
fof(lameq_f318,axiom,
! [A_27b,A_27a,V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ! [V1x] : ap(f318(A_27b,A_27a,V0r),V1x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V1x),ap(c_2Ebool_2E_3F(A_27b),f317(A_27a,A_27b,V1x,V0r))) ) ).
fof(ax_thm_2Eset__relation_2Edomain__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ap(c_2Eset__relation_2Edomain(A_27a,A_27b),V0r) = ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f318(A_27b,A_27a,V0r)) ) ) ) ).
fof(lameq_f319,axiom,
! [A_27a,A_27b,V1y] :
( mem(V1y,A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27b,A_27a),bool))
=> ! [V2x] : ap(f319(A_27a,A_27b,V1y,V0r),V2x) = ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27b,A_27a)),ap(ap(c_2Epair_2E_2C(A_27b,A_27a),V2x),V1y)),V0r) ) ) ).
fof(lameq_f320,axiom,
! [A_27b,A_27a,V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27b,A_27a),bool))
=> ! [V1y] : ap(f320(A_27b,A_27a,V0r),V1y) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V1y),ap(c_2Ebool_2E_3F(A_27b),f319(A_27a,A_27b,V1y,V0r))) ) ).
fof(ax_thm_2Eset__relation_2Erange__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27b,A_27a),bool))
=> ap(c_2Eset__relation_2Erange(A_27a,A_27b),V0r) = ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f320(A_27b,A_27a,V0r)) ) ) ) ).
fof(conj_thm_2Eset__relation_2Ein__domain,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1r] :
( mem(V1r,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Eset__relation_2Edomain(A_27a,A_27b),V1r)))
<=> ? [V2y] :
( mem(V2y,A_27b)
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V0x),V2y)),V1r)) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Ein__range,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0y] :
( mem(V0y,A_27a)
=> ! [V1r] :
( mem(V1r,arr(ty_2Epair_2Eprod(A_27b,A_27a),bool))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0y),ap(c_2Eset__relation_2Erange(A_27a,A_27b),V1r)))
<=> ? [V2x] :
( mem(V2x,A_27b)
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27b,A_27a)),ap(ap(c_2Epair_2E_2C(A_27b,A_27a),V2x),V0y)),V1r)) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Ein__dom__rg,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27b)
=> ! [V2r] :
( mem(V2r,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V0x),V1y)),V2r))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(c_2Eset__relation_2Edomain(A_27a,A_27b),V2r)))
& p(ap(ap(c_2Ebool_2EIN(A_27b),V1y),ap(c_2Eset__relation_2Erange(A_27b,A_27a),V2r))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Edomain__mono,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ! [V1r_27] :
( mem(V1r_27,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Epair_2Eprod(A_27a,A_27b)),V0r),V1r_27))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Eset__relation_2Edomain(A_27a,A_27b),V0r)),ap(c_2Eset__relation_2Edomain(A_27a,A_27b),V1r_27))) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Erange__mono,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ! [V1r_27] :
( mem(V1r_27,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Epair_2Eprod(A_27a,A_27b)),V0r),V1r_27))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),ap(c_2Eset__relation_2Erange(A_27b,A_27a),V0r)),ap(c_2Eset__relation_2Erange(A_27b,A_27a),V1r_27))) ) ) ) ) ) ).
fof(lameq_f321,axiom,
! [A_27a,V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] : ap(f321(A_27a,V1s,V0r,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(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2x),V3y)),V0r)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)),ap(ap(c_2Ebool_2EIN(A_27a),V3y),V1s)))) ) ) ) ).
fof(lameq_f322,axiom,
! [A_27a,V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V2x] : ap(f322(A_27a,V1s,V0r),V2x) = f321(A_27a,V1s,V0r,V2x) ) ) ).
fof(ax_thm_2Eset__relation_2Errestrict__def,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))
=> ap(ap(c_2Eset__relation_2Errestrict(A_27a),V0r),V1s) = 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)),f322(A_27a,V1s,V0r))) ) ) ) ).
fof(conj_thm_2Eset__relation_2Ein__rrestrict,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ! [V2r] :
( mem(V2r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V3s] :
( mem(V3s,arr(A_27a,bool))
=> ( 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(ap(c_2Eset__relation_2Errestrict(A_27a),V2r),V3s)))
<=> ( 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)),V2r))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V3s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V1y),V3s)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Ein__rrestrict__alt,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,ty_2Epair_2Eprod(A_27a,A_27a))
=> ! [V1r] :
( mem(V1r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),V0x),ap(ap(c_2Eset__relation_2Errestrict(A_27a),V1r),V2s)))
<=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),V0x),V1r))
& p(ap(ap(c_2Ebool_2EIN(A_27a),ap(c_2Epair_2EFST(A_27a,A_27a),V0x)),V2s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),ap(c_2Epair_2ESND(A_27a,A_27a),V0x)),V2s)) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_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_2Eset__relation_2Errestrict__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))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ap(ap(c_2Eset__relation_2Errestrict(A_27a),ap(ap(c_2Epred__set_2EUNION(ty_2Epair_2Eprod(A_27a,A_27a)),V0r1),V1r2)),V2s) = ap(ap(c_2Epred__set_2EUNION(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Eset__relation_2Errestrict(A_27a),V0r1),V2s)),ap(ap(c_2Eset__relation_2Errestrict(A_27a),V1r2),V2s)) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Errestrict__rrestrict,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1x] :
( mem(V1x,arr(A_27a,bool))
=> ! [V2y] :
( mem(V2y,arr(A_27a,bool))
=> ap(ap(c_2Eset__relation_2Errestrict(A_27a),ap(ap(c_2Eset__relation_2Errestrict(A_27a),V0r),V1x)),V2y) = ap(ap(c_2Eset__relation_2Errestrict(A_27a),V0r),ap(ap(c_2Epred__set_2EINTER(A_27a),V1x),V2y)) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Edomain__rrestrict__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(A_27a),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),ap(ap(c_2Eset__relation_2Errestrict(A_27a),V0r),V1s))),V1s)) ) ) ) ).
fof(conj_thm_2Eset__relation_2Erange__rrestrict__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(A_27a),ap(c_2Eset__relation_2Erange(A_27a,A_27a),ap(ap(c_2Eset__relation_2Errestrict(A_27a),V0r),V1s))),V1s)) ) ) ) ).
