ITP001 Axioms: ITP012^5.ax
%------------------------------------------------------------------------------
% File : ITP012^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : sum^2.ax [Gau20]
% : HL4012^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 62 ( 1 unt; 13 typ; 0 def)
% Number of atoms : 614 ( 49 equ; 0 cnn)
% Maximal formula atoms : 35 ( 9 avg)
% Number of connectives : 1523 ( 9 ~; 4 |; 31 &;1340 @)
% ( 19 <=>; 120 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 12 avg;1340 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 29 ( 29 >; 0 *; 0 +; 0 <<)
% Number of symbols : 30 ( 29 usr; 16 con; 0-4 aty)
% Number of variables : 247 ( 13 ^ 225 !; 9 ?; 247 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Esum_2Esum,type,
ty_2Esum_2Esum: del > del > del ).
thf(tp_c_2Esum_2E_2B_2B,type,
c_2Esum_2E_2B_2B: del > del > del > del > $i ).
thf(mem_c_2Esum_2E_2B_2B,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : ( mem @ ( c_2Esum_2E_2B_2B @ A_27a @ A_27b @ A_27c @ A_27d ) @ ( arr @ ( arr @ A_27a @ A_27c ) @ ( arr @ ( arr @ A_27b @ A_27d ) @ ( arr @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ ( ty_2Esum_2Esum @ A_27c @ A_27d ) ) ) ) ) ).
thf(tp_c_2Esum_2EABS__sum,type,
c_2Esum_2EABS__sum: del > del > $i ).
thf(mem_c_2Esum_2EABS__sum,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Esum_2EABS__sum @ A_27a @ A_27b ) @ ( arr @ ( arr @ bool @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) ) ).
thf(tp_c_2Esum_2EINL,type,
c_2Esum_2EINL: del > del > $i ).
thf(mem_c_2Esum_2EINL,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) ) ).
thf(tp_c_2Esum_2EINR,type,
c_2Esum_2EINR: del > del > $i ).
thf(mem_c_2Esum_2EINR,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ ( arr @ A_27b @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) ) ).
thf(tp_c_2Esum_2EISL,type,
c_2Esum_2EISL: del > del > $i ).
thf(mem_c_2Esum_2EISL,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Esum_2EISL @ A_27a @ A_27b ) @ ( arr @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ bool ) ) ).
thf(tp_c_2Esum_2EISR,type,
c_2Esum_2EISR: del > del > $i ).
thf(mem_c_2Esum_2EISR,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Esum_2EISR @ A_27a @ A_27b ) @ ( arr @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ bool ) ) ).
thf(tp_c_2Esum_2EIS__SUM__REP,type,
c_2Esum_2EIS__SUM__REP: del > del > $i ).
thf(mem_c_2Esum_2EIS__SUM__REP,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Esum_2EIS__SUM__REP @ A_27a @ A_27b ) @ ( arr @ ( arr @ bool @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) @ bool ) ) ).
thf(tp_c_2Esum_2EOUTL,type,
c_2Esum_2EOUTL: del > del > $i ).
thf(mem_c_2Esum_2EOUTL,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Esum_2EOUTL @ A_27a @ A_27b ) @ ( arr @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ A_27a ) ) ).
thf(tp_c_2Esum_2EOUTR,type,
c_2Esum_2EOUTR: del > del > $i ).
thf(mem_c_2Esum_2EOUTR,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Esum_2EOUTR @ A_27a @ A_27b ) @ ( arr @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ A_27b ) ) ).
thf(tp_c_2Esum_2EREP__sum,type,
c_2Esum_2EREP__sum: del > del > $i ).
thf(mem_c_2Esum_2EREP__sum,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Esum_2EREP__sum @ A_27a @ A_27b ) @ ( arr @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ ( arr @ bool @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) ) ) ).
thf(tp_c_2Esum_2ESUM__ALL,type,
c_2Esum_2ESUM__ALL: del > del > $i ).
thf(mem_c_2Esum_2ESUM__ALL,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Esum_2ESUM__ALL @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ( arr @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ bool ) ) ) ) ).
thf(tp_c_2Esum_2Esum__CASE,type,
c_2Esum_2Esum__CASE: del > del > del > $i ).
