ITP001 Axioms: ITP028^5.ax
%------------------------------------------------------------------------------
% File : ITP028^5 : 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 : fixedPoint^2.ax [Gau20]
% : HL4028^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 38 ( 1 unt; 7 typ; 0 def)
% Number of atoms : 561 ( 12 equ; 0 cnn)
% Maximal formula atoms : 36 ( 14 avg)
% Number of connectives : 1017 ( 0 ~; 0 |; 12 &; 935 @)
% ( 3 <=>; 67 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 13 avg; 935 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 10 ( 10 >; 0 *; 0 +; 0 <<)
% Number of symbols : 24 ( 23 usr; 16 con; 0-2 aty)
% Number of variables : 92 ( 3 ^ 89 !; 0 ?; 92 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2EfixedPoint_2Eclosed,type,
c_2EfixedPoint_2Eclosed: del > $i ).
thf(mem_c_2EfixedPoint_2Eclosed,axiom,
! [A_27a: del] : ( mem @ ( c_2EfixedPoint_2Eclosed @ A_27a ) @ ( arr @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2EfixedPoint_2Edense,type,
c_2EfixedPoint_2Edense: del > $i ).
thf(mem_c_2EfixedPoint_2Edense,axiom,
! [A_27a: del] : ( mem @ ( c_2EfixedPoint_2Edense @ A_27a ) @ ( arr @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2EfixedPoint_2Eempty,type,
c_2EfixedPoint_2Eempty: del > del > $i ).
thf(mem_c_2EfixedPoint_2Eempty,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EfixedPoint_2Eempty @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) ).
thf(tp_c_2EfixedPoint_2Efnsum,type,
c_2EfixedPoint_2Efnsum: del > del > $i ).
thf(mem_c_2EfixedPoint_2Efnsum,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EfixedPoint_2Efnsum @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) ) ) ) ).
thf(tp_c_2EfixedPoint_2Egfp,type,
c_2EfixedPoint_2Egfp: del > $i ).
thf(mem_c_2EfixedPoint_2Egfp,axiom,
! [A_27a: del] : ( mem @ ( c_2EfixedPoint_2Egfp @ A_27a ) @ ( arr @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2EfixedPoint_2Elfp,type,
c_2EfixedPoint_2Elfp: del > $i ).
thf(mem_c_2EfixedPoint_2Elfp,axiom,
! [A_27a: del] : ( mem @ ( c_2EfixedPoint_2Elfp @ A_27a ) @ ( arr @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2EfixedPoint_2Emonotone,type,
c_2EfixedPoint_2Emonotone: del > del > $i ).
thf(mem_c_2EfixedPoint_2Emonotone,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27b @ bool ) ) @ bool ) ) ).
thf(ax_thm_2EfixedPoint_2Emonotone__def,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27b @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27b ) @ V0f ) )
<=> ! [V1X: $i] :
( ( mem @ V1X @ ( arr @ A_27a @ bool ) )
=> ! [V2Y: $i] :
( ( mem @ V2Y @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1X ) @ V2Y ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ ( ap @ V0f @ V1X ) ) @ ( ap @ V0f @ V2Y ) ) ) ) ) ) ) ) ).
thf(ax_thm_2EfixedPoint_2Elfp__def,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( c_2EfixedPoint_2Elfp @ A_27a ) @ V0f )
= ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V1X: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ V1X ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ V0f @ V1X ) ) @ V1X ) ) ) ) ) ) ) ).
thf(ax_thm_2EfixedPoint_2Egfp__def,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( c_2EfixedPoint_2Egfp @ A_27a ) @ V0f )
= ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V1X: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ V1X ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1X ) @ ( ap @ V0f @ V1X ) ) ) ) ) ) ) ) ).
thf(ax_thm_2EfixedPoint_2Eclosed__def,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) )
=> ! [V1X: $i] :
( ( mem @ V1X @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EfixedPoint_2Eclosed @ A_27a ) @ V0f ) @ V1X ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ V0f @ V1X ) ) @ V1X ) ) ) ) ) ).
thf(ax_thm_2EfixedPoint_2Edense__def,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) )
=> ! [V1X: $i] :
( ( mem @ V1X @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EfixedPoint_2Edense @ A_27a ) @ V0f ) @ V1X ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1X ) @ ( ap @ V0f @ V1X ) ) ) ) ) ) ).
