ITP001 Axioms: ITP028^5.ax


%------------------------------------------------------------------------------
% File     : ITP028^5 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 set theory export, chainy mode
% Version  : [BG+19] axioms.
% English  :

% Refs     : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
%          : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source   : [BG+19]
% Names    : 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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