ITP001 Axioms: ITP032^7.ax
%------------------------------------------------------------------------------
% File : ITP032^7 : 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 : res_quan.ax [Gau19]
% : HL4032^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 91 ( 26 unt; 27 typ; 0 def)
% Number of atoms : 225 ( 36 equ; 13 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 638 ( 13 ~; 5 |; 10 &; 566 @)
% ( 29 <=>; 15 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 8 avg; 566 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 199 ( 199 >; 0 *; 0 +; 0 <<)
% Number of symbols : 28 ( 26 usr; 3 con; 0-5 aty)
% Number of variables : 323 ( 67 ^ 231 !; 5 ?; 323 :)
% ( 20 !>; 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(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
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_2Ebool_2E_3F_21,type,
c_2Ebool_2E_3F_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Emin_2E_40,type,
c_2Emin_2E_40:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
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_2EDIFF,type,
c_2Epred__set_2EDIFF:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EEMPTY,type,
c_2Epred__set_2EEMPTY:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Ebool_2EF,type,
c_2Ebool_2EF: $o ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EINSERT,type,
c_2Epred__set_2EINSERT:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Ebool_2ERES__ABSTRACT,type,
c_2Ebool_2ERES__ABSTRACT:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Ebool_2ERES__EXISTS,type,
c_2Ebool_2ERES__EXISTS:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2ERES__EXISTS__UNIQUE,type,
c_2Ebool_2ERES__EXISTS__UNIQUE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2ERES__FORALL,type,
c_2Ebool_2ERES__FORALL:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2ERES__SELECT,type,
c_2Ebool_2ERES__SELECT:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a ) ).
thf(c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2ET,type,
c_2Ebool_2ET: $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_2Epred__set_2EUNIV,type,
c_2Epred__set_2EUNIV:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $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_2Eres__quan_2ERES__SELECT__UNIV,axiom,
! [A_27a: $tType,V0p: A_27a > $o] :
( ( c_2Ebool_2ERES__SELECT @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) @ V0p )
= ( c_2Emin_2E_40 @ A_27a @ V0p ) ) ).
thf(thm_2Eres__quan_2ERES__SELECT__EMPTY,axiom,
! [A_27a: $tType,V0p: A_27a > $o] :
( ( c_2Ebool_2ERES__SELECT @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V0p )
= ( c_2Emin_2E_40 @ A_27a
@ ^ [V1x: A_27a] : c_2Ebool_2EF ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__UNIQUE__ALT,axiom,
! [A_27a: $tType,V0p: A_27a > $o,V1m: A_27a > $o] :
( ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a @ V0p @ V1m )
= ( c_2Ebool_2ERES__EXISTS @ A_27a @ V0p
@ ^ [V2x: A_27a] :
( c_2Ebool_2E_2F_5C @ ( V1m @ V2x )
@ ( c_2Ebool_2ERES__FORALL @ A_27a @ V0p
@ ^ [V3y: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( V1m @ V3y ) @ ( c_2Emin_2E_3D @ A_27a @ V3y @ V2x ) ) ) ) ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__UNIQUE__SING,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0P: A_27a,V1s: A_27b > $o,V2x: A_27c] :
( ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27b @ V1s
@ ^ [V3x: A_27b] : c_2Ebool_2ET )
<=> ? [V4y: A_27b] :
( V1s
= ( c_2Epred__set_2EINSERT @ A_27b @ V4y @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__UNIQUE__NULL,axiom,
! [A_27a: $tType,V0p: A_27a > $o,V1m: $o] :
( ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a @ V0p
@ ^ [V2x: A_27a] : V1m )
<=> ( ? [V3x: A_27a] :
( V0p
