ITP001 Axioms: ITP078^7.ax
%------------------------------------------------------------------------------
% File : ITP078^7 : 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 : update.ax [Gau19]
% : HL4078^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 64 ( 13 unt; 39 typ; 0 def)
% Number of atoms : 60 ( 32 equ; 4 cnn)
% Maximal formula atoms : 8 ( 0 avg)
% Number of connectives : 782 ( 4 ~; 1 |; 13 &; 749 @)
% ( 8 <=>; 7 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 8 avg; 749 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% Number of symbols : 40 ( 38 usr; 1 con; 0-6 aty)
% Number of variables : 197 ( 17 ^ 135 !; 1 ?; 197 :)
% ( 44 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Elist_2Elist,type,
tyop_2Elist_2Elist: $tType > $tType ).
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Eoption_2Eoption,type,
tyop_2Eoption_2Eoption: $tType > $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
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_2Emin_2E_40,type,
c_2Emin_2E_40:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Elist_2EALL__DISTINCT,type,
c_2Elist_2EALL__DISTINCT:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
thf(c_2Elist_2EAPPEND,type,
c_2Elist_2EAPPEND:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Elist_2ECONS,type,
c_2Elist_2ECONS:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Elist_2EFILTER,type,
c_2Elist_2EFILTER:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Eupdate_2EFIND,type,
c_2Eupdate_2EFIND:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Epair_2EFST,type,
c_2Epair_2EFST:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27a ) ).
thf(c_2Ecombin_2EI,type,
c_2Ecombin_2EI:
!>[A_27a: $tType] : ( A_27a > A_27a ) ).
thf(c_2Eupdate_2ELIST__UPDATE,type,
c_2Eupdate_2ELIST__UPDATE:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) > ( A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Elist_2EMAP,type,
c_2Elist_2EMAP:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27b ) ) ).
thf(c_2Elist_2ENIL,type,
c_2Elist_2ENIL:
!>[A_27a: $tType] : ( tyop_2Elist_2Elist @ A_27a ) ).
thf(c_2Eoption_2ENONE,type,
c_2Eoption_2ENONE:
!>[A_27a: $tType] : ( tyop_2Eoption_2Eoption @ A_27a ) ).
thf(c_2Eupdate_2EOVERRIDE,type,
c_2Eupdate_2EOVERRIDE:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) > ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ).
thf(c_2Esorting_2EPERM,type,
c_2Esorting_2EPERM:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
thf(c_2Esorting_2EQSORT,type,
c_2Esorting_2EQSORT:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Epair_2ESND,type,
c_2Epair_2ESND:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27b ) ).
thf(c_2Elist_2ESNOC,type,
c_2Elist_2ESNOC:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Eoption_2ESOME,type,
c_2Eoption_2ESOME:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
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_2Erelation_2EWF,type,
c_2Erelation_2EWF:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2EWFREC,type,
c_2Erelation_2EWFREC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( ( A_27a > A_27b ) > 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_2Elist_2Elist__CASE,type,
c_2Elist_2Elist__CASE:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > A_27b > ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > A_27b ) > 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_2Eoption_2Eoption__CASE,type,
c_2Eoption_2Eoption__CASE:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Eoption_2Eoption @ A_27a ) > A_27b > ( A_27a > A_27b ) > A_27b ) ).
thf(c_2Epair_2Epair__CASE,type,
c_2Epair_2Epair__CASE:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) > ( A_27b > A_27c > A_27a ) > A_27a ) ).
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_2Eupdate_2EFIND__def,axiom,
! [A_27a: $tType] :
( ! [V0P: A_27a > $o] :
( ( c_2Eupdate_2EFIND @ A_27a @ V0P @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ! [V1P: A_27a > $o,V2h: A_27a,V3t: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Eupdate_2EFIND @ A_27a @ V1P @ ( c_2Elist_2ECONS @ A_27a @ V2h @ V3t ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( V1P @ V2h ) @ ( c_2Eoption_2ESOME @ A_27a @ V2h ) @ ( c_2Eupdate_2EFIND @ A_27a @ V1P @ V3t ) ) ) ) ).
thf(thm_2Eupdate_2EOVERRIDE__primitive__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Eupdate_2EOVERRIDE @ A_27a @ A_27b )
= ( c_2Erelation_2EWFREC @ ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( c_2Emin_2E_40 @ ( ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) > ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) > $o )
@ ^ [V0R: ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) > ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) > $o] :
( c_2Ebool_2E_2F_5C @ ( c_2Erelation_2EWF @ ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0R )
@ ( c_2Ebool_2E_21 @ ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ^ [V1t: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( c_2Ebool_2E_21 @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V2x: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( V0R
@ ( c_2Elist_2EFILTER @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V3y: tyop_2Epair_2Eprod @ A_27a @ A_27b] : ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V2x ) @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V3y ) ) )
@ V1t )
@ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V2x @ V1t ) ) ) ) ) )
@ ^ [V4OVERRIDE: ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) > ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ),V5a: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( c_2Elist_2Elist__CASE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V5a @ ( c_2Ecombin_2EI @ ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) )
@ ^ [V6x: tyop_2Epair_2Eprod @ A_27a @ A_27b,V7t: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( c_2Ecombin_2EI @ ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V6x
@ ( V4OVERRIDE
@ ( c_2Elist_2EFILTER @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V8y: tyop_2Epair_2Eprod @ A_27a @ A_27b] : ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V6x ) @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V8y ) ) )
@ V7t ) ) ) ) ) ) ) ).
