let s be State of SCMPDS; :: thesis: for k1 being Integer
for a being Int_position holds
( (Exec ((saveIC (a,k1)),s)) . (IC ) = 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 ) = 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 ) = 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 CARD_3:107;
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_1:44;
set DL = DataLoc ((s . a),k1);
A1: I = [i,{},<*da,k1*>] ;
then A2: ( I P21address = da & I P22const = k1 ) by SCMPDS_1:20;
A3: Exec ((saveIC (a,k1)),s) = SCM-Exec-Res (I,S) by SCMPDS_1:def 23
.= SCM-Chg ((SCM-Chg (S,(Address_Add (S,(I P21address),(I P22const))),m)),(succ (IC S))) by A1, SCMPDS_1:def 22 ;
hence (Exec ((saveIC (a,k1)),s)) . (IC ) = succ (IC s) by Th6, SCMPDS_1:13; :: 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_1:14
.= IC s by Th6, SCMPDS_1:16 ; :: 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_1:14
.= s . b by A2, A4, SCMPDS_1:17 ; :: thesis: verum