thf(mem_c_2Esum_2Esum__CASE,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Esum_2Esum__CASE @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27c ) @ ( arr @ ( arr @ A_27b @ A_27c ) @ A_27c ) ) ) ) ).
thf(ax_thm_2Esum_2EIS__SUM__REP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ bool @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) )
=> ( ( p @ ( ap @ ( c_2Esum_2EIS__SUM__REP @ A_27a @ A_27b ) @ V0f ) )
<=> ? [V1v1: $i] :
( ( mem @ V1v1 @ A_27a )
& ? [V2v2: $i] :
( ( mem @ V2v2 @ A_27b )
& ( ( V0f
= ( lam @ bool
@ ^ [V3b: $i] :
( lam @ A_27a
@ ^ [V4x: $i] :
( lam @ A_27b
@ ^ [V5y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V4x ) @ V1v1 ) ) @ V3b ) ) ) ) )
| ( V0f
= ( lam @ bool
@ ^ [V6b: $i] :
( lam @ A_27a
@ ^ [V7x: $i] :
( lam @ A_27b
@ ^ [V8y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ V8y ) @ V2v2 ) ) @ ( ap @ c_2Ebool_2E_7E @ V6b ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Esum_2Esum__TY__DEF,axiom,
! [A_27a: del,A_27b: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ ( arr @ bool @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( arr @ bool @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( c_2Esum_2EIS__SUM__REP @ A_27a @ A_27b ) ) @ V0rep ) ) ) ).
thf(ax_thm_2Esum_2Esum__ISO__DEF,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
=> ( ( ap @ ( c_2Esum_2EABS__sum @ A_27a @ A_27b ) @ ( ap @ ( c_2Esum_2EREP__sum @ A_27a @ A_27b ) @ V0a ) )
= V0a ) )
& ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ bool @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) )
=> ( ( p @ ( ap @ ( c_2Esum_2EIS__SUM__REP @ A_27a @ A_27b ) @ V1r ) )
<=> ( ( ap @ ( c_2Esum_2EREP__sum @ A_27a @ A_27b ) @ ( ap @ ( c_2Esum_2EABS__sum @ A_27a @ A_27b ) @ V1r ) )
= V1r ) ) ) ) ).
thf(ax_thm_2Esum_2EINL__DEF,axiom,
! [A_27a: del,A_27b: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ( ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V0e )
= ( ap @ ( c_2Esum_2EABS__sum @ A_27a @ A_27b )
@ ( lam @ bool
@ ^ [V1b: $i] :
( lam @ A_27a
@ ^ [V2x: $i] :
( lam @ A_27b
@ ^ [V3y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2x ) @ V0e ) ) @ V1b ) ) ) ) ) ) ) ).
thf(ax_thm_2Esum_2EINR__DEF,axiom,
! [A_27a: del,A_27b: del,V0e: $i] :
( ( mem @ V0e @ A_27b )
=> ( ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V0e )
= ( ap @ ( c_2Esum_2EABS__sum @ A_27a @ A_27b )
@ ( lam @ bool
@ ^ [V1b: $i] :
( lam @ A_27a
@ ^ [V2x: $i] :
( lam @ A_27b
@ ^ [V3y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ V3y ) @ V0e ) ) @ ( ap @ c_2Ebool_2E_7E @ V1b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esum_2EINL__11,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ( ( ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V0x )
= ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V1y ) )
<=> ( V0x = V1y ) ) ) ) ).
thf(conj_thm_2Esum_2EINR__11,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27b )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27b )
=> ( ( ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V0x )
= ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V1y ) )
<=> ( V0x = V1y ) ) ) ) ).
thf(conj_thm_2Esum_2EINR__INL__11,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0y: $i] :
( ( mem @ V0y @ A_27a )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V1x )
= ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V0y ) )
<=> ( V1x = V0y ) ) ) )
& ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27b )
=> ( ( ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V3x )
= ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V2y ) )
<=> ( V3x = V2y ) ) ) ) ) ).
thf(conj_thm_2Esum_2EINR__neq__INL,axiom,
! [A_27a: del,A_27b: del,V0v1: $i] :
( ( mem @ V0v1 @ A_27a )
=> ! [V1v2: $i] :
( ( mem @ V1v2 @ A_27b )
=> ( ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V1v2 )
!= ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V0v1 ) ) ) ) ).
