ITP001 Axioms: ITP033^5.ax
%------------------------------------------------------------------------------
% File : ITP033^5 : 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 : set_relation^2.ax [Gau20]
% : HL4033^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 233 ( 4 unt; 31 typ; 0 def)
% Number of atoms : 5470 ( 96 equ; 0 cnn)
% Maximal formula atoms : 154 ( 23 avg)
% Number of connectives : 10096 ( 23 ~; 8 |; 188 &;9214 @)
% ( 52 <=>; 611 =>; 0 <=; 0 <~>)
% Maximal formula depth : 63 ( 16 avg;9214 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 39 ( 39 >; 0 *; 0 +; 0 <<)
% Number of symbols : 104 ( 103 usr; 72 con; 0-3 aty)
% Number of variables : 821 ( 64 ^ 730 !; 27 ?; 821 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Eset__relation_2ERREFL__EXP,type,
c_2Eset__relation_2ERREFL__EXP: del > $i ).
thf(mem_c_2Eset__relation_2ERREFL__EXP,axiom,
! [A_27a: del] : ( 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 ) ) ) ) ) ).
thf(tp_c_2Eset__relation_2ERRUNIV,type,
c_2Eset__relation_2ERRUNIV: del > $i ).
thf(mem_c_2Eset__relation_2ERRUNIV,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2ERRUNIV @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Eset__relation_2Eacyclic,type,
c_2Eset__relation_2Eacyclic: del > $i ).
thf(mem_c_2Eset__relation_2Eacyclic,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ bool ) ) ).
thf(tp_c_2Eset__relation_2Eall__choices,type,
c_2Eset__relation_2Eall__choices: del > $i ).
thf(mem_c_2Eset__relation_2Eall__choices,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Eall__choices @ A_27a ) @ ( arr @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Eantisym,type,
c_2Eset__relation_2Eantisym: del > $i ).
thf(mem_c_2Eset__relation_2Eantisym,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ bool ) ) ).
thf(tp_c_2Eset__relation_2Echain,type,
c_2Eset__relation_2Echain: del > $i ).
thf(mem_c_2Eset__relation_2Echain,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Echain @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Edomain,type,
c_2Eset__relation_2Edomain: del > del > $i ).
thf(mem_c_2Eset__relation_2Edomain,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Efchains,type,
c_2Eset__relation_2Efchains: del > $i ).
thf(mem_c_2Eset__relation_2Efchains,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Efchains @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Efinite__prefixes,type,
c_2Eset__relation_2Efinite__prefixes: del > del > $i ).
thf(mem_c_2Eset__relation_2Efinite__prefixes,axiom,
! [A_27a: del,A_27b: del] : ( 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 ) ) ) ).
thf(tp_c_2Eset__relation_2Eget__min,type,
c_2Eset__relation_2Eget__min: del > $i ).
thf(mem_c_2Eset__relation_2Eget__min,axiom,
! [A_27a: del] : ( 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 ) ) ) ) ).
thf(tp_c_2Eset__relation_2Eirreflexive,type,
c_2Eset__relation_2Eirreflexive: del > $i ).
thf(mem_c_2Eset__relation_2Eirreflexive,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Eirreflexive @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Elinear__order,type,
c_2Eset__relation_2Elinear__order: del > $i ).
thf(mem_c_2Eset__relation_2Elinear__order,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Emaximal__elements,type,
c_2Eset__relation_2Emaximal__elements: del > $i ).
thf(mem_c_2Eset__relation_2Emaximal__elements,axiom,
! [A_27a: del] : ( 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 ) ) ) ) ).
thf(tp_c_2Eset__relation_2Eminimal__elements,type,
c_2Eset__relation_2Eminimal__elements: del > $i ).
thf(mem_c_2Eset__relation_2Eminimal__elements,axiom,
! [A_27a: del] : ( 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 ) ) ) ) ).
thf(tp_c_2Eset__relation_2Enth__min,type,
c_2Eset__relation_2Enth__min: del > $i ).
thf(mem_c_2Eset__relation_2Enth__min,axiom,
! [A_27a: del] : ( 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 ) ) ) ) ) ).
thf(tp_c_2Eset__relation_2Enum__order,type,
c_2Eset__relation_2Enum__order: del > $i ).
thf(mem_c_2Eset__relation_2Enum__order,axiom,
! [A_27a: del] : ( 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 ) ) ) ) ).
thf(tp_c_2Eset__relation_2Epartial__order,type,
c_2Eset__relation_2Epartial__order: del > $i ).
thf(mem_c_2Eset__relation_2Epartial__order,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Eper,type,
c_2Eset__relation_2Eper: del > $i ).
thf(mem_c_2Eset__relation_2Eper,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Eper @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Eper__restrict,type,
c_2Eset__relation_2Eper__restrict: del > $i ).
thf(mem_c_2Eset__relation_2Eper__restrict,axiom,
! [A_27a: del] : ( 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 ) ) ) ) ).
thf(tp_c_2Eset__relation_2Erange,type,
c_2Eset__relation_2Erange: del > del > $i ).
thf(mem_c_2Eset__relation_2Erange,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eset__relation_2Erange @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ bool ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Ercomp,type,
c_2Eset__relation_2Ercomp: del > del > del > $i ).
thf(mem_c_2Eset__relation_2Ercomp,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( 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 ) ) ) ) ).
thf(tp_c_2Eset__relation_2Ereflexive,type,
c_2Eset__relation_2Ereflexive: del > $i ).