fof(lameq_f323,axiom,
! [A_27b,A_27a,A_27c,V2x] :
( mem(V2x,A_27a)
=> ! [V0r1] :
( mem(V0r1,arr(ty_2Epair_2Eprod(A_27a,A_27c),bool))
=> ! [V3y] :
( mem(V3y,A_27b)
=> ! [V1r2] :
( mem(V1r2,arr(ty_2Epair_2Eprod(A_27c,A_27b),bool))
=> ! [V4z] : ap(f323(A_27b,A_27a,A_27c,V2x,V0r1,V3y,V1r2),V4z) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27c)),ap(ap(c_2Epair_2E_2C(A_27a,A_27c),V2x),V4z)),V0r1)),ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27c,A_27b)),ap(ap(c_2Epair_2E_2C(A_27c,A_27b),V4z),V3y)),V1r2)) ) ) ) ) ).
fof(lameq_f324,axiom,
! [A_27a,A_27c,A_27b,V0r1] :
( mem(V0r1,arr(ty_2Epair_2Eprod(A_27a,A_27c),bool))
=> ! [V1r2] :
( mem(V1r2,arr(ty_2Epair_2Eprod(A_27c,A_27b),bool))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] : ap(f324(A_27a,A_27c,A_27b,V0r1,V1r2,V2x),V3y) = ap(ap(c_2Epair_2E_2C(ty_2Epair_2Eprod(A_27a,A_27b),bool),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V2x),V3y)),ap(c_2Ebool_2E_3F(A_27c),f323(A_27b,A_27a,A_27c,V2x,V0r1,V3y,V1r2))) ) ) ) ).
fof(lameq_f325,axiom,
! [A_27b,A_27c,A_27a,V0r1] :
( mem(V0r1,arr(ty_2Epair_2Eprod(A_27a,A_27c),bool))
=> ! [V1r2] :
( mem(V1r2,arr(ty_2Epair_2Eprod(A_27c,A_27b),bool))
=> ! [V2x] : ap(f325(A_27b,A_27c,A_27a,V0r1,V1r2),V2x) = f324(A_27a,A_27c,A_27b,V0r1,V1r2,V2x) ) ) ).
fof(ax_thm_2Eset__relation_2Ercomp__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0r1] :
( mem(V0r1,arr(ty_2Epair_2Eprod(A_27a,A_27c),bool))
=> ! [V1r2] :
( mem(V1r2,arr(ty_2Epair_2Eprod(A_27c,A_27b),bool))
=> ap(ap(c_2Eset__relation_2Ercomp(A_27a,A_27b,A_27c),V0r1),V1r2) = ap(c_2Epred__set_2EGSPEC(ty_2Epair_2Eprod(A_27a,A_27b),ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,ty_2Epair_2Eprod(ty_2Epair_2Eprod(A_27a,A_27b),bool)),f325(A_27b,A_27c,A_27a,V0r1,V1r2))) ) ) ) ) ) ).
fof(lameq_f326,axiom,
! [A_27a,V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] : ap(f326(A_27a,V0r,V1x),V2y) = ap(ap(c_2Epair_2E_2C(ty_2Epair_2Eprod(A_27a,A_27a),bool),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1x),V2y)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1x),V2y)),V0r)),ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(A_27a),V1x),V2y)))) ) ) ).
fof(lameq_f327,axiom,
! [A_27a,V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1x] : ap(f327(A_27a,V0r),V1x) = f326(A_27a,V0r,V1x) ) ).
fof(ax_thm_2Eset__relation_2Estrict__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ap(c_2Eset__relation_2Estrict(A_27a),V0r) = 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)),f327(A_27a,V0r))) ) ) ).
fof(conj_thm_2Eset__relation_2Estrict__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))
=> ap(c_2Eset__relation_2Estrict(A_27a),ap(ap(c_2Eset__relation_2Errestrict(A_27a),V0r),V1s)) = ap(ap(c_2Eset__relation_2Errestrict(A_27a),ap(c_2Eset__relation_2Estrict(A_27a),V0r)),V1s) ) ) ) ).
fof(lameq_f328,axiom,
! [A_27a,V0xs] :
( mem(V0xs,arr(A_27a,bool))
=> ! [V1x1] :
( mem(V1x1,A_27a)
=> ! [V2x2] : ap(f328(A_27a,V0xs,V1x1),V2x2) = ap(ap(c_2Epair_2E_2C(ty_2Epair_2Eprod(A_27a,A_27a),bool),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1x1),V2x2)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V1x1),V0xs)),ap(ap(c_2Ebool_2EIN(A_27a),V2x2),V0xs))) ) ) ).
fof(lameq_f329,axiom,
! [A_27a,V0xs] :
( mem(V0xs,arr(A_27a,bool))
=> ! [V1x1] : ap(f329(A_27a,V0xs),V1x1) = f328(A_27a,V0xs,V1x1) ) ).
fof(ax_thm_2Eset__relation_2Euniv__reln__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0xs] :
( mem(V0xs,arr(A_27a,bool))
=> ap(c_2Eset__relation_2Euniv__reln(A_27a),V0xs) = 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)),f329(A_27a,V0xs))) ) ) ).
fof(lameq_f330,axiom,
! [A_27b,A_27a,V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ! [V2e] :
( mem(V2e,A_27b)
=> ! [V3e_27] : ap(f330(A_27b,A_27a,V0r,V2e),V3e_27) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V3e_27),ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V3e_27),V2e)),V0r)) ) ) ).