thf(conj_thm_2Esum_2Esum__axiom,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27c ) )
=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ ( arr @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ A_27c ) )
@ ( lam @ ( arr @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ A_27c )
@ ^ [V2h: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ V2h ) @ ( c_2Esum_2EINL @ A_27a @ A_27b ) ) ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27b @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ A_27c @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ V2h ) @ ( c_2Esum_2EINR @ A_27a @ A_27b ) ) ) @ V1g ) ) ) ) ) ) ) ).
thf(conj_thm_2Esum_2Esum__INDUCT,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ bool ) )
=> ( ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V1x ) ) ) )
& ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V2y ) ) ) ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
=> ( p @ ( ap @ V0P @ V3s ) ) ) ) ) ).
thf(conj_thm_2Esum_2EFORALL__SUM,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ bool ) )
=> ( ! [V1s: $i] :
( ( mem @ V1s @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
=> ( p @ ( ap @ V0P @ V1s ) ) )
<=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V2x ) ) ) )
& ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V3y ) ) ) ) ) ) ) ).
thf(conj_thm_2Esum_2EEXISTS__SUM,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ bool ) )
=> ( ? [V1s: $i] :
( ( mem @ V1s @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
& ( p @ ( ap @ V0P @ V1s ) ) )
<=> ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ V0P @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V2x ) ) ) )
| ? [V3y: $i] :
( ( mem @ V3y @ A_27b )
& ( p @ ( ap @ V0P @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V3y ) ) ) ) ) ) ) ).
thf(conj_thm_2Esum_2Esum__Axiom,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27c ) )
=> ? [V2h: $i] :
( ( mem @ V2h @ ( arr @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ A_27c ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ap @ V2h @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V3x ) )
= ( ap @ V0f @ V3x ) ) )
& ! [V4y: $i] :
( ( mem @ V4y @ A_27b )
=> ( ( ap @ V2h @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V4y ) )
= ( ap @ V1g @ V4y ) ) ) ) ) ) ).
thf(conj_thm_2Esum_2Esum__CASES,axiom,
! [A_27a: del,A_27b: del,V0ss: $i] :
( ( mem @ V0ss @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
=> ( ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ( V0ss
= ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V1x ) ) )
| ? [V2y: $i] :
( ( mem @ V2y @ A_27b )
& ( V0ss
= ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V2y ) ) ) ) ) ).
thf(conj_thm_2Esum_2Esum__distinct,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27b )
=> ( ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V0x )
!= ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V1y ) ) ) ) ).
thf(conj_thm_2Esum_2Esum__distinct1,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27b )
=> ( ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V1y )
!= ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V0x ) ) ) ) ).
thf(ax_thm_2Esum_2EISL,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( p @ ( ap @ ( c_2Esum_2EISL @ A_27a @ A_27b ) @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V0x ) ) ) )
& ! [V1y: $i] :
( ( mem @ V1y @ A_27b )
=> ~ ( p @ ( ap @ ( c_2Esum_2EISL @ A_27a @ A_27b ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V1y ) ) ) ) ) ).
thf(ax_thm_2Esum_2EISR,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0x: $i] :
( ( mem @ V0x @ A_27b )
=> ( p @ ( ap @ ( c_2Esum_2EISR @ A_27a @ A_27b ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V0x ) ) ) )
& ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ~ ( p @ ( ap @ ( c_2Esum_2EISR @ A_27a @ A_27b ) @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V1y ) ) ) ) ) ).
thf(ax_thm_2Esum_2EOUTL,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( c_2Esum_2EOUTL @ A_27a @ A_27b ) @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V0x ) )
= V0x ) ) ).
thf(ax_thm_2Esum_2EOUTR,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27b )
=> ( ( ap @ ( c_2Esum_2EOUTR @ A_27a @ A_27b ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V0x ) )
= V0x ) ) ).
thf(conj_thm_2Esum_2EISL__OR__ISR,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( c_2Esum_2EISL @ A_27a @ A_27b ) @ V0x ) )
| ( p @ ( ap @ ( c_2Esum_2EISR @ A_27a @ A_27b ) @ V0x ) ) ) ) ).
