ITP001 Axioms: ITP056^7.ax
%------------------------------------------------------------------------------
% File : ITP056^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : transfer.ax [Gau19]
% : HL4056^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 37 ( 12 unt; 19 typ; 0 def)
% Number of atoms : 30 ( 4 equ; 1 cnn)
% Maximal formula atoms : 3 ( 0 avg)
% Number of connectives : 169 ( 1 ~; 1 |; 7 &; 138 @)
% ( 15 <=>; 7 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 10 avg; 138 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 92 ( 92 >; 0 *; 0 +; 0 <<)
% Number of symbols : 20 ( 18 usr; 1 con; 0-8 aty)
% Number of variables : 113 ( 2 ^ 83 !; 3 ?; 113 :)
% ( 25 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Etransfer_2EFUN__REL,type,
c_2Etransfer_2EFUN__REL:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( A_27a > A_27b > $o ) > ( A_27c > A_27d > $o ) > ( A_27a > A_27c ) > ( A_27b > A_27d ) > $o ) ).
thf(c_2Etransfer_2EPAIR__REL,type,
c_2Etransfer_2EPAIR__REL:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( A_27a > A_27b > $o ) > ( A_27c > A_27d > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) > ( tyop_2Epair_2Eprod @ A_27b @ A_27d ) > $o ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Etransfer_2Ebi__unique,type,
c_2Etransfer_2Ebi__unique:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > $o ) ).
thf(c_2Etransfer_2Ebitotal,type,
c_2Etransfer_2Ebitotal:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > $o ) ).
thf(c_2Etransfer_2Eleft__unique,type,
c_2Etransfer_2Eleft__unique:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > $o ) ).
thf(c_2Etransfer_2Eright__unique,type,
c_2Etransfer_2Eright__unique:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > $o ) ).
thf(c_2Etransfer_2Esurj,type,
c_2Etransfer_2Esurj:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > $o ) ).
thf(c_2Etransfer_2Etotal,type,
c_2Etransfer_2Etotal:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > $o ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Etransfer_2Eright__unique__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o] :
( ( c_2Etransfer_2Eright__unique @ A_27a @ A_27b @ V0R )
<=> ! [V1a: A_27a,V2b1: A_27b,V3b2: A_27b] :
( ( ( V0R @ V1a @ V2b1 )
& ( V0R @ V1a @ V3b2 ) )
=> ( V2b1 = V3b2 ) ) ) ).
thf(thm_2Etransfer_2Eleft__unique__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o] :
( ( c_2Etransfer_2Eleft__unique @ A_27a @ A_27b @ V0R )
<=> ! [V1a1: A_27a,V2a2: A_27a,V3b: A_27b] :
( ( ( V0R @ V1a1 @ V3b )
& ( V0R @ V2a2 @ V3b ) )
=> ( V1a1 = V2a2 ) ) ) ).
thf(thm_2Etransfer_2Ebi__unique__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o] :
( ( c_2Etransfer_2Ebi__unique @ A_27a @ A_27b @ V0R )
<=> ( ( c_2Etransfer_2Eleft__unique @ A_27a @ A_27b @ V0R )
& ( c_2Etransfer_2Eright__unique @ A_27a @ A_27b @ V0R ) ) ) ).
thf(thm_2Etransfer_2Etotal__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o] :
( ( c_2Etransfer_2Etotal @ A_27a @ A_27b @ V0R )
<=> ! [V1x: A_27a] :
? [V2y: A_27b] : ( V0R @ V1x @ V2y ) ) ).
thf(thm_2Etransfer_2Esurj__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o] :
( ( c_2Etransfer_2Esurj @ A_27a @ A_27b @ V0R )
<=> ! [V1y: A_27b] :
? [V2x: A_27a] : ( V0R @ V2x @ V1y ) ) ).
thf(thm_2Etransfer_2Ebitotal__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o] :
( ( c_2Etransfer_2Ebitotal @ A_27a @ A_27b @ V0R )
<=> ( ( c_2Etransfer_2Etotal @ A_27a @ A_27b @ V0R )
& ( c_2Etransfer_2Esurj @ A_27a @ A_27b @ V0R ) ) ) ).
thf(thm_2Etransfer_2EFUN__REL__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0AB: A_27a > A_27b > $o,V1CD: A_27c > A_27d > $o,V2f: A_27a > A_27c,V3g: A_27b > A_27d] :
( ( c_2Etransfer_2EFUN__REL @ A_27a @ A_27b @ A_27c @ A_27d @ V0AB @ V1CD @ V2f @ V3g )
<=> ! [V4a: A_27a,V5b: A_27b] :
( ( V0AB @ V4a @ V5b )
=> ( V1CD @ ( V2f @ V4a ) @ ( V3g @ V5b ) ) ) ) ).
thf(thm_2Etransfer_2EPAIR__REL__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0AB: A_27a > A_27b > $o,V1CD: A_27c > A_27d > $o,V2a: A_27a,V3c: A_27c,V4b: A_27b,V5d: A_27d] :
( ( c_2Etransfer_2EPAIR__REL @ A_27a @ A_27b @ A_27c @ A_27d @ V0AB @ V1CD @ ( c_2Epair_2E_2C @ A_27a @ A_27c @ V2a @ V3c ) @ ( c_2Epair_2E_2C @ A_27b @ A_27d @ V4b @ V5d ) )
<=> ( ( V0AB @ V2a @ V4b )
& ( V1CD @ V3c @ V5d ) ) ) ).
thf(thm_2Etransfer_2EFUN__REL__COMB,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0g: A_27b > A_27d,V1f: A_27a > A_27c,V2b: A_27b,V3a: A_27a,V4CD: A_27c > A_27d > $o,V5AB: A_27a > A_27b > $o] :
( ( ( c_2Etransfer_2EFUN__REL @ A_27a @ A_27b @ A_27c @ A_27d @ V5AB @ V4CD @ V1f @ V0g )
& ( V5AB @ V3a @ V2b ) )
=> ( V4CD @ ( V1f @ V3a ) @ ( V0g @ V2b ) ) ) ).
thf(thm_2Etransfer_2EFUN__REL__ABS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0g: A_27b > A_27d,V1f: A_27a > A_27c,V2CD: A_27c > A_27d > $o,V3AB: A_27a > A_27b > $o] :
( ! [V4a: A_27a,V5b: A_27b] :
( ( V3AB @ V4a @ V5b )
=> ( V2CD @ ( V1f @ V4a ) @ ( V0g @ V5b ) ) )
=> ( c_2Etransfer_2EFUN__REL @ A_27a @ A_27b @ A_27c @ A_27d @ V3AB @ V2CD
@ ^ [V6a: A_27a] : ( V1f @ V6a )
@ ^ [V7b: A_27b] : ( V0g @ V7b ) ) ) ).
thf(thm_2Etransfer_2EFUN__REL__EQ2,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( 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 @ ( A_27a > A_27b ) ) ) ).
%------------------------------------------------------------------------------