thf(mem_c_2Eset__relation_2Ereflexive,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Ereflexive @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Erel__to__reln,type,
c_2Eset__relation_2Erel__to__reln: del > del > $i ).
thf(mem_c_2Eset__relation_2Erel__to__reln,axiom,
! [A_27a: del,A_27b: del] : ( 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 ) ) ) ).
thf(tp_c_2Eset__relation_2Ereln__to__rel,type,
c_2Eset__relation_2Ereln__to__rel: del > del > $i ).
thf(mem_c_2Eset__relation_2Ereln__to__rel,axiom,
! [A_27a: del,A_27b: del] : ( 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 ) ) ) ) ).
thf(tp_c_2Eset__relation_2Errestrict,type,
c_2Eset__relation_2Errestrict: del > $i ).
thf(mem_c_2Eset__relation_2Errestrict,axiom,
! [A_27a: del] : ( 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 ) ) ) ) ).
thf(tp_c_2Eset__relation_2Estrict,type,
c_2Eset__relation_2Estrict: del > $i ).
thf(mem_c_2Eset__relation_2Estrict,axiom,
! [A_27a: del] : ( 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 ) ) ) ).
thf(tp_c_2Eset__relation_2Estrict__linear__order,type,
c_2Eset__relation_2Estrict__linear__order: del > $i ).
thf(mem_c_2Eset__relation_2Estrict__linear__order,axiom,
! [A_27a: del] : ( 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 ) ) ) ).
thf(tp_c_2Eset__relation_2Etc,type,
c_2Eset__relation_2Etc: del > $i ).
thf(mem_c_2Eset__relation_2Etc,axiom,
! [A_27a: del] : ( 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 ) ) ) ).
thf(tp_c_2Eset__relation_2Etransitive,type,
c_2Eset__relation_2Etransitive: del > $i ).
thf(mem_c_2Eset__relation_2Etransitive,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ bool ) ) ).
thf(tp_c_2Eset__relation_2Euniv__reln,type,
c_2Eset__relation_2Euniv__reln: del > $i ).
thf(mem_c_2Eset__relation_2Euniv__reln,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Euniv__reln @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Eupper__bounds,type,
c_2Eset__relation_2Eupper__bounds: del > del > $i ).
thf(mem_c_2Eset__relation_2Eupper__bounds,axiom,
! [A_27a: del,A_27b: del] : ( 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 ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Erextension,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ( ( V0s = V1t )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Edomain__def,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( 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 )
@ ( lam @ A_27a
@ ^ [V1x: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V1x )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27b )
@ ( lam @ A_27b
@ ^ [V2y: $i] : ( 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 ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Erange__def,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( 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 )
@ ( lam @ A_27a
@ ^ [V1y: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V1y )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27b )
@ ( lam @ A_27b
@ ^ [V2x: $i] : ( 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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Ein__domain,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1r: $i] :
( ( 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: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Ein__range,axiom,
! [A_27a: del,A_27b: del,V0y: $i] :
( ( mem @ V0y @ A_27a )
=> ! [V1r: $i] :
( ( 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: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Ein__dom__rg,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27b )
=> ! [V2r: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Edomain__mono,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1r_27: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Erange__mono,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1r_27: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Errestrict__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( 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 ) )
@ ( lam @ A_27a
@ ^ [V2x: $i] :
( lam @ A_27a
@ ^ [V3y: $i] : ( 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 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Ein__rrestrict,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2r: $i] :
( ( mem @ V2r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V3s: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Ein__rrestrict__alt,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V2s: $i] :
( ( 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 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Errestrict__SUBSET,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( 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 ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Errestrict__union,axiom,
! [A_27a: del,V0r1: $i] :
( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1r2: $i] :
( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V2s: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Errestrict__rrestrict,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ bool ) )
=> ! [V2y: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Edomain__rrestrict__SUBSET,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( 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 ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Erange__rrestrict__SUBSET,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( 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 ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Ercomp__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0r1: $i] :
( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ bool ) )
=> ! [V1r2: $i] :
( ( 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 ) )
@ ( lam @ A_27a
@ ^ [V2x: $i] :
( lam @ A_27b
@ ^ [V3y: $i] :
( 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 )
@ ( lam @ A_27c
@ ^ [V4z: $i] : ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Estrict__def,axiom,
! [A_27a: del,V0r: $i] :
( ( 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 ) )
@ ( lam @ A_27a
@ ^ [V1x: $i] :
( lam @ A_27a
@ ^ [V2y: $i] : ( 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 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Estrict__rrestrict,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( 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 ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Euniv__reln__def,axiom,
! [A_27a: del,V0xs: $i] :
( ( 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 ) )
@ ( lam @ A_27a
@ ^ [V1x1: $i] :
( lam @ A_27a
@ ^ [V2x2: $i] : ( 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 ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Efinite__prefixes__def,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V0r ) @ V1s ) )
<=> ! [V2e: $i] :
( ( 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 )
@ ( lam @ A_27a
@ ^ [V3e_27: $i] : ( 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 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__prefixes__subset__s,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2s_27: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__prefixes__subset__r,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1r_27: $i] :
( ( mem @ V1r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V2s: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__prefixes__subset__rs,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2r_27: $i] :
( ( mem @ V2r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V3s_27: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__prefixes__subset,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2s_27: $i] :
( ( 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 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__prefixes__union,axiom,
! [A_27a: del,A_27b: del,V0r1: $i] :
( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1r2: $i] :
( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V2s1: $i] :
( ( mem @ V2s1 @ ( arr @ A_27b @ bool ) )
=> ! [V3s2: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__prefixes__comp,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0r1: $i] :
( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1r2: $i] :
( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) @ bool ) )
=> ! [V2s1: $i] :
( ( mem @ V2s1 @ ( arr @ A_27b @ bool ) )
=> ! [V3s2: $i] :
( ( 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 )
@ ( lam @ A_27b
@ ^ [V4x: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ V4x )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27c )
@ ( lam @ A_27c
@ ^ [V5y: $i] : ( 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 ) ) ) ) ) ) ) )
@ 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 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__prefixes__inj__image,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( 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 ) )
@ ( lam @ A_27a
@ ^ [V5x: $i] :
( lam @ A_27a
@ ^ [V6y: $i] : ( 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 ) ) ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__prefixes__range,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2t: $i] :
( ( 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 ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Etc__def,axiom,
! [A_27a: del] :
( ( c_2Eset__relation_2Etc @ A_27a )
= ( lam @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool )
@ ^ [V0r: $i] :
( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27a )
@ ^ [V1a0: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
@ ( lam @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool )
@ ^ [V2tc_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27a )
@ ^ [V3a0: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap
@ ( ap @ c_2Ebool_2E_5C_2F
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V5y: $i] : ( 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 ) ) ) ) ) ) ) )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V6x: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V7y: $i] :
( 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 )
@ ( lam @ A_27a
@ ^ [V8z: $i] : ( 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 ) ) ) ) ) ) ) ) ) ) ) )
@ ( ap @ V2tc_27 @ V3a0 ) ) ) ) )
@ ( ap @ V2tc_27 @ V1a0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__rules,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( 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: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ? [V5z: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__cases,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( 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: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__ind,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1tc_27: $i] :