fof(ax_thm_2Eset__relation_2Efinite__prefixes__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27b,bool))
=> ( p(ap(ap(c_2Eset__relation_2Efinite__prefixes(A_27a,A_27b),V0r),V1s))
<=> ! [V2e] :
( mem(V2e,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V2e),V1s))
=> p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f330(A_27b,A_27a,V0r,V2e)))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Efinite__prefixes__subset__s,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27b,bool))
=> ! [V2s_27] :
( mem(V2s_27,arr(A_27b,bool))
=> ( ( p(ap(ap(c_2Eset__relation_2Efinite__prefixes(A_27a,A_27b),V0r),V1s))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),V2s_27),V1s)) )
=> p(ap(ap(c_2Eset__relation_2Efinite__prefixes(A_27a,A_27b),V0r),V2s_27)) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Efinite__prefixes__subset__r,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ! [V1r_27] :
( mem(V1r_27,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ! [V2s] :
( mem(V2s,arr(A_27b,bool))
=> ( ( p(ap(ap(c_2Eset__relation_2Efinite__prefixes(A_27a,A_27b),V0r),V2s))
& p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Epair_2Eprod(A_27a,A_27b)),V1r_27),V0r)) )
=> p(ap(ap(c_2Eset__relation_2Efinite__prefixes(A_27a,A_27b),V1r_27),V2s)) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Efinite__prefixes__subset__rs,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27b,bool))
=> ! [V2r_27] :
( mem(V2r_27,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ! [V3s_27] :
( mem(V3s_27,arr(A_27b,bool))
=> ( p(ap(ap(c_2Eset__relation_2Efinite__prefixes(A_27a,A_27b),V0r),V1s))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Epair_2Eprod(A_27a,A_27b)),V2r_27),V0r))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),V3s_27),V1s))
=> p(ap(ap(c_2Eset__relation_2Efinite__prefixes(A_27a,A_27b),V2r_27),V3s_27)) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Efinite__prefixes__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))
=> ! [V2s_27] :
( mem(V2s_27,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Eset__relation_2Efinite__prefixes(A_27a,A_27a),V0r),V1s))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2s_27),V1s)) )
=> ( p(ap(ap(c_2Eset__relation_2Efinite__prefixes(A_27a,A_27a),V0r),V2s_27))
& p(ap(ap(c_2Eset__relation_2Efinite__prefixes(A_27a,A_27a),ap(ap(c_2Eset__relation_2Errestrict(A_27a),V0r),V2s_27)),V2s_27)) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Efinite__prefixes__union,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0r1] :
( mem(V0r1,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ! [V1r2] :
( mem(V1r2,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ! [V2s1] :
( mem(V2s1,arr(A_27b,bool))
=> ! [V3s2] :
( mem(V3s2,arr(A_27b,bool))
=> ( ( p(ap(ap(c_2Eset__relation_2Efinite__prefixes(A_27a,A_27b),V0r1),V2s1))
& p(ap(ap(c_2Eset__relation_2Efinite__prefixes(A_27a,A_27b),V1r2),V3s2)) )
=> p(ap(ap(c_2Eset__relation_2Efinite__prefixes(A_27a,A_27b),ap(ap(c_2Epred__set_2EUNION(ty_2Epair_2Eprod(A_27a,A_27b)),V0r1),V1r2)),ap(ap(c_2Epred__set_2EINTER(A_27b),V2s1),V3s2))) ) ) ) ) ) ) ) ).
fof(lameq_f331,axiom,
! [A_27b,A_27c,V1r2] :
( mem(V1r2,arr(ty_2Epair_2Eprod(A_27b,A_27c),bool))
=> ! [V4x] :
( mem(V4x,A_27b)
=> ! [V3s2] :
( mem(V3s2,arr(A_27c,bool))
=> ! [V5y] : ap(f331(A_27b,A_27c,V1r2,V4x,V3s2),V5y) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27c),V5y),V3s2)),ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27b,A_27c)),ap(ap(c_2Epair_2E_2C(A_27b,A_27c),V4x),V5y)),V1r2)) ) ) ) ).
fof(lameq_f332,axiom,
! [A_27c,A_27b,V1r2] :
( mem(V1r2,arr(ty_2Epair_2Eprod(A_27b,A_27c),bool))
=> ! [V3s2] :
( mem(V3s2,arr(A_27c,bool))
=> ! [V4x] : ap(f332(A_27c,A_27b,V1r2,V3s2),V4x) = ap(ap(c_2Epair_2E_2C(A_27b,bool),V4x),ap(c_2Ebool_2E_3F(A_27c),f331(A_27b,A_27c,V1r2,V4x,V3s2))) ) ) ).
fof(conj_thm_2Eset__relation_2Efinite__prefixes__comp,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0r1] :
( mem(V0r1,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ! [V1r2] :
( mem(V1r2,arr(ty_2Epair_2Eprod(A_27b,A_27c),bool))
=> ! [V2s1] :
( mem(V2s1,arr(A_27b,bool))
=> ! [V3s2] :
( mem(V3s2,arr(A_27c,bool))
=> ( ( p(ap(ap(c_2Eset__relation_2Efinite__prefixes(A_27a,A_27b),V0r1),V2s1))
& p(ap(ap(c_2Eset__relation_2Efinite__prefixes(A_27b,A_27c),V1r2),V3s2))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),ap(c_2Epred__set_2EGSPEC(A_27b,A_27b),f332(A_27c,A_27b,V1r2,V3s2))),V2s1)) )
=> p(ap(ap(c_2Eset__relation_2Efinite__prefixes(A_27a,A_27c),ap(ap(c_2Eset__relation_2Ercomp(A_27a,A_27c,A_27b),V0r1),V1r2)),V3s2)) ) ) ) ) ) ) ) ) ).
fof(lameq_f333,axiom,
! [A_27b,A_27a,V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V5x] :
( mem(V5x,A_27a)
=> ! [V1r] :
( mem(V1r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V6y] : ap(f333(A_27b,A_27a,V0f,V5x,V1r),V6y) = ap(ap(c_2Epair_2E_2C(ty_2Epair_2Eprod(A_27b,A_27b),bool),ap(ap(c_2Epair_2E_2C(A_27b,A_27b),ap(V0f,V5x)),ap(V0f,V6y))),ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V5x),V6y)),V1r)) ) ) ) ).
fof(lameq_f334,axiom,
! [A_27b,A_27a,V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1r] :
( mem(V1r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V5x] : ap(f334(A_27b,A_27a,V0f,V1r),V5x) = f333(A_27b,A_27a,V0f,V5x,V1r) ) ) ).