thf(conj_thm_2Esum_2EINL,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( c_2Esum_2EISL @ A_27a @ A_27b ) @ V0x ) )
=> ( ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ ( ap @ ( c_2Esum_2EOUTL @ A_27a @ A_27b ) @ V0x ) )
= V0x ) ) ) ).
thf(conj_thm_2Esum_2EINR,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( c_2Esum_2EISR @ A_27a @ A_27b ) @ V0x ) )
=> ( ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ ( ap @ ( c_2Esum_2EOUTR @ A_27a @ A_27b ) @ V0x ) )
= V0x ) ) ) ).
thf(ax_thm_2Esum_2Esum__case__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del] :
( ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27c ) )
=> ! [V2f1: $i] :
( ( mem @ V2f1 @ ( arr @ A_27b @ A_27c ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esum_2Esum__CASE @ A_27a @ A_27b @ A_27c ) @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V0x ) ) @ V1f ) @ V2f1 )
= ( ap @ V1f @ V0x ) ) ) ) )
& ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ A_27c ) )
=> ! [V5f1: $i] :
( ( mem @ V5f1 @ ( arr @ A_27b @ A_27c ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esum_2Esum__CASE @ A_27a @ A_27b @ A_27c ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V3y ) ) @ V4f ) @ V5f1 )
= ( ap @ V5f1 @ V3y ) ) ) ) ) ) ).
thf(conj_thm_2Esum_2Esum__case__cong,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f_27: $i] :
( ( mem @ V0f_27 @ ( arr @ A_27a @ A_27c ) )
=> ! [V1f1_27: $i] :
( ( mem @ V1f1_27 @ ( arr @ A_27b @ A_27c ) )
=> ! [V2M: $i] :
( ( mem @ V2M @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
=> ! [V3M_27: $i] :
( ( mem @ V3M_27 @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ A_27c ) )
=> ! [V5f1: $i] :
( ( mem @ V5f1 @ ( arr @ A_27b @ A_27c ) )
=> ( ( ( V2M = V3M_27 )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ( ( V3M_27
= ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V6x ) )
=> ( ( ap @ V4f @ V6x )
= ( ap @ V0f_27 @ V6x ) ) ) )
& ! [V7y: $i] :
( ( mem @ V7y @ A_27b )
=> ( ( V3M_27
= ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V7y ) )
=> ( ( ap @ V5f1 @ V7y )
= ( ap @ V1f1_27 @ V7y ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esum_2Esum__CASE @ A_27a @ A_27b @ A_27c ) @ V2M ) @ V4f ) @ V5f1 )
= ( ap @ ( ap @ ( ap @ ( c_2Esum_2Esum__CASE @ A_27a @ A_27b @ A_27c ) @ V3M_27 ) @ V0f_27 ) @ V1f1_27 ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Esum_2ESUM__MAP__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27d ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esum_2E_2B_2B @ A_27a @ A_27b @ A_27c @ A_27d ) @ V0f ) @ V1g ) @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V2a ) )
= ( ap @ ( c_2Esum_2EINL @ A_27c @ A_27d ) @ ( ap @ V0f @ V2a ) ) ) ) ) )
& ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ A_27c ) )
=> ! [V4g: $i] :
( ( mem @ V4g @ ( arr @ A_27b @ A_27d ) )
=> ! [V5b: $i] :
( ( mem @ V5b @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esum_2E_2B_2B @ A_27a @ A_27b @ A_27c @ A_27d ) @ V3f ) @ V4g ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V5b ) )
= ( ap @ ( c_2Esum_2EINR @ A_27c @ A_27d ) @ ( ap @ V4g @ V5b ) ) ) ) ) ) ) ).