( ( mem @ V1tc_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( 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: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ? [V6z: $i] :
( ( 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: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8y: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__closure,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Esubset__tc,axiom,
! [A_27a: del,V0r: $i] :
( ( 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 ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__idemp,axiom,
! [A_27a: del,V0r: $i] :
( ( 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 ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__mono,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__strongind,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1tc_27: $i] :
( ( mem @ V1tc_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( 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: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ? [V6z: $i] :
( ( 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: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8y: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__cases__right,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( 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: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__cases__left,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( 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: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__ind__left,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1tc_27: $i] :
( ( mem @ V1tc_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( 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: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ? [V6z: $i] :
( ( 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: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8y: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__strongind__left,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1tc_27: $i] :
( ( mem @ V1tc_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( 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: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ? [V6z: $i] :
( ( 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: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8y: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__ind__right,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1tc_27: $i] :
( ( mem @ V1tc_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( 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: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ? [V6z: $i] :
( ( 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: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8y: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Ertc__ind__right,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1tc_27: $i] :
( ( mem @ V1tc_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( 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: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ? [V5z: $i] :
( ( 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: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7y: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__strongind__right,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1tc_27: $i] :
( ( mem @ V1tc_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( 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: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ? [V6z: $i] :
( ( 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: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8y: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__union,axiom,
! [A_27a: del,V0r1: $i] :
( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( 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: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__implication,axiom,
! [A_27a: del,V0r1: $i] :
( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1r2: $i] :
( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( 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: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__empty,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( 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 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__empty__eqn,axiom,
! [A_27a: del] :
( ( 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 ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__domain__range,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Errestrict__tc,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ bool ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ! [V3e_27: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__SWAP,axiom,
! [A_27a: del,V0r: $i] :
( ( 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 ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Eacyclic__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V0r ) )
<=> ! [V1x: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eacyclic__subset,axiom,
! [A_27a: del,V0r1: $i] :
( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1r2: $i] :
( ( 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 ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eacyclic__rrestrict,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eacyclic__irreflexive,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1x: $i] :
( ( 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 ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eacyclic__SWAP,axiom,
! [A_27a: del,V0r: $i] :
( ( 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 ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eacyclic__bigunion,axiom,
! [A_27a: del,V0rs: $i] :
( ( mem @ V0rs @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ bool ) )
=> ( ( ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V2r_27: $i] :
( ( 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: $i] :
( ( 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 ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eacyclic__union,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1r_27: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Ereflexive__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Ereflexive @ A_27a ) @ V0r ) @ V1s ) )
<=> ! [V2x: $i] :
( ( 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 ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Eirreflexive__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Eirreflexive @ A_27a ) @ V0r ) @ V1s ) )
<=> ! [V2x: $i] :
( ( 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 ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Etransitive__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ V0r ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etransitive__tc,axiom,
! [A_27a: del,V0r: $i] :
( ( 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 ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__transitive,axiom,
! [A_27a: del,V0r: $i] :
( ( 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 ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Eantisym__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ V0r ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Epartial__order__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eantisym__subset,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Epartial__order__dom__rng,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Epartial__order__subset,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2s_27: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Estrict__partial__order,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( 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 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Estrict__partial__order__acyclic,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Elinear__order__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( 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: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Elinear__order__subset,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2s_27: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Epartial__order__linear__order,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( 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 ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Estrict__linear__order__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( 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: $i] :
( ( 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: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Estrict__linear__order__dom__rng,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Elinear__order__dom__rng,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Ereln__to__rel__def,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V0r )
= ( lam @ A_27a
@ ^ [V1x: $i] :
( lam @ A_27b
@ ^ [V2y: $i] : ( 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 ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Erel__to__reln__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( 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 ) )
@ ( lam @ A_27a
@ ^ [V1x: $i] :
( lam @ A_27b
@ ^ [V2y: $i] : ( 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 ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2ERRUNIV__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2ERRUNIV @ A_27a ) @ V0s )
= ( lam @ A_27a
@ ^ [V1x: $i] :
( lam @ A_27a
@ ^ [V2y: $i] : ( 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 ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2ERREFL__EXP__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( 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 )
@ ( lam @ A_27a
@ ^ [V2x: $i] :
( lam @ A_27a
@ ^ [V3y: $i] : ( 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 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2ERREFL__EXP__RSUBSET,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( 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 ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2ERREFL__EXP__UNIV,axiom,
! [A_27a: del,V0R: $i] :
( ( 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 ) ) ).
