ITP001 Axioms: ITP012^7.ax
%------------------------------------------------------------------------------
% File : ITP012^7 : 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 : sum.ax [Gau19]
% : HL4012^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 72 ( 13 unt; 28 typ; 0 def)
% Number of atoms : 112 ( 55 equ; 10 cnn)
% Maximal formula atoms : 10 ( 1 avg)
% Number of connectives : 684 ( 10 ~; 5 |; 23 &; 611 @)
% ( 21 <=>; 14 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg; 611 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 125 ( 125 >; 0 *; 0 +; 0 <<)
% Number of symbols : 29 ( 27 usr; 1 con; 0-7 aty)
% Number of variables : 275 ( 13 ^ 213 !; 10 ?; 275 :)
% ( 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(tyop_2Esum_2Esum,type,
tyop_2Esum_2Esum: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Esum_2E_2B_2B,type,
c_2Esum_2E_2B_2B:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( A_27a > A_27c ) > ( A_27b > A_27d ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) > ( tyop_2Esum_2Esum @ A_27c @ A_27d ) ) ).
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_2Esum_2EABS__sum,type,
c_2Esum_2EABS__sum:
!>[A_27a: $tType,A_27b: $tType] : ( ( $o > A_27a > A_27b > $o ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Ebool_2EDATATYPE,type,
c_2Ebool_2EDATATYPE:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Ecombin_2EI,type,
c_2Ecombin_2EI:
!>[A_27a: $tType] : ( A_27a > A_27a ) ).
thf(c_2Esum_2EINL,type,
c_2Esum_2EINL:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) ).
thf(c_2Esum_2EINR,type,
c_2Esum_2EINR:
!>[A_27a: $tType,A_27b: $tType] : ( A_27b > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) ).
thf(c_2Esum_2EISL,type,
c_2Esum_2EISL:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Esum_2Esum @ A_27a @ A_27b ) > $o ) ).
thf(c_2Esum_2EISR,type,
c_2Esum_2EISR:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Esum_2Esum @ A_27a @ A_27b ) > $o ) ).
thf(c_2Esum_2EIS__SUM__REP,type,
c_2Esum_2EIS__SUM__REP:
!>[A_27a: $tType,A_27b: $tType] : ( ( $o > A_27a > A_27b > $o ) > $o ) ).
thf(c_2Esum_2EOUTL,type,
c_2Esum_2EOUTL:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Esum_2Esum @ A_27a @ A_27b ) > A_27a ) ).
thf(c_2Esum_2EOUTR,type,
c_2Esum_2EOUTR:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Esum_2Esum @ A_27a @ A_27b ) > A_27b ) ).
thf(c_2Esum_2EREP__sum,type,
c_2Esum_2EREP__sum:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Esum_2Esum @ A_27a @ A_27b ) > $o > A_27a > A_27b > $o ) ).
thf(c_2Esum_2ESUM__ALL,type,
c_2Esum_2ESUM__ALL:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) > $o ) ).
thf(c_2Ebool_2ETYPE__DEFINITION,type,
c_2Ebool_2ETYPE__DEFINITION:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
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_2Esum_2Esum__CASE,type,
c_2Esum_2Esum__CASE:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Esum_2Esum @ A_27a @ A_27b ) > ( A_27a > A_27c ) > ( A_27b > A_27c ) > A_27c ) ).
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_2Esum_2EIS__SUM__REP,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: $o > A_27a > A_27b > $o] :
( ( c_2Esum_2EIS__SUM__REP @ A_27a @ A_27b @ V0f )
<=> ? [V1v1: A_27a,V2v2: A_27b] :
( ( V0f
= ( ^ [V3b: $o,V4x: A_27a,V5y: A_27b] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ A_27a @ V4x @ V1v1 ) @ V3b ) ) )
| ( V0f
= ( ^ [V6b: $o,V7x: A_27a,V8y: A_27b] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ A_27b @ V8y @ V2v2 ) @ ( c_2Ebool_2E_7E @ V6b ) ) ) ) ) ) ).
