let a, b be Data-Location ; :: thesis: for s being State of SCM holds
( (Exec ((a := b),s)) . (IC ) = succ (IC s) & (Exec ((a := b),s)) . a = s . b & ( for c being Data-Location st c <> a holds
(Exec ((a := b),s)) . c = s . c ) )
let s be State of SCM; :: thesis: ( (Exec ((a := b),s)) . (IC ) = succ (IC s) & (Exec ((a := b),s)) . a = s . b & ( for c being Data-Location st c <> a holds
(Exec ((a := b),s)) . c = s . c ) )
reconsider S = s as SCM-State by CARD_3:107;
reconsider mk = a, ml = b as Element of SCM-Data-Loc by Def2;
reconsider I = a := b as Element of SCM-Instr ;
set S1 = SCM-Chg (S,(I address_1),(S . (I address_2)));
reconsider i = 1 as Element of Segm 9 by NAT_1:44;
A1: I = [i,{},<*mk,ml*>] ;
then A2: I address_1 = mk by AMI_2:17;
A3: I address_2 = ml by A1, AMI_2:17;
A4: Exec ((a := b),s) = SCM-Exec-Res (I,S) by AMI_2:def 14
.= SCM-Chg ((SCM-Chg (S,(I address_1),(S . (I address_2)))),(succ (IC S))) by A1, AMI_2:def 13 ;
hence (Exec ((a := b),s)) . (IC ) = succ (IC s) by Th4, AMI_2:11; :: thesis: ( (Exec ((a := b),s)) . a = s . b & ( for c being Data-Location st c <> a holds
(Exec ((a := b),s)) . c = s . c ) )
thus (Exec ((a := b),s)) . a = (SCM-Chg (S,(I address_1),(S . (I address_2)))) . mk by A4, AMI_2:12
.= s . b by A2, A3, AMI_2:15 ; :: thesis: for c being Data-Location st c <> a holds
(Exec ((a := b),s)) . c = s . c
let c be Data-Location ; :: thesis: ( c <> a implies (Exec ((a := b),s)) . c = s . c )
reconsider mn = c as Element of SCM-Data-Loc by Def2;
assume A5: c <> a ; :: thesis: (Exec ((a := b),s)) . c = s . c
thus (Exec ((a := b),s)) . c = (SCM-Chg (S,(I address_1),(S . (I address_2)))) . mn by A4, AMI_2:12
.= s . c by A2, A5, AMI_2:16 ; :: thesis: verum