thf(conj_thm_2Eset__relation_2EREL__RESTRICT__UNIV,axiom,
! [A_27a: del,V0R: $i] :
( ( 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 ) ) ).
thf(conj_thm_2Eset__relation_2Ein__rel__to__reln,axiom,
! [A_27a: del,A_27b: del,V0xy: $i] :
( ( mem @ V0xy @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V1R: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Ereln__to__rel__app,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Erel__to__reln__IS__UNCURRY,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b )
= ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ bool ) ) ).
thf(conj_thm_2Eset__relation_2Ereln__to__rel__IS__CURRY,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b )
= ( c_2Epair_2ECURRY @ A_27a @ A_27b @ bool ) ) ).
thf(conj_thm_2Eset__relation_2Erel__to__reln__inv,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( 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 ) ) ).
thf(conj_thm_2Eset__relation_2Ereln__to__rel__inv,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V0r ) )
= V0r ) ) ).
thf(conj_thm_2Eset__relation_2Ereln__to__rel__11,axiom,
! [A_27a: del,A_27b: del,V0r1: $i] :
( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1r2: $i] :
( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ( ( ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V0r1 )
= ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V1r2 ) )
<=> ( V0r1 = V1r2 ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Erel__to__reln__11,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ V0R1 )
= ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ V1R2 ) )
<=> ( V0R1 = V1R2 ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Erel__to__reln__swap,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( V0r
= ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ V1R ) )
<=> ( ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V0r )
= V1R ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Edomain__to__rel__conv,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27b ) @ V0r )
= ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27b ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V0r ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Erange__to__rel__conv,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27b ) @ V0r )
= ( ap @ ( c_2Erelation_2ERRANGE @ A_27b @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27b @ A_27a ) @ V0r ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Estrict__to__rel__conv,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V0r )
= ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a ) @ ( ap @ ( c_2Erelation_2ESTRORD @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Errestrict__to__rel__conv,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s )
= ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Ercomp__to__rel__conv,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0r1: $i] :
( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ bool ) )
=> ! [V1r2: $i] :
( ( 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_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Erelation_2EO @ A_27a @ A_27c @ A_27b ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27c @ A_27b ) @ V1r2 ) ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27c ) @ V0r1 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Euniv__reln__to__rel__conv,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2Euniv__reln @ A_27a ) @ V0s )
= ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a ) @ ( ap @ ( c_2Eset__relation_2ERRUNIV @ A_27a ) @ V0s ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__to__rel__conv,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r )
= ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eacyclic__reln__to__rel__conv,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V0r ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eirreflexive__reln__to__rel__conv,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Eirreflexive @ A_27a ) @ V0r ) @ V1s ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eirreflexive__reln__to__rel__conv__UNIV,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Eirreflexive @ A_27a ) @ V0r ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Ereflexive__reln__to__rel__conv,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Ereflexive @ A_27a ) @ V0r ) @ V1s ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2ERREFL__EXP @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Ereflexive__reln__to__rel__conv__UNIV,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Ereflexive @ A_27a ) @ V0r ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etransitive__reln__to__rel__conv,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ V0r ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eantisym__reln__to__rel__conv,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ V0r ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Eantisymmetric @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Epartial__order__reln__to__rel__conv,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V0r ) @ V1s ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Eset__relation_2ERRUNIV @ A_27a ) @ V1s ) ) )
& ( p @ ( ap @ ( c_2Erelation_2EWeakOrder @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2ERREFL__EXP @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Epartial__order__reln__to__rel__conv__UNIV,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V0r ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
<=> ( p @ ( ap @ ( c_2Erelation_2EWeakOrder @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Elinear__order__reln__to__rel__conv__UNIV,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0r ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
<=> ( p @ ( ap @ ( c_2Erelation_2EWeakLinearOrder @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Estrict__linear__order__reln__to__rel__conv__UNIV,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ V0r ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
<=> ( p @ ( ap @ ( c_2Erelation_2EStrongLinearOrder @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Ereln__rel__conv__thms,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0xy: $i] :
( ( mem @ V0xy @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2r: $i] :
( ( mem @ V2r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27b )
=> ! [V5r1: $i] :
( ( mem @ V5r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V6r2: $i] :
( ( mem @ V6r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V7R1: $i] :
( ( mem @ V7R1 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V8R2: $i] :
( ( mem @ V8R2 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V9R: $i] :
( ( mem @ V9R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V10r: $i] :
( ( mem @ V10r @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ bool ) )
=> ! [V11r: $i] :
( ( mem @ V11r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V12s: $i] :
( ( mem @ V12s @ ( arr @ A_27a @ bool ) )
=> ! [V13r1: $i] :
( ( mem @ V13r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ bool ) )
=> ! [V14r2: $i] :
( ( mem @ V14r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27c @ 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 ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V2r ) @ V3x ) @ V4y ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3x ) @ V4y ) ) @ V2r ) ) )
& ( ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ V1R ) )
= V1R )
& ( ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V2r ) )
= V2r )
& ( ( ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V5r1 )
= ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V6r2 ) )
<=> ( V5r1 = V6r2 ) )
& ( ( ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ V7R1 )
= ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ V8R2 ) )
<=> ( V7R1 = V8R2 ) )
& ( ( ap @ ( ap @ ( c_2Eset__relation_2ERREFL__EXP @ A_27a ) @ V9R ) @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= V9R )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a ) @ V9R ) @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= V9R )
& ( ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27b ) @ V2r )
= ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27b ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V2r ) ) )
& ( ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27b ) @ V10r )
= ( ap @ ( c_2Erelation_2ERRANGE @ A_27b @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27b @ A_27a ) @ V10r ) ) )
& ( ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V11r )
= ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a ) @ ( ap @ ( c_2Erelation_2ESTRORD @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V11r ) ) ) )
& ( ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V11r ) @ V12s )
= ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V11r ) ) @ V12s ) ) )
& ( ( ap @ ( ap @ ( c_2Eset__relation_2Ercomp @ A_27a @ A_27b @ A_27c ) @ V13r1 ) @ V14r2 )
= ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Erelation_2EO @ A_27a @ A_27c @ A_27b ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27c @ A_27b ) @ V14r2 ) ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27c ) @ V13r1 ) ) ) )
& ( ( ap @ ( c_2Eset__relation_2Euniv__reln @ A_27a ) @ V12s )
= ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a ) @ ( ap @ ( c_2Eset__relation_2ERRUNIV @ A_27a ) @ V12s ) ) )
& ( ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V11r )
= ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V11r ) ) ) )
& ( ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V11r ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V11r ) ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Eirreflexive @ A_27a ) @ V11r ) @ V12s ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V11r ) ) @ V12s ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Ereflexive @ A_27a ) @ V11r ) @ V12s ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2ERREFL__EXP @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V11r ) ) @ V12s ) ) ) )
& ( ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ V11r ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V11r ) ) ) )
& ( ( p @ ( ap @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ V11r ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Eantisymmetric @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V11r ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V11r ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
<=> ( p @ ( ap @ ( c_2Erelation_2EWeakOrder @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V11r ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V11r ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
<=> ( p @ ( ap @ ( c_2Erelation_2EWeakLinearOrder @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V11r ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ V11r ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
<=> ( p @ ( ap @ ( c_2Erelation_2EStrongLinearOrder @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V11r ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eacyclic__WF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V1r ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V1r ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V1r ) ) @ V0s ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V1r ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Emaximal__elements__def,axiom,
! [A_27a: del,V0xs: $i] :
( ( mem @ V0xs @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V0xs ) @ V1r )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0xs ) )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x_27: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x_27 ) @ V0xs ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3x_27 ) ) @ V1r ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2x ) @ V3x_27 ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Eminimal__elements__def,axiom,
! [A_27a: del,V0xs: $i] :
( ( mem @ V0xs @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0xs ) @ V1r )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0xs ) )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x_27: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x_27 ) @ V0xs ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3x_27 ) @ V2x ) ) @ V1r ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2x ) @ V3x_27 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eminimal__elements__subset,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1lo: $i] :
( ( mem @ V1lo @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0s ) @ V1lo ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eminimal__elements__SWAP,axiom,
! [A_27a: del,V0xs: $i] :
( ( mem @ V0xs @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0xs ) @ ( 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 ) ) @ V1r ) )
= ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V0xs ) @ V1r ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Emaximal__union,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2r1: $i] :
( ( mem @ V2r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V3r2: $i] :
( ( mem @ V3r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V1s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V2r1 ) @ V3r2 ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V1s ) @ V2r1 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V1s ) @ V3r2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eminimal__union,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2r1: $i] :
( ( mem @ V2r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V3r2: $i] :
( ( mem @ V3r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V1s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V2r1 ) @ V3r2 ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V1s ) @ V2r1 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V1s ) @ V3r2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eminimal__elements__mono,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1r_27: $i] :
( ( mem @ V1r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V2xs: $i] :
( ( mem @ V2xs @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r ) @ V1r_27 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V2xs ) @ V1r_27 ) ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V2xs ) @ V0r ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eminimal__elements__rrestrict,axiom,
! [A_27a: del,V0xs: $i] :
( ( mem @ V0xs @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0xs ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V1r ) @ V0xs ) )
= ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0xs ) @ V1r ) ) ) ) ).
