let R be good Ring; :: thesis: for a, b being Data-Location of R
for s being State of (SCM R) holds
( (Exec (a := b),s) . (IC (SCM R)) = succ (IC s) & (Exec (a := b),s) . a = s . b & ( for c being Data-Location of R st c <> a holds
(Exec (a := b),s) . c = s . c ) )
let a, b be Data-Location of R; :: thesis: for s being State of (SCM R) holds
( (Exec (a := b),s) . (IC (SCM R)) = succ (IC s) & (Exec (a := b),s) . a = s . b & ( for c being Data-Location of R st c <> a holds
(Exec (a := b),s) . c = s . c ) )
let s be State of (SCM R); :: thesis: ( (Exec (a := b),s) . (IC (SCM R)) = succ (IC s) & (Exec (a := b),s) . a = s . b & ( for c being Data-Location of R st c <> a holds
(Exec (a := b),s) . c = s . c ) )
A1: a is Element of SCM-Data-Loc by Th1;
Y: the Object-Kind of (SCM R) = SCM-OK R by Def1;
reconsider S = s as SCM-State of R by Y, PBOOLE:155;
reconsider I = a := b as Element of SCM-Instr R by Def1;
set S1 = SCM-Chg S,(I address_1 ),(S . (I address_2 ));
reconsider i = 1 as Element of Segm 8 by NAT_1:45;
A2: IC s = IC S by Def1;
A3: b is Element of SCM-Data-Loc by Th1;
A4: Exec (a := b),s = SCM-Exec-Res I,S by Th12
.= SCM-Chg (SCM-Chg S,(I address_1 ),(S . (I address_2 ))),(succ (IC S)) by A1, A3, SCMRING1:def 14 ;
A5: I = [i,<*a,b*>] ;
then A6: I address_1 = a by A1, A3, SCMRING1:17;
A7: I address_2 = b by A5, A1, A3, SCMRING1:17;
thus (Exec (a := b),s) . (IC (SCM R)) = (Exec (a := b),s) . NAT by Def1
.= succ (IC s) by A2, A4, SCMRING1:10 ; :: thesis: ( (Exec (a := b),s) . a = s . b & ( for c being Data-Location of R 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 ))) . a by A1, A4, SCMRING1:11
.= s . b by A6, A7, SCMRING1:14 ; :: thesis: for c being Data-Location of R st c <> a holds
(Exec (a := b),s) . c = s . c
let c be Data-Location of R; :: thesis: ( c <> a implies (Exec (a := b),s) . c = s . c )
assume A8: c <> a ; :: thesis: (Exec (a := b),s) . c = s . c
A9: c is Element of SCM-Data-Loc by Th1;
hence (Exec (a := b),s) . c = (SCM-Chg S,(I address_1 ),(S . (I address_2 ))) . c by A4, SCMRING1:11
.= s . c by A6, A8, A9, SCMRING1:15 ;
:: thesis: verum