ITP001 Axioms: ITP078^5.ax
%------------------------------------------------------------------------------
% File : ITP078^5 : TPTP v9.0.0. 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 : update^2.ax [Gau20]
% : HL4078^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 24 ( 1 unt; 3 typ; 0 def)
% Number of atoms : 681 ( 31 equ; 0 cnn)
% Maximal formula atoms : 56 ( 28 avg)
% Number of connectives : 1356 ( 3 ~; 0 |; 12 &;1257 @)
% ( 1 <=>; 83 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 13 avg;1257 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 5 ( 5 >; 0 *; 0 +; 0 <<)
% Number of symbols : 42 ( 41 usr; 38 con; 0-2 aty)
% Number of variables : 142 ( 17 ^ 125 !; 0 ?; 142 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Eupdate_2EFIND,type,
c_2Eupdate_2EFIND: del > $i ).
thf(mem_c_2Eupdate_2EFIND,axiom,
! [A_27a: del] : ( mem @ ( c_2Eupdate_2EFIND @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ) ).
thf(tp_c_2Eupdate_2ELIST__UPDATE,type,
c_2Eupdate_2ELIST__UPDATE: del > del > $i ).
thf(mem_c_2Eupdate_2ELIST__UPDATE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) ) ) ).
thf(tp_c_2Eupdate_2EOVERRIDE,type,
c_2Eupdate_2EOVERRIDE: del > del > $i ).
thf(mem_c_2Eupdate_2EOVERRIDE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eupdate_2EOVERRIDE @ A_27a @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ) ).
thf(ax_thm_2Eupdate_2EFIND__def,axiom,
! [A_27a: del] :
( ! [V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Eupdate_2EFIND @ A_27a ) @ V0P ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Eoption_2ENONE @ A_27a ) ) )
& ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Eupdate_2EFIND @ A_27a ) @ V1P ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ V1P @ V2h ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2h ) ) @ ( ap @ ( ap @ ( c_2Eupdate_2EFIND @ A_27a ) @ V1P ) @ V3t ) ) ) ) ) ) ) ).
thf(ax_thm_2Eupdate_2EOVERRIDE__primitive__def,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Eupdate_2EOVERRIDE @ A_27a @ A_27b )
= ( ap
@ ( ap @ ( c_2Erelation_2EWFREC @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
@ ( ap @ ( c_2Emin_2E_40 @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ bool ) ) )
@ ( lam @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ bool ) )
@ ^ [V0R: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( c_2Erelation_2EWF @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) @ V0R ) )
@ ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
@ ( lam @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ^ [V1t: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V2x: $i] :
( ap
@ ( ap @ V0R
@ ( ap
@ ( ap @ ( c_2Elist_2EFILTER @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V3y: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V2x ) ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V3y ) ) ) ) )
@ V1t ) )
@ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V2x ) @ V1t ) ) ) ) ) ) ) ) ) )
@ ( lam @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
@ ^ [V4OVERRIDE: $i] :
( lam @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ^ [V5a: $i] :
( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) @ V5a ) @ ( ap @ ( c_2Ecombin_2EI @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V6x: $i] :
( lam @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ^ [V7t: $i] :
( ap @ ( c_2Ecombin_2EI @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
@ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V6x )
@ ( ap @ V4OVERRIDE
@ ( ap
@ ( ap @ ( c_2Elist_2EFILTER @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V8y: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V6x ) ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V8y ) ) ) ) )
@ V7t ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eupdate_2EOVERRIDE__ind,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) )
& ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ( p
@ ( ap @ V0P
@ ( ap
@ ( ap @ ( c_2Elist_2EFILTER @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V3y: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V1x ) ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V3y ) ) ) ) )
@ V2t ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V1x ) @ V2t ) ) ) ) ) ) )
=> ! [V4v: $i] :
( ( mem @ V4v @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( p @ ( ap @ V0P @ V4v ) ) ) ) ) ).
thf(conj_thm_2Eupdate_2EOVERRIDE__def,axiom,
! [A_27a: del,A_27b: del] :
( ( ( ap @ ( c_2Eupdate_2EOVERRIDE @ A_27a @ A_27b ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
= ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
& ! [V0x: $i] :
( ( mem @ V0x @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ( ap @ ( c_2Eupdate_2EOVERRIDE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0x ) @ V1t ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0x )
@ ( ap @ ( c_2Eupdate_2EOVERRIDE @ A_27a @ A_27b )
@ ( ap
@ ( ap @ ( c_2Elist_2EFILTER @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V2y: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0x ) ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V2y ) ) ) ) )
@ V1t ) ) ) ) ) ) ) ).