thf(thm_2Eupdate_2ELIST__UPDATE__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) )
= ( c_2Ecombin_2EI @ ( A_27a > A_27b ) ) )
& ! [V0h: tyop_2Epair_2Eprod @ A_27a @ A_27b,V1t: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0h @ V1t ) )
= ( c_2Ecombin_2Eo @ ( A_27a > A_27b ) @ ( A_27a > A_27b ) @ ( A_27a > A_27b ) @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V0h ) @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V0h ) ) @ ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b @ V1t ) ) ) ) ).
thf(thm_2Eupdate_2EOVERRIDE__ind,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) > $o] :
( ( ( V0P @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) )
& ! [V1x: tyop_2Epair_2Eprod @ A_27a @ A_27b,V2t: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( V0P
@ ( c_2Elist_2EFILTER @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V3y: tyop_2Epair_2Eprod @ A_27a @ A_27b] : ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V1x ) @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V3y ) ) )
@ V2t ) )
=> ( V0P @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V1x @ V2t ) ) ) )
=> ! [V4v: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] : ( V0P @ V4v ) ) ).
thf(thm_2Eupdate_2EOVERRIDE__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( ( c_2Eupdate_2EOVERRIDE @ A_27a @ A_27b @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) )
= ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) )
& ! [V0x: tyop_2Epair_2Eprod @ A_27a @ A_27b,V1t: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( c_2Eupdate_2EOVERRIDE @ A_27a @ A_27b @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0x @ V1t ) )
= ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0x
@ ( c_2Eupdate_2EOVERRIDE @ A_27a @ A_27b
@ ( c_2Elist_2EFILTER @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V2y: tyop_2Epair_2Eprod @ A_27a @ A_27b] : ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V0x ) @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V2y ) ) )
@ V1t ) ) ) ) ) ).
thf(thm_2Eupdate_2ELIST__UPDATE__LOOKUP,axiom,
! [A_27a: $tType,A_27b: $tType,V0l: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1f: A_27a > A_27b,V2i: A_27a] :
( ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b @ V0l @ V1f @ V2i )
= ( c_2Eoption_2Eoption__CASE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27b
@ ( c_2Eupdate_2EFIND @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V3x: tyop_2Epair_2Eprod @ A_27a @ A_27b] : ( c_2Emin_2E_3D @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V3x ) @ V2i )
@ V0l )
@ ( V1f @ V2i )
@ ^ [V4v: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( c_2Epair_2Epair__CASE @ A_27b @ A_27a @ A_27b @ V4v
@ ^ [V5v1: A_27a,V6e: A_27b] : V6e ) ) ) ).
thf(thm_2Eupdate_2ELIST__UPDATE__OVERRIDE,axiom,
! [A_27a: $tType,A_27b: $tType,V0l: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b @ V0l )
= ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b @ ( c_2Eupdate_2EOVERRIDE @ A_27a @ A_27b @ V0l ) ) ) ).
thf(thm_2Eupdate_2ELIST__UPDATE__ALL__DISTINCT,axiom,
! [A_27a: $tType,A_27b: $tType,V0l1: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V1l2: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( ( c_2Elist_2EALL__DISTINCT @ A_27a @ ( c_2Elist_2EMAP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V1l2 ) )
& ( c_2Esorting_2EPERM @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0l1 @ V1l2 ) )
=> ( ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b @ V0l1 )
= ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b @ V1l2 ) ) ) ).
thf(thm_2Eupdate_2ELIST__UPDATE__SORT__OVERRIDE,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V1l: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b @ V1l )
= ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b @ ( c_2Esorting_2EQSORT @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0R @ ( c_2Eupdate_2EOVERRIDE @ A_27a @ A_27b @ V1l ) ) ) ) ).
