ITP001 Axioms: ITP056^5.ax
%------------------------------------------------------------------------------
% File : ITP056^5 : 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 : transfer^2.ax [Gau20]
% : HL4056^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 27 ( 1 unt; 8 typ; 0 def)
% Number of atoms : 259 ( 3 equ; 0 cnn)
% Maximal formula atoms : 31 ( 9 avg)
% Number of connectives : 537 ( 0 ~; 0 |; 8 &; 477 @)
% ( 8 <=>; 44 =>; 0 <=; 0 <~>)
% Maximal formula depth : 32 ( 15 avg; 477 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 20 ( 20 >; 0 *; 0 +; 0 <<)
% Number of symbols : 19 ( 18 usr; 10 con; 0-4 aty)
% Number of variables : 92 ( 2 ^ 88 !; 2 ?; 92 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Etransfer_2EFUN__REL,type,
c_2Etransfer_2EFUN__REL: del > del > del > del > $i ).
thf(mem_c_2Etransfer_2EFUN__REL,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : ( mem @ ( c_2Etransfer_2EFUN__REL @ A_27a @ A_27b @ A_27c @ A_27d ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( arr @ A_27c @ ( arr @ A_27d @ bool ) ) @ ( arr @ ( arr @ A_27a @ A_27c ) @ ( arr @ ( arr @ A_27b @ A_27d ) @ bool ) ) ) ) ) ).
thf(tp_c_2Etransfer_2EPAIR__REL,type,
c_2Etransfer_2EPAIR__REL: del > del > del > del > $i ).
thf(mem_c_2Etransfer_2EPAIR__REL,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : ( mem @ ( c_2Etransfer_2EPAIR__REL @ A_27a @ A_27b @ A_27c @ A_27d ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( arr @ A_27c @ ( arr @ A_27d @ bool ) ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27d ) @ bool ) ) ) ) ) ).
thf(tp_c_2Etransfer_2Ebi__unique,type,
c_2Etransfer_2Ebi__unique: del > del > $i ).
thf(mem_c_2Etransfer_2Ebi__unique,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Etransfer_2Ebi__unique @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ bool ) ) ).
thf(tp_c_2Etransfer_2Ebitotal,type,
c_2Etransfer_2Ebitotal: del > del > $i ).
thf(mem_c_2Etransfer_2Ebitotal,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Etransfer_2Ebitotal @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ bool ) ) ).
thf(tp_c_2Etransfer_2Eleft__unique,type,
c_2Etransfer_2Eleft__unique: del > del > $i ).
thf(mem_c_2Etransfer_2Eleft__unique,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Etransfer_2Eleft__unique @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ bool ) ) ).
thf(tp_c_2Etransfer_2Eright__unique,type,
c_2Etransfer_2Eright__unique: del > del > $i ).
thf(mem_c_2Etransfer_2Eright__unique,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Etransfer_2Eright__unique @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ bool ) ) ).
thf(tp_c_2Etransfer_2Esurj,type,
c_2Etransfer_2Esurj: del > del > $i ).
thf(mem_c_2Etransfer_2Esurj,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Etransfer_2Esurj @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ bool ) ) ).
thf(tp_c_2Etransfer_2Etotal,type,
c_2Etransfer_2Etotal: del > del > $i ).
thf(mem_c_2Etransfer_2Etotal,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Etransfer_2Etotal @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ bool ) ) ).
thf(ax_thm_2Etransfer_2Eright__unique__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Etransfer_2Eright__unique @ A_27a @ A_27b ) @ V0R ) )
<=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2b1: $i] :
( ( mem @ V2b1 @ A_27b )
=> ! [V3b2: $i] :
( ( mem @ V3b2 @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V1a ) @ V2b1 ) )
& ( p @ ( ap @ ( ap @ V0R @ V1a ) @ V3b2 ) ) )
=> ( V2b1 = V3b2 ) ) ) ) ) ) ) ).
thf(ax_thm_2Etransfer_2Eleft__unique__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Etransfer_2Eleft__unique @ A_27a @ A_27b ) @ V0R ) )
<=> ! [V1a1: $i] :
( ( mem @ V1a1 @ A_27a )
=> ! [V2a2: $i] :
( ( mem @ V2a2 @ A_27a )
=> ! [V3b: $i] :
( ( mem @ V3b @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V1a1 ) @ V3b ) )
& ( p @ ( ap @ ( ap @ V0R @ V2a2 ) @ V3b ) ) )
=> ( V1a1 = V2a2 ) ) ) ) ) ) ) ).
thf(ax_thm_2Etransfer_2Ebi__unique__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Etransfer_2Ebi__unique @ A_27a @ A_27b ) @ V0R ) )
<=> ( ( p @ ( ap @ ( c_2Etransfer_2Eleft__unique @ A_27a @ A_27b ) @ V0R ) )
& ( p @ ( ap @ ( c_2Etransfer_2Eright__unique @ A_27a @ A_27b ) @ V0R ) ) ) ) ) ).