= ( c_2Epred__set_2EINSERT @ A_27a @ V3x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& V1m ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__UNIQUE__UNIV,axiom,
! [A_27a: $tType,V0p: A_27a > $o] :
( ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) @ V0p )
= ( c_2Ebool_2E_3F_21 @ A_27a @ V0p ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__UNIQUE__NOT__EMPTY,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1s: A_27a > $o] :
( ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a @ V1s @ V0P )
=> ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__UNIQUE__EMPTY,axiom,
! [A_27a: $tType,V0p: A_27a > $o] : ( (~) @ ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V0p ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__UNIQUE__T,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0P: A_27a,V1s: A_27b > $o,V2x: A_27c] :
( ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27b @ V1s
@ ^ [V3x: A_27b] : c_2Ebool_2ET )
= ( c_2Ebool_2E_3F_21 @ A_27b
@ ^ [V4x: A_27b] : ( c_2Ebool_2EIN @ A_27b @ V4x @ V1s ) ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__UNIQUE__F,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0P: A_27a,V1s: A_27b > $o,V2x: A_27c] :
( (~)
@ ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27b @ V1s
@ ^ [V3x: A_27b] : c_2Ebool_2EF ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__UNIQUE__EXISTS,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1s: A_27a > $o] :
( ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a @ V0P @ V1s )
=> ( c_2Ebool_2ERES__EXISTS @ A_27a @ V0P @ V1s ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__UNIQUE__ELIM,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1s: A_27a > $o] :
( ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a @ V1s
@ ^ [V2x: A_27a] : ( V0P @ V2x ) )
= ( c_2Ebool_2E_3F_21 @ A_27a
@ ^ [V3x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s ) @ ( V0P @ V3x ) ) ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__BIGINTER,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1sos: ( A_27a > $o ) > $o] :
( ( c_2Ebool_2ERES__EXISTS @ A_27a @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V1sos )
@ ^ [V2x: A_27a] : ( V0P @ V2x ) )
<=> ? [V3x: A_27a] :
( ( c_2Ebool_2ERES__FORALL @ ( A_27a > $o ) @ V1sos
@ ^ [V4s: A_27a > $o] : ( c_2Ebool_2EIN @ A_27a @ V3x @ V4s ) )
& ( V0P @ V3x ) ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__BIGUNION,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1sos: ( A_27a > $o ) > $o] :
( ( c_2Ebool_2ERES__EXISTS @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1sos )
@ ^ [V2x: A_27a] : ( V0P @ V2x ) )
= ( c_2Ebool_2ERES__EXISTS @ ( A_27a > $o ) @ V1sos
@ ^ [V3s: A_27a > $o] :
( c_2Ebool_2ERES__EXISTS @ A_27a @ V3s
@ ^ [V4x: A_27a] : ( V0P @ V4x ) ) ) ) ).
thf(thm_2Eres__quan_2EIN__BIGUNION__RES__EXISTS,axiom,
! [A_27a: $tType,V0x: A_27a,V1sos: ( A_27a > $o ) > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1sos ) )
= ( c_2Ebool_2ERES__EXISTS @ ( A_27a > $o ) @ V1sos
@ ^ [V2s: A_27a > $o] : ( c_2Ebool_2EIN @ A_27a @ V0x @ V2s ) ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__DIFF,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1s: A_27a > $o,V2t: A_27a > $o,V3x: A_27b] :
( ( c_2Ebool_2ERES__EXISTS @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ V1s @ V2t )
@ ^ [V4x: A_27a] : ( V0P @ V4x ) )
= ( c_2Ebool_2ERES__EXISTS @ A_27a @ V1s
@ ^ [V5x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2E_7E @ ( c_2Ebool_2EIN @ A_27a @ V5x @ V2t ) ) @ ( V0P @ V5x ) ) ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__UNION,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Ebool_2ERES__EXISTS @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) @ V0P )
<=> ( ( c_2Ebool_2ERES__EXISTS @ A_27a @ V1s @ V0P )
