ITP001 Axioms: ITP008^7.ax
%------------------------------------------------------------------------------
% File : ITP008^7 : 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 : combin.ax [Gau19]
% : HL4008^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 90 ( 37 unt; 30 typ; 0 def)
% Number of atoms : 110 ( 65 equ; 6 cnn)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 586 ( 6 ~; 2 |; 10 &; 545 @)
% ( 17 <=>; 6 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 6 avg; 545 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 169 ( 169 >; 0 *; 0 +; 0 <<)
% Number of symbols : 31 ( 29 usr; 3 con; 0-6 aty)
% Number of variables : 326 ( 25 ^ 261 !; 1 ?; 326 :)
% ( 39 !>; 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_2Ecombin_2E_3A_3E,type,
c_2Ecombin_2E_3A_3E:
!>[A_27a: $tType,A_27b: $tType] : ( A_27b > ( A_27b > A_27a ) > A_27a ) ).
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_2Ecombin_2EASSOC,type,
c_2Ecombin_2EASSOC:
!>[A_27a: $tType] : ( ( A_27a > A_27a > A_27a ) > $o ) ).
thf(c_2Emarker_2EAbbrev,type,
c_2Emarker_2EAbbrev: $o > $o ).
thf(c_2Ecombin_2EC,type,
c_2Ecombin_2EC:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > A_27b > A_27a > A_27c ) ).
thf(c_2Ecombin_2ECOMM,type,
c_2Ecombin_2ECOMM:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > A_27b ) > $o ) ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Ebool_2EF,type,
c_2Ebool_2EF: $o ).
thf(c_2Ecombin_2EFAIL,type,
c_2Ecombin_2EFAIL:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > A_27a ) ).
thf(c_2Ecombin_2EFCOMM,type,
c_2Ecombin_2EFCOMM:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27a ) > ( A_27c > A_27a > A_27a ) > $o ) ).
thf(c_2Ecombin_2EI,type,
c_2Ecombin_2EI:
!>[A_27a: $tType] : ( A_27a > A_27a ) ).
thf(c_2Ecombin_2EK,type,
c_2Ecombin_2EK:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > A_27a ) ).
thf(c_2Ecombin_2ELEFT__ID,type,
c_2Ecombin_2ELEFT__ID:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > A_27b ) > A_27a > $o ) ).
thf(c_2Ebool_2ELET,type,
c_2Ebool_2ELET:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Ecombin_2EMONOID,type,
c_2Ecombin_2EMONOID:
!>[A_27a: $tType] : ( ( A_27a > A_27a > A_27a ) > A_27a > $o ) ).
thf(c_2Ecombin_2ERIGHT__ID,type,
c_2Ecombin_2ERIGHT__ID:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > A_27a ) > A_27b > $o ) ).
thf(c_2Ecombin_2ES,type,
c_2Ecombin_2ES:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( A_27a > A_27b ) > A_27a > A_27c ) ).
thf(c_2Ebool_2ET,type,
c_2Ebool_2ET: $o ).
thf(c_2Ecombin_2EUPDATE,type,
c_2Ecombin_2EUPDATE:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Ecombin_2EW,type,
c_2Ecombin_2EW:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Ebool_2Eliteral__case,type,
c_2Ebool_2Eliteral__case:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Ecombin_2Eo,type,
c_2Ecombin_2Eo:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > A_27c ) > A_27a > A_27b ) ).
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_2Ecombin_2EK__DEF,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Ecombin_2EK @ A_27a @ A_27b )
= ( ^ [V0x: A_27a,V1y: A_27b] : V0x ) ) ).
thf(thm_2Ecombin_2ES__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType] :
( ( c_2Ecombin_2ES @ A_27a @ A_27b @ A_27c )
= ( ^ [V0f: A_27a > A_27b > A_27c,V1g: A_27a > A_27b,V2x: A_27a] : ( V0f @ V2x @ ( V1g @ V2x ) ) ) ) ).
