let R be good Ring; :: thesis: for a, b being Data-Location of R
for s being State of (SCM R) holds
( (Exec ((MultBy (a,b)),s)) . (IC (SCM R)) = succ (IC s) & (Exec ((MultBy (a,b)),s)) . a = (s . a) * (s . b) & ( for c being Data-Location of R st c <> a holds
(Exec ((MultBy (a,b)),s)) . c = s . c ) )
let a, b be Data-Location of R; :: thesis: for s being State of (SCM R) holds
( (Exec ((MultBy (a,b)),s)) . (IC (SCM R)) = succ (IC s) & (Exec ((MultBy (a,b)),s)) . a = (s . a) * (s . b) & ( for c being Data-Location of R st c <> a holds
(Exec ((MultBy (a,b)),s)) . c = s . c ) )
let s be State of (SCM R); :: thesis: ( (Exec ((MultBy (a,b)),s)) . (IC (SCM R)) = succ (IC s) & (Exec ((MultBy (a,b)),s)) . a = (s . a) * (s . b) & ( for c being Data-Location of R st c <> a holds
(Exec ((MultBy (a,b)),s)) . c = s . c ) )
A1: a is Element of Data-Locations SCM 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 = MultBy (a,b) as Element of SCM-Instr R by Def1;
set S1 = SCM-Chg (S,(I address_1),((S . (I address_1)) * (S . (I address_2))));
reconsider i = 4 as Element of Segm 8 by NAT_1:45;
A2: IC s = IC S by Def1;
A3: b is Element of Data-Locations SCM by Th1;
A4: Exec ((MultBy (a,b)),s) = SCM-Exec-Res (I,S) by Th12
.= SCM-Chg ((SCM-Chg (S,(I address_1),((S . (I address_1)) * (S . (I address_2))))),(succ (IC S))) by A1, A3, AMI_3:72, SCMRING1:def 14 ;
A5: I = [i,{},<*a,b*>] ;
then A6: I address_1 = a by A1, A3, AMI_3:72, SCMRING1:17;
A7: I address_2 = b by A5, A1, A3, AMI_3:72, SCMRING1:17;
thus (Exec ((MultBy (a,b)),s)) . (IC (SCM R)) = (Exec ((MultBy (a,b)),s)) . NAT by Def1
.= succ (IC s) by A2, A4, SCMRING1:10 ; :: thesis: ( (Exec ((MultBy (a,b)),s)) . a = (s . a) * (s . b) & ( for c being Data-Location of R st c <> a holds
(Exec ((MultBy (a,b)),s)) . c = s . c ) )
thus (Exec ((MultBy (a,b)),s)) . a = (SCM-Chg (S,(I address_1),((S . (I address_1)) * (S . (I address_2))))) . a by A1, A4, AMI_3:72, SCMRING1:11
.= (s . a) * (s . b) by A6, A7, SCMRING1:14 ; :: thesis: for c being Data-Location of R st c <> a holds
(Exec ((MultBy (a,b)),s)) . c = s . c
let c be Data-Location of R; :: thesis: ( c <> a implies (Exec ((MultBy (a,b)),s)) . c = s . c )
assume A8: c <> a ; :: thesis: (Exec ((MultBy (a,b)),s)) . c = s . c
A9: c is Element of Data-Locations SCM by Th1;
hence (Exec ((MultBy (a,b)),s)) . c = (SCM-Chg (S,(I address_1),((S . (I address_1)) * (S . (I address_2))))) . c by A4, AMI_3:72, SCMRING1:11
.= s . c by A6, A8, A9, AMI_3:72, SCMRING1:15 ;
:: thesis: verum