| ( c_2Ebool_2ERES__EXISTS @ A_27a @ V2t @ V0P ) ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__SUBSET,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ V2t )
=> ( ( c_2Ebool_2ERES__EXISTS @ A_27a @ V1s @ V0P )
=> ( c_2Ebool_2ERES__EXISTS @ A_27a @ V2t @ V0P ) ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__NOT__EMPTY,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1s: A_27a > $o] :
( ( c_2Ebool_2ERES__EXISTS @ A_27a @ V1s @ V0P )
=> ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Eres__quan_2ENOT__RES__EXISTS,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1s: A_27a > $o] :
( ( (~)
@ ( c_2Ebool_2ERES__EXISTS @ A_27a @ V1s
@ ^ [V2x: A_27a] : ( V0P @ V2x ) ) )
<=> ( c_2Ebool_2ERES__FORALL @ A_27a @ V1s
@ ^ [V3x: A_27a] : ( c_2Ebool_2E_7E @ ( V0P @ V3x ) ) ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__ALT,axiom,
! [A_27a: $tType,V0p: A_27a > $o,V1m: A_27a > $o] :
( ( c_2Ebool_2ERES__EXISTS @ A_27a @ V0p @ V1m )
<=> ( ( c_2Ebool_2EIN @ A_27a @ ( c_2Ebool_2ERES__SELECT @ A_27a @ V0p @ V1m ) @ V0p )
& ( V1m @ ( c_2Ebool_2ERES__SELECT @ A_27a @ V0p @ V1m ) ) ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__NULL,axiom,
! [A_27a: $tType,V0p: A_27a > $o,V1m: $o] :
( ( c_2Ebool_2ERES__EXISTS @ A_27a @ V0p
@ ^ [V2x: A_27a] : V1m )
<=> ( ( (~)
@ ( V0p
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& V1m ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__UNIV,axiom,
! [A_27a: $tType,V0p: A_27a > $o] :
( ( c_2Ebool_2ERES__EXISTS @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) @ V0p )
= ( c_2Ebool_2E_3F @ A_27a @ V0p ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__EMPTY,axiom,
! [A_27a: $tType,V0p: A_27a > $o] : ( (~) @ ( c_2Ebool_2ERES__EXISTS @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V0p ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__T,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0P: A_27a,V1s: A_27b > $o,V2x: A_27c] :
( ( c_2Ebool_2ERES__EXISTS @ A_27b @ V1s
@ ^ [V3x: A_27b] : c_2Ebool_2ET )
<=> ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__F,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0P: A_27a,V1s: A_27b,V2x: A_27c > $o] :
( (~)
@ ( c_2Ebool_2ERES__EXISTS @ A_27c @ V2x
@ ^ [V3s: A_27c] : c_2Ebool_2EF ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__REORDER,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1Q: A_27b > $o,V2R: A_27a > A_27b > $o] :
( ( c_2Ebool_2ERES__EXISTS @ A_27a @ V0P
@ ^ [V3i: A_27a] :
( c_2Ebool_2ERES__EXISTS @ A_27b @ V1Q
@ ^ [V4j: A_27b] : ( V2R @ V3i @ V4j ) ) )
= ( c_2Ebool_2ERES__EXISTS @ A_27b @ V1Q
@ ^ [V5j: A_27b] :
( c_2Ebool_2ERES__EXISTS @ A_27a @ V0P
@ ^ [V6i: A_27a] : ( V2R @ V6i @ V5j ) ) ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__EQUAL,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1j: A_27a] :
( ( c_2Ebool_2ERES__EXISTS @ A_27a @ ( c_2Emin_2E_3D @ A_27a @ V1j )
@ ^ [V2i: A_27a] : ( V0P @ V2i ) )
= ( V0P @ V1j ) ) ).
thf(thm_2Eres__quan_2ERES__DISJ__EXISTS__DIST,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o,V2R: A_27a > $o] :
( ( c_2Ebool_2ERES__EXISTS @ A_27a
@ ^ [V3i: A_27a] : ( c_2Ebool_2E_5C_2F @ ( V0P @ V3i ) @ ( V1Q @ V3i ) )
@ ^ [V4i: A_27a] : ( V2R @ V4i ) )
<=> ( ( c_2Ebool_2ERES__EXISTS @ A_27a @ V0P
@ ^ [V5i: A_27a] : ( V2R @ V5i ) )
| ( c_2Ebool_2ERES__EXISTS @ A_27a @ V1Q
@ ^ [V6i: A_27a] : ( V2R @ V6i ) ) ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__DISJ__DIST,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o,V2R: A_27a > $o] :
( ( c_2Ebool_2ERES__EXISTS @ A_27a @ V0P
@ ^ [V3i: A_27a] : ( c_2Ebool_2E_5C_2F @ ( V1Q @ V3i ) @ ( V2R @ V3i ) ) )
<=> ( ( c_2Ebool_2ERES__EXISTS @ A_27a @ V0P
@ ^ [V4i: A_27a] : ( V1Q @ V4i ) )
| ( c_2Ebool_2ERES__EXISTS @ A_27a @ V0P
@ ^ [V5i: A_27a] : ( V2R @ V5i ) ) ) ) ).