thf(thm_2Esum_2Esum__TY__DEF,axiom,
! [A_27a: $tType,A_27b: $tType] :
? [V0rep: ( tyop_2Esum_2Esum @ A_27a @ A_27b ) > $o > A_27a > A_27b > $o] : ( c_2Ebool_2ETYPE__DEFINITION @ ( $o > A_27a > A_27b > $o ) @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ ( c_2Esum_2EIS__SUM__REP @ A_27a @ A_27b ) @ V0rep ) ).
thf(thm_2Esum_2Esum__ISO__DEF,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0a: tyop_2Esum_2Esum @ A_27a @ A_27b] :
( ( c_2Esum_2EABS__sum @ A_27a @ A_27b @ ( c_2Esum_2EREP__sum @ A_27a @ A_27b @ V0a ) )
= V0a )
& ! [V1r: $o > A_27a > A_27b > $o] :
( ( c_2Esum_2EIS__SUM__REP @ A_27a @ A_27b @ V1r )
<=> ( ( c_2Esum_2EREP__sum @ A_27a @ A_27b @ ( c_2Esum_2EABS__sum @ A_27a @ A_27b @ V1r ) )
= V1r ) ) ) ).
thf(thm_2Esum_2EINL__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0e: A_27a] :
( ( c_2Esum_2EINL @ A_27a @ A_27b @ V0e )
= ( c_2Esum_2EABS__sum @ A_27a @ A_27b
@ ^ [V1b: $o,V2x: A_27a,V3y: A_27b] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ A_27a @ V2x @ V0e ) @ V1b ) ) ) ).
thf(thm_2Esum_2EINR__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0e: A_27b] :
( ( c_2Esum_2EINR @ A_27a @ A_27b @ V0e )
= ( c_2Esum_2EABS__sum @ A_27a @ A_27b
@ ^ [V1b: $o,V2x: A_27a,V3y: A_27b] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ A_27b @ V3y @ V0e ) @ ( c_2Ebool_2E_7E @ V1b ) ) ) ) ).
thf(thm_2Esum_2EISL,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0x: A_27a] : ( c_2Esum_2EISL @ A_27a @ A_27b @ ( c_2Esum_2EINL @ A_27a @ A_27b @ V0x ) )
& ! [V1y: A_27b] : ( (~) @ ( c_2Esum_2EISL @ A_27a @ A_27b @ ( c_2Esum_2EINR @ A_27a @ A_27b @ V1y ) ) ) ) ).
thf(thm_2Esum_2EISR,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0x: A_27b] : ( c_2Esum_2EISR @ A_27a @ A_27b @ ( c_2Esum_2EINR @ A_27a @ A_27b @ V0x ) )
& ! [V1y: A_27a] : ( (~) @ ( c_2Esum_2EISR @ A_27a @ A_27b @ ( c_2Esum_2EINL @ A_27a @ A_27b @ V1y ) ) ) ) ).
thf(thm_2Esum_2EOUTL,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a] :
( ( c_2Esum_2EOUTL @ A_27a @ A_27b @ ( c_2Esum_2EINL @ A_27a @ A_27b @ V0x ) )
= V0x ) ).
thf(thm_2Esum_2EOUTR,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27b] :
( ( c_2Esum_2EOUTR @ A_27a @ A_27b @ ( c_2Esum_2EINR @ A_27a @ A_27b @ V0x ) )
= V0x ) ).
thf(thm_2Esum_2Esum__case__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType] :
( ! [V0x: A_27a,V1f: A_27a > A_27c,V2f1: A_27b > A_27c] :
( ( c_2Esum_2Esum__CASE @ A_27a @ A_27b @ A_27c @ ( c_2Esum_2EINL @ A_27a @ A_27b @ V0x ) @ V1f @ V2f1 )
= ( V1f @ V0x ) )
& ! [V3y: A_27b,V4f: A_27a > A_27c,V5f1: A_27b > A_27c] :
( ( c_2Esum_2Esum__CASE @ A_27a @ A_27b @ A_27c @ ( c_2Esum_2EINR @ A_27a @ A_27b @ V3y ) @ V4f @ V5f1 )
= ( V5f1 @ V3y ) ) ) ).
