let s be State of SCMPDS; for k1, k2 being Integer
for a, b being Int_position holds
( (Exec (((a,k1) := (b,k2)),s)) . (IC ) = (IC s) + 1 & (Exec (((a,k1) := (b,k2)),s)) . (DataLoc ((s . a),k1)) = s . (DataLoc ((s . b),k2)) & ( for c being Int_position st c <> DataLoc ((s . a),k1) holds
(Exec (((a,k1) := (b,k2)),s)) . c = s . c ) )
let k1, k2 be Integer; for a, b being Int_position holds
( (Exec (((a,k1) := (b,k2)),s)) . (IC ) = (IC s) + 1 & (Exec (((a,k1) := (b,k2)),s)) . (DataLoc ((s . a),k1)) = s . (DataLoc ((s . b),k2)) & ( for c being Int_position st c <> DataLoc ((s . a),k1) holds
(Exec (((a,k1) := (b,k2)),s)) . c = s . c ) )
let a, b be Int_position; ( (Exec (((a,k1) := (b,k2)),s)) . (IC ) = (IC s) + 1 & (Exec (((a,k1) := (b,k2)),s)) . (DataLoc ((s . a),k1)) = s . (DataLoc ((s . b),k2)) & ( for c being Int_position st c <> DataLoc ((s . a),k1) holds
(Exec (((a,k1) := (b,k2)),s)) . c = s . c ) )
reconsider S = s as SCM-State by CARD_3:107;
reconsider da = a, db = b as Element of SCM-Data-Loc by AMI_2:def 16;
reconsider I = (a,k1) := (b,k2) as Element of SCMPDS-Instr ;
set A2 = Address_Add (S,(I P41address),(I P43const));
set A4 = Address_Add (S,(I P42address),(I P44const));
set S1 = SCM-Chg (S,(Address_Add (S,(I P41address),(I P43const))),(S . (Address_Add (S,(I P42address),(I P44const)))));
reconsider i = 13 as Element of Segm 15 by NAT_1:44;
A1:
I = [i,{},<*da,db,k1,k2*>]
;
then A2:
( I P42address = db & I P44const = k2 )
by SCMPDS_I:7;
A3:
InsCode I = 13
;
A4: Exec (((a,k1) := (b,k2)),s) =
SCM-Exec-Res (I,S)
by SCMPDS_1:def 23
.=
SCM-Chg ((SCM-Chg (S,(Address_Add (S,(I P41address),(I P43const))),(S . (Address_Add (S,(I P42address),(I P44const)))))),((IC S) + 1))
by A3, SCMPDS_1:def 22
;
hence
(Exec (((a,k1) := (b,k2)),s)) . (IC ) = (IC s) + 1
by Th1, AMI_2:11; ( (Exec (((a,k1) := (b,k2)),s)) . (DataLoc ((s . a),k1)) = s . (DataLoc ((s . b),k2)) & ( for c being Int_position st c <> DataLoc ((s . a),k1) holds
(Exec (((a,k1) := (b,k2)),s)) . c = s . c ) )
A5:
( I P41address = da & I P43const = k1 )
by A1, SCMPDS_I:7;
hence (Exec (((a,k1) := (b,k2)),s)) . (DataLoc ((s . a),k1)) =
(SCM-Chg (S,(Address_Add (S,(I P41address),(I P43const))),(S . (Address_Add (S,(I P42address),(I P44const)))))) . (Address_Add (S,(I P41address),(I P43const)))
by A4, AMI_2:12
.=
s . (DataLoc ((s . b),k2))
by A2, AMI_2:15
;
for c being Int_position st c <> DataLoc ((s . a),k1) holds
(Exec (((a,k1) := (b,k2)),s)) . c = s . c
let c be Int_position; ( c <> DataLoc ((s . a),k1) implies (Exec (((a,k1) := (b,k2)),s)) . c = s . c )
reconsider mn = c as Element of SCM-Data-Loc by AMI_2:def 16;
assume A6:
c <> DataLoc ((s . a),k1)
; (Exec (((a,k1) := (b,k2)),s)) . c = s . c
thus (Exec (((a,k1) := (b,k2)),s)) . c =
(SCM-Chg (S,(Address_Add (S,(I P41address),(I P43const))),(S . (Address_Add (S,(I P42address),(I P44const)))))) . mn
by A4, AMI_2:12
.=
s . c
by A5, A6, AMI_2:16
; verum