thf(thm_2Eres__quan_2ERES__FORALL__BIGINTER,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1sos: ( A_27a > $o ) > $o] :
( ( c_2Ebool_2ERES__FORALL @ A_27a @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V1sos )
@ ^ [V2x: A_27a] : ( V0P @ V2x ) )
<=> ! [V3x: A_27a] :
( ( c_2Ebool_2ERES__FORALL @ ( A_27a > $o ) @ V1sos
@ ^ [V4s: A_27a > $o] : ( c_2Ebool_2EIN @ A_27a @ V3x @ V4s ) )
=> ( V0P @ V3x ) ) ) ).
thf(thm_2Eres__quan_2ERES__FORALL__BIGUNION,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1sos: ( A_27a > $o ) > $o] :
( ( c_2Ebool_2ERES__FORALL @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1sos )
@ ^ [V2x: A_27a] : ( V0P @ V2x ) )
= ( c_2Ebool_2ERES__FORALL @ ( A_27a > $o ) @ V1sos
@ ^ [V3s: A_27a > $o] :
( c_2Ebool_2ERES__FORALL @ A_27a @ V3s
@ ^ [V4x: A_27a] : ( V0P @ V4x ) ) ) ) ).
thf(thm_2Eres__quan_2EIN__BIGINTER__RES__FORALL,axiom,
! [A_27a: $tType,V0x: A_27a,V1sos: ( A_27a > $o ) > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V1sos ) )
= ( c_2Ebool_2ERES__FORALL @ ( A_27a > $o ) @ V1sos
@ ^ [V2s: A_27a > $o] : ( c_2Ebool_2EIN @ A_27a @ V0x @ V2s ) ) ) ).
thf(thm_2Eres__quan_2ERES__FORALL__DIFF,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1s: A_27a > $o,V2t: A_27a > $o,V3x: A_27b] :
( ( c_2Ebool_2ERES__FORALL @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ V1s @ V2t )
@ ^ [V4x: A_27a] : ( V0P @ V4x ) )
= ( c_2Ebool_2ERES__FORALL @ A_27a @ V1s
@ ^ [V5x: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_7E @ ( c_2Ebool_2EIN @ A_27a @ V5x @ V2t ) ) @ ( V0P @ V5x ) ) ) ) ).
thf(thm_2Eres__quan_2ERES__FORALL__UNION,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Ebool_2ERES__FORALL @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) @ V0P )
<=> ( ( c_2Ebool_2ERES__FORALL @ A_27a @ V1s @ V0P )
& ( c_2Ebool_2ERES__FORALL @ A_27a @ V2t @ V0P ) ) ) ).
thf(thm_2Eres__quan_2ERES__FORALL__SUBSET,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ V2t )
=> ( ( c_2Ebool_2ERES__FORALL @ A_27a @ V2t @ V0P )
=> ( c_2Ebool_2ERES__FORALL @ A_27a @ V1s @ V0P ) ) ) ).
thf(thm_2Eres__quan_2ERES__FORALL__NOT__EMPTY,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1s: A_27a > $o] :
( ( (~) @ ( c_2Ebool_2ERES__FORALL @ A_27a @ V1s @ V0P ) )
=> ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Eres__quan_2ENOT__RES__FORALL,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1s: A_27a > $o] :
( ( (~)
@ ( c_2Ebool_2ERES__FORALL @ A_27a @ V1s
@ ^ [V2x: A_27a] : ( V0P @ V2x ) ) )
<=> ( c_2Ebool_2ERES__EXISTS @ A_27a @ V1s
@ ^ [V3x: A_27a] : ( c_2Ebool_2E_7E @ ( V0P @ V3x ) ) ) ) ).
