thus SCM R is IC-good :: thesis:

proof

let I be Instruction of (SCM R); :: according to AMISTD_2:def 19 :: thesis:

per cases by SCMRING2:8;

suppose I = [0,{},{}] ; :: thesis:

then I = halt (SCM R) by SCMRING2:30;
hence I is IC-good ; :: thesis:

end;

suppose ex a, b being Data-Location of R st I = a := b ; :: thesis:

hence I is IC-good ; :: thesis:

end;

suppose ex a, b being Data-Location of R st I = AddTo (a,b) ; :: thesis:

hence I is IC-good ; :: thesis:

end;

suppose ex a, b being Data-Location of R st I = SubFrom (a,b) ; :: thesis:

hence I is IC-good ; :: thesis:

end;

suppose ex a, b being Data-Location of R st I = MultBy (a,b) ; :: thesis:

hence I is IC-good ; :: thesis:

end;

suppose A1: ex i1 being Element of NAT st I = goto (i1,R) ; :: thesis:

let k be natural number ; :: according to AMISTD_2:def 18 :: thesis:
let s1 be State of (SCM R); :: thesis:
set s2 = IncrIC (s1,k);
consider i1 being Element of NAT such that
A2: I = goto (i1,R) by A1;
IC (Exec (I,s1)) = i1 by A2, SCMRING2:17;
hence (IC (Exec (I,s1))) + k = IC (Exec ((goto ((i1 + k),R)),(IncrIC (s1,k)))) by SCMRING2:17
.= IC (Exec ((IncAddr (I,k)),(IncrIC (s1,k)))) by A2, Th69 ;
:: thesis:

end;

suppose ex a being Data-Location of R ex i1 being Element of NAT st I = a =0_goto i1 ; :: thesis:

then consider a being Data-Location of R, i1 being Element of NAT such that
A3: I = a =0_goto i1 ;
let k be natural number ; :: according to AMISTD_2:def 18 :: thesis:
let s1 be State of (SCM R); :: thesis:
set s2 = IncrIC (s1,k);
( a <> IC (SCM R) & dom ((IC (SCM R)) .--> ((IC s1) + k)) = {(IC (SCM R))} ) by Th3, FUNCOP_1:19;
then not a in dom ((IC (SCM R)) .--> ((IC s1) + k)) by TARSKI:def 1;
then A5: s1 . a = (IncrIC (s1,k)) . a by FUNCT_4:12;

now

per cases ;

suppose A6: s1 . a = 0. R ; :: thesis:

then IC (Exec (I,s1)) = i1 by A3, SCMRING2:18;
hence (IC (Exec (I,s1))) + k = IC (Exec ((a =0_goto (i1 + k)),(IncrIC (s1,k)))) by A5, A6, SCMRING2:18
.= IC (Exec ((IncAddr (I,k)),(IncrIC (s1,k)))) by A3, Th70 ;
:: thesis:

end;

suppose A7: s1 . a <> 0. R ; :: thesis:

dom ((IC (SCM R)) .--> ((IC s1) + k)) = {(IC (SCM R))} by FUNCOP_1:19;
then IC (SCM R) in dom ((IC (SCM R)) .--> ((IC s1) + k)) by TARSKI:def 1;
then A8: IC (IncrIC (s1,k)) = ((IC (SCM R)) .--> ((IC s1) + k)) . (IC (SCM R)) by FUNCT_4:14
.= (IC s1) + k by FUNCOP_1:87 ;
A9: IC (Exec (I,s1)) = succ (IC s1) by A3, A7, SCMRING2:18
.= (IC s1) + 1 ;
IC (Exec (I,(IncrIC (s1,k)))) = succ (IC (IncrIC (s1,k))) by A3, A5, A7, SCMRING2:18
.= ((IC s1) + k) + 1 by A8 ;
hence (IC (Exec (I,s1))) + k = succ (IC (IncrIC (s1,k))) by A3, A5, A7, A9, SCMRING2:18
.= IC (Exec ((a =0_goto (i1 + k)),(IncrIC (s1,k)))) by A5, A7, SCMRING2:18
.= IC (Exec ((IncAddr (I,k)),(IncrIC (s1,k)))) by A3, Th70 ;
:: thesis:

end;

end;

end;

hence (IC (Exec (I,s1))) + k = IC (Exec ((IncAddr (I,k)),(IncrIC (s1,k)))) ; :: thesis:

end;