thf(conj_thm_2Esum_2ESUM__MAP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27d ) )
=> ! [V2z: $i] :
( ( mem @ V2z @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esum_2E_2B_2B @ A_27a @ A_27b @ A_27c @ A_27d ) @ V0f ) @ V1g ) @ V2z )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esum_2Esum @ A_27c @ A_27d ) ) @ ( ap @ ( c_2Esum_2EISL @ A_27a @ A_27b ) @ V2z ) ) @ ( ap @ ( c_2Esum_2EINL @ A_27c @ A_27d ) @ ( ap @ V0f @ ( ap @ ( c_2Esum_2EOUTL @ A_27a @ A_27b ) @ V2z ) ) ) ) @ ( ap @ ( c_2Esum_2EINR @ A_27c @ A_27d ) @ ( ap @ V1g @ ( ap @ ( c_2Esum_2EOUTR @ A_27a @ A_27b ) @ V2z ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esum_2ESUM__MAP__CASE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27d ) )
=> ! [V2z: $i] :
( ( mem @ V2z @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esum_2E_2B_2B @ A_27a @ A_27b @ A_27c @ A_27d ) @ V0f ) @ V1g ) @ V2z )
= ( ap @ ( ap @ ( ap @ ( c_2Esum_2Esum__CASE @ A_27a @ A_27b @ ( ty_2Esum_2Esum @ A_27c @ A_27d ) ) @ V2z ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ( ty_2Esum_2Esum @ A_27c @ A_27d ) @ A_27c ) @ ( c_2Esum_2EINL @ A_27c @ A_27d ) ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esum_2Esum @ A_27c @ A_27d ) @ A_27d ) @ ( c_2Esum_2EINR @ A_27c @ A_27d ) ) @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Esum_2ESUM__MAP__I,axiom,
! [A_27a: del,A_27b: del] :
( ( ap @ ( ap @ ( 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 @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) ) ).
thf(conj_thm_2Esum_2Econd__sum__expand,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,A_27g: del,A_27h: del,V0P: $i] :
( ( mem @ V0P @ bool )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esum_2Esum @ A_27b @ A_27a ) ) @ V0P ) @ ( ap @ ( c_2Esum_2EINR @ A_27b @ A_27a ) @ V1x ) ) @ ( ap @ ( c_2Esum_2EINL @ A_27b @ A_27a ) @ V2y ) )
= ( ap @ ( c_2Esum_2EINR @ A_27b @ A_27a ) @ V3z ) )
<=> ( ( p @ V0P )
& ( V3z = V1x ) ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27c )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27d )
=> ! [V6z: $i] :
( ( mem @ V6z @ A_27d )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esum_2Esum @ A_27d @ A_27c ) ) @ V0P ) @ ( ap @ ( c_2Esum_2EINR @ A_27d @ A_27c ) @ V4x ) ) @ ( ap @ ( c_2Esum_2EINL @ A_27d @ A_27c ) @ V5y ) )
= ( ap @ ( c_2Esum_2EINL @ A_27d @ A_27c ) @ V6z ) )
<=> ( ~ ( p @ V0P )
& ( V6z = V5y ) ) ) ) ) )
& ! [V7x: $i] :
( ( mem @ V7x @ A_27e )
=> ! [V8y: $i] :
( ( mem @ V8y @ A_27f )
=> ! [V9z: $i] :
( ( mem @ V9z @ A_27e )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esum_2Esum @ A_27e @ A_27f ) ) @ V0P ) @ ( ap @ ( c_2Esum_2EINL @ A_27e @ A_27f ) @ V7x ) ) @ ( ap @ ( c_2Esum_2EINR @ A_27e @ A_27f ) @ V8y ) )
= ( ap @ ( c_2Esum_2EINL @ A_27e @ A_27f ) @ V9z ) )
<=> ( ( p @ V0P )
& ( V9z = V7x ) ) ) ) ) )
& ! [V10x: $i] :
( ( mem @ V10x @ A_27g )
=> ! [V11y: $i] :
( ( mem @ V11y @ A_27h )
=> ! [V12z: $i] :
( ( mem @ V12z @ A_27h )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esum_2Esum @ A_27g @ A_27h ) ) @ V0P ) @ ( ap @ ( c_2Esum_2EINL @ A_27g @ A_27h ) @ V10x ) ) @ ( ap @ ( c_2Esum_2EINR @ A_27g @ A_27h ) @ V11y ) )
= ( ap @ ( c_2Esum_2EINR @ A_27g @ A_27h ) @ V12z ) )
<=> ( ~ ( p @ V0P )
& ( V12z = V11y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esum_2ENOT__ISL__ISR,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
=> ( ~ ( p @ ( ap @ ( c_2Esum_2EISL @ A_27a @ A_27b ) @ V0x ) )
<=> ( p @ ( ap @ ( c_2Esum_2EISR @ A_27a @ A_27b ) @ V0x ) ) ) ) ).