fof(conj_thm_2Eset__relation_2Efinite__prefixes__inj__image,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1r] :
( mem(V1r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( mem(V4y,A_27a)
=> ( ap(V0f,V3x) = ap(V0f,V4y)
=> V3x = V4y ) ) )
& p(ap(ap(c_2Eset__relation_2Efinite__prefixes(A_27a,A_27a),V1r),V2s)) )
=> p(ap(ap(c_2Eset__relation_2Efinite__prefixes(A_27b,A_27b),ap(c_2Epred__set_2EGSPEC(ty_2Epair_2Eprod(A_27b,A_27b),ty_2Epair_2Eprod(A_27a,A_27a)),ap(c_2Epair_2EUNCURRY(A_27a,A_27a,ty_2Epair_2Eprod(ty_2Epair_2Eprod(A_27b,A_27b),bool)),f334(A_27b,A_27a,V0f,V1r)))),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s))) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Efinite__prefixes__range,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27b,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ( ( p(ap(ap(c_2Eset__relation_2Efinite__prefixes(A_27a,A_27b),V0r),V1s))
& p(ap(ap(c_2Epred__set_2EDISJOINT(A_27b),V2t),ap(c_2Eset__relation_2Erange(A_27b,A_27a),V0r))) )
=> p(ap(ap(c_2Eset__relation_2Efinite__prefixes(A_27a,A_27b),V0r),ap(ap(c_2Epred__set_2EUNION(A_27b),V1s),V2t))) ) ) ) ) ) ) ).
fof(lameq_f335,axiom,
! [A_27a,V3a0] :
( mem(V3a0,ty_2Epair_2Eprod(A_27a,A_27a))
=> ! [V4x] :
( mem(V4x,A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V5y] : ap(f335(A_27a,V3a0,V4x,V0r),V5y) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(ty_2Epair_2Eprod(A_27a,A_27a)),V3a0),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V4x),V5y))),ap(V0r,ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V4x),V5y))) ) ) ) ).
fof(lameq_f336,axiom,
! [A_27a,V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V3a0] :
( mem(V3a0,ty_2Epair_2Eprod(A_27a,A_27a))
=> ! [V4x] : ap(f336(A_27a,V0r,V3a0),V4x) = ap(c_2Ebool_2E_3F(A_27a),f335(A_27a,V3a0,V4x,V0r)) ) ) ).
fof(lameq_f337,axiom,
! [A_27a,V6x] :
( mem(V6x,A_27a)
=> ! [V7y] :
( mem(V7y,A_27a)
=> ! [V2tc_27] :
( mem(V2tc_27,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V8z] : ap(f337(A_27a,V6x,V7y,V2tc_27),V8z) = ap(ap(c_2Ebool_2E_2F_5C,ap(V2tc_27,ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V6x),V8z))),ap(V2tc_27,ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V8z),V7y))) ) ) ) ).
fof(lameq_f338,axiom,
! [A_27a,V3a0] :
( mem(V3a0,ty_2Epair_2Eprod(A_27a,A_27a))
=> ! [V2tc_27] :
( mem(V2tc_27,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V6x] :
( mem(V6x,A_27a)
=> ! [V7y] : ap(f338(A_27a,V3a0,V2tc_27,V6x),V7y) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(ty_2Epair_2Eprod(A_27a,A_27a)),V3a0),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V6x),V7y))),ap(c_2Ebool_2E_3F(A_27a),f337(A_27a,V6x,V7y,V2tc_27))) ) ) ) ).
fof(lameq_f339,axiom,
! [A_27a,V2tc_27] :
( mem(V2tc_27,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V3a0] :
( mem(V3a0,ty_2Epair_2Eprod(A_27a,A_27a))
=> ! [V6x] : ap(f339(A_27a,V2tc_27,V3a0),V6x) = ap(c_2Ebool_2E_3F(A_27a),f338(A_27a,V3a0,V2tc_27,V6x)) ) ) ).
fof(lameq_f340,axiom,
! [A_27a,V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V2tc_27] :
( mem(V2tc_27,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V3a0] : ap(f340(A_27a,V0r,V2tc_27),V3a0) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2E_5C_2F,ap(c_2Ebool_2E_3F(A_27a),f336(A_27a,V0r,V3a0))),ap(c_2Ebool_2E_3F(A_27a),f339(A_27a,V2tc_27,V3a0)))),ap(V2tc_27,V3a0)) ) ) ).
fof(lameq_f341,axiom,
! [A_27a,V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1a0] :
( mem(V1a0,ty_2Epair_2Eprod(A_27a,A_27a))
=> ! [V2tc_27] : ap(f341(A_27a,V0r,V1a0),V2tc_27) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(c_2Ebool_2E_21(ty_2Epair_2Eprod(A_27a,A_27a)),f340(A_27a,V0r,V2tc_27))),ap(V2tc_27,V1a0)) ) ) ).
fof(lameq_f342,axiom,
! [A_27a,V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1a0] : ap(f342(A_27a,V0r),V1a0) = ap(c_2Ebool_2E_21(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool)),f341(A_27a,V0r,V1a0)) ) ).
fof(lameq_f343,axiom,
! [A_27a,V0r] : ap(f343(A_27a),V0r) = f342(A_27a,V0r) ).
fof(ax_thm_2Eset__relation_2Etc__def,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Eset__relation_2Etc(A_27a) = f343(A_27a) ) ).