suppose ex a being Data-Location of R ex r being Element of R st I = a := r ; :: thesis:

hence I is IC-good ; :: thesis:

end;

end;

end;

let I be Instruction of (SCM R); :: according to AMISTD_2:def 21 :: thesis:
let s1, s2 be State of (SCM R); :: according to AMISTD_2:def 20 :: thesis:
assume A11: s1,s2 equal_outside NAT ; :: thesis:
A12: IC s1 = IC s2 by A11, COMPOS_1:24;
A13: dom (Exec (I,s1)) = the carrier of (SCM R) by PARTFUN1:def 4;
then A14: dom (Exec (I,s1)) = dom (Exec (I,s2)) by PARTFUN1:def 4;

per cases by SCMRING2:8;

suppose I = [0,{},{}] ; :: thesis:

then I = halt (SCM R) by SCMRING2:30;
hence Exec (I,s1), Exec (I,s2) equal_outside NAT by A11, AMISTD_2:def 20; :: thesis:

end;

suppose ex a, b being Data-Location of R st I = a := b ; :: thesis:

then consider a, b being Data-Location of R such that
A15: I = a := b ;
for x being set st x in (dom (Exec (I,s1))) \ NAT holds
(Exec (I,s1)) . x = (Exec (I,s2)) . x

proof

let x be set ; :: thesis:
assume A16: x in (dom (Exec (I,s1))) \ NAT ; :: thesis:
then A17: not x in NAT by XBOOLE_0:def 5;

per cases by A13, A16, A17, Th5;

suppose A18: x = IC (SCM R) ; :: thesis:

hence (Exec (I,s1)) . x = succ (IC s1) by A15, SCMRING2:13
.= (Exec (I,s2)) . x by A12, A15, A18, SCMRING2:13 ;
:: thesis:

end;

suppose A19: x = a ; :: thesis:

hence (Exec (I,s1)) . x = s1 . b by A15, SCMRING2:13
.= s2 . b by A11, Th7
.= (Exec (I,s2)) . x by A15, A19, SCMRING2:13 ;
:: thesis:

end;

suppose that A20: x is Data-Location of R and
A21: x <> a ; :: thesis:

thus (Exec (I,s1)) . x = s1 . x by A15, A20, A21, SCMRING2:13
.= s2 . x by A11, A20, Th7
.= (Exec (I,s2)) . x by A15, A20, A21, SCMRING2:13 ; :: thesis:

end;

end;

end;

hence (Exec (I,s1)) | ((dom (Exec (I,s1))) \ NAT) = (Exec (I,s2)) | ((dom (Exec (I,s2))) \ NAT) by A14, FUNCT_1:165; :: according to FUNCT_7:def 2 :: thesis:

end;

suppose ex a, b being Data-Location of R st I = AddTo (a,b) ; :: thesis:

then consider a, b being Data-Location of R such that
A22: I = AddTo (a,b) ;
for x being set st x in (dom (Exec (I,s1))) \ NAT holds
(Exec (I,s1)) . x = (Exec (I,s2)) . x

proof

let x be set ; :: thesis:
assume A23: x in (dom (Exec (I,s1))) \ NAT ; :: thesis:
then A24: not x in NAT by XBOOLE_0:def 5;

per cases by A13, A23, A24, Th5;

suppose A25: x = IC (SCM R) ; :: thesis:

hence (Exec (I,s1)) . x = succ (IC s1) by A22, SCMRING2:14
.= (Exec (I,s2)) . x by A12, A22, A25, SCMRING2:14 ;
:: thesis:

end;

suppose A26: x = a ; :: thesis:

hence (Exec (I,s1)) . x = (s1 . a) + (s1 . b) by A22, SCMRING2:14
.= (s1 . a) + (s2 . b) by A11, Th7
.= (s2 . a) + (s2 . b) by A11, Th7
.= (Exec (I,s2)) . x by A22, A26, SCMRING2:14 ;
:: thesis:

end;

suppose that A27: x is Data-Location of R and
A28: x <> a ; :: thesis:

thus (Exec (I,s1)) . x = s1 . x by A22, A27, A28, SCMRING2:14
.= s2 . x by A11, A27, Th7
.= (Exec (I,s2)) . x by A22, A27, A28, SCMRING2:14 ; :: thesis:

end;

end;

end;

hence (Exec (I,s1)) | ((dom (Exec (I,s1))) \ NAT) = (Exec (I,s2)) | ((dom (Exec (I,s2))) \ NAT) by A14, FUNCT_1:165; :: according to FUNCT_7:def 2 :: thesis:

end;

suppose ex a, b being Data-Location of R st I = SubFrom (a,b) ; :: thesis:

then consider a, b being Data-Location of R such that
A29: I = SubFrom (a,b) ;
for x being set st x in (dom (Exec (I,s1))) \ NAT holds
(Exec (I,s1)) . x = (Exec (I,s2)) . x

proof

let x be set ; :: thesis:
assume A30: x in (dom (Exec (I,s1))) \ NAT ; :: thesis:
then A31: not x in NAT by XBOOLE_0:def 5;

per cases by A13, A30, A31, Th5;

suppose A32: x = IC (SCM R) ; :: thesis:

hence (Exec (I,s1)) . x = succ (IC s1) by A29, SCMRING2:15
.= (Exec (I,s2)) . x by A12, A29, A32, SCMRING2:15 ;
:: thesis:

end;

suppose A33: x = a ; :: thesis:

hence (Exec (I,s1)) . x = (s1 . a) - (s1 . b) by A29, SCMRING2:15
.= (s1 . a) - (s2 . b) by A11, Th7
.= (s2 . a) - (s2 . b) by A11, Th7
.= (Exec (I,s2)) . x by A29, A33, SCMRING2:15 ;
:: thesis:

end;

suppose that A34: x is Data-Location of R and
A35: x <> a ; :: thesis:

thus (Exec (I,s1)) . x = s1 . x by A29, A34, A35, SCMRING2:15
.= s2 . x by A11, A34, Th7
.= (Exec (I,s2)) . x by A29, A34, A35, SCMRING2:15 ; :: thesis:

end;

end;

end;

hence (Exec (I,s1)) | ((dom (Exec (I,s1))) \ NAT) = (Exec (I,s2)) | ((dom (Exec (I,s2))) \ NAT) by A14, FUNCT_1:165; :: according to FUNCT_7:def 2 :: thesis:

end;

suppose ex a, b being Data-Location of R st I = MultBy (a,b) ; :: thesis:

then consider a, b being Data-Location of R such that
A36: I = MultBy (a,b) ;
for x being set st x in (dom (Exec (I,s1))) \ NAT holds
(Exec (I,s1)) . x = (Exec (I,s2)) . x

proof

let x be set ; :: thesis:
assume A37: x in (dom (Exec (I,s1))) \ NAT ; :: thesis:
then A38: not x in NAT by XBOOLE_0:def 5;

per cases by A13, A37, A38, Th5;

suppose A39: x = IC (SCM R) ; :: thesis:

hence (Exec (I,s1)) . x = succ (IC s1) by A36, SCMRING2:16
.= (Exec (I,s2)) . x by A12, A36, A39, SCMRING2:16 ;
:: thesis:

end;

suppose A40: x = a ; :: thesis:

hence (Exec (I,s1)) . x = (s1 . a) * (s1 . b) by A36, SCMRING2:16
.= (s1 . a) * (s2 . b) by A11, Th7
.= (s2 . a) * (s2 . b) by A11, Th7
.= (Exec (I,s2)) . x by A36, A40, SCMRING2:16 ;
:: thesis:

end;

suppose that A41: x is Data-Location of R and
A42: x <> a ; :: thesis:

thus (Exec (I,s1)) . x = s1 . x by A36, A41, A42, SCMRING2:16
.= s2 . x by A11, A41, Th7
.= (Exec (I,s2)) . x by A36, A41, A42, SCMRING2:16 ; :: thesis:

end;

end;

end;

hence (Exec (I,s1)) | ((dom (Exec (I,s1))) \ NAT) = (Exec (I,s2)) | ((dom (Exec (I,s2))) \ NAT) by A14, FUNCT_1:165; :: according to FUNCT_7:def 2 :: thesis:

end;

suppose ex i1 being Element of NAT st I = goto (i1,R) ; :: thesis:

then consider i1 being Element of NAT such that
A43: I = goto (i1,R) ;
for x being set st x in (dom (Exec (I,s1))) \ NAT holds
(Exec (I,s1)) . x = (Exec (I,s2)) . x

proof

let x be set ; :: thesis:
assume A44: x in (dom (Exec (I,s1))) \ NAT ; :: thesis:
then A45: not x in NAT by XBOOLE_0:def 5;

per cases by A13, A44, A45, Th5;