thf(conj_thm_2Esum_2ENOT__ISR__ISL,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
=> ( ~ ( p @ ( ap @ ( c_2Esum_2EISR @ A_27a @ A_27b ) @ V0x ) )
<=> ( p @ ( ap @ ( c_2Esum_2EISL @ A_27a @ A_27b ) @ V0x ) ) ) ) ).
thf(ax_thm_2Esum_2ESUM__ALL__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27b @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Esum_2ESUM__ALL @ A_27a @ A_27b ) @ V0P ) @ V1Q ) @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V2x ) ) )
<=> ( p @ ( ap @ V0P @ V2x ) ) ) ) ) )
& ! [V3P: $i] :
( ( mem @ V3P @ ( arr @ A_27a @ bool ) )
=> ! [V4Q: $i] :
( ( mem @ V4Q @ ( arr @ A_27b @ bool ) )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Esum_2ESUM__ALL @ A_27a @ A_27b ) @ V3P ) @ V4Q ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V5y ) ) )
<=> ( p @ ( ap @ V4Q @ V5y ) ) ) ) ) ) ) ).
thf(conj_thm_2Esum_2ESUM__ALL__MONO,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1P_27: $i] :
( ( mem @ V1P_27 @ ( arr @ A_27a @ bool ) )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ ( arr @ A_27b @ bool ) )
=> ! [V3Q_27: $i] :
( ( mem @ V3Q_27 @ ( arr @ A_27b @ bool ) )
=> ! [V4s: $i] :
( ( mem @ V4s @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
=> ( ( ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( p @ ( ap @ V0P @ V5x ) )
=> ( p @ ( ap @ V1P_27 @ V5x ) ) ) )
& ! [V6y: $i] :
( ( mem @ V6y @ A_27b )
=> ( ( p @ ( ap @ V2Q @ V6y ) )
=> ( p @ ( ap @ V3Q_27 @ V6y ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Esum_2ESUM__ALL @ A_27a @ A_27b ) @ V0P ) @ V2Q ) @ V4s ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Esum_2ESUM__ALL @ A_27a @ A_27b ) @ V1P_27 ) @ V3Q_27 ) @ V4s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esum_2ESUM__ALL__CONG,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
=> ! [V1s_27: $i] :
( ( mem @ V1s_27 @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
=> ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ A_27a @ bool ) )
=> ! [V3P_27: $i] :
( ( mem @ V3P_27 @ ( arr @ A_27a @ bool ) )
=> ! [V4Q: $i] :
( ( mem @ V4Q @ ( arr @ A_27b @ bool ) )
=> ! [V5Q_27: $i] :
( ( mem @ V5Q_27 @ ( arr @ A_27b @ bool ) )
=> ( ( ( V0s = V1s_27 )
& ! [V6a: $i] :
( ( mem @ V6a @ A_27a )
=> ( ( V1s_27
= ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V6a ) )
=> ( ( p @ ( ap @ V2P @ V6a ) )
<=> ( p @ ( ap @ V3P_27 @ V6a ) ) ) ) )
& ! [V7b: $i] :
( ( mem @ V7b @ A_27b )
=> ( ( V1s_27
= ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V7b ) )
=> ( ( p @ ( ap @ V4Q @ V7b ) )
<=> ( p @ ( ap @ V5Q_27 @ V7b ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Esum_2ESUM__ALL @ A_27a @ A_27b ) @ V2P ) @ V4Q ) @ V0s ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Esum_2ESUM__ALL @ A_27a @ A_27b ) @ V3P_27 ) @ V5Q_27 ) @ V1s_27 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esum_2Edatatype__sum,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0sum: $i] :
( ( mem @ V0sum @ ( arr @ ( arr @ A_27a @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( arr @ ( arr @ A_27b @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ A_27c ) ) )
=> ( p @ ( ap @ ( c_2Ebool_2EDATATYPE @ A_27c ) @ ( ap @ ( ap @ V0sum @ ( c_2Esum_2EINL @ A_27a @ A_27b ) ) @ ( c_2Esum_2EINR @ A_27a @ A_27b ) ) ) ) ) ).
%------------------------------------------------------------------------------