ITP001 Axioms: ITP103^5.ax
%------------------------------------------------------------------------------
% File : ITP103^5 : 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 : tc^2.ax [Gau20]
% : HL4103^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 50 ( 0 unt; 8 typ; 0 def)
% Number of atoms : 815 ( 36 equ; 0 cnn)
% Maximal formula atoms : 59 ( 16 avg)
% Number of connectives : 1471 ( 2 ~; 5 |; 20 &;1348 @)
% ( 6 <=>; 90 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 12 avg;1348 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 8 ( 8 >; 0 *; 0 +; 0 <<)
% Number of symbols : 38 ( 37 usr; 29 con; 0-2 aty)
% Number of variables : 128 ( 0 ^ 124 !; 4 ?; 128 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Etc_2EFMAP__TO__RELN,type,
c_2Etc_2EFMAP__TO__RELN: del > $i ).
thf(mem_c_2Etc_2EFMAP__TO__RELN,axiom,
! [A_27a: del] : ( mem @ ( c_2Etc_2EFMAP__TO__RELN @ A_27a ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Etc_2ERELN__TO__FMAP,type,
c_2Etc_2ERELN__TO__FMAP: del > $i ).
thf(mem_c_2Etc_2ERELN__TO__FMAP,axiom,
! [A_27a: del] : ( mem @ ( c_2Etc_2ERELN__TO__FMAP @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Etc_2ETC__ITER,type,
c_2Etc_2ETC__ITER: del > $i ).
thf(mem_c_2Etc_2ETC__ITER,axiom,
! [A_27a: del] : ( mem @ ( c_2Etc_2ETC__ITER @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ) ).
thf(tp_c_2Etc_2ETC__MOD,type,
c_2Etc_2ETC__MOD: del > $i ).
thf(mem_c_2Etc_2ETC__MOD,axiom,
! [A_27a: del] : ( mem @ ( c_2Etc_2ETC__MOD @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) ) ) ) ).
thf(tp_c_2Etc_2E_5E_7C,type,
c_2Etc_2E_5E_7C: del > $i ).
thf(mem_c_2Etc_2E_5E_7C,axiom,
! [A_27a: del] : ( mem @ ( c_2Etc_2E_5E_7C @ 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_2Etc_2E_5E_7C_5E,type,
c_2Etc_2E_5E_7C_5E: del > $i ).
thf(mem_c_2Etc_2E_5E_7C_5E,axiom,
! [A_27a: del] : ( mem @ ( c_2Etc_2E_5E_7C_5E @ 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_2Etc_2EsubTC,type,
c_2Etc_2EsubTC: del > $i ).
thf(mem_c_2Etc_2EsubTC,axiom,
! [A_27a: del] : ( mem @ ( c_2Etc_2EsubTC @ 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_2Etc_2E_7C_5E,type,
c_2Etc_2E_7C_5E: del > $i ).
thf(mem_c_2Etc_2E_7C_5E,axiom,
! [A_27a: del] : ( mem @ ( c_2Etc_2E_7C_5E @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ) ).
thf(ax_thm_2Etc_2EDRESTR,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3b: $i] :
( ( mem @ V3b @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Etc_2E_5E_7C @ A_27a ) @ V0R ) @ V1s ) @ V2a ) @ V3b ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2a ) @ V1s ) )
& ( p @ ( ap @ ( ap @ V0R @ V2a ) @ V3b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etc_2EDRESTR__IN,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Etc_2E_5E_7C @ A_27a ) @ V0R ) @ V1s ) @ V2a )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2a ) @ V1s ) ) @ ( ap @ V0R @ V2a ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ).