thf(thm_2Esum_2ESUM__MAP__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] :
( ! [V0f: A_27a > A_27c,V1g: A_27b > A_27d,V2a: A_27a] :
( ( c_2Esum_2E_2B_2B @ A_27a @ A_27b @ A_27c @ A_27d @ V0f @ V1g @ ( c_2Esum_2EINL @ A_27a @ A_27b @ V2a ) )
= ( c_2Esum_2EINL @ A_27c @ A_27d @ ( V0f @ V2a ) ) )
& ! [V3f: A_27a > A_27c,V4g: A_27b > A_27d,V5b: A_27b] :
( ( c_2Esum_2E_2B_2B @ A_27a @ A_27b @ A_27c @ A_27d @ V3f @ V4g @ ( c_2Esum_2EINR @ A_27a @ A_27b @ V5b ) )
= ( c_2Esum_2EINR @ A_27c @ A_27d @ ( V4g @ V5b ) ) ) ) ).
thf(thm_2Esum_2ESUM__ALL__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0P: A_27a > $o,V1Q: A_27b > $o,V2x: A_27a] :
( ( c_2Esum_2ESUM__ALL @ A_27a @ A_27b @ V0P @ V1Q @ ( c_2Esum_2EINL @ A_27a @ A_27b @ V2x ) )
= ( V0P @ V2x ) )
& ! [V3P: A_27a > $o,V4Q: A_27b > $o,V5y: A_27b] :
( ( c_2Esum_2ESUM__ALL @ A_27a @ A_27b @ V3P @ V4Q @ ( c_2Esum_2EINR @ A_27a @ A_27b @ V5y ) )
= ( V4Q @ V5y ) ) ) ).
thf(thm_2Esum_2EINL__11,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27a,V1x: A_27a] :
( ( ( c_2Esum_2EINL @ A_27a @ A_27b @ V1x )
= ( c_2Esum_2EINL @ A_27a @ A_27b @ V0y ) )
<=> ( V1x = V0y ) ) ).
thf(thm_2Esum_2EINR__11,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27b,V1x: A_27b] :
( ( ( c_2Esum_2EINR @ A_27a @ A_27b @ V1x )
= ( c_2Esum_2EINR @ A_27a @ A_27b @ V0y ) )
<=> ( V1x = V0y ) ) ).
thf(thm_2Esum_2EINR__INL__11,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0y: A_27a,V1x: A_27a] :
( ( ( c_2Esum_2EINL @ A_27a @ A_27b @ V1x )
= ( c_2Esum_2EINL @ A_27a @ A_27b @ V0y ) )
<=> ( V1x = V0y ) )
& ! [V2y: A_27b,V3x: A_27b] :
( ( ( c_2Esum_2EINR @ A_27a @ A_27b @ V3x )
= ( c_2Esum_2EINR @ A_27a @ A_27b @ V2y ) )
<=> ( V3x = V2y ) ) ) ).
thf(thm_2Esum_2EINR__neq__INL,axiom,
! [A_27a: $tType,A_27b: $tType,V0v1: A_27a,V1v2: A_27b] :
( (~)
@ ( ( c_2Esum_2EINR @ A_27a @ A_27b @ V1v2 )
= ( c_2Esum_2EINL @ A_27a @ A_27b @ V0v1 ) ) ) ).
thf(thm_2Esum_2Esum__axiom,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27c,V1g: A_27b > A_27c] :
( c_2Ebool_2E_3F_21 @ ( ( tyop_2Esum_2Esum @ A_27a @ A_27b ) > A_27c )
@ ^ [V2h: ( tyop_2Esum_2Esum @ A_27a @ A_27b ) > A_27c] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ ( A_27a > A_27c ) @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ V2h @ ( c_2Esum_2EINL @ A_27a @ A_27b ) ) @ V0f ) @ ( c_2Emin_2E_3D @ ( A_27b > A_27c ) @ ( c_2Ecombin_2Eo @ A_27b @ A_27c @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ V2h @ ( c_2Esum_2EINR @ A_27a @ A_27b ) ) @ V1g ) ) ) ).
thf(thm_2Esum_2Esum__INDUCT,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: ( tyop_2Esum_2Esum @ A_27a @ A_27b ) > $o] :
( ( ! [V1x: A_27a] : ( V0P @ ( c_2Esum_2EINL @ A_27a @ A_27b @ V1x ) )
& ! [V2y: A_27b] : ( V0P @ ( c_2Esum_2EINR @ A_27a @ A_27b @ V2y ) ) )
=> ! [V3s: tyop_2Esum_2Esum @ A_27a @ A_27b] : ( V0P @ V3s ) ) ).
