ITP001 Axioms: ITP028^7.ax
%------------------------------------------------------------------------------
% File : ITP028^7 : 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 : fixedPoint.ax [Gau19]
% : HL4028^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 57 ( 13 unt; 26 typ; 0 def)
% Number of atoms : 125 ( 15 equ; 1 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 382 ( 1 ~; 1 |; 13 &; 337 @)
% ( 8 <=>; 22 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 8 avg; 337 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 188 ( 188 >; 0 *; 0 +; 0 <<)
% Number of symbols : 27 ( 25 usr; 1 con; 0-6 aty)
% Number of variables : 122 ( 3 ^ 94 !; 1 ?; 122 :)
% ( 24 !>; 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_2Epred__set_2EBIGINTER,type,
c_2Epred__set_2EBIGINTER:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EBIGUNION,type,
c_2Epred__set_2EBIGUNION:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EEMPTY,type,
c_2Epred__set_2EEMPTY:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Epred__set_2EGSPEC,type,
c_2Epred__set_2EGSPEC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ) ) > A_27a > $o ) ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EINTER,type,
c_2Epred__set_2EINTER:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EUNION,type,
c_2Epred__set_2EUNION:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2EfixedPoint_2Eclosed,type,
c_2EfixedPoint_2Eclosed:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2EfixedPoint_2Edense,type,
c_2EfixedPoint_2Edense:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2EfixedPoint_2Eempty,type,
c_2EfixedPoint_2Eempty:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > $o ) ).
thf(c_2EfixedPoint_2Efnsum,type,
c_2EfixedPoint_2Efnsum:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > A_27a > $o ) > ( A_27b > A_27a > $o ) > A_27b > A_27a > $o ) ).
thf(c_2EfixedPoint_2Egfp,type,
c_2EfixedPoint_2Egfp:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > A_27a > $o ) > A_27a > $o ) ).
thf(c_2EfixedPoint_2Elfp,type,
c_2EfixedPoint_2Elfp:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > A_27a > $o ) > A_27a > $o ) ).
thf(c_2EfixedPoint_2Emonotone,type,
c_2EfixedPoint_2Emonotone:
!>[A_27a: $tType,A_27b: $tType] : ( ( ( A_27a > $o ) > 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_2EfixedPoint_2Emonotone__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: ( A_27a > $o ) > A_27b > $o] :
( ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27b @ V0f )
<=> ! [V1X: A_27a > $o,V2Y: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V1X @ V2Y )
=> ( c_2Epred__set_2ESUBSET @ A_27b @ ( V0f @ V1X ) @ ( V0f @ V2Y ) ) ) ) ).
thf(thm_2EfixedPoint_2Elfp__def,axiom,
! [A_27a: $tType,V0f: ( A_27a > $o ) > A_27a > $o] :
( ( c_2EfixedPoint_2Elfp @ A_27a @ V0f )
= ( c_2Epred__set_2EBIGINTER @ A_27a
@ ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V1X: A_27a > $o] : ( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ V1X @ ( c_2Epred__set_2ESUBSET @ A_27a @ ( V0f @ V1X ) @ V1X ) ) ) ) ) ).
thf(thm_2EfixedPoint_2Egfp__def,axiom,
! [A_27a: $tType,V0f: ( A_27a > $o ) > A_27a > $o] :
( ( c_2EfixedPoint_2Egfp @ A_27a @ V0f )
= ( c_2Epred__set_2EBIGUNION @ A_27a
@ ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V1X: A_27a > $o] : ( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ V1X @ ( c_2Epred__set_2ESUBSET @ A_27a @ V1X @ ( V0f @ V1X ) ) ) ) ) ) ).
thf(thm_2EfixedPoint_2Eclosed__def,axiom,
! [A_27a: $tType,V0f: ( A_27a > $o ) > A_27a > $o,V1X: A_27a > $o] :
( ( c_2EfixedPoint_2Eclosed @ A_27a @ V0f @ V1X )
= ( c_2Epred__set_2ESUBSET @ A_27a @ ( V0f @ V1X ) @ V1X ) ) ).
thf(thm_2EfixedPoint_2Edense__def,axiom,
! [A_27a: $tType,V0f: ( A_27a > $o ) > A_27a > $o,V1X: A_27a > $o] :
( ( c_2EfixedPoint_2Edense @ A_27a @ V0f @ V1X )
= ( c_2Epred__set_2ESUBSET @ A_27a @ V1X @ ( V0f @ V1X ) ) ) ).
thf(thm_2EfixedPoint_2Efnsum__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1: A_27b > A_27a > $o,V1f2: A_27b > A_27a > $o,V2X: A_27b] :
( ( c_2EfixedPoint_2Efnsum @ A_27a @ A_27b @ V0f1 @ V1f2 @ V2X )
= ( c_2Epred__set_2EUNION @ A_27a @ ( V0f1 @ V2X ) @ ( V1f2 @ V2X ) ) ) ).