thf(ax_thm_2Etc_2ERRESTR,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3b: $i] :
( ( mem @ V3b @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Etc_2E_7C_5E @ A_27a ) @ V0R ) @ V1s ) @ V2a ) @ V3b ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3b ) @ V1s ) )
& ( p @ ( ap @ ( ap @ V0R @ V2a ) @ V3b ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Etc_2EBRESTR,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_2Etc_2E_5E_7C_5E @ A_27a ) @ V0R ) @ V1s )
= ( ap @ ( ap @ ( c_2Etc_2E_7C_5E @ A_27a ) @ ( ap @ ( ap @ ( c_2Etc_2E_5E_7C @ A_27a ) @ V0R ) @ V1s ) ) @ V1s ) ) ) ) ).
thf(conj_thm_2Etc_2EDRESTR__EMPTY,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Etc_2E_5E_7C @ A_27a ) @ V0R ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Erelation_2EEMPTY__REL @ A_27a ) ) ) ).
thf(conj_thm_2Etc_2EDRESTR__RDOM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Etc_2E_5E_7C @ A_27a ) @ V0R ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a ) @ V0R ) )
= V0R ) ) ).
thf(conj_thm_2Etc_2EREMPTY__RRESTR,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Etc_2E_7C_5E @ A_27a ) @ ( c_2Erelation_2EEMPTY__REL @ A_27a ) ) @ V0s )
= ( c_2Erelation_2EEMPTY__REL @ A_27a ) ) ) ).
thf(conj_thm_2Etc_2EO__REMPTY__O,axiom,
! [A_27a: del] :
( ! [V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Erelation_2EO @ A_27a @ A_27a @ A_27a ) @ V0R ) @ ( c_2Erelation_2EEMPTY__REL @ A_27a ) )
= ( c_2Erelation_2EEMPTY__REL @ A_27a ) ) )
& ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Erelation_2EO @ A_27a @ A_27a @ A_27a ) @ ( c_2Erelation_2EEMPTY__REL @ A_27a ) ) @ V1R )
= ( c_2Erelation_2EEMPTY__REL @ A_27a ) ) ) ) ).
thf(ax_thm_2Etc_2EsubTC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ 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 @ ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ V1s ) @ V2x ) @ V3y ) )
<=> ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) )
| ? [V4a: $i] :
( ( mem @ V4a @ A_27a )
& ? [V5b: $i] :
( ( mem @ V5b @ A_27a )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ ( ap @ ( ap @ ( c_2Etc_2E_5E_7C_5E @ A_27a ) @ V0R ) @ V1s ) ) @ V4a ) @ V5b ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4a ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5b ) @ V1s ) )
& ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V4a ) )
& ( p @ ( ap @ ( ap @ V0R @ V5b ) @ V3y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etc_2EsubTC__thm,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_2Etc_2EsubTC @ A_27a ) @ V0R ) @ V1s )
= ( ap @ ( ap @ ( c_2Erelation_2ERUNION @ A_27a @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Erelation_2EO @ A_27a @ A_27a @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Erelation_2EO @ A_27a @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Etc_2E_5E_7C @ A_27a ) @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ ( ap @ ( ap @ ( c_2Etc_2E_5E_7C_5E @ A_27a ) @ V0R ) @ V1s ) ) ) @ V1s ) ) @ V0R ) ) ) ) ) ) ).
thf(conj_thm_2Etc_2EsubTC__EMPTY,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= V0R ) ) ).