thf(thm_2Esum_2EFORALL__SUM,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: ( tyop_2Esum_2Esum @ A_27a @ A_27b ) > $o] :
( ! [V1s: tyop_2Esum_2Esum @ A_27a @ A_27b] : ( V0P @ V1s )
<=> ( ! [V2x: A_27a] : ( V0P @ ( c_2Esum_2EINL @ A_27a @ A_27b @ V2x ) )
& ! [V3y: A_27b] : ( V0P @ ( c_2Esum_2EINR @ A_27a @ A_27b @ V3y ) ) ) ) ).
thf(thm_2Esum_2EEXISTS__SUM,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: ( tyop_2Esum_2Esum @ A_27a @ A_27b ) > $o] :
( ? [V1s: tyop_2Esum_2Esum @ A_27a @ A_27b] : ( V0P @ V1s )
<=> ( ? [V2x: A_27a] : ( V0P @ ( c_2Esum_2EINL @ A_27a @ A_27b @ V2x ) )
| ? [V3y: A_27b] : ( V0P @ ( c_2Esum_2EINR @ A_27a @ A_27b @ V3y ) ) ) ) ).
thf(thm_2Esum_2Esum__Axiom,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27c,V1g: A_27b > A_27c] :
? [V2h: ( tyop_2Esum_2Esum @ A_27a @ A_27b ) > A_27c] :
( ! [V3x: A_27a] :
( ( V2h @ ( c_2Esum_2EINL @ A_27a @ A_27b @ V3x ) )
= ( V0f @ V3x ) )
& ! [V4y: A_27b] :
( ( V2h @ ( c_2Esum_2EINR @ A_27a @ A_27b @ V4y ) )
= ( V1g @ V4y ) ) ) ).
thf(thm_2Esum_2Esum__CASES,axiom,
! [A_27a: $tType,A_27b: $tType,V0ss: tyop_2Esum_2Esum @ A_27a @ A_27b] :
( ? [V1x: A_27a] :
( V0ss
= ( c_2Esum_2EINL @ A_27a @ A_27b @ V1x ) )
| ? [V2y: A_27b] :
( V0ss
= ( c_2Esum_2EINR @ A_27a @ A_27b @ V2y ) ) ) ).
thf(thm_2Esum_2Esum__distinct,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1y: A_27b] :
( (~)
@ ( ( c_2Esum_2EINL @ A_27a @ A_27b @ V0x )
= ( c_2Esum_2EINR @ A_27a @ A_27b @ V1y ) ) ) ).
thf(thm_2Esum_2Esum__distinct1,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1y: A_27b] :
( (~)
@ ( ( c_2Esum_2EINR @ A_27a @ A_27b @ V1y )
= ( c_2Esum_2EINL @ A_27a @ A_27b @ V0x ) ) ) ).
thf(thm_2Esum_2EISL__OR__ISR,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Esum_2Esum @ A_27a @ A_27b] :
( ( c_2Esum_2EISL @ A_27a @ A_27b @ V0x )
| ( c_2Esum_2EISR @ A_27a @ A_27b @ V0x ) ) ).
thf(thm_2Esum_2EINL,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Esum_2Esum @ A_27a @ A_27b] :
( ( c_2Esum_2EISL @ A_27a @ A_27b @ V0x )
=> ( ( c_2Esum_2EINL @ A_27a @ A_27b @ ( c_2Esum_2EOUTL @ A_27a @ A_27b @ V0x ) )
= V0x ) ) ).
thf(thm_2Esum_2EINR,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Esum_2Esum @ A_27a @ A_27b] :
( ( c_2Esum_2EISR @ A_27a @ A_27b @ V0x )
=> ( ( c_2Esum_2EINR @ A_27a @ A_27b @ ( c_2Esum_2EOUTR @ A_27a @ A_27b @ V0x ) )
= V0x ) ) ).