fof(conj_thm_2Eset__relation_2Etc__rules,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ( ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,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),V1x),V2y)),V0r))
=> 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_2Etc(A_27a),V0r))) ) ) )
& ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( mem(V4y,A_27a)
=> ( ? [V5z] :
( mem(V5z,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),V5z)),ap(c_2Eset__relation_2Etc(A_27a),V0r)))
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V5z),V4y)),ap(c_2Eset__relation_2Etc(A_27a),V0r))) )
=> 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_2Etc(A_27a),V0r))) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Etc__cases,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,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),V1x),V2y)),ap(c_2Eset__relation_2Etc(A_27a),V0r)))
<=> ( 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)),V0r))
| ? [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),V1x),V3z)),ap(c_2Eset__relation_2Etc(A_27a),V0r)))
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V3z),V2y)),ap(c_2Eset__relation_2Etc(A_27a),V0r))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Etc__ind,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1tc_27] :
( mem(V1tc_27,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [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),V2x),V3y)),V0r))
=> p(ap(ap(V1tc_27,V2x),V3y)) ) ) )
& ! [V4x] :
( mem(V4x,A_27a)
=> ! [V5y] :
( mem(V5y,A_27a)
=> ( ? [V6z] :
( mem(V6z,A_27a)
& p(ap(ap(V1tc_27,V4x),V6z))
& p(ap(ap(V1tc_27,V6z),V5y)) )
=> p(ap(ap(V1tc_27,V4x),V5y)) ) ) ) )
=> ! [V7x] :
( mem(V7x,A_27a)
=> ! [V8y] :
( mem(V8y,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),V7x),V8y)),ap(c_2Eset__relation_2Etc(A_27a),V0r)))
=> p(ap(ap(V1tc_27,V7x),V8y)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Etc__closure,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1s] :
( mem(V1s,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Epair_2Eprod(A_27a,A_27a)),V0r),ap(c_2Eset__relation_2Etc(A_27a),V1s)))
=> p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Epair_2Eprod(A_27a,A_27a)),ap(c_2Eset__relation_2Etc(A_27a),V0r)),ap(c_2Eset__relation_2Etc(A_27a),V1s))) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Esubset__tc,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Epair_2Eprod(A_27a,A_27a)),V0r),ap(c_2Eset__relation_2Etc(A_27a),V0r))) ) ) ).
fof(conj_thm_2Eset__relation_2Etc__idemp,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ap(c_2Eset__relation_2Etc(A_27a),ap(c_2Eset__relation_2Etc(A_27a),V0r)) = ap(c_2Eset__relation_2Etc(A_27a),V0r) ) ) ).
fof(conj_thm_2Eset__relation_2Etc__mono,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1s] :
( mem(V1s,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Epair_2Eprod(A_27a,A_27a)),V0r),V1s))
=> p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Epair_2Eprod(A_27a,A_27a)),ap(c_2Eset__relation_2Etc(A_27a),V0r)),ap(c_2Eset__relation_2Etc(A_27a),V1s))) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Etc__strongind,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1tc_27] :
( mem(V1tc_27,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [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),V2x),V3y)),V0r))
=> p(ap(ap(V1tc_27,V2x),V3y)) ) ) )
& ! [V4x] :
( mem(V4x,A_27a)
=> ! [V5y] :
( mem(V5y,A_27a)
=> ( ? [V6z] :
( mem(V6z,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),V6z)),ap(c_2Eset__relation_2Etc(A_27a),V0r)))
& p(ap(ap(V1tc_27,V4x),V6z))
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V6z),V5y)),ap(c_2Eset__relation_2Etc(A_27a),V0r)))
& p(ap(ap(V1tc_27,V6z),V5y)) )
=> p(ap(ap(V1tc_27,V4x),V5y)) ) ) ) )
=> ! [V7x] :
( mem(V7x,A_27a)
=> ! [V8y] :
( mem(V8y,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),V7x),V8y)),ap(c_2Eset__relation_2Etc(A_27a),V0r)))
=> p(ap(ap(V1tc_27,V7x),V8y)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Etc__cases__right,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,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),V1x),V2y)),ap(c_2Eset__relation_2Etc(A_27a),V0r)))
<=> ( 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)),V0r))
| ? [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),V1x),V3z)),ap(c_2Eset__relation_2Etc(A_27a),V0r)))
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V3z),V2y)),V0r)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Etc__cases__left,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,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),V1x),V2y)),ap(c_2Eset__relation_2Etc(A_27a),V0r)))
<=> ( 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)),V0r))
| ? [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),V1x),V3z)),V0r))
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V3z),V2y)),ap(c_2Eset__relation_2Etc(A_27a),V0r))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Etc__ind__left,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1tc_27] :
( mem(V1tc_27,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [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),V2x),V3y)),V0r))
=> p(ap(ap(V1tc_27,V2x),V3y)) ) ) )
& ! [V4x] :
( mem(V4x,A_27a)
=> ! [V5y] :
( mem(V5y,A_27a)
=> ( ? [V6z] :
( mem(V6z,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),V6z)),V0r))
& p(ap(ap(V1tc_27,V6z),V5y)) )
=> p(ap(ap(V1tc_27,V4x),V5y)) ) ) ) )
=> ! [V7x] :
( mem(V7x,A_27a)
=> ! [V8y] :
( mem(V8y,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),V7x),V8y)),ap(c_2Eset__relation_2Etc(A_27a),V0r)))
=> p(ap(ap(V1tc_27,V7x),V8y)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Etc__strongind__left,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1tc_27] :
( mem(V1tc_27,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [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),V2x),V3y)),V0r))
=> p(ap(ap(V1tc_27,V2x),V3y)) ) ) )
& ! [V4x] :
( mem(V4x,A_27a)
=> ! [V5y] :
( mem(V5y,A_27a)
=> ( ? [V6z] :
( mem(V6z,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),V6z)),V0r))
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V6z),V5y)),ap(c_2Eset__relation_2Etc(A_27a),V0r)))
& p(ap(ap(V1tc_27,V6z),V5y)) )
=> p(ap(ap(V1tc_27,V4x),V5y)) ) ) ) )
=> ! [V7x] :
( mem(V7x,A_27a)
=> ! [V8y] :
( mem(V8y,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),V7x),V8y)),ap(c_2Eset__relation_2Etc(A_27a),V0r)))
=> p(ap(ap(V1tc_27,V7x),V8y)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Etc__ind__right,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1tc_27] :
( mem(V1tc_27,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [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),V2x),V3y)),V0r))
=> p(ap(ap(V1tc_27,V2x),V3y)) ) ) )
& ! [V4x] :
( mem(V4x,A_27a)
=> ! [V5y] :
( mem(V5y,A_27a)
=> ( ? [V6z] :
( mem(V6z,A_27a)
& p(ap(ap(V1tc_27,V4x),V6z))
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V6z),V5y)),V0r)) )