thf(conj_thm_2Eset__relation_2EWF__has__minimal__path,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V2s ) )
=> ? [V3y: $i] :
( ( mem @ V3y @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V2s ) @ 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 ) @ V1x ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
| ( V3y = V1x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__acyclic__has__maximal,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V1r ) )
=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__acyclic__has__minimal,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V1r ) )
=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Emaximal__TC,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V1r ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V1r ) ) @ V0s ) ) )
=> ( ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V0s ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V1r ) )
= ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eminimal__TC,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V1r ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V1r ) ) @ V0s ) ) )
=> ( ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0s ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V1r ) )
= ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__acyclic__has__minimal__path,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V1r ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) )
=> ? [V3y: $i] :
( ( mem @ V3y @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0s ) @ V1r ) ) )
& ( 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_2Etc @ A_27a ) @ V1r ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__acyclic__has__maximal__path,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V1r ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) )
=> ? [V3y: $i] :
( ( mem @ V3y @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V0s ) @ V1r ) ) )
& ( 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_2Etc @ A_27a ) @ V1r ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__prefix__po__has__minimal__path,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3s_27: $i] :
( ( mem @ V3s_27 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V0r ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a ) @ V0r ) @ V1s ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V3s_27 ) @ V0r ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V3s_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3s_27 ) @ V1s ) ) )
=> ? [V4x_27: $i] :
( ( mem @ V4x_27 @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x_27 ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V3s_27 ) @ V0r ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4x_27 ) @ V2x ) ) @ V0r ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eempty__strict__linear__order,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ V0r ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
<=> ( V0r
= ( c_2Epred__set_2EEMPTY @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eempty__linear__order,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0r ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
<=> ( V0r
= ( c_2Epred__set_2EEMPTY @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Elinear__order__restrict,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V2s_27: $i] :
( ( mem @ V2s_27 @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V1r ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V1r ) @ V2s_27 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V2s_27 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Estrict__linear__order__restrict,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V2s_27: $i] :
( ( mem @ V2s_27 @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ V1r ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V1r ) @ V2s_27 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V2s_27 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Elinear__order__dom__rg,axiom,
! [A_27a: del,V0lo: $i] :
( ( mem @ V0lo @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1X: $i] :
( ( mem @ V1X @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0lo ) @ V1X ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0lo ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0lo ) )
= V1X ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Elinear__order__refl,axiom,
! [A_27a: del,V0lo: $i] :
( ( mem @ V0lo @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1X: $i] :
( ( mem @ V1X @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0lo ) @ V1X ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1X ) )
=> ( 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 ) ) @ V0lo ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Elinear__order__in__set,axiom,
! [A_27a: del,V0lo: $i] :
( ( mem @ V0lo @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1X: $i] :
( ( mem @ V1X @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0lo ) @ V1X ) )
=> ( ( 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 ) ) @ V0lo ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1X ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1X ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2EIN__MIN__LO,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1X: $i] :
( ( mem @ V1X @ ( arr @ A_27a @ bool ) )
=> ! [V2lo: $i] :
( ( mem @ V2lo @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V1X ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V2lo ) @ V1X ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V1X ) @ V2lo ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3y ) @ V0x ) ) @ V2lo ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eextend__linear__order,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0r ) @ V1s ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a )
@ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ A_27a )
@ ( lam @ A_27a
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3y ) @ V2x ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Estrict__linear__order__acyclic,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ V0r ) @ V1s ) )
=> ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V0r ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Estrict__linear__order__union__acyclic,axiom,
! [A_27a: del,V0r1: $i] :
( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1r2: $i] :
( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ V0r1 ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V1r2 ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V1r2 ) ) ) @ V2s ) ) )
=> ( ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r1 ) @ V1r2 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V1r2 ) @ V0r1 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Estrict__linear__order,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0r ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V0r ) ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Elinear__order,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ V0r ) @ V1s ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a )
@ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( 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 ) @ V2x ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) ) ) ) ) )
@ V1s ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__strict__linear__order__has__maximal,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ V1r ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__strict__linear__order__has__minimal,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ V1r ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__linear__order__has__maximal,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V1r ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__linear__order__has__minimal,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V1r ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Emaximal__linear__order,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V1r ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) )
=> ( 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 ) ) @ V1r ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eminimal__linear__order,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V1r ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) )
=> ( 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 ) ) @ V1r ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eminimal__linear__order__unique,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2s_27: $i] :
( ( mem @ V2s_27 @ ( arr @ A_27a @ bool ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0r ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V2s_27 ) @ V0r ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V2s_27 ) @ V0r ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2s_27 ) @ V1s ) ) )
=> ( V3x = V4y ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__prefix__linear__order__has__unique__minimal,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3s_27: $i] :