thf(thm_2Esum_2Esum__case__cong,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f1_27: A_27b > A_27c,V1f_27: A_27a > A_27c,V2M: tyop_2Esum_2Esum @ A_27a @ A_27b,V3M_27: tyop_2Esum_2Esum @ A_27a @ A_27b,V4f: A_27a > A_27c,V5f1: A_27b > A_27c] :
( ( ( V2M = V3M_27 )
& ! [V6x: A_27a] :
( ( V3M_27
= ( c_2Esum_2EINL @ A_27a @ A_27b @ V6x ) )
=> ( ( V4f @ V6x )
= ( V1f_27 @ V6x ) ) )
& ! [V7y: A_27b] :
( ( V3M_27
= ( c_2Esum_2EINR @ A_27a @ A_27b @ V7y ) )
=> ( ( V5f1 @ V7y )
= ( V0f1_27 @ V7y ) ) ) )
=> ( ( c_2Esum_2Esum__CASE @ A_27a @ A_27b @ A_27c @ V2M @ V4f @ V5f1 )
= ( c_2Esum_2Esum__CASE @ A_27a @ A_27b @ A_27c @ V3M_27 @ V1f_27 @ V0f1_27 ) ) ) ).
thf(thm_2Esum_2ESUM__MAP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0f: A_27a > A_27c,V1g: A_27b > A_27d,V2z: tyop_2Esum_2Esum @ A_27a @ A_27b] :
( ( c_2Esum_2E_2B_2B @ A_27a @ A_27b @ A_27c @ A_27d @ V0f @ V1g @ V2z )
= ( c_2Ebool_2ECOND @ ( tyop_2Esum_2Esum @ A_27c @ A_27d ) @ ( c_2Esum_2EISL @ A_27a @ A_27b @ V2z ) @ ( c_2Esum_2EINL @ A_27c @ A_27d @ ( V0f @ ( c_2Esum_2EOUTL @ A_27a @ A_27b @ V2z ) ) ) @ ( c_2Esum_2EINR @ A_27c @ A_27d @ ( V1g @ ( c_2Esum_2EOUTR @ A_27a @ A_27b @ V2z ) ) ) ) ) ).
thf(thm_2Esum_2ESUM__MAP__CASE,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0f: A_27a > A_27c,V1g: A_27b > A_27d,V2z: tyop_2Esum_2Esum @ A_27a @ A_27b] :
( ( c_2Esum_2E_2B_2B @ A_27a @ A_27b @ A_27c @ A_27d @ V0f @ V1g @ V2z )
= ( c_2Esum_2Esum__CASE @ A_27a @ A_27b @ ( tyop_2Esum_2Esum @ A_27c @ A_27d ) @ V2z @ ( c_2Ecombin_2Eo @ A_27a @ ( tyop_2Esum_2Esum @ A_27c @ A_27d ) @ A_27c @ ( c_2Esum_2EINL @ A_27c @ A_27d ) @ V0f ) @ ( c_2Ecombin_2Eo @ A_27b @ ( tyop_2Esum_2Esum @ A_27c @ A_27d ) @ A_27d @ ( c_2Esum_2EINR @ A_27c @ A_27d ) @ V1g ) ) ) ).
thf(thm_2Esum_2ESUM__MAP__I,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Esum_2E_2B_2B @ A_27a @ A_27b @ A_27a @ A_27b @ ( c_2Ecombin_2EI @ A_27a ) @ ( c_2Ecombin_2EI @ A_27b ) )
= ( c_2Ecombin_2EI @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) ) ).