=> p(ap(ap(V1tc_27,V4x),V5y)) ) ) ) )
=> ! [V7x] :
( mem(V7x,A_27a)
=> ! [V8y] :
( mem(V8y,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),V7x),V8y)),ap(c_2Eset__relation_2Etc(A_27a),V0r)))
=> p(ap(ap(V1tc_27,V7x),V8y)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Ertc__ind__right,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1tc_27] :
( mem(V1tc_27,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [V2x] :
( mem(V2x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),V0r)))
| p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Eset__relation_2Erange(A_27a,A_27a),V0r))) )
=> p(ap(ap(V1tc_27,V2x),V2x)) ) )
& ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( mem(V4y,A_27a)
=> ( ? [V5z] :
( mem(V5z,A_27a)
& p(ap(ap(V1tc_27,V3x),V5z))
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V5z),V4y)),V0r)) )
=> p(ap(ap(V1tc_27,V3x),V4y)) ) ) ) )
=> ! [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_2Etc(A_27a),V0r)))
=> p(ap(ap(V1tc_27,V6x),V7y)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Etc__strongind__right,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1tc_27] :
( mem(V1tc_27,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [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),V2x),V3y)),V0r))
=> p(ap(ap(V1tc_27,V2x),V3y)) ) ) )
& ! [V4x] :
( mem(V4x,A_27a)
=> ! [V5y] :
( mem(V5y,A_27a)
=> ( ? [V6z] :
( mem(V6z,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),V6z)),ap(c_2Eset__relation_2Etc(A_27a),V0r)))
& p(ap(ap(V1tc_27,V4x),V6z))
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V6z),V5y)),V0r)) )
=> p(ap(ap(V1tc_27,V4x),V5y)) ) ) ) )
=> ! [V7x] :
( mem(V7x,A_27a)
=> ! [V8y] :
( mem(V8y,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),V7x),V8y)),ap(c_2Eset__relation_2Etc(A_27a),V0r)))
=> p(ap(ap(V1tc_27,V7x),V8y)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Etc__union,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r1] :
( mem(V0r1,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,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),V1x),V2y)),ap(c_2Eset__relation_2Etc(A_27a),V0r1)))
=> ! [V3r2] :
( mem(V3r2,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> 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_2Etc(A_27a),ap(ap(c_2Epred__set_2EUNION(ty_2Epair_2Eprod(A_27a,A_27a)),V0r1),V3r2)))) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Etc__implication,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))
=> ( ! [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),V2x),V3y)),V0r1))
=> 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)),V1r2)) ) ) )
=> ! [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_2Etc(A_27a),V0r1)))
=> 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_2Etc(A_27a),V1r2))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Etc__empty,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_2Etc(A_27a),c_2Epred__set_2EEMPTY(ty_2Epair_2Eprod(A_27a,A_27a))))) ) ) ) ).
fof(conj_thm_2Eset__relation_2Etc__empty__eqn,axiom,
! [A_27a] :
( ne(A_27a)
=> ap(c_2Eset__relation_2Etc(A_27a),c_2Epred__set_2EEMPTY(ty_2Epair_2Eprod(A_27a,A_27a))) = c_2Epred__set_2EEMPTY(ty_2Epair_2Eprod(A_27a,A_27a)) ) ).
fof(conj_thm_2Eset__relation_2Etc__domain__range,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,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),V1x),V2y)),ap(c_2Eset__relation_2Etc(A_27a),V0r)))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),V0r)))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V2y),ap(c_2Eset__relation_2Erange(A_27a,A_27a),V0r))) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Errestrict__tc,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1x] :
( mem(V1x,arr(A_27a,bool))
=> ! [V2e] :
( mem(V2e,A_27a)
=> ! [V3e_27] :
( mem(V3e_27,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),V2e),V3e_27)),ap(c_2Eset__relation_2Etc(A_27a),ap(ap(c_2Eset__relation_2Errestrict(A_27a),V0r),V1x))))
=> p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2e),V3e_27)),ap(c_2Eset__relation_2Etc(A_27a),V0r))) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Etc__SWAP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ap(c_2Eset__relation_2Etc(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Epair_2Eprod(A_27a,A_27a),ty_2Epair_2Eprod(A_27a,A_27a)),c_2Epair_2ESWAP(A_27a,A_27a)),V0r)) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Epair_2Eprod(A_27a,A_27a),ty_2Epair_2Eprod(A_27a,A_27a)),c_2Epair_2ESWAP(A_27a,A_27a)),ap(c_2Eset__relation_2Etc(A_27a),V0r)) ) ) ).
fof(ax_thm_2Eset__relation_2Eacyclic__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ( p(ap(c_2Eset__relation_2Eacyclic(A_27a),V0r))
<=> ! [V1x] :
( mem(V1x,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),V1x),V1x)),ap(c_2Eset__relation_2Etc(A_27a),V0r))) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Eacyclic__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(c_2Eset__relation_2Eacyclic(A_27a),V0r1))
& p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Epair_2Eprod(A_27a,A_27a)),V1r2),V0r1)) )
=> p(ap(c_2Eset__relation_2Eacyclic(A_27a),V1r2)) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Eacyclic__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_2Eacyclic(A_27a),V0r))
=> p(ap(c_2Eset__relation_2Eacyclic(A_27a),ap(ap(c_2Eset__relation_2Errestrict(A_27a),V0r),V1s))) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Eacyclic__irreflexive,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ( p(ap(c_2Eset__relation_2Eacyclic(A_27a),V0r))
=> ~ p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1x),V1x)),V0r)) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Eacyclic__SWAP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ( p(ap(c_2Eset__relation_2Eacyclic(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Epair_2Eprod(A_27a,A_27a),ty_2Epair_2Eprod(A_27a,A_27a)),c_2Epair_2ESWAP(A_27a,A_27a)),V0r)))
<=> p(ap(c_2Eset__relation_2Eacyclic(A_27a),V0r)) ) ) ) ).
fof(conj_thm_2Eset__relation_2Eacyclic__bigunion,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0rs] :
( mem(V0rs,arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool),bool))
=> ( ( ! [V1r] :
( mem(V1r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V2r_27] :
( mem(V2r_27,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool)),V1r),V0rs))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool)),V2r_27),V0rs))
& V1r != V2r_27 )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),V1r)),ap(c_2Eset__relation_2Erange(A_27a,A_27a),V1r))),ap(ap(c_2Epred__set_2EUNION(A_27a),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),V2r_27)),ap(c_2Eset__relation_2Erange(A_27a,A_27a),V2r_27)))) ) ) )
& ! [V3r] :
( mem(V3r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Epair_2Eprod(A_27a,A_27a),bool)),V3r),V0rs))
=> p(ap(c_2Eset__relation_2Eacyclic(A_27a),V3r)) ) ) )
=> p(ap(c_2Eset__relation_2Eacyclic(A_27a),ap(c_2Epred__set_2EBIGUNION(ty_2Epair_2Eprod(A_27a,A_27a)),V0rs))) ) ) ) ).