thf(conj_thm_2Etc_2ERTC__INSERT,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3w: $i] :
( ( mem @ V3w @ A_27a )
=> ! [V4z: $i] :
( ( mem @ V4z @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ ( ap @ ( ap @ ( c_2Etc_2E_5E_7C_5E @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2a ) @ V1s ) ) ) @ V3w ) @ V4z ) )
<=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ ( ap @ ( ap @ ( c_2Etc_2E_5E_7C_5E @ A_27a ) @ V0R ) @ V1s ) ) @ V3w ) @ V4z ) )
| ( ( ( V2a = V3w )
| ? [V5x: $i] :
( ( mem @ V5x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ ( ap @ ( ap @ ( c_2Etc_2E_5E_7C_5E @ A_27a ) @ V0R ) @ V1s ) ) @ V3w ) @ V5x ) )
& ( p @ ( ap @ ( ap @ V0R @ V5x ) @ V2a ) ) ) )
& ( ( V2a = V4z )
| ? [V6y: $i] :
( ( mem @ V6y @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6y ) @ V1s ) )
& ( p @ ( ap @ ( ap @ V0R @ V2a ) @ V6y ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ ( ap @ ( ap @ ( c_2Etc_2E_5E_7C_5E @ A_27a ) @ V0R ) @ V1s ) ) @ V6y ) @ V4z ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etc_2EsubTC__INSERT,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2q: $i] :
( ( mem @ V2q @ A_27a )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2q ) @ V1s ) ) @ V3x ) @ V4y ) )
<=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ V1s ) @ V3x ) @ V4y ) )
| ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ V1s ) @ V3x ) @ V2q ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ V1s ) @ V2q ) @ V4y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etc_2EsubTC__RDOM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) ) ) ).
thf(conj_thm_2Etc_2EsubTC__INSERT__COR,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ V1s ) ) @ V3a )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ V1s ) @ V3a ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ V1s ) @ V3a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ V1s ) @ V2x ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ V1s ) @ V3a ) ) ) ) ) ) ) ).
thf(ax_thm_2Etc_2EFMAP__TO__RELN,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Etc_2EFMAP__TO__RELN @ A_27a ) @ V0f ) @ V1x )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( arr @ A_27a @ bool ) ) @ V0f ) ) ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ ( arr @ A_27a @ bool ) ) @ V0f ) @ V1x ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(ax_thm_2Etc_2ERELN__TO__FMAP,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( c_2Etc_2ERELN__TO__FMAP @ A_27a ) @ V0R )
= ( ap @ ( ap @ ( c_2Efinite__map_2EFUN__FMAP @ A_27a @ ( arr @ A_27a @ bool ) ) @ V0R ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Etc_2ERDOM__SUBSET__FDOM,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a ) @ ( ap @ ( c_2Etc_2EFMAP__TO__RELN @ A_27a ) @ V0f ) ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( arr @ A_27a @ bool ) ) @ V0f ) ) ) ) ).
thf(conj_thm_2Etc_2EFINITE__RDOM,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a ) @ ( ap @ ( c_2Etc_2EFMAP__TO__RELN @ A_27a ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Etc_2EFDOM__RDOM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a ) @ V0R ) ) )
=> ( ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Etc_2ERELN__TO__FMAP @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Etc_2ERELN__TO__FMAP__TO__RELN__ID,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a ) @ V0R ) ) )
=> ( ( ap @ ( c_2Etc_2EFMAP__TO__RELN @ A_27a ) @ ( ap @ ( c_2Etc_2ERELN__TO__FMAP @ A_27a ) @ V0R ) )
= V0R ) ) ) ).
thf(conj_thm_2Etc_2ERDOM__subTC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ V1s ) )
= ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Etc_2ENOT__IN__RDOM,axiom,
! [A_27a: del,V0Q: $i] :
( ( mem @ V0Q @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( ( ap @ V0Q @ V1x )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
<=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a ) @ V0Q ) ) ) ) ) ) ).
thf(ax_thm_2Etc_2ETC__MOD,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1rx: $i] :
( ( mem @ V1rx @ ( arr @ A_27a @ bool ) )
=> ! [V2ra: $i] :
( ( mem @ V2ra @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Etc_2ETC__MOD @ A_27a ) @ V0x ) @ V1rx ) @ V2ra )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V2ra ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V2ra ) @ V1rx ) ) @ V2ra ) ) ) ) ) ).
thf(conj_thm_2Etc_2ETC__MOD__EMPTY__ID,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1ra: $i] :
( ( mem @ V1ra @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Etc_2ETC__MOD @ A_27a ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Ecombin_2EI @ ( arr @ A_27a @ bool ) ) ) ) ) ).