thf(thm_2EfixedPoint_2Eempty__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2EfixedPoint_2Eempty @ A_27a @ A_27b )
= ( ^ [V0X: A_27a] : ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ).
thf(thm_2EfixedPoint_2Elfp__least__closed,axiom,
! [A_27a: $tType,V0f: ( A_27a > $o ) > A_27a > $o] :
( ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a @ V0f )
=> ( ( c_2EfixedPoint_2Eclosed @ A_27a @ V0f @ ( c_2EfixedPoint_2Elfp @ A_27a @ V0f ) )
& ! [V1X: A_27a > $o] :
( ( c_2EfixedPoint_2Eclosed @ A_27a @ V0f @ V1X )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2EfixedPoint_2Elfp @ A_27a @ V0f ) @ V1X ) ) ) ) ).
thf(thm_2EfixedPoint_2Egfp__greatest__dense,axiom,
! [A_27a: $tType,V0f: ( A_27a > $o ) > A_27a > $o] :
( ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a @ V0f )
=> ( ( c_2EfixedPoint_2Edense @ A_27a @ V0f @ ( c_2EfixedPoint_2Egfp @ A_27a @ V0f ) )
& ! [V1X: A_27a > $o] :
( ( c_2EfixedPoint_2Edense @ A_27a @ V0f @ V1X )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ V1X @ ( c_2EfixedPoint_2Egfp @ A_27a @ V0f ) ) ) ) ) ).
thf(thm_2EfixedPoint_2Elfp__fixedpoint,axiom,
! [A_27a: $tType,V0f: ( A_27a > $o ) > A_27a > $o] :
( ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a @ V0f )
=> ( ( ( c_2EfixedPoint_2Elfp @ A_27a @ V0f )
= ( V0f @ ( c_2EfixedPoint_2Elfp @ A_27a @ V0f ) ) )
& ! [V1X: A_27a > $o] :
( ( V1X
= ( V0f @ V1X ) )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2EfixedPoint_2Elfp @ A_27a @ V0f ) @ V1X ) ) ) ) ).
thf(thm_2EfixedPoint_2Egfp__greatest__fixedpoint,axiom,
! [A_27a: $tType,V0f: ( A_27a > $o ) > A_27a > $o] :
( ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a @ V0f )
=> ( ( ( c_2EfixedPoint_2Egfp @ A_27a @ V0f )
= ( V0f @ ( c_2EfixedPoint_2Egfp @ A_27a @ V0f ) ) )
& ! [V1X: A_27a > $o] :
( ( V1X
= ( V0f @ V1X ) )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ V1X @ ( c_2EfixedPoint_2Egfp @ A_27a @ V0f ) ) ) ) ) ).
thf(thm_2EfixedPoint_2Elfp__induction,axiom,
! [A_27a: $tType,V0f: ( A_27a > $o ) > A_27a > $o] :
( ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a @ V0f )
=> ! [V1X: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ ( V0f @ V1X ) @ V1X )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2EfixedPoint_2Elfp @ A_27a @ V0f ) @ V1X ) ) ) ).
thf(thm_2EfixedPoint_2Egfp__coinduction,axiom,
! [A_27a: $tType,V0f: ( A_27a > $o ) > A_27a > $o] :
( ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a @ V0f )
=> ! [V1X: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V1X @ ( V0f @ V1X ) )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ V1X @ ( c_2EfixedPoint_2Egfp @ A_27a @ V0f ) ) ) ) ).
thf(thm_2EfixedPoint_2Elfp__strong__induction,axiom,
! [A_27a: $tType,V0f: ( A_27a > $o ) > A_27a > $o] :
( ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a @ V0f )
=> ! [V1X: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ ( V0f @ ( c_2Epred__set_2EINTER @ A_27a @ V1X @ ( c_2EfixedPoint_2Elfp @ A_27a @ V0f ) ) ) @ V1X )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2EfixedPoint_2Elfp @ A_27a @ V0f ) @ V1X ) ) ) ).
thf(thm_2EfixedPoint_2Egfp__strong__coinduction,axiom,
! [A_27a: $tType,V0f: ( A_27a > $o ) > A_27a > $o] :
( ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a @ V0f )
=> ! [V1X: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V1X @ ( V0f @ ( c_2Epred__set_2EUNION @ A_27a @ V1X @ ( c_2EfixedPoint_2Egfp @ A_27a @ V0f ) ) ) )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ V1X @ ( c_2EfixedPoint_2Egfp @ A_27a @ V0f ) ) ) ) ).