fof(conj_thm_2Eset__relation_2Eacyclic__union,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1r_27] :
( mem(V1r_27,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ( ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),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))),ap(ap(c_2Epred__set_2EUNION(A_27a),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),V1r_27)),ap(c_2Eset__relation_2Erange(A_27a,A_27a),V1r_27))))
& p(ap(c_2Eset__relation_2Eacyclic(A_27a),V0r))
& p(ap(c_2Eset__relation_2Eacyclic(A_27a),V1r_27)) )
=> p(ap(c_2Eset__relation_2Eacyclic(A_27a),ap(ap(c_2Epred__set_2EUNION(ty_2Epair_2Eprod(A_27a,A_27a)),V0r),V1r_27))) ) ) ) ) ).
fof(ax_thm_2Eset__relation_2Ereflexive__def,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),V1s))
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2x),V2x)),V0r)) ) ) ) ) ) ) ).
fof(ax_thm_2Eset__relation_2Eirreflexive__def,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_2Eirreflexive(A_27a),V0r),V1s))
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> ~ p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2x),V2x)),V0r)) ) ) ) ) ) ) ).
fof(ax_thm_2Eset__relation_2Etransitive__def,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))
<=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,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),V1x),V2y)),V0r))
& p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2y),V3z)),V0r)) )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1x),V3z)),V0r)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Etransitive__tc,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))
=> ap(c_2Eset__relation_2Etc(A_27a),V0r) = V0r ) ) ) ).
fof(conj_thm_2Eset__relation_2Etc__transitive,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),ap(c_2Eset__relation_2Etc(A_27a),V0r))) ) ) ).
fof(ax_thm_2Eset__relation_2Eantisym__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ( p(ap(c_2Eset__relation_2Eantisym(A_27a),V0r))
<=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,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),V1x),V2y)),V0r))
& 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)),V0r)) )
=> V1x = V2y ) ) ) ) ) ) ).
fof(ax_thm_2Eset__relation_2Epartial__order__def,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_2Epartial__order(A_27a),V0r),V1s))
<=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),V0r)),V1s))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Eset__relation_2Erange(A_27a,A_27a),V0r)),V1s))
& p(ap(c_2Eset__relation_2Etransitive(A_27a),V0r))
& p(ap(ap(c_2Eset__relation_2Ereflexive(A_27a),V0r),V1s))
& p(ap(c_2Eset__relation_2Eantisym(A_27a),V0r)) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Eantisym__subset,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ! [V1s] :
( mem(V1s,arr(ty_2Epair_2Eprod(A_27a,A_27a),bool))
=> ( p(ap(c_2Eset__relation_2Eantisym(A_27a),V0t))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Epair_2Eprod(A_27a,A_27a)),V1s),V0t))
=> p(ap(c_2Eset__relation_2Eantisym(A_27a),V1s)) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Epartial__order__dom__rng,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))
=> ! [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),V2x),V3y)),V0r))
& p(ap(ap(c_2Eset__relation_2Epartial__order(A_27a),V0r),V1s)) )
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V3y),V1s)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Epartial__order__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))
=> ! [V2s_27] :
( mem(V2s_27,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Eset__relation_2Epartial__order(A_27a),V0r),V1s))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2s_27),V1s)) )
=> p(ap(ap(c_2Eset__relation_2Epartial__order(A_27a),ap(ap(c_2Eset__relation_2Errestrict(A_27a),V0r),V2s_27)),V2s_27)) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Estrict__partial__order,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_2Epartial__order(A_27a),V0r),V1s))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),ap(c_2Eset__relation_2Estrict(A_27a),V0r))),V1s))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Eset__relation_2Erange(A_27a,A_27a),ap(c_2Eset__relation_2Estrict(A_27a),V0r))),V1s))
& p(ap(c_2Eset__relation_2Etransitive(A_27a),ap(c_2Eset__relation_2Estrict(A_27a),V0r)))
& p(ap(c_2Eset__relation_2Eantisym(A_27a),ap(c_2Eset__relation_2Estrict(A_27a),V0r))) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Estrict__partial__order__acyclic,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_2Epartial__order(A_27a),V0r),V1s))
=> p(ap(c_2Eset__relation_2Eacyclic(A_27a),ap(c_2Eset__relation_2Estrict(A_27a),V0r))) ) ) ) ) ).
fof(ax_thm_2Eset__relation_2Elinear__order__def,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),V1s))
<=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),V0r)),V1s))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Eset__relation_2Erange(A_27a,A_27a),V0r)),V1s))
& p(ap(c_2Eset__relation_2Etransitive(A_27a),V0r))
& p(ap(c_2Eset__relation_2Eantisym(A_27a),V0r))
& ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( mem(V3y,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V3y),V1s)) )
=> ( 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)),V0r))
| 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)),V0r)) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Elinear__order__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))
=> ! [V2s_27] :
( mem(V2s_27,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Eset__relation_2Elinear__order(A_27a),V0r),V1s))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2s_27),V1s)) )
=> p(ap(ap(c_2Eset__relation_2Elinear__order(A_27a),ap(ap(c_2Eset__relation_2Errestrict(A_27a),V0r),V2s_27)),V2s_27)) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Epartial__order__linear__order,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),V1s))
=> p(ap(ap(c_2Eset__relation_2Epartial__order(A_27a),V0r),V1s)) ) ) ) ) ).