thf(thm_2Ecombin_2EI__DEF,axiom,
! [A_27a: $tType] :
( ( c_2Ecombin_2EI @ A_27a )
= ( c_2Ecombin_2ES @ A_27a @ ( A_27a > A_27a ) @ A_27a @ ( c_2Ecombin_2EK @ A_27a @ ( A_27a > A_27a ) ) @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) ) ) ).
thf(thm_2Ecombin_2EC__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType] :
( ( c_2Ecombin_2EC @ A_27a @ A_27b @ A_27c )
= ( ^ [V0f: A_27a > A_27b > A_27c,V1x: A_27b,V2y: A_27a] : ( V0f @ V2y @ V1x ) ) ) ).
thf(thm_2Ecombin_2EW__DEF,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Ecombin_2EW @ A_27a @ A_27b )
= ( ^ [V0f: A_27a > A_27a > A_27b,V1x: A_27a] : ( V0f @ V1x @ V1x ) ) ) ).
thf(thm_2Ecombin_2Eo__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27c > A_27b,V1g: A_27a > A_27c] :
( ( c_2Ecombin_2Eo @ A_27a @ A_27b @ A_27c @ V0f @ V1g )
= ( ^ [V2x: A_27a] : ( V0f @ ( V1g @ V2x ) ) ) ) ).
thf(thm_2Ecombin_2EAPP__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27b,V1f: A_27b > A_27a] :
( ( c_2Ecombin_2E_3A_3E @ A_27a @ A_27b @ V0x @ V1f )
= ( V1f @ V0x ) ) ).
thf(thm_2Ecombin_2EUPDATE__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0a: A_27a,V1b: A_27b] :
( ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V0a @ V1b )
= ( ^ [V2f: A_27a > A_27b,V3c: A_27a] : ( c_2Ebool_2ECOND @ A_27b @ ( c_2Emin_2E_3D @ A_27a @ V0a @ V3c ) @ V1b @ ( V2f @ V3c ) ) ) ) ).
thf(thm_2Ecombin_2EASSOC__DEF,axiom,
! [A_27a: $tType,V0f: A_27a > A_27a > A_27a] :
( ( c_2Ecombin_2EASSOC @ A_27a @ V0f )
<=> ! [V1x: A_27a,V2y: A_27a,V3z: A_27a] :
( ( V0f @ V1x @ ( V0f @ V2y @ V3z ) )
= ( V0f @ ( V0f @ V1x @ V2y ) @ V3z ) ) ) ).
thf(thm_2Ecombin_2ECOMM__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27a > A_27b] :
( ( c_2Ecombin_2ECOMM @ A_27a @ A_27b @ V0f )
<=> ! [V1x: A_27a,V2y: A_27a] :
( ( V0f @ V1x @ V2y )
= ( V0f @ V2y @ V1x ) ) ) ).
thf(thm_2Ecombin_2EFCOMM__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b > A_27a,V1g: A_27c > A_27a > A_27a] :
( ( c_2Ecombin_2EFCOMM @ A_27a @ A_27b @ A_27c @ V0f @ V1g )
<=> ! [V2x: A_27c,V3y: A_27a,V4z: A_27b] :
( ( V1g @ V2x @ ( V0f @ V3y @ V4z ) )
= ( V0f @ ( V1g @ V2x @ V3y ) @ V4z ) ) ) ).
thf(thm_2Ecombin_2ERIGHT__ID__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > A_27a,V1e: A_27b] :
( ( c_2Ecombin_2ERIGHT__ID @ A_27a @ A_27b @ V0f @ V1e )
<=> ! [V2x: A_27a] :
( ( V0f @ V2x @ V1e )
= V2x ) ) ).
thf(thm_2Ecombin_2ELEFT__ID__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > A_27b,V1e: A_27a] :
( ( c_2Ecombin_2ELEFT__ID @ A_27a @ A_27b @ V0f @ V1e )
<=> ! [V2x: A_27b] :
( ( V0f @ V1e @ V2x )
= V2x ) ) ).