( ( mem @ V3s_27 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V1r ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a ) @ V1r ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V3s_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3s_27 ) @ V2s ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2ESING @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V3s_27 ) @ V1r ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Enat__order__iso__thm,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V2n: tp__ty_2Enum_2Enum,V3m: tp__ty_2Enum_2Enum] :
( ( ( ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3m ) )
= ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
& ( ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3m ) )
!= ( c_2Eoption_2ENONE @ A_27a ) ) )
=> ( V3m = V2n ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V1s ) )
=> ? [V5m: tp__ty_2Enum_2Enum] :
( ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V5m ) )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V4x ) ) ) )
& ! [V6m: tp__ty_2Enum_2Enum,V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ( ( ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V6m ) )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V7x ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V7x ) @ V1s ) ) ) ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2Eset__relation_2Elinear__order @ 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 ) )
@ ( lam @ A_27a
@ ^ [V8x: $i] :
( lam @ A_27a
@ ^ [V9y: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V8x ) @ V9y ) )
@ ( ap @ ( c_2Ebool_2E_3F @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V10m: $i] :
( ap @ ( c_2Ebool_2E_3F @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V11n: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ V10m ) @ V11n ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ V0f @ V10m ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V8x ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ V0f @ V11n ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V9y ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ V1s ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ 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 ) )
@ ( lam @ A_27a
@ ^ [V12x: $i] :
( lam @ A_27a
@ ^ [V13y: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V12x ) @ V13y ) )
@ ( ap @ ( c_2Ebool_2E_3F @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V14m: $i] :
( ap @ ( c_2Ebool_2E_3F @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V15n: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ V14m ) @ V15n ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ V0f @ V14m ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V12x ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ V0f @ V15n ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V13y ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ V1s ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Echain__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Echain @ A_27a ) @ V0s ) @ V1r ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V0s ) ) )
=> ( ( 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 ) ) @ V1r ) )
| ( 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 ) ) @ V1r ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Eupper__bounds__def,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27b @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27b ) @ V0s ) @ V1r )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27b ) @ V1r ) ) )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V3y: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3y ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27a ) @ V3y ) @ V2x ) ) @ V1r ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eupper__bounds__lem,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2x1: $i] :
( ( mem @ V2x1 @ A_27a )
=> ! [V3x2: $i] :
( ( mem @ V3x2 @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ V0r ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x1 ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27a ) @ V1s ) @ V0r ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x1 ) @ V3x2 ) ) @ V0r ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x2 ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27a ) @ V1s ) @ V0r ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Efchains__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2Efchains @ A_27a ) @ V0r )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V1k: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ V1k )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Eset__relation_2Echain @ A_27a ) @ V1k ) @ V0r ) )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ bool ) ) @ V1k ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
@ ( ap @ ( c_2Ebool_2E_21 @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V2C: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Eset__relation_2Echain @ A_27a ) @ V2C ) @ V0r ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2C ) @ V1k ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27a ) @ V2C ) @ V0r ) ) @ V2C ) ) @ V1k ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27a ) @ V2C ) @ V0r ) ) @ V2C ) ) ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27a ) @ V2C ) @ V0r ) ) @ V2C ) ) @ V1k ) ) @ V0r ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Ezorns__lemma,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( V1s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V0r ) @ V1s ) )
& ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Echain @ A_27a ) @ V2t ) @ V0r ) )
=> ( ( ap @ ( ap @ ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27a ) @ V2t ) @ V0r )
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) )
=> ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V1s ) @ V0r ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Eper__def,axiom,
! [A_27a: del,V0xs: $i] :
( ( mem @ V0xs @ ( arr @ A_27a @ bool ) )
=> ! [V1xss: $i] :
( ( mem @ V1xss @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Eper @ A_27a ) @ V0xs ) @ V1xss ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1xss ) ) @ V0xs ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V1xss ) )
& ! [V2xs1: $i] :
( ( mem @ V2xs1 @ ( arr @ A_27a @ bool ) )
=> ! [V3xs2: $i] :
( ( mem @ V3xs2 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2xs1 ) @ V1xss ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3xs2 ) @ V1xss ) )
& ( V2xs1 != V3xs2 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V2xs1 ) @ V3xs2 ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Eper__restrict__def,axiom,
! [A_27a: del,V0xss: $i] :
( ( mem @ V0xss @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1xs: $i] :
( ( mem @ V1xs @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Eset__relation_2Eper__restrict @ A_27a ) @ V0xss ) @ V1xs )
= ( ap
@ ( ap @ ( c_2Epred__set_2EDELETE @ ( arr @ A_27a @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V2xs_27: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V2xs_27 ) @ V1xs ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2xs_27 ) @ V0xss ) ) ) ) )
@ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eper__delete,axiom,
! [A_27a: del,V0xs: $i] :
( ( mem @ V0xs @ ( arr @ A_27a @ bool ) )
=> ! [V1xss: $i] :
( ( mem @ V1xss @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Eper @ A_27a ) @ V0xs ) @ V1xss ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Eset__relation_2Eper @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V0xs ) @ V2e ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V3es: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ V3es )
@ ( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3es )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V4es: $i] : ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V4es ) @ V2e ) ) )
@ V1xss ) ) )
@ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ bool ) ) @ V3es ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eper__restrict__per,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2s_27: $i] :
( ( mem @ V2s_27 @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Eper @ A_27a ) @ V1s ) @ V0r ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Eper @ A_27a ) @ V2s_27 ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eper__restrict @ A_27a ) @ V0r ) @ V2s_27 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Ecountable__per,axiom,
! [A_27a: del,V0xs: $i] :
( ( mem @ V0xs @ ( arr @ A_27a @ bool ) )
=> ! [V1xss: $i] :
( ( mem @ V1xss @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0xs ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Eper @ A_27a ) @ V0xs ) @ V1xss ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( arr @ A_27a @ bool ) ) @ V1xss ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Eall__choices__def,axiom,
! [A_27a: del,V0xss: $i] :
( ( mem @ V0xss @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2Eall__choices @ A_27a ) @ V0xss )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ A_27a ) )
@ ( lam @ ( arr @ ( arr @ A_27a @ bool ) @ A_27a )
@ ^ [V1choice: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( arr @ A_27a @ bool ) @ A_27a ) @ V1choice ) @ V0xss ) )
