let s be State of SCMPDS; :: thesis: for k1, k2 being Integer
for a being Int_position holds
( (Exec ((AddTo (a,k1,k2)),s)) . (IC ) = succ (IC s) & (Exec ((AddTo (a,k1,k2)),s)) . (DataLoc ((s . a),k1)) = (s . (DataLoc ((s . a),k1))) + k2 & ( for b being Int_position st b <> DataLoc ((s . a),k1) holds
(Exec ((AddTo (a,k1,k2)),s)) . b = s . b ) )
let k1, k2 be Integer; :: thesis: for a being Int_position holds
( (Exec ((AddTo (a,k1,k2)),s)) . (IC ) = succ (IC s) & (Exec ((AddTo (a,k1,k2)),s)) . (DataLoc ((s . a),k1)) = (s . (DataLoc ((s . a),k1))) + k2 & ( for b being Int_position st b <> DataLoc ((s . a),k1) holds
(Exec ((AddTo (a,k1,k2)),s)) . b = s . b ) )
let a be Int_position ; :: thesis: ( (Exec ((AddTo (a,k1,k2)),s)) . (IC ) = succ (IC s) & (Exec ((AddTo (a,k1,k2)),s)) . (DataLoc ((s . a),k1)) = (s . (DataLoc ((s . a),k1))) + k2 & ( for b being Int_position st b <> DataLoc ((s . a),k1) holds
(Exec ((AddTo (a,k1,k2)),s)) . b = s . b ) )
reconsider S = s as SCMPDS-State by CARD_3:107;
reconsider mk = a as Element of SCM-Data-Loc by Def2;
reconsider I = AddTo (a,k1,k2) as Element of SCMPDS-Instr ;
set A2 = Address_Add (S,(I P31address),(I P32const));
set S1 = SCM-Chg (S,(Address_Add (S,(I P31address),(I P32const))),((S . (Address_Add (S,(I P31address),(I P32const)))) + (I P33const)));
reconsider i = 8 as Element of Segm 14 by NAT_1:44;
A1: I = [i,{},<*mk,k1,k2*>] ;
then A2: I P33const = k2 by SCMPDS_1:21;
A3: Exec ((AddTo (a,k1,k2)),s) = SCM-Exec-Res (I,S) by SCMPDS_1:def 23
.= SCM-Chg ((SCM-Chg (S,(Address_Add (S,(I P31address),(I P32const))),((S . (Address_Add (S,(I P31address),(I P32const)))) + (I P33const)))),(succ (IC S))) by A1, SCMPDS_1:def 22 ;
hence (Exec ((AddTo (a,k1,k2)),s)) . (IC ) = succ (IC s) by Th6, SCMPDS_1:13; :: thesis: ( (Exec ((AddTo (a,k1,k2)),s)) . (DataLoc ((s . a),k1)) = (s . (DataLoc ((s . a),k1))) + k2 & ( for b being Int_position st b <> DataLoc ((s . a),k1) holds
(Exec ((AddTo (a,k1,k2)),s)) . b = s . b ) )
A4: ( I P31address = mk & I P32const = k1 ) by A1, SCMPDS_1:21;
hence (Exec ((AddTo (a,k1,k2)),s)) . (DataLoc ((s . a),k1)) = (SCM-Chg (S,(Address_Add (S,(I P31address),(I P32const))),((S . (Address_Add (S,(I P31address),(I P32const)))) + (I P33const)))) . (Address_Add (S,(I P31address),(I P32const))) by A3, SCMPDS_1:14
.= (s . (DataLoc ((s . a),k1))) + k2 by A4, A2, SCMPDS_1:16 ;
:: thesis: for b being Int_position st b <> DataLoc ((s . a),k1) holds
(Exec ((AddTo (a,k1,k2)),s)) . b = s . b
let c be Int_position ; :: thesis: ( c <> DataLoc ((s . a),k1) implies (Exec ((AddTo (a,k1,k2)),s)) . c = s . c )
reconsider mn = c as Element of SCM-Data-Loc by Def2;
assume A5: c <> DataLoc ((s . a),k1) ; :: thesis: (Exec ((AddTo (a,k1,k2)),s)) . c = s . c
thus (Exec ((AddTo (a,k1,k2)),s)) . c = (SCM-Chg (S,(Address_Add (S,(I P31address),(I P32const))),((S . (Address_Add (S,(I P31address),(I P32const)))) + (I P33const)))) . mn by A3, SCMPDS_1:14
.= s . c by A4, A5, SCMPDS_1:17 ; :: thesis: verum