thf(thm_2EfixedPoint_2Efnsum__monotone,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1: ( A_27a > $o ) > A_27b > $o,V1f2: ( A_27a > $o ) > A_27b > $o] :
( ( ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27b @ V0f1 )
& ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27b @ V1f2 ) )
=> ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27b @ ( c_2EfixedPoint_2Efnsum @ A_27b @ ( A_27a > $o ) @ V0f1 @ V1f2 ) ) ) ).
thf(thm_2EfixedPoint_2Eempty__monotone,axiom,
! [A_27a: $tType,A_27b: $tType] : ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27b @ ( c_2EfixedPoint_2Eempty @ ( A_27a > $o ) @ A_27b ) ) ).
thf(thm_2EfixedPoint_2Efnsum__empty,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > $o] :
( ( ( c_2EfixedPoint_2Efnsum @ A_27b @ A_27a @ V0f @ ( c_2EfixedPoint_2Eempty @ A_27a @ A_27b ) )
= V0f )
& ( ( c_2EfixedPoint_2Efnsum @ A_27b @ A_27a @ ( c_2EfixedPoint_2Eempty @ A_27a @ A_27b ) @ V0f )
= V0f ) ) ).
thf(thm_2EfixedPoint_2Efnsum__ASSOC,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > $o,V1g: A_27a > A_27b > $o,V2h: A_27a > A_27b > $o] :
( ( c_2EfixedPoint_2Efnsum @ A_27b @ A_27a @ V0f @ ( c_2EfixedPoint_2Efnsum @ A_27b @ A_27a @ V1g @ V2h ) )
= ( c_2EfixedPoint_2Efnsum @ A_27b @ A_27a @ ( c_2EfixedPoint_2Efnsum @ A_27b @ A_27a @ V0f @ V1g ) @ V2h ) ) ).
thf(thm_2EfixedPoint_2Efnsum__COMM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > $o,V1g: A_27a > A_27b > $o] :
( ( c_2EfixedPoint_2Efnsum @ A_27b @ A_27a @ V0f @ V1g )
= ( c_2EfixedPoint_2Efnsum @ A_27b @ A_27a @ V1g @ V0f ) ) ).
thf(thm_2EfixedPoint_2Efnsum__SUBSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > $o,V1g: A_27a > A_27b > $o,V2X: A_27a] :
( ( c_2Epred__set_2ESUBSET @ A_27b @ ( V0f @ V2X ) @ ( c_2EfixedPoint_2Efnsum @ A_27b @ A_27a @ V0f @ V1g @ V2X ) )
& ( c_2Epred__set_2ESUBSET @ A_27b @ ( V1g @ V2X ) @ ( c_2EfixedPoint_2Efnsum @ A_27b @ A_27a @ V0f @ V1g @ V2X ) ) ) ).
thf(thm_2EfixedPoint_2Elfp__fnsum,axiom,
! [A_27a: $tType,V0f1: ( A_27a > $o ) > A_27a > $o,V1f2: ( A_27a > $o ) > A_27a > $o] :
( ( ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a @ V0f1 )
& ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a @ V1f2 ) )
=> ( ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2EfixedPoint_2Elfp @ A_27a @ V0f1 ) @ ( c_2EfixedPoint_2Elfp @ A_27a @ ( c_2EfixedPoint_2Efnsum @ A_27a @ ( A_27a > $o ) @ V0f1 @ V1f2 ) ) )
& ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2EfixedPoint_2Elfp @ A_27a @ V1f2 ) @ ( c_2EfixedPoint_2Elfp @ A_27a @ ( c_2EfixedPoint_2Efnsum @ A_27a @ ( A_27a > $o ) @ V0f1 @ V1f2 ) ) ) ) ) ).
thf(thm_2EfixedPoint_2Elfp__rule__applied,axiom,
! [A_27a: $tType,V0f: ( A_27a > $o ) > A_27a > $o,V1X: A_27a > $o,V2y: A_27a] :
( ( ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a @ V0f )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V1X @ ( c_2EfixedPoint_2Elfp @ A_27a @ V0f ) )
& ( c_2Ebool_2EIN @ A_27a @ V2y @ ( V0f @ V1X ) ) )
=> ( c_2Ebool_2EIN @ A_27a @ V2y @ ( c_2EfixedPoint_2Elfp @ A_27a @ V0f ) ) ) ).
thf(thm_2EfixedPoint_2Elfp__empty,axiom,
! [A_27a: $tType,V0f: ( A_27a > $o ) > A_27a > $o,V1x: A_27a] :
( ( ( c_2EfixedPoint_2Emonotone @ A_27a @ A_27a @ V0f )
& ( c_2Ebool_2EIN @ A_27a @ V1x @ ( V0f @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
=> ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2EfixedPoint_2Elfp @ A_27a @ V0f ) ) ) ).
%------------------------------------------------------------------------------