let s be State of SCMPDS ; :: thesis: for k1 being Integer
for a being Int_position holds
( (Exec (saveIC a,k1),s) . (IC SCMPDS ) = succ (IC s) & (Exec (saveIC a,k1),s) . (DataLoc (s . a),k1) = IC s & ( for b being Int_position st DataLoc (s . a),k1 <> b holds
(Exec (saveIC a,k1),s) . b = s . b ) )
let k1 be Integer; :: thesis: for a being Int_position holds
( (Exec (saveIC a,k1),s) . (IC SCMPDS ) = succ (IC s) & (Exec (saveIC a,k1),s) . (DataLoc (s . a),k1) = IC s & ( for b being Int_position st DataLoc (s . a),k1 <> b holds
(Exec (saveIC a,k1),s) . b = s . b ) )
let a be Int_position ; :: thesis: ( (Exec (saveIC a,k1),s) . (IC SCMPDS ) = succ (IC s) & (Exec (saveIC a,k1),s) . (DataLoc (s . a),k1) = IC s & ( for b being Int_position st DataLoc (s . a),k1 <> b holds
(Exec (saveIC a,k1),s) . b = s . b ) )
reconsider S = s as SCMPDS-State by PBOOLE:155;
reconsider m = IC S as Element of NAT ;
reconsider da = a as Element of SCM-Data-Loc by Def2;
reconsider I = saveIC a,k1 as Element of SCMPDS-Instr ;
set A1 = Address_Add S,(I P21address ),(I P22const );
set S1 = SCM-Chg S,(Address_Add S,(I P21address ),(I P22const )),m;
reconsider i = 3 as Element of Segm 14 by NAT_1:45;
set DL = DataLoc (s . a),k1;
A1: I = [i,<*da,k1*>] ;
then A2: ( I P21address = da & I P22const = k1 ) by SCMPDS_1:35;
A3: Exec (saveIC a,k1),s = SCM-Exec-Res I,S by SCMPDS_1:def 25
.= SCM-Chg (SCM-Chg S,(Address_Add S,(I P21address ),(I P22const )),m),(succ (IC S)) by A1, SCMPDS_1:def 24 ;
hence (Exec (saveIC a,k1),s) . (IC SCMPDS ) = succ (IC s) by Th6, SCMPDS_1:26; :: thesis: ( (Exec (saveIC a,k1),s) . (DataLoc (s . a),k1) = IC s & ( for b being Int_position st DataLoc (s . a),k1 <> b holds
(Exec (saveIC a,k1),s) . b = s . b ) )
thus (Exec (saveIC a,k1),s) . (DataLoc (s . a),k1) = (SCM-Chg S,(Address_Add S,(I P21address ),(I P22const )),m) . (Address_Add S,(I P21address ),(I P22const )) by A3, A2, SCMPDS_1:27
.= IC s by Th6, SCMPDS_1:30 ; :: thesis: for b being Int_position st DataLoc (s . a),k1 <> b holds
(Exec (saveIC a,k1),s) . b = s . b
let b be Int_position ; :: thesis: ( DataLoc (s . a),k1 <> b implies (Exec (saveIC a,k1),s) . b = s . b )
reconsider mn = b as Element of SCM-Data-Loc by Def2;
assume A4: DataLoc (s . a),k1 <> b ; :: thesis: (Exec (saveIC a,k1),s) . b = s . b
thus (Exec (saveIC a,k1),s) . b = (SCM-Chg S,(Address_Add S,(I P21address ),(I P22const )),m) . mn by A3, SCMPDS_1:27
.= s . b by A2, A4, SCMPDS_1:31 ; :: thesis: verum