thf(ax_thm_2Etransfer_2Etotal__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Etransfer_2Etotal @ A_27a @ A_27b ) @ V0R ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ? [V2y: $i] :
( ( mem @ V2y @ A_27b )
& ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) ) ) ) ) ) ).
thf(ax_thm_2Etransfer_2Esurj__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Etransfer_2Esurj @ A_27a @ A_27b ) @ V0R ) )
<=> ! [V1y: $i] :
( ( mem @ V1y @ A_27b )
=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V1y ) ) ) ) ) ) ).
thf(ax_thm_2Etransfer_2Ebitotal__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Etransfer_2Ebitotal @ A_27a @ A_27b ) @ V0R ) )
<=> ( ( p @ ( ap @ ( c_2Etransfer_2Etotal @ A_27a @ A_27b ) @ V0R ) )
& ( p @ ( ap @ ( c_2Etransfer_2Esurj @ A_27a @ A_27b ) @ V0R ) ) ) ) ) ).
thf(ax_thm_2Etransfer_2EFUN__REL__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0AB: $i] :
( ( mem @ V0AB @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1CD: $i] :
( ( mem @ V1CD @ ( arr @ A_27c @ ( arr @ A_27d @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27c ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27b @ A_27d ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Etransfer_2EFUN__REL @ A_27a @ A_27b @ A_27c @ A_27d ) @ V0AB ) @ V1CD ) @ V2f ) @ V3g ) )
<=> ! [V4a: $i] :
( ( mem @ V4a @ A_27a )
=> ! [V5b: $i] :
( ( mem @ V5b @ A_27b )
=> ( ( p @ ( ap @ ( ap @ V0AB @ V4a ) @ V5b ) )
=> ( p @ ( ap @ ( ap @ V1CD @ ( ap @ V2f @ V4a ) ) @ ( ap @ V3g @ V5b ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransfer_2EFUN__REL__COMB,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0AB: $i] :
( ( mem @ V0AB @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1CD: $i] :
( ( mem @ V1CD @ ( arr @ A_27c @ ( arr @ A_27d @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27c ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27b @ A_27d ) )
=> ! [V4a: $i] :
( ( mem @ V4a @ A_27a )
=> ! [V5b: $i] :
( ( mem @ V5b @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Etransfer_2EFUN__REL @ A_27a @ A_27b @ A_27c @ A_27d ) @ V0AB ) @ V1CD ) @ V2f ) @ V3g ) )
& ( p @ ( ap @ ( ap @ V0AB @ V4a ) @ V5b ) ) )
=> ( p @ ( ap @ ( ap @ V1CD @ ( ap @ V2f @ V4a ) ) @ ( ap @ V3g @ V5b ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransfer_2EFUN__REL__ABS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0AB: $i] :
( ( mem @ V0AB @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1CD: $i] :
( ( mem @ V1CD @ ( arr @ A_27c @ ( arr @ A_27d @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27c ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27b @ A_27d ) )
=> ( ! [V4a: $i] :
( ( mem @ V4a @ A_27a )
=> ! [V5b: $i] :
( ( mem @ V5b @ A_27b )
=> ( ( p @ ( ap @ ( ap @ V0AB @ V4a ) @ V5b ) )
=> ( p @ ( ap @ ( ap @ V1CD @ ( ap @ V2f @ V4a ) ) @ ( ap @ V3g @ V5b ) ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( ap @ ( ap @ ( c_2Etransfer_2EFUN__REL @ A_27a @ A_27b @ A_27c @ A_27d ) @ V0AB ) @ V1CD )
@ ( lam @ A_27a
@ ^ [V6a: $i] : ( ap @ V2f @ V6a ) ) )
@ ( lam @ A_27b
@ ^ [V7b: $i] : ( ap @ V3g @ V7b ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransfer_2EFUN__REL__EQ2,axiom,
! [A_27a: del,A_27b: del] :
( ( ap @ ( ap @ ( c_2Etransfer_2EFUN__REL @ A_27a @ A_27a @ A_27b @ A_27b ) @ ( c_2Emin_2E_3D @ A_27a ) ) @ ( c_2Emin_2E_3D @ A_27b ) )
= ( c_2Emin_2E_3D @ ( arr @ A_27a @ A_27b ) ) ) ).
thf(ax_thm_2Etransfer_2EPAIR__REL__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0AB: $i] :
( ( mem @ V0AB @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1CD: $i] :
( ( mem @ V1CD @ ( arr @ A_27c @ ( arr @ A_27d @ bool ) ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3c: $i] :
( ( mem @ V3c @ A_27c )
=> ! [V4b: $i] :
( ( mem @ V4b @ A_27b )
=> ! [V5d: $i] :
( ( mem @ V5d @ A_27d )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Etransfer_2EPAIR__REL @ A_27a @ A_27b @ A_27c @ A_27d ) @ V0AB ) @ V1CD ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27c ) @ V2a ) @ V3c ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27d ) @ V4b ) @ V5d ) ) )
<=> ( ( p @ ( ap @ ( ap @ V0AB @ V2a ) @ V4b ) )
& ( p @ ( ap @ ( ap @ V1CD @ V3c ) @ V5d ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------