thf(conj_thm_2EfixedPoint_2Elfp__least__closed,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a ) @ V0f ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EfixedPoint_2Eclosed @ A_27a ) @ V0f ) @ ( ap @ ( c_2EfixedPoint_2Elfp @ A_27a ) @ V0f ) ) )
& ! [V1X: $i] :
( ( mem @ V1X @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EfixedPoint_2Eclosed @ A_27a ) @ V0f ) @ V1X ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2EfixedPoint_2Elfp @ A_27a ) @ V0f ) ) @ V1X ) ) ) ) ) ) ) ).
thf(conj_thm_2EfixedPoint_2Egfp__greatest__dense,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a ) @ V0f ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EfixedPoint_2Edense @ A_27a ) @ V0f ) @ ( ap @ ( c_2EfixedPoint_2Egfp @ A_27a ) @ V0f ) ) )
& ! [V1X: $i] :
( ( mem @ V1X @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EfixedPoint_2Edense @ A_27a ) @ V0f ) @ V1X ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1X ) @ ( ap @ ( c_2EfixedPoint_2Egfp @ A_27a ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EfixedPoint_2Elfp__fixedpoint,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a ) @ V0f ) )
=> ( ( ( ap @ ( c_2EfixedPoint_2Elfp @ A_27a ) @ V0f )
= ( ap @ V0f @ ( ap @ ( c_2EfixedPoint_2Elfp @ A_27a ) @ V0f ) ) )
& ! [V1X: $i] :
( ( mem @ V1X @ ( arr @ A_27a @ bool ) )
=> ( ( V1X
= ( ap @ V0f @ V1X ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2EfixedPoint_2Elfp @ A_27a ) @ V0f ) ) @ V1X ) ) ) ) ) ) ) ).
thf(conj_thm_2EfixedPoint_2Egfp__greatest__fixedpoint,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a ) @ V0f ) )
=> ( ( ( ap @ ( c_2EfixedPoint_2Egfp @ A_27a ) @ V0f )
= ( ap @ V0f @ ( ap @ ( c_2EfixedPoint_2Egfp @ A_27a ) @ V0f ) ) )
& ! [V1X: $i] :
( ( mem @ V1X @ ( arr @ A_27a @ bool ) )
=> ( ( V1X
= ( ap @ V0f @ V1X ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1X ) @ ( ap @ ( c_2EfixedPoint_2Egfp @ A_27a ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EfixedPoint_2Elfp__induction,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a ) @ V0f ) )
=> ! [V1X: $i] :
( ( mem @ V1X @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ V0f @ V1X ) ) @ V1X ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2EfixedPoint_2Elfp @ A_27a ) @ V0f ) ) @ V1X ) ) ) ) ) ) ).
thf(conj_thm_2EfixedPoint_2Egfp__coinduction,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a ) @ V0f ) )
=> ! [V1X: $i] :
( ( mem @ V1X @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1X ) @ ( ap @ V0f @ V1X ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1X ) @ ( ap @ ( c_2EfixedPoint_2Egfp @ A_27a ) @ V0f ) ) ) ) ) ) ) ).
thf(conj_thm_2EfixedPoint_2Elfp__strong__induction,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a ) @ V0f ) )
=> ! [V1X: $i] :
( ( mem @ V1X @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ V0f @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1X ) @ ( ap @ ( c_2EfixedPoint_2Elfp @ A_27a ) @ V0f ) ) ) ) @ V1X ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2EfixedPoint_2Elfp @ A_27a ) @ V0f ) ) @ V1X ) ) ) ) ) ) ).
thf(conj_thm_2EfixedPoint_2Egfp__strong__coinduction,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a ) @ V0f ) )
=> ! [V1X: $i] :
( ( mem @ V1X @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1X ) @ ( ap @ V0f @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1X ) @ ( ap @ ( c_2EfixedPoint_2Egfp @ A_27a ) @ V0f ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1X ) @ ( ap @ ( c_2EfixedPoint_2Egfp @ A_27a ) @ V0f ) ) ) ) ) ) ) ).
thf(ax_thm_2EfixedPoint_2Efnsum__def,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) )
=> ! [V2X: $i] :
( ( mem @ V2X @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EfixedPoint_2Efnsum @ A_27a @ A_27b ) @ V0f1 ) @ V1f2 ) @ V2X )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ V0f1 @ V2X ) ) @ ( ap @ V1f2 @ V2X ) ) ) ) ) ) ).