thf(conj_thm_2Etc_2EI__o__f,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efinite__map_2Efmap @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27a @ A_27b @ A_27b ) @ ( c_2Ecombin_2EI @ A_27b ) ) @ V0f )
= V0f ) ) ).
thf(conj_thm_2Etc_2EsubTC__MAX__RDOM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ 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 ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a ) @ V0R ) ) )
=> ( ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ V1s ) )
= ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Etc_2EsubTC__SUPERSET__RDOM,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 @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a ) @ V0R ) ) @ V1s ) )
= ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a ) @ V0R ) ) ) ) ) ) ).
thf(conj_thm_2Etc_2EsubTC__FDOM,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V1R ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a ) @ V1R ) )
= ( ap @ ( c_2Etc_2EFMAP__TO__RELN @ A_27a ) @ V0g ) )
=> ( ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V1R ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( arr @ A_27a @ bool ) ) @ V0g ) )
= ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V1R ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a ) @ V1R ) ) ) ) ) ) ).
thf(conj_thm_2Etc_2ESUBSET__FDOM__LEM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ V1s )
= ( ap @ ( c_2Etc_2EFMAP__TO__RELN @ A_27a ) @ V2f ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a ) @ V0R ) ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( arr @ A_27a @ bool ) ) @ V2f ) ) ) ) ) ) ) ).
thf(conj_thm_2Etc_2EsubTC__FDOM__RDOM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( arr @ A_27a @ bool ) ) @ V1f ) )
= ( ap @ ( c_2Etc_2EFMAP__TO__RELN @ A_27a ) @ V1f ) )
=> ( ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Etc_2EFMAP__TO__RELN @ A_27a ) @ V1f ) ) ) ) ) ).
thf(conj_thm_2Etc_2ETC__MOD__LEM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( arr @ A_27a @ bool ) ) @ V3f ) ) )
& ( ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ V1s )
= ( ap @ ( c_2Etc_2EFMAP__TO__RELN @ A_27a ) @ V3f ) ) )
=> ( ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ V1s ) )
= ( ap @ ( c_2Etc_2EFMAP__TO__RELN @ A_27a ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27a @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Etc_2ETC__MOD @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ ( arr @ A_27a @ bool ) ) @ V3f ) @ V2x ) ) ) @ V3f ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Etc_2ETC__ITER,axiom,
! [A_27a: del] :
( ! [V0r: $i] :
( ( mem @ V0r @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Etc_2ETC__ITER @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V0r )
= V0r ) )
& ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2d: $i] :
( ( mem @ V2d @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3r: $i] :
( ( mem @ V3r @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Etc_2ETC__ITER @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ V2d ) ) @ V3r )
= ( ap @ ( ap @ ( c_2Etc_2ETC__ITER @ A_27a ) @ V2d ) @ ( ap @ ( ap @ ( c_2Efinite__map_2Eo__f @ A_27a @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Etc_2ETC__MOD @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ ( arr @ A_27a @ bool ) ) @ V3r ) @ V1x ) ) ) @ V3r ) ) ) ) ) ) ) ).
thf(conj_thm_2Etc_2ETC__ITER__THM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1d: $i] :
( ( mem @ V1d @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( ty_2Efinite__map_2Efmap @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ( ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V3s ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1d ) )
= ( ap @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( arr @ A_27a @ bool ) ) @ V2f ) )
& ( ( ap @ ( ap @ ( c_2Etc_2EsubTC @ A_27a ) @ V0R ) @ V3s )
= ( ap @ ( c_2Etc_2EFMAP__TO__RELN @ A_27a ) @ V2f ) ) )
=> ( ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R )
= ( ap @ ( c_2Etc_2EFMAP__TO__RELN @ A_27a ) @ ( ap @ ( ap @ ( c_2Etc_2ETC__ITER @ A_27a ) @ V1d ) @ V2f ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------