thf(thm_2Eres__quan_2ERES__FORALL__NULL,axiom,
! [A_27a: $tType,V0p: A_27a > $o,V1m: $o] :
( ( c_2Ebool_2ERES__FORALL @ A_27a @ V0p
@ ^ [V2x: A_27a] : V1m )
<=> ( ( V0p
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
| V1m ) ) ).
thf(thm_2Eres__quan_2ERES__FORALL__UNIV,axiom,
! [A_27a: $tType,V0p: A_27a > $o] :
( ( c_2Ebool_2ERES__FORALL @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) @ V0p )
= ( c_2Ebool_2E_21 @ A_27a @ V0p ) ) ).
thf(thm_2Eres__quan_2ERES__FORALL__EMPTY,axiom,
! [A_27a: $tType,V0p: A_27a > $o] : ( c_2Ebool_2ERES__FORALL @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V0p ) ).
thf(thm_2Eres__quan_2ERES__FORALL__F,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0P: A_27a,V1s: A_27b > $o,V2x: A_27c] :
( ( c_2Ebool_2ERES__FORALL @ A_27b @ V1s
@ ^ [V3x: A_27b] : c_2Ebool_2EF )
<=> ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ).
thf(thm_2Eres__quan_2ERES__FORALL__T,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0P: A_27a,V1s: A_27b > $o,V2x: A_27c] :
( c_2Ebool_2ERES__FORALL @ A_27b @ V1s
@ ^ [V3x: A_27b] : c_2Ebool_2ET ) ).
thf(thm_2Eres__quan_2ERES__FORALL__REORDER,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1Q: A_27b > $o,V2R: A_27a > A_27b > $o] :
( ( c_2Ebool_2ERES__FORALL @ A_27a @ V0P
@ ^ [V3i: A_27a] :
( c_2Ebool_2ERES__FORALL @ A_27b @ V1Q
@ ^ [V4j: A_27b] : ( V2R @ V3i @ V4j ) ) )
= ( c_2Ebool_2ERES__FORALL @ A_27b @ V1Q
@ ^ [V5j: A_27b] :
( c_2Ebool_2ERES__FORALL @ A_27a @ V0P
@ ^ [V6i: A_27a] : ( V2R @ V6i @ V5j ) ) ) ) ).
thf(thm_2Eres__quan_2ERES__FORALL__FORALL,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1R: A_27a > A_27b > $o,V2x: A_27b] :
( ! [V3x: A_27b] :
( c_2Ebool_2ERES__FORALL @ A_27a @ V0P
@ ^ [V4i: A_27a] : ( V1R @ V4i @ V3x ) )
<=> ( c_2Ebool_2ERES__FORALL @ A_27a @ V0P
@ ^ [V5i: A_27a] :
( c_2Ebool_2E_21 @ A_27b
@ ^ [V6x: A_27b] : ( V1R @ V5i @ V6x ) ) ) ) ).
thf(thm_2Eres__quan_2ERES__FORALL__UNIQUE,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1j: A_27a] :
( ( c_2Ebool_2ERES__FORALL @ A_27a @ ( c_2Emin_2E_3D @ A_27a @ V1j )
@ ^ [V2i: A_27a] : ( V0P @ V2i ) )
= ( V0P @ V1j ) ) ).
thf(thm_2Eres__quan_2ERES__FORALL__DISJ__DIST,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o,V2R: A_27a > $o] :
( ( c_2Ebool_2ERES__FORALL @ A_27a
@ ^ [V3j: A_27a] : ( c_2Ebool_2E_5C_2F @ ( V0P @ V3j ) @ ( V1Q @ V3j ) )
@ ^ [V4i: A_27a] : ( V2R @ V4i ) )
<=> ( ( c_2Ebool_2ERES__FORALL @ A_27a @ V0P
@ ^ [V5i: A_27a] : ( V2R @ V5i ) )
& ( c_2Ebool_2ERES__FORALL @ A_27a @ V1Q
@ ^ [V6i: A_27a] : ( V2R @ V6i ) ) ) ) ).