thf(thm_2Eupdate_2ELIST__UPDATE__THMS,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] :
( ! [V0l1: A_27a,V1l2: A_27a,V2r1: A_27b,V3r2: A_27b] :
( ( c_2Ecombin_2Eo @ ( A_27a > A_27b ) @ ( A_27a > A_27b ) @ ( A_27a > A_27b ) @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V0l1 @ V2r1 ) @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V1l2 @ V3r2 ) )
= ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V0l1 @ V2r1 ) @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1l2 @ V3r2 ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ) ) )
& ! [V4l: A_27c,V5r: A_27d,V6t: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d )] :
( ( c_2Ecombin_2Eo @ ( A_27c > A_27d ) @ ( A_27c > A_27d ) @ ( A_27c > A_27d ) @ ( c_2Ecombin_2EUPDATE @ A_27c @ A_27d @ V4l @ V5r ) @ ( c_2Eupdate_2ELIST__UPDATE @ A_27c @ A_27d @ V6t ) )
= ( c_2Eupdate_2ELIST__UPDATE @ A_27c @ A_27d @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d ) @ ( c_2Epair_2E_2C @ A_27c @ A_27d @ V4l @ V5r ) @ V6t ) ) )
& ! [V7l1: A_27e,V8l2: A_27e,V9r1: A_27f,V10r2: A_27f,V11f: A_27e > A_27f] :
( ( c_2Ecombin_2EUPDATE @ A_27e @ A_27f @ V7l1 @ V9r1 @ ( c_2Ecombin_2EUPDATE @ A_27e @ A_27f @ V8l2 @ V10r2 @ V11f ) )
= ( c_2Eupdate_2ELIST__UPDATE @ A_27e @ A_27f @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ A_27e @ A_27f ) @ ( c_2Epair_2E_2C @ A_27e @ A_27f @ V7l1 @ V9r1 ) @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ A_27e @ A_27f ) @ ( c_2Epair_2E_2C @ A_27e @ A_27f @ V8l2 @ V10r2 ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ A_27e @ A_27f ) ) ) ) @ V11f ) )
& ! [V12l: A_27g,V13r: A_27h,V14t: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27g @ A_27h ),V15f: A_27g > A_27h] :
( ( c_2Ecombin_2EUPDATE @ A_27g @ A_27h @ V12l @ V13r @ ( c_2Eupdate_2ELIST__UPDATE @ A_27g @ A_27h @ V14t @ V15f ) )
= ( c_2Eupdate_2ELIST__UPDATE @ A_27g @ A_27h @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ A_27g @ A_27h ) @ ( c_2Epair_2E_2C @ A_27g @ A_27h @ V12l @ V13r ) @ V14t ) @ V15f ) )
& ! [V16l1: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ),V17l2: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )] :
( ( c_2Ecombin_2Eo @ ( A_27a > A_27b ) @ ( A_27a > A_27b ) @ ( A_27a > A_27b ) @ ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b @ V16l1 ) @ ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b @ V17l2 ) )
= ( c_2Eupdate_2ELIST__UPDATE @ A_27a @ A_27b @ ( c_2Elist_2EAPPEND @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V16l1 @ V17l2 ) ) )
& ! [V18l1: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d ),V19l2: A_27c,V20r: A_27d] :
( ( c_2Ecombin_2Eo @ ( A_27c > A_27d ) @ ( A_27c > A_27d ) @ ( A_27c > A_27d ) @ ( c_2Eupdate_2ELIST__UPDATE @ A_27c @ A_27d @ V18l1 ) @ ( c_2Ecombin_2EUPDATE @ A_27c @ A_27d @ V19l2 @ V20r ) )
= ( c_2Eupdate_2ELIST__UPDATE @ A_27c @ A_27d @ ( c_2Elist_2ESNOC @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d ) @ ( c_2Epair_2E_2C @ A_27c @ A_27d @ V19l2 @ V20r ) @ V18l1 ) ) )
& ! [V21l1: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27e @ A_27f ),V22l2: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27e @ A_27f ),V23f: A_27e > A_27f] :
( ( c_2Eupdate_2ELIST__UPDATE @ A_27e @ A_27f @ V21l1 @ ( c_2Eupdate_2ELIST__UPDATE @ A_27e @ A_27f @ V22l2 @ V23f ) )
= ( c_2Eupdate_2ELIST__UPDATE @ A_27e @ A_27f @ ( c_2Elist_2EAPPEND @ ( tyop_2Epair_2Eprod @ A_27e @ A_27f ) @ V21l1 @ V22l2 ) @ V23f ) )
& ! [V24l1: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ A_27g @ A_27h ),V25l2: A_27g,V26r: A_27h,V27f: A_27g > A_27h] :
( ( c_2Eupdate_2ELIST__UPDATE @ A_27g @ A_27h @ V24l1 @ ( c_2Ecombin_2EUPDATE @ A_27g @ A_27h @ V25l2 @ V26r @ V27f ) )
= ( c_2Eupdate_2ELIST__UPDATE @ A_27g @ A_27h @ ( c_2Elist_2ESNOC @ ( tyop_2Epair_2Eprod @ A_27g @ A_27h ) @ ( c_2Epair_2E_2C @ A_27g @ A_27h @ V25l2 @ V26r ) @ V24l1 ) @ V27f ) ) ) ).
thf(thm_2Eupdate_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_2Eupdate_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_2Eupdate_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_2Eupdate_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_2Eupdate_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_2Eupdate_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_2Eupdate_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_2Eupdate_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 ) ) ) ) ).
%------------------------------------------------------------------------------