thf(thm_2Ecombin_2EMONOID__DEF,axiom,
! [A_27a: $tType,V0f: A_27a > A_27a > A_27a,V1e: A_27a] :
( ( c_2Ecombin_2EMONOID @ A_27a @ V0f @ V1e )
<=> ( ( c_2Ecombin_2EASSOC @ A_27a @ V0f )
& ( c_2Ecombin_2ERIGHT__ID @ A_27a @ A_27a @ V0f @ V1e )
& ( c_2Ecombin_2ELEFT__ID @ A_27a @ A_27a @ V0f @ V1e ) ) ) ).
thf(thm_2Ecombin_2EFAIL__DEF,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Ecombin_2EFAIL @ A_27a @ A_27b )
= ( ^ [V0x: A_27a,V1y: A_27b] : V0x ) ) ).
thf(thm_2Ecombin_2Eo__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b,V1g: A_27c > A_27a,V2x: A_27c] :
( ( c_2Ecombin_2Eo @ A_27c @ A_27b @ A_27a @ V0f @ V1g @ V2x )
= ( V0f @ ( V1g @ V2x ) ) ) ).
thf(thm_2Ecombin_2Eo__ASSOC,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0f: A_27a > A_27b,V1g: A_27c > A_27a,V2h: A_27d > A_27c] :
( ( c_2Ecombin_2Eo @ A_27d @ A_27b @ A_27a @ V0f @ ( c_2Ecombin_2Eo @ A_27d @ A_27a @ A_27c @ V1g @ V2h ) )
= ( c_2Ecombin_2Eo @ A_27d @ A_27b @ A_27c @ ( c_2Ecombin_2Eo @ A_27c @ A_27b @ A_27a @ V0f @ V1g ) @ V2h ) ) ).
thf(thm_2Ecombin_2Eo__ABS__L,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0g: A_27b > A_27a,V1f: A_27a > A_27c] :
( ( c_2Ecombin_2Eo @ A_27b @ A_27c @ A_27a
@ ^ [V2x: A_27a] : ( V1f @ V2x )
@ V0g )
= ( ^ [V3x: A_27b] : ( V1f @ ( V0g @ V3x ) ) ) ) ).
thf(thm_2Ecombin_2Eo__ABS__R,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0g: A_27a > A_27c,V1f: A_27c > A_27b] :
( ( c_2Ecombin_2Eo @ A_27a @ A_27b @ A_27c @ V1f
@ ^ [V2x: A_27a] : ( V0g @ V2x ) )
= ( ^ [V3x: A_27a] : ( V1f @ ( V0g @ V3x ) ) ) ) ).
thf(thm_2Ecombin_2EK__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1y: A_27b] :
( ( c_2Ecombin_2EK @ A_27a @ A_27b @ V0x @ V1y )
= V0x ) ).
thf(thm_2Ecombin_2ES__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b > A_27c,V1g: A_27a > A_27b,V2x: A_27a] :
( ( c_2Ecombin_2ES @ A_27a @ A_27b @ A_27c @ V0f @ V1g @ V2x )
= ( V0f @ V2x @ ( V1g @ V2x ) ) ) ).
thf(thm_2Ecombin_2ES__ABS__L,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0g: A_27a > A_27c,V1f: A_27a > A_27c > A_27b] :
( ( c_2Ecombin_2ES @ A_27a @ A_27c @ A_27b
@ ^ [V2x: A_27a] : ( V1f @ V2x )
@ V0g )
= ( ^ [V3x: A_27a] : ( V1f @ V3x @ ( V0g @ V3x ) ) ) ) ).
thf(thm_2Ecombin_2ES__ABS__R,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0g: A_27a > A_27c,V1f: A_27a > A_27c > A_27b] :
( ( c_2Ecombin_2ES @ A_27a @ A_27c @ A_27b @ V1f
@ ^ [V2x: A_27a] : ( V0g @ V2x ) )
= ( ^ [V3x: A_27a] : ( V1f @ V3x @ ( V0g @ V3x ) ) ) ) ).
thf(thm_2Ecombin_2EC__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b > A_27c,V1x: A_27b,V2y: A_27a] :
( ( c_2Ecombin_2EC @ A_27a @ A_27b @ A_27c @ V0f @ V1x @ V2y )
= ( V0f @ V2y @ V1x ) ) ).