fof(ax_thm_2Eset__relation_2Estrict__linear__order__def,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_2Estrict__linear__order(A_27a),V0r),V1s))
<=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Eset__relation_2Edomain(A_27a,A_27a),V0r)),V1s))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Eset__relation_2Erange(A_27a,A_27a),V0r)),V1s))
& p(ap(c_2Eset__relation_2Etransitive(A_27a),V0r))
& ! [V2x] :
( mem(V2x,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),V2x)),V0r)) )
& ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( mem(V4y,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V4y),V1s))
& V3x != V4y )
=> ( 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)),V0r))
| p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27a)),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V4y),V3x)),V0r)) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Estrict__linear__order__dom__rng,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))
=> ! [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),V2x),V3y)),V0r))
& p(ap(ap(c_2Eset__relation_2Estrict__linear__order(A_27a),V0r),V1s)) )
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V3y),V1s)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Elinear__order__dom__rng,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))
=> ! [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),V2x),V3y)),V0r))
& p(ap(ap(c_2Eset__relation_2Elinear__order(A_27a),V0r),V1s)) )
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V3y),V1s)) ) ) ) ) ) ) ) ).
fof(lameq_f344,axiom,
! [A_27b,A_27a,V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ! [V1x] : ap(f344(A_27b,A_27a,V0r),V1x) = f317(A_27a,A_27b,V1x,V0r) ) ).
fof(ax_thm_2Eset__relation_2Ereln__to__rel__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ap(c_2Eset__relation_2Ereln__to__rel(A_27a,A_27b),V0r) = f344(A_27b,A_27a,V0r) ) ) ) ).
fof(lameq_f345,axiom,
! [A_27a,A_27b,V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] : ap(f345(A_27a,A_27b,V0R,V1x),V2y) = ap(ap(c_2Epair_2E_2C(ty_2Epair_2Eprod(A_27a,A_27b),bool),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V1x),V2y)),ap(ap(V0R,V1x),V2y)) ) ) ).
fof(lameq_f346,axiom,
! [A_27b,A_27a,V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1x] : ap(f346(A_27b,A_27a,V0R),V1x) = f345(A_27a,A_27b,V0R,V1x) ) ).
fof(ax_thm_2Eset__relation_2Erel__to__reln__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ap(c_2Eset__relation_2Erel__to__reln(A_27a,A_27b),V0R) = ap(c_2Epred__set_2EGSPEC(ty_2Epair_2Eprod(A_27a,A_27b),ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,ty_2Epair_2Eprod(ty_2Epair_2Eprod(A_27a,A_27b),bool)),f346(A_27b,A_27a,V0R))) ) ) ) ).
fof(lameq_f347,axiom,
! [A_27a,V1x] :
( mem(V1x,A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V2y] : ap(f347(A_27a,V1x,V0s),V2y) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)),ap(ap(c_2Ebool_2EIN(A_27a),V2y),V0s)) ) ) ).
fof(lameq_f348,axiom,
! [A_27a,V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1x] : ap(f348(A_27a,V0s),V1x) = f347(A_27a,V1x,V0s) ) ).
fof(ax_thm_2Eset__relation_2ERRUNIV__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ap(c_2Eset__relation_2ERRUNIV(A_27a),V0s) = f348(A_27a,V0s) ) ) ).
fof(lameq_f349,axiom,
! [A_27a,V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] : ap(f349(A_27a,V1s,V2x),V3y) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(A_27a),V2x),V3y)),ap(c_2Ebool_2E_7E,ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))) ) ) ).
fof(lameq_f350,axiom,
! [A_27a,V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x] : ap(f350(A_27a,V1s),V2x) = f349(A_27a,V1s,V2x) ) ).
fof(ax_thm_2Eset__relation_2ERREFL__EXP__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ap(ap(c_2Eset__relation_2ERREFL__EXP(A_27a),V0R),V1s) = ap(ap(c_2Erelation_2ERUNION(A_27a,A_27a),V0R),f350(A_27a,V1s)) ) ) ) ).
fof(conj_thm_2Eset__relation_2ERREFL__EXP__RSUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> p(ap(ap(c_2Erelation_2ERSUBSET(A_27a,A_27a),V0R),ap(ap(c_2Eset__relation_2ERREFL__EXP(A_27a),V0R),V1s))) ) ) ) ).
fof(conj_thm_2Eset__relation_2ERREFL__EXP__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ap(ap(c_2Eset__relation_2ERREFL__EXP(A_27a),V0R),c_2Epred__set_2EUNIV(A_27a)) = V0R ) ) ).
fof(conj_thm_2Eset__relation_2EREL__RESTRICT__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ap(ap(c_2Epred__set_2EREL__RESTRICT(A_27a),V0R),c_2Epred__set_2EUNIV(A_27a)) = V0R ) ) ).
fof(conj_thm_2Eset__relation_2Ein__rel__to__reln,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0xy] :
( mem(V0xy,ty_2Epair_2Eprod(A_27a,A_27b))
=> ! [V1R] :
( mem(V1R,arr(A_27a,arr(A_27b,bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27b)),V0xy),ap(c_2Eset__relation_2Erel__to__reln(A_27a,A_27b),V1R)))
<=> p(ap(ap(V1R,ap(c_2Epair_2EFST(A_27a,A_27b),V0xy)),ap(c_2Epair_2ESND(A_27a,A_27b),V0xy))) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Ereln__to__rel__app,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0r] :
( mem(V0r,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,A_27b)
=> ( p(ap(ap(ap(c_2Eset__relation_2Ereln__to__rel(A_27a,A_27b),V0r),V1x),V2y))
<=> p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V1x),V2y)),V0r)) ) ) ) ) ) ) ).
fof(conj_thm_2Eset__relation_2Erel__to__reln__IS__UNCURRY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> c_2Eset__relation_2Erel__to__reln(A_27a,A_27b) = c_2Epair_2EUNCURRY(A_27a,A_27b,bool) ) ) ).
fof(conj_thm_2Eset__relation_2Ereln__to__rel__IS__CURRY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> c_2Eset__relation_2Ereln__to__rel(A_27a,A_27b) = c_2Epair_2ECURRY(A_27a,A_27b,bool) ) ) ).
fof(conj_thm_2Eset__relation_2Erel__to__reln__inv,conjecture,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ap(c_2Eset__relation_2Ereln__to__rel(A_27a,A_27b),ap(c_2Eset__relation_2Erel__to__reln(A_27a,A_27b),V0R)) = V0R ) ) ) ).
%------------------------------------------------------------------------------