suppose A46: x = IC (SCM R) ; :: thesis:

hence (Exec (I,s1)) . x = i1 by A43, SCMRING2:17
.= (Exec (I,s2)) . x by A43, A46, SCMRING2:17 ;
:: thesis:

end;

suppose A47: x is Data-Location of R ; :: thesis:

hence (Exec (I,s1)) . x = s1 . x by A43, SCMRING2:17
.= s2 . x by A11, A47, Th7
.= (Exec (I,s2)) . x by A43, A47, SCMRING2:17 ;
:: thesis:

end;

end;

end;

hence (Exec (I,s1)) | ((dom (Exec (I,s1))) \ NAT) = (Exec (I,s2)) | ((dom (Exec (I,s2))) \ NAT) by A14, FUNCT_1:165; :: according to FUNCT_7:def 2 :: thesis:

end;

suppose ex a being Data-Location of R ex i1 being Element of NAT st I = a =0_goto i1 ; :: thesis:

then consider a being Data-Location of R, i1 being Element of NAT such that
A48: I = a =0_goto i1 ;
for x being set st x in (dom (Exec (I,s1))) \ NAT holds
(Exec (I,s1)) . x = (Exec (I,s2)) . x

proof

let x be set ; :: thesis:
assume A49: x in (dom (Exec (I,s1))) \ NAT ; :: thesis:
then A50: not x in NAT by XBOOLE_0:def 5;
A51: s1 . a = s2 . a by A11, Th7;

per cases by A13, A49, A50, Th5;

suppose A52: ( x = IC (SCM R) & s1 . a = 0. R ) ; :: thesis:

thus (Exec (I,s1)) . x = i1 by A48, A52, SCMRING2:18
.= (Exec (I,s2)) . x by A48, A51, A52, SCMRING2:18 ; :: thesis:

end;

suppose A53: ( x = IC (SCM R) & s1 . a <> 0. R ) ; :: thesis:

thus (Exec (I,s1)) . x = succ (IC s1) by A48, A53, SCMRING2:18
.= (Exec (I,s2)) . x by A12, A48, A51, A53, SCMRING2:18 ; :: thesis:

end;

suppose A54: x is Data-Location of R ; :: thesis:

hence (Exec (I,s1)) . x = s1 . x by A48, SCMRING2:18
.= s2 . x by A11, A54, Th7
.= (Exec (I,s2)) . x by A48, A54, SCMRING2:18 ;
:: thesis:

end;

end;

end;

hence (Exec (I,s1)) | ((dom (Exec (I,s1))) \ NAT) = (Exec (I,s2)) | ((dom (Exec (I,s2))) \ NAT) by A14, FUNCT_1:165; :: according to FUNCT_7:def 2 :: thesis:

end;

suppose ex a being Data-Location of R ex r being Element of R st I = a := r ; :: thesis:

then consider a being Data-Location of R, r being Element of R such that
A55: I = a := r ;
for x being set st x in (dom (Exec (I,s1))) \ NAT holds
(Exec (I,s1)) . x = (Exec (I,s2)) . x

proof

let x be set ; :: thesis:
assume A56: x in (dom (Exec (I,s1))) \ NAT ; :: thesis:
then A57: not x in NAT by XBOOLE_0:def 5;

per cases by A13, A56, A57, Th5;

suppose A58: x = IC (SCM R) ; :: thesis:

hence (Exec (I,s1)) . x = succ (IC s1) by A55, SCMRING2:19
.= (Exec (I,s2)) . x by A12, A55, A58, SCMRING2:19 ;
:: thesis:

end;

suppose A59: x = a ; :: thesis:

hence (Exec (I,s1)) . x = r by A55, SCMRING2:19
.= (Exec (I,s2)) . x by A55, A59, SCMRING2:19 ;
:: thesis:

end;

suppose that A60: x is Data-Location of R and
A61: x <> a ; :: thesis:

thus (Exec (I,s1)) . x = s1 . x by A55, A60, A61, SCMRING2:19
.= s2 . x by A11, A60, Th7
.= (Exec (I,s2)) . x by A55, A60, A61, SCMRING2:19 ; :: thesis:

end;

end;

end;

hence (Exec (I,s1)) | ((dom (Exec (I,s1))) \ NAT) = (Exec (I,s2)) | ((dom (Exec (I,s2))) \ NAT) by A14, FUNCT_1:165; :: according to FUNCT_7:def 2 :: thesis:

end;

end;