thf(thm_2Eres__quan_2ERES__FORALL__CONJ__DIST,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o,V2R: A_27a > $o] :
( ( c_2Ebool_2ERES__FORALL @ A_27a @ V0P
@ ^ [V3i: A_27a] : ( c_2Ebool_2E_2F_5C @ ( V1Q @ V3i ) @ ( V2R @ V3i ) ) )
<=> ( ( c_2Ebool_2ERES__FORALL @ A_27a @ V0P
@ ^ [V4i: A_27a] : ( V1Q @ V4i ) )
& ( c_2Ebool_2ERES__FORALL @ A_27a @ V0P
@ ^ [V5i: A_27a] : ( V2R @ V5i ) ) ) ) ).
thf(thm_2Eres__quan_2ERES__FORALL,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1f: A_27a > $o] :
( ( c_2Ebool_2ERES__FORALL @ A_27a @ V0P @ V1f )
<=> ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0P )
=> ( V1f @ V2x ) ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1f: A_27a > $o] :
( ( c_2Ebool_2ERES__EXISTS @ A_27a @ V0P @ V1f )
<=> ? [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0P )
& ( V1f @ V2x ) ) ) ).
thf(thm_2Eres__quan_2ERES__EXISTS__UNIQUE,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1f: A_27a > $o] :
( ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a @ V0P @ V1f )
<=> ( ( c_2Ebool_2ERES__EXISTS @ A_27a @ V0P
@ ^ [V2x: A_27a] : ( V1f @ V2x ) )
& ( c_2Ebool_2ERES__FORALL @ A_27a @ V0P
@ ^ [V3x: A_27a] :
( c_2Ebool_2ERES__FORALL @ A_27a @ V0P
@ ^ [V4y: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_2F_5C @ ( V1f @ V3x ) @ ( V1f @ V4y ) ) @ ( c_2Emin_2E_3D @ A_27a @ V3x @ V4y ) ) ) ) ) ) ).
thf(thm_2Eres__quan_2ERES__SELECT,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1f: A_27a > $o] :
( ( c_2Ebool_2ERES__SELECT @ A_27a @ V0P @ V1f )
= ( c_2Emin_2E_40 @ A_27a
@ ^ [V2x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0P ) @ ( V1f @ V2x ) ) ) ) ).
thf(thm_2Eres__quan_2ERES__ABSTRACT,axiom,
! [A_27a: $tType,A_27b: $tType,V0p: A_27a > $o,V1m: A_27a > A_27b,V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0p )
=> ( ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ V0p @ V1m @ V2x )
= ( V1m @ V2x ) ) ) ).
thf(thm_2Eres__quan_2ERES__ABSTRACT__EQUAL,axiom,
! [A_27a: $tType,A_27b: $tType,V0p: A_27a > $o,V1m1: A_27a > A_27b,V2m2: A_27a > A_27b] :
( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0p )
=> ( ( V1m1 @ V3x )
= ( V2m2 @ V3x ) ) )
=> ( ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ V0p @ V1m1 )
= ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ V0p @ V2m2 ) ) ) ).
thf(thm_2Eres__quan_2ERES__ABSTRACT__IDEMPOT,axiom,
! [A_27a: $tType,A_27b: $tType,V0p: A_27a > $o,V1m: A_27a > A_27b] :
( ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ V0p @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ V0p @ V1m ) )
= ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ V0p @ V1m ) ) ).
thf(thm_2Eres__quan_2ERES__ABSTRACT__EQUAL__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0p: A_27a > $o,V1m1: A_27a > A_27b,V2m2: A_27a > A_27b] :
( ( ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ V0p @ V1m1 )
= ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ V0p @ V2m2 ) )
<=> ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0p )
=> ( ( V1m1 @ V3x )
= ( V2m2 @ V3x ) ) ) ) ).
thf(thm_2Eres__quan_2ERES__ABSTRACT__UNIV,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: A_27a > A_27b] :
( ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ ( c_2Epred__set_2EUNIV @ A_27a ) @ V0m )
= V0m ) ).
%------------------------------------------------------------------------------