thf(thm_2Ecombin_2EC__ABS__L,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0y: A_27c,V1f: A_27a > A_27c > A_27b] :
( ( c_2Ecombin_2EC @ A_27a @ A_27c @ A_27b
@ ^ [V2x: A_27a] : ( V1f @ V2x )
@ V0y )
= ( ^ [V3x: A_27a] : ( V1f @ V3x @ V0y ) ) ) ).
thf(thm_2Ecombin_2EW__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27a > A_27b,V1x: A_27a] :
( ( c_2Ecombin_2EW @ A_27a @ A_27b @ V0f @ V1x )
= ( V0f @ V1x @ V1x ) ) ).
thf(thm_2Ecombin_2EI__THM,axiom,
! [A_27a: $tType,V0x: A_27a] :
( ( c_2Ecombin_2EI @ A_27a @ V0x )
= V0x ) ).
thf(thm_2Ecombin_2EI__o__ID,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b] :
( ( ( c_2Ecombin_2Eo @ A_27a @ A_27b @ A_27b @ ( c_2Ecombin_2EI @ A_27b ) @ V0f )
= V0f )
& ( ( c_2Ecombin_2Eo @ A_27a @ A_27b @ A_27a @ V0f @ ( c_2Ecombin_2EI @ A_27a ) )
= V0f ) ) ).
thf(thm_2Ecombin_2EK__o__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType] :
( ! [V0f: A_27a > A_27b,V1v: A_27c] :
( ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b @ ( c_2Ecombin_2EK @ A_27c @ A_27b @ V1v ) @ V0f )
= ( c_2Ecombin_2EK @ A_27c @ A_27a @ V1v ) )
& ! [V2f: A_27d > A_27e,V3v: A_27d] :
( ( c_2Ecombin_2Eo @ A_27f @ A_27e @ A_27d @ V2f @ ( c_2Ecombin_2EK @ A_27d @ A_27f @ V3v ) )
= ( c_2Ecombin_2EK @ A_27e @ A_27f @ ( V2f @ V3v ) ) ) ) ).
thf(thm_2Ecombin_2EUPDATE__APPLY,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] :
( ! [V0a: A_27a,V1x: A_27b,V2f: A_27a > A_27b] :
( ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V0a @ V1x @ V2f @ V0a )
= V1x )
& ! [V3a: A_27c,V4b: A_27c,V5x: A_27d,V6f: A_27c > A_27d] :
( ( (~) @ ( V3a = V4b ) )
=> ( ( c_2Ecombin_2EUPDATE @ A_27c @ A_27d @ V3a @ V5x @ V6f @ V4b )
= ( V6f @ V4b ) ) ) ) ).
thf(thm_2Ecombin_2EAPPLY__UPDATE__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1a: A_27a,V2b: A_27b,V3c: A_27a] :
( ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V1a @ V2b @ V0f @ V3c )
= ( c_2Ebool_2ECOND @ A_27b @ ( c_2Emin_2E_3D @ A_27a @ V1a @ V3c ) @ V2b @ ( V0f @ V3c ) ) ) ).
thf(thm_2Ecombin_2EUPDATE__COMMUTES,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1a: A_27a,V2b: A_27a,V3c: A_27b,V4d: A_27b] :
( ( (~) @ ( V1a = V2b ) )
=> ( ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V1a @ V3c @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V2b @ V4d @ V0f ) )
= ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V2b @ V4d @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V1a @ V3c @ V0f ) ) ) ) ).
thf(thm_2Ecombin_2EUPDATE__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1a: A_27a,V2b: A_27b,V3c: A_27b] :
( ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V1a @ V3c @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V1a @ V2b @ V0f ) )
= ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V1a @ V3c @ V0f ) ) ).
thf(thm_2Ecombin_2EUPDATE__APPLY__ID,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1a: A_27a,V2b: A_27b] :
( ( ( V0f @ V1a )
= V2b )
<=> ( ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V1a @ V2b @ V0f )
= V0f ) ) ).