thf(thm_2Esum_2Econd__sum__expand,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,A_27h: $tType,V0P: $o] :
( ! [V1x: A_27a,V2y: A_27b,V3z: A_27a] :
( ( ( c_2Ebool_2ECOND @ ( tyop_2Esum_2Esum @ A_27b @ A_27a ) @ V0P @ ( c_2Esum_2EINR @ A_27b @ A_27a @ V1x ) @ ( c_2Esum_2EINL @ A_27b @ A_27a @ V2y ) )
= ( c_2Esum_2EINR @ A_27b @ A_27a @ V3z ) )
<=> ( V0P
& ( V3z = V1x ) ) )
& ! [V4x: A_27c,V5y: A_27d,V6z: A_27d] :
( ( ( c_2Ebool_2ECOND @ ( tyop_2Esum_2Esum @ A_27d @ A_27c ) @ V0P @ ( c_2Esum_2EINR @ A_27d @ A_27c @ V4x ) @ ( c_2Esum_2EINL @ A_27d @ A_27c @ V5y ) )
= ( c_2Esum_2EINL @ A_27d @ A_27c @ V6z ) )
<=> ( ( (~) @ V0P )
& ( V6z = V5y ) ) )
& ! [V7x: A_27e,V8y: A_27f,V9z: A_27e] :
( ( ( c_2Ebool_2ECOND @ ( tyop_2Esum_2Esum @ A_27e @ A_27f ) @ V0P @ ( c_2Esum_2EINL @ A_27e @ A_27f @ V7x ) @ ( c_2Esum_2EINR @ A_27e @ A_27f @ V8y ) )
= ( c_2Esum_2EINL @ A_27e @ A_27f @ V9z ) )
<=> ( V0P
& ( V9z = V7x ) ) )
& ! [V10x: A_27g,V11y: A_27h,V12z: A_27h] :
( ( ( c_2Ebool_2ECOND @ ( tyop_2Esum_2Esum @ A_27g @ A_27h ) @ V0P @ ( c_2Esum_2EINL @ A_27g @ A_27h @ V10x ) @ ( c_2Esum_2EINR @ A_27g @ A_27h @ V11y ) )
= ( c_2Esum_2EINR @ A_27g @ A_27h @ V12z ) )
<=> ( ( (~) @ V0P )
& ( V12z = V11y ) ) ) ) ).
thf(thm_2Esum_2ENOT__ISL__ISR,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Esum_2Esum @ A_27a @ A_27b] :
( ( (~) @ ( c_2Esum_2EISL @ A_27a @ A_27b @ V0x ) )
<=> ( c_2Esum_2EISR @ A_27a @ A_27b @ V0x ) ) ).
thf(thm_2Esum_2ENOT__ISR__ISL,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Esum_2Esum @ A_27a @ A_27b] :
( ( (~) @ ( c_2Esum_2EISR @ A_27a @ A_27b @ V0x ) )
<=> ( c_2Esum_2EISL @ A_27a @ A_27b @ V0x ) ) ).
thf(thm_2Esum_2ESUM__ALL__MONO,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: tyop_2Esum_2Esum @ A_27a @ A_27b,V1Q_27: A_27b > $o,V2Q: A_27b > $o,V3P_27: A_27a > $o,V4P: A_27a > $o] :
( ( ! [V5x: A_27a] :
( ( V4P @ V5x )
=> ( V3P_27 @ V5x ) )
& ! [V6y: A_27b] :
( ( V2Q @ V6y )
=> ( V1Q_27 @ V6y ) ) )
=> ( ( c_2Esum_2ESUM__ALL @ A_27a @ A_27b @ V4P @ V2Q @ V0s )
=> ( c_2Esum_2ESUM__ALL @ A_27a @ A_27b @ V3P_27 @ V1Q_27 @ V0s ) ) ) ).
thf(thm_2Esum_2ESUM__ALL__CONG,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: tyop_2Esum_2Esum @ A_27a @ A_27b,V1s_27: tyop_2Esum_2Esum @ A_27a @ A_27b,V2P: A_27a > $o,V3P_27: A_27a > $o,V4Q: A_27b > $o,V5Q_27: A_27b > $o] :
( ( ( V0s = V1s_27 )
& ! [V6a: A_27a] :
( ( V1s_27
= ( c_2Esum_2EINL @ A_27a @ A_27b @ V6a ) )
=> ( ( V2P @ V6a )
= ( V3P_27 @ V6a ) ) )
& ! [V7b: A_27b] :
( ( V1s_27
= ( c_2Esum_2EINR @ A_27a @ A_27b @ V7b ) )
=> ( ( V4Q @ V7b )
= ( V5Q_27 @ V7b ) ) ) )
=> ( ( c_2Esum_2ESUM__ALL @ A_27a @ A_27b @ V2P @ V4Q @ V0s )
= ( c_2Esum_2ESUM__ALL @ A_27a @ A_27b @ V3P_27 @ V5Q_27 @ V1s_27 ) ) ) ).
thf(thm_2Esum_2Edatatype__sum,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0sum: ( A_27a > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) > ( A_27b > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) > A_27c] : ( c_2Ebool_2EDATATYPE @ A_27c @ ( V0sum @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ ( c_2Esum_2EINR @ A_27a @ A_27b ) ) ) ).
%------------------------------------------------------------------------------