thf(ax_thm_2Eupdate_2ELIST__UPDATE__def,axiom,
! [A_27a: del,A_27b: del] :
( ( ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
= ( c_2Ecombin_2EI @ ( arr @ A_27a @ A_27b ) ) )
& ! [V0h: $i] :
( ( mem @ V0h @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0h ) @ V1t ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0h ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V0h ) ) ) @ ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Eupdate_2ELIST__UPDATE__LOOKUP,axiom,
! [A_27a: del,A_27b: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2i: $i] :
( ( mem @ V2i @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b ) @ V0l ) @ V1f ) @ V2i )
= ( ap
@ ( ap
@ ( ap @ ( c_2Eoption_2Eoption__CASE @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27b )
@ ( ap
@ ( ap @ ( c_2Eupdate_2EFIND @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V3x ) ) @ V2i ) ) )
@ V0l ) )
@ ( ap @ V1f @ V2i ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V4v: $i] :
( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ A_27b @ A_27a @ A_27b ) @ V4v )
@ ( lam @ A_27a
@ ^ [V5v1: $i] :
( lam @ A_27b
@ ^ [V6e: $i] : V6e ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eupdate_2ELIST__UPDATE__OVERRIDE,axiom,
! [A_27a: del,A_27b: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b ) @ V0l )
= ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b ) @ ( ap @ ( c_2Eupdate_2EOVERRIDE @ A_27a @ A_27b ) @ V0l ) ) ) ) ).
thf(conj_thm_2Eupdate_2ELIST__UPDATE__ALL__DISTINCT,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) @ ( c_2Epair_2EFST @ A_27a @ A_27b ) ) @ V1l2 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0l1 ) @ V1l2 ) ) )
=> ( ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b ) @ V0l1 )
= ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b ) @ V1l2 ) ) ) ) ) ).
thf(conj_thm_2Eupdate_2ELIST__UPDATE__SORT__OVERRIDE,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b ) @ V1l )
= ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Esorting_2EQSORT @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0R ) @ ( ap @ ( c_2Eupdate_2EOVERRIDE @ A_27a @ A_27b ) @ V1l ) ) ) ) ) ) ).
thf(conj_thm_2Eupdate_2ELIST__UPDATE__THMS,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] :
( ! [V0l1: $i] :
( ( mem @ V0l1 @ A_27a )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ A_27a )
=> ! [V2r1: $i] :
( ( mem @ V2r1 @ A_27b )
=> ! [V3r2: $i] :
( ( mem @ V3r2 @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V0l1 ) @ V2r1 ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V1l2 ) @ V3r2 ) )
= ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0l1 ) @ V2r1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1l2 ) @ V3r2 ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ) ) ) ) ) ) )
& ! [V4l: $i] :
( ( mem @ V4l @ A_27c )
=> ! [V5r: $i] :
( ( mem @ V5r @ A_27d )
=> ! [V6t: $i] :
( ( mem @ V6t @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27c @ A_27d ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( arr @ A_27c @ A_27d ) @ ( arr @ A_27c @ A_27d ) @ ( arr @ A_27c @ A_27d ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27c @ A_27d ) @ V4l ) @ V5r ) ) @ ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27c @ A_27d ) @ V6t ) )
= ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27c @ A_27d ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ A_27c @ A_27d ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27c @ A_27d ) @ V4l ) @ V5r ) ) @ V6t ) ) ) ) ) )
& ! [V7l1: $i] :
( ( mem @ V7l1 @ A_27e )
=> ! [V8l2: $i] :
( ( mem @ V8l2 @ A_27e )
=> ! [V9r1: $i] :
( ( mem @ V9r1 @ A_27f )
=> ! [V10r2: $i] :
( ( mem @ V10r2 @ A_27f )
=> ! [V11f: $i] :
( ( mem @ V11f @ ( arr @ A_27e @ A_27f ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27e @ A_27f ) @ V7l1 ) @ V9r1 ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27e @ A_27f ) @ V8l2 ) @ V10r2 ) @ V11f ) )
= ( ap @ ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27e @ A_27f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ A_27e @ A_27f ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27e @ A_27f ) @ V7l1 ) @ V9r1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ A_27e @ A_27f ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27e @ A_27f ) @ V8l2 ) @ V10r2 ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ A_27e @ A_27f ) ) ) ) ) @ V11f ) ) ) ) ) ) )
& ! [V12l: $i] :
( ( mem @ V12l @ A_27g )
=> ! [V13r: $i] :
( ( mem @ V13r @ A_27h )
=> ! [V14t: $i] :
( ( mem @ V14t @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27g @ A_27h ) ) )
=> ! [V15f: $i] :
( ( mem @ V15f @ ( arr @ A_27g @ A_27h ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27g @ A_27h ) @ V12l ) @ V13r ) @ ( ap @ ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27g @ A_27h ) @ V14t ) @ V15f ) )
= ( ap @ ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27g @ A_27h ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ A_27g @ A_27h ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27g @ A_27h ) @ V12l ) @ V13r ) ) @ V14t ) ) @ V15f ) ) ) ) ) )
& ! [V16l1: $i] :
( ( mem @ V16l1 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ! [V17l2: $i] :