thf(thm_2Ecombin_2EUPDATE__APPLY__IMP__ID,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1b: A_27b,V2a: A_27a] :
( ( ( V0f @ V2a )
= V1b )
=> ( ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V2a @ V1b @ V0f )
= V0f ) ) ).
thf(thm_2Ecombin_2EAPPLY__UPDATE__ID,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1a: A_27a] :
( ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V1a @ ( V0f @ V1a ) @ V0f )
= V0f ) ).
thf(thm_2Ecombin_2EUPD11__SAME__BASE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1a: A_27a,V2b: A_27a,V3c: A_27b,V4d: A_27b] :
( ( ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V1a @ V3c @ V0f )
= ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V2b @ V4d @ V0f ) )
<=> ( ( ( V1a = V2b )
& ( V3c = V4d ) )
| ( ( (~) @ ( V1a = V2b ) )
& ( ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V1a @ V3c @ V0f )
= V0f )
& ( ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V2b @ V4d @ V0f )
= V0f ) ) ) ) ).
thf(thm_2Ecombin_2ESAME__KEY__UPDATE__DIFFER,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0f: A_27c > A_27d,V1f1: A_27a,V2f2: A_27b,V3a: A_27c,V4b: A_27d,V5c: A_27d] :
( ( (~) @ ( V4b = V5c ) )
=> ( (~)
@ ( ( c_2Ecombin_2EUPDATE @ A_27c @ A_27d @ V3a @ V4b @ V0f )
= ( c_2Ecombin_2EUPDATE @ A_27c @ A_27d @ V3a @ V5c @ V0f ) ) ) ) ).
thf(thm_2Ecombin_2EUPD11__SAME__KEY__AND__BASE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1a: A_27a,V2b: A_27b,V3c: A_27b] :
( ( ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V1a @ V2b @ V0f )
= ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V1a @ V3c @ V0f ) )
<=> ( V2b = V3c ) ) ).
thf(thm_2Ecombin_2EUPD__SAME__KEY__UNWIND,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1: A_27a > A_27b,V1f2: A_27a > A_27b,V2a: A_27a,V3b: A_27b,V4c: A_27b] :
( ( ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V2a @ V3b @ V0f1 )
= ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V2a @ V4c @ V1f2 ) )
=> ( ( V3b = V4c )
& ! [V5v: A_27b] :
( ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V2a @ V5v @ V0f1 )
= ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V2a @ V5v @ V1f2 ) ) ) ) ).
thf(thm_2Ecombin_2EGEN__LET__RAND,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0v: A_27c,V1f: A_27c > A_27b,V2P: A_27b > A_27a] :
( ( V2P @ ( c_2Ebool_2ELET @ A_27c @ A_27b @ V1f @ V0v ) )
= ( c_2Ebool_2ELET @ A_27c @ A_27a @ ( c_2Ecombin_2Eo @ A_27c @ A_27a @ A_27b @ V2P @ V1f ) @ V0v ) ) ).
thf(thm_2Ecombin_2EGEN__LET__RATOR,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0x: A_27c,V1v: A_27b,V2f: A_27b > A_27c > A_27a] :
( ( c_2Ebool_2ELET @ A_27b @ ( A_27c > A_27a ) @ V2f @ V1v @ V0x )
= ( c_2Ebool_2ELET @ A_27b @ A_27a @ ( c_2Ecombin_2EC @ A_27b @ A_27c @ A_27a @ V2f @ V0x ) @ V1v ) ) ).
thf(thm_2Ecombin_2ELET__FORALL__ELIM,axiom,
! [A_27a: $tType,V0v: A_27a,V1f: A_27a > $o] :
( ( c_2Ebool_2ELET @ A_27a @ $o @ V1f @ V0v )
= ( c_2Ebool_2E_21 @ A_27a @ ( c_2Ecombin_2ES @ A_27a @ $o @ $o @ ( c_2Ecombin_2Eo @ A_27a @ ( $o > $o ) @ $o @ c_2Emin_2E_3D_3D_3E @ ( c_2Ecombin_2Eo @ A_27a @ $o @ $o @ c_2Emarker_2EAbbrev @ ( c_2Ecombin_2EC @ A_27a @ A_27a @ $o @ ( c_2Emin_2E_3D @ A_27a ) @ V0v ) ) ) @ V1f ) ) ) ).