@ ( ap @ ( c_2Ebool_2E_21 @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V2xs: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2xs ) @ V0xss ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ V1choice @ V2xs ) ) @ V2xs ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eall__choices__thm,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V0x ) @ ( ap @ ( c_2Eset__relation_2Eall__choices @ A_27a ) @ V1s ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2y ) @ V0x ) ) )
=> ? [V3z: $i] :
( ( mem @ V3z @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3z ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2y ) @ V3z ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Enum__order__def,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Eset__relation_2Enum__order @ A_27a ) @ V0f ) @ 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 ) )
@ ( lam @ A_27a
@ ^ [V2x: $i] :
( lam @ A_27a
@ ^ [V3y: $i] : ( 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 @ A_27a ) @ V2x ) @ V1s ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ V0f @ V2x ) ) @ ( ap @ V0f @ V3y ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Elinear__order__num__order,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ ty_2Enum_2Enum ) @ V0f ) @ V1s ) @ V2t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Enum__order @ A_27a ) @ V0f ) @ V1s ) ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Enum__order__finite__prefix,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ ty_2Enum_2Enum ) @ V0f ) @ V1s ) @ V2t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Enum__order @ A_27a ) @ V0f ) @ V1s ) ) @ V1s ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Eget__min__def,axiom,
! [A_27a: del,V0r_27: $i] :
( ( mem @ V0r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2r: $i] :
( ( mem @ V2r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Eset__relation_2Eget__min @ A_27a ) @ V0r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V1s ) @ V2r ) )
= ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( arr @ A_27a @ bool ) @ ( ty_2Eoption_2Eoption @ A_27a ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V3mins: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( c_2Epred__set_2ESING @ A_27a ) @ V3mins ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ V3mins ) ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) )
@ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V1s ) @ V2r ) ) @ V0r_27 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Enth__min__ind,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( 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 @ bool ) ) ) )
=> ( ( ! [V1r_27: $i] :
( ( mem @ V1r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3r: $i] :
( ( mem @ V3r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V1r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V2s ) @ V3r ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
& ! [V4r_27: $i] :
( ( mem @ V4r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V5s: $i] :
( ( mem @ V5s @ ( arr @ A_27a @ bool ) )
=> ! [V6r: $i] :
( ( mem @ V6r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V7n: tp__ty_2Enum_2Enum] :
( ! [V8min: $i] :
( ( mem @ V8min @ ( ty_2Eoption_2Eoption @ A_27a ) )
=> ( ( ( V8min
= ( ap @ ( ap @ ( c_2Eset__relation_2Eget__min @ A_27a ) @ V4r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V5s ) @ V6r ) ) )
& ( V8min
!= ( c_2Eoption_2ENONE @ A_27a ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V4r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V5s ) @ ( ap @ ( c_2Eoption_2ETHE @ A_27a ) @ V8min ) ) ) @ V6r ) ) @ ( inj__ty_2Enum_2Enum @ V7n ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V4r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V5s ) @ V6r ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V7n ) ) ) ) ) ) ) ) )
=> ! [V9v: $i] :
( ( mem @ V9v @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V10v1: $i] :
( ( mem @ V10v1 @ ( arr @ A_27a @ bool ) )
=> ! [V11v2: $i] :
( ( mem @ V11v2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V12v3: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( ap @ V0P @ V9v ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V10v1 ) @ V11v2 ) ) @ ( inj__ty_2Enum_2Enum @ V12v3 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Enth__min__def,axiom,
! [A_27a: del] :
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r_27: $i] :
( ( mem @ V1r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V2r: $i] :
( ( mem @ V2r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eset__relation_2Enth__min @ A_27a ) @ V1r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V0s ) @ V2r ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= ( ap @ ( ap @ ( c_2Eset__relation_2Eget__min @ A_27a ) @ V1r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V0s ) @ V2r ) ) ) ) ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ! [V4r_27: $i] :
( ( mem @ V4r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V5r: $i] :
( ( mem @ V5r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V6n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( ap @ ( c_2Eset__relation_2Enth__min @ A_27a ) @ V4r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V3s ) @ V5r ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V6n ) ) )
= ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( ty_2Eoption_2Eoption @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27a ) )
@ ( lam @ ( ty_2Eoption_2Eoption @ A_27a )
@ ^ [V7min: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ V7min ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eset__relation_2Enth__min @ A_27a ) @ V4r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V3s ) @ ( ap @ ( c_2Eoption_2ETHE @ A_27a ) @ V7min ) ) ) @ V5r ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Eset__relation_2Eget__min @ A_27a ) @ V4r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V3s ) @ V5r ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Enth__min__def__compute,axiom,
! [A_27a: del] :
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r_27: $i] :
( ( mem @ V1r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V2r: $i] :
( ( mem @ V2r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eset__relation_2Enth__min @ A_27a ) @ V1r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V0s ) @ V2r ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= ( ap @ ( ap @ ( c_2Eset__relation_2Eget__min @ A_27a ) @ V1r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V0s ) @ V2r ) ) ) ) ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ! [V4r_27: $i] :
( ( mem @ V4r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V5r: $i] :
( ( mem @ V5r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V6n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( ap @ ( c_2Eset__relation_2Enth__min @ A_27a ) @ V4r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V3s ) @ V5r ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) )
= ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( ty_2Eoption_2Eoption @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27a ) )
@ ( lam @ ( ty_2Eoption_2Eoption @ A_27a )
@ ^ [V7min: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ V7min ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eset__relation_2Enth__min @ A_27a ) @ V4r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V3s ) @ ( ap @ ( c_2Eoption_2ETHE @ A_27a ) @ V7min ) ) ) @ V5r ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Eset__relation_2Eget__min @ A_27a ) @ V4r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V3s ) @ V5r ) ) ) ) ) ) )
& ! [V8s: $i] :
( ( mem @ V8s @ ( arr @ A_27a @ bool ) )
=> ! [V9r_27: $i] :
( ( mem @ V9r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V10r: $i] :
( ( mem @ V10r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V11n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( ap @ ( c_2Eset__relation_2Enth__min @ A_27a ) @ V9r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V8s ) @ V10r ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V11n ) ) ) )
= ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( ty_2Eoption_2Eoption @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27a ) )
@ ( lam @ ( ty_2Eoption_2Eoption @ A_27a )
@ ^ [V12min: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ V12min ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eset__relation_2Enth__min @ A_27a ) @ V9r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V8s ) @ ( ap @ ( c_2Eoption_2ETHE @ A_27a ) @ V12min ) ) ) @ V10r ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V11n ) ) ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Eset__relation_2Eget__min @ A_27a ) @ V9r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V8s ) @ V10r ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Elinear__order__of__countable__po,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V0r ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a ) @ V0r ) @ V1s ) ) )
=> ? [V2r_27: $i] :
( ( mem @ V2r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V2r_27 ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a ) @ V2r_27 ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r ) @ V2r_27 ) ) ) ) ) ) ).
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