let s be State of SCMPDS ; :: thesis: for a being Int_position holds
( (Exec (return a),s) . (IC SCMPDS ) = (abs (s . (DataLoc (s . a),RetIC ))) + 2 & (Exec (return a),s) . a = s . (DataLoc (s . a),RetSP ) & ( for b being Int_position st a <> b holds
(Exec (return a),s) . b = s . b ) )
let a be Int_position ; :: thesis: ( (Exec (return a),s) . (IC SCMPDS ) = (abs (s . (DataLoc (s . a),RetIC ))) + 2 & (Exec (return a),s) . a = s . (DataLoc (s . a),RetSP ) & ( for b being Int_position st a <> b holds
(Exec (return a),s) . b = s . b ) )
reconsider S = s as SCMPDS-State by PBOOLE:155;
reconsider da = a as Element of SCM-Data-Loc by Def2;
reconsider I = return a as Element of SCMPDS-Instr ;
set A1 = Address_Add S,(I address_1 ),RetSP ;
set S1 = SCM-Chg S,(I address_1 ),(S . (Address_Add S,(I address_1 ),RetSP ));
set A2 = Address_Add S,(I address_1 ),RetIC ;
set lc = PopInstrLoc S,(Address_Add S,(I address_1 ),RetIC );
reconsider i = 1 as Element of Segm 14 by NAT_1:45;
A1: I = [i,{} ,<*da*>] ;
then A2: I address_1 = da by SCMPDS_1:33;
A3: Exec (return a),s = SCM-Exec-Res I,S by SCMPDS_1:def 25
.= SCM-Chg (SCM-Chg S,(I address_1 ),(S . (Address_Add S,(I address_1 ),RetSP ))),(PopInstrLoc S,(Address_Add S,(I address_1 ),RetIC )) by A1, SCMPDS_1:def 24 ;
hence (Exec (return a),s) . (IC SCMPDS ) = (abs (s . (DataLoc (s . a),RetIC ))) + 2 by A2, Th6, SCMPDS_1:26; :: thesis: ( (Exec (return a),s) . a = s . (DataLoc (s . a),RetSP ) & ( for b being Int_position st a <> b holds
(Exec (return a),s) . b = s . b ) )
thus (Exec (return a),s) . a = (SCM-Chg S,(I address_1 ),(S . (Address_Add S,(I address_1 ),RetSP ))) . da by A3, SCMPDS_1:27
.= s . (DataLoc (s . a),RetSP ) by A2, SCMPDS_1:30 ; :: thesis: for b being Int_position st a <> b holds
(Exec (return a),s) . b = s . b
let b be Int_position ; :: thesis: ( a <> b implies (Exec (return a),s) . b = s . b )
reconsider mn = b as Element of SCM-Data-Loc by Def2;
assume A4: b <> a ; :: thesis: (Exec (return a),s) . b = s . b
thus (Exec (return a),s) . b = (SCM-Chg S,(I address_1 ),(S . (Address_Add S,(I address_1 ),RetSP ))) . mn by A3, SCMPDS_1:27
.= s . b by A2, A4, SCMPDS_1:31 ; :: thesis: verum