thf(thm_2Ecombin_2EGEN__literal__case__RAND,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0v: A_27c,V1f: A_27c > A_27b,V2P: A_27b > A_27a] :
( ( V2P @ ( c_2Ebool_2Eliteral__case @ A_27c @ A_27b @ V1f @ V0v ) )
= ( c_2Ebool_2Eliteral__case @ A_27c @ A_27a @ ( c_2Ecombin_2Eo @ A_27c @ A_27a @ A_27b @ V2P @ V1f ) @ V0v ) ) ).
thf(thm_2Ecombin_2EGEN__literal__case__RATOR,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0x: A_27c,V1v: A_27b,V2f: A_27b > A_27c > A_27a] :
( ( c_2Ebool_2Eliteral__case @ A_27b @ ( A_27c > A_27a ) @ V2f @ V1v @ V0x )
= ( c_2Ebool_2Eliteral__case @ A_27b @ A_27a @ ( c_2Ecombin_2EC @ A_27b @ A_27c @ A_27a @ V2f @ V0x ) @ V1v ) ) ).
thf(thm_2Ecombin_2Eliteral__case__FORALL__ELIM,axiom,
! [A_27a: $tType,V0v: A_27a,V1f: A_27a > $o] :
( ( c_2Ebool_2Eliteral__case @ A_27a @ $o @ V1f @ V0v )
= ( c_2Ebool_2E_21 @ A_27a @ ( c_2Ecombin_2ES @ A_27a @ $o @ $o @ ( c_2Ecombin_2Eo @ A_27a @ ( $o > $o ) @ $o @ c_2Emin_2E_3D_3D_3E @ ( c_2Ecombin_2Eo @ A_27a @ $o @ $o @ c_2Emarker_2EAbbrev @ ( c_2Ecombin_2EC @ A_27a @ A_27a @ $o @ ( c_2Emin_2E_3D @ A_27a ) @ V0v ) ) ) @ V1f ) ) ) ).
thf(thm_2Ecombin_2EASSOC__CONJ,axiom,
c_2Ecombin_2EASSOC @ $o @ c_2Ebool_2E_2F_5C ).
thf(thm_2Ecombin_2EASSOC__SYM,axiom,
! [A_27a: $tType,V0f: A_27a > A_27a > A_27a] :
( ( c_2Ecombin_2EASSOC @ A_27a @ V0f )
<=> ! [V1x: A_27a,V2y: A_27a,V3z: A_27a] :
( ( V0f @ ( V0f @ V1x @ V2y ) @ V3z )
= ( V0f @ V1x @ ( V0f @ V2y @ V3z ) ) ) ) ).
thf(thm_2Ecombin_2EASSOC__DISJ,axiom,
c_2Ecombin_2EASSOC @ $o @ c_2Ebool_2E_5C_2F ).
thf(thm_2Ecombin_2EFCOMM__ASSOC,axiom,
! [A_27a: $tType,V0f: A_27a > A_27a > A_27a] :
( ( c_2Ecombin_2EFCOMM @ A_27a @ A_27a @ A_27a @ V0f @ V0f )
= ( c_2Ecombin_2EASSOC @ A_27a @ V0f ) ) ).
thf(thm_2Ecombin_2EMONOID__CONJ__T,axiom,
c_2Ecombin_2EMONOID @ $o @ c_2Ebool_2E_2F_5C @ c_2Ebool_2ET ).
thf(thm_2Ecombin_2EMONOID__DISJ__F,axiom,
c_2Ecombin_2EMONOID @ $o @ c_2Ebool_2E_5C_2F @ c_2Ebool_2EF ).
thf(thm_2Ecombin_2EFAIL__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27b,V1x: A_27a] :
( ( c_2Ecombin_2EFAIL @ A_27a @ A_27b @ V1x @ V0y )
= V1x ) ).
%------------------------------------------------------------------------------