thf(conj_thm_2EfixedPoint_2Efnsum__monotone,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27b @ bool ) ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27b @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27b ) @ V0f1 ) )
& ( p @ ( ap @ ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27b ) @ V1f2 ) ) )
=> ( p @ ( ap @ ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2EfixedPoint_2Efnsum @ A_27b @ ( arr @ A_27a @ bool ) ) @ V0f1 ) @ V1f2 ) ) ) ) ) ) ).
thf(ax_thm_2EfixedPoint_2Eempty__def,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2EfixedPoint_2Eempty @ A_27a @ A_27b )
= ( lam @ A_27a
@ ^ [V0X: $i] : ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ).
thf(conj_thm_2EfixedPoint_2Eempty__monotone,axiom,
! [A_27a: del,A_27b: del] : ( p @ ( ap @ ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27b ) @ ( c_2EfixedPoint_2Eempty @ ( arr @ A_27a @ bool ) @ A_27b ) ) ) ).
thf(conj_thm_2EfixedPoint_2Efnsum__empty,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ( ap @ ( ap @ ( c_2EfixedPoint_2Efnsum @ A_27b @ A_27a ) @ V0f ) @ ( c_2EfixedPoint_2Eempty @ A_27a @ A_27b ) )
= V0f )
& ( ( ap @ ( ap @ ( c_2EfixedPoint_2Efnsum @ A_27b @ A_27a ) @ ( c_2EfixedPoint_2Eempty @ A_27a @ A_27b ) ) @ V0f )
= V0f ) ) ) ).
thf(conj_thm_2EfixedPoint_2Efnsum__ASSOC,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2EfixedPoint_2Efnsum @ A_27b @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2EfixedPoint_2Efnsum @ A_27b @ A_27a ) @ V1g ) @ V2h ) )
= ( ap @ ( ap @ ( c_2EfixedPoint_2Efnsum @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2EfixedPoint_2Efnsum @ A_27b @ A_27a ) @ V0f ) @ V1g ) ) @ V2h ) ) ) ) ) ).
thf(conj_thm_2EfixedPoint_2Efnsum__COMM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2EfixedPoint_2Efnsum @ A_27b @ A_27a ) @ V0f ) @ V1g )
= ( ap @ ( ap @ ( c_2EfixedPoint_2Efnsum @ A_27b @ A_27a ) @ V1g ) @ V0f ) ) ) ) ).
thf(conj_thm_2EfixedPoint_2Efnsum__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2X: $i] :
( ( mem @ V2X @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ ( ap @ V0f @ V2X ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EfixedPoint_2Efnsum @ A_27b @ A_27a ) @ V0f ) @ V1g ) @ V2X ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ ( ap @ V1g @ V2X ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EfixedPoint_2Efnsum @ A_27b @ A_27a ) @ V0f ) @ V1g ) @ V2X ) ) ) ) ) ) ) ).
thf(conj_thm_2EfixedPoint_2Elfp__fnsum,axiom,
! [A_27a: del,V0f1: $i] :
( ( mem @ V0f1 @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) )
=> ! [V1f2: $i] :
( ( mem @ V1f2 @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a ) @ V0f1 ) )
& ( p @ ( ap @ ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a ) @ V1f2 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2EfixedPoint_2Elfp @ A_27a ) @ V0f1 ) ) @ ( ap @ ( c_2EfixedPoint_2Elfp @ A_27a ) @ ( ap @ ( ap @ ( c_2EfixedPoint_2Efnsum @ A_27a @ ( arr @ A_27a @ bool ) ) @ V0f1 ) @ V1f2 ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2EfixedPoint_2Elfp @ A_27a ) @ V1f2 ) ) @ ( ap @ ( c_2EfixedPoint_2Elfp @ A_27a ) @ ( ap @ ( ap @ ( c_2EfixedPoint_2Efnsum @ A_27a @ ( arr @ A_27a @ bool ) ) @ V0f1 ) @ V1f2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EfixedPoint_2Elfp__rule__applied,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) )
=> ! [V1X: $i] :
( ( mem @ V1X @ ( arr @ A_27a @ bool ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a ) @ V0f ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1X ) @ ( ap @ ( c_2EfixedPoint_2Elfp @ A_27a ) @ V0f ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2y ) @ ( ap @ V0f @ V1X ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2y ) @ ( ap @ ( c_2EfixedPoint_2Elfp @ A_27a ) @ V0f ) ) ) ) ) ) ) ).
thf(conj_thm_2EfixedPoint_2Elfp__empty,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a ) @ V0f ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ V0f @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2EfixedPoint_2Elfp @ A_27a ) @ V0f ) ) ) ) ) ) ).
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