( ( mem @ V17l2 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b ) @ V16l1 ) ) @ ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b ) @ V17l2 ) )
= ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V16l1 ) @ V17l2 ) ) ) ) )
& ! [V18l1: $i] :
( ( mem @ V18l1 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27c @ A_27d ) ) )
=> ! [V19l2: $i] :
( ( mem @ V19l2 @ A_27c )
=> ! [V20r: $i] :
( ( mem @ V20r @ A_27d )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( arr @ A_27c @ A_27d ) @ ( arr @ A_27c @ A_27d ) @ ( arr @ A_27c @ A_27d ) ) @ ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27c @ A_27d ) @ V18l1 ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27c @ A_27d ) @ V19l2 ) @ V20r ) )
= ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27c @ A_27d ) @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ ( ty_2Epair_2Eprod @ A_27c @ A_27d ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27c @ A_27d ) @ V19l2 ) @ V20r ) ) @ V18l1 ) ) ) ) ) )
& ! [V21l1: $i] :
( ( mem @ V21l1 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27e @ A_27f ) ) )
=> ! [V22l2: $i] :
( ( mem @ V22l2 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27e @ A_27f ) ) )
=> ! [V23f: $i] :
( ( mem @ V23f @ ( arr @ A_27e @ A_27f ) )
=> ( ( ap @ ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27e @ A_27f ) @ V21l1 ) @ ( ap @ ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27e @ A_27f ) @ V22l2 ) @ V23f ) )
= ( ap @ ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27e @ A_27f ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ ( ty_2Epair_2Eprod @ A_27e @ A_27f ) ) @ V21l1 ) @ V22l2 ) ) @ V23f ) ) ) ) )
& ! [V24l1: $i] :
( ( mem @ V24l1 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ A_27g @ A_27h ) ) )
=> ! [V25l2: $i] :
( ( mem @ V25l2 @ A_27g )
=> ! [V26r: $i] :
( ( mem @ V26r @ A_27h )
=> ! [V27f: $i] :
( ( mem @ V27f @ ( arr @ A_27g @ A_27h ) )
=> ( ( ap @ ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27g @ A_27h ) @ V24l1 ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27g @ A_27h ) @ V25l2 ) @ V26r ) @ V27f ) )
= ( ap @ ( ap @ ( c_2Eupdate_2ELIST__UPDATE @ A_27g @ A_27h ) @ ( ap @ ( ap @ ( c_2Elist_2ESNOC @ ( ty_2Epair_2Eprod @ A_27g @ A_27h ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27g @ A_27h ) @ V25l2 ) @ V26r ) ) @ V24l1 ) ) @ V27f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eupdate_2EAPPLY__UPDATE__ID,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V1a ) @ ( ap @ V0f @ V1a ) ) @ V0f )
= V0f ) ) ) ).
thf(conj_thm_2Eupdate_2EAPPLY__UPDATE__THM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27b )
=> ! [V3c: $i] :
( ( mem @ V3c @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V1a ) @ V2b ) @ V0f ) @ V3c )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1a ) @ V3c ) ) @ V2b ) @ ( ap @ V0f @ V3c ) ) ) ) ) ) ) ).
thf(conj_thm_2Eupdate_2ESAME__KEY__UPDATE__DIFFER,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27c @ A_27d ) )
=> ! [V1f1: $i] :
( ( mem @ V1f1 @ A_27a )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ A_27b )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27c )
=> ! [V4b: $i] :
( ( mem @ V4b @ A_27d )
=> ! [V5c: $i] :
( ( mem @ V5c @ A_27d )
=> ( ( V4b != V5c )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27c @ A_27d ) @ V3a ) @ V4b ) @ V0f )
!= ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27c @ A_27d ) @ V3a ) @ V5c ) @ V0f ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eupdate_2EUPDATE__APPLY__ID,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27b )
=> ( ( ( ap @ V0f @ V1a )
= V2b )
<=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V1a ) @ V2b ) @ V0f )
= V0f ) ) ) ) ) ).
thf(conj_thm_2Eupdate_2EUPDATE__APPLY__IMP__ID,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ A_27b )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( ( ap @ V0f @ V2a )
= V1b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V2a ) @ V1b ) @ V0f )
= V0f ) ) ) ) ) ).
thf(conj_thm_2Eupdate_2EUPDATE__COMMUTES,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27a )
=> ! [V3c: $i] :
( ( mem @ V3c @ A_27b )
=> ! [V4d: $i] :
( ( mem @ V4d @ A_27b )
=> ( ( V1a != V2b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V1a ) @ V3c ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V2b ) @ V4d ) @ V0f ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V2b ) @ V4d ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V1a ) @ V3c ) @ V0f ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eupdate_2EUPDATE__EQ,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27b )
=> ! [V3c: $i] :
( ( mem @ V3c @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V1a ) @ V3c ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V1a ) @ V2b ) @ V0f ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V1a ) @ V3c ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eupdate_2EUPDATE__def,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1b: $i] :
( ( mem @ V1b @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V0a ) @ V1b )
= ( lam @ ( arr @ A_27a @ A_27b )
@ ^ [V2f: $i] :
( lam @ A_27a
@ ^ [V3c: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V0a ) @ V3c ) ) @ V1b ) @ ( ap @ V2f @ V3c ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------