let s be State of SCM+FSA; :: thesis: for p being Instruction-Sequence of SCM+FSA

for I being really-closed InitHalting Program of SCM+FSA

for f being FinSeq-Location st not f in UsedI*Loc I holds

(IExec (I,p,s)) . f = s . f

let p be Instruction-Sequence of SCM+FSA; :: thesis: for I being really-closed InitHalting Program of SCM+FSA

for f being FinSeq-Location st not f in UsedI*Loc I holds

(IExec (I,p,s)) . f = s . f

let I be really-closed InitHalting Program of SCM+FSA; :: thesis: for f being FinSeq-Location st not f in UsedI*Loc I holds

(IExec (I,p,s)) . f = s . f

let f be FinSeq-Location ; :: thesis: ( not f in UsedI*Loc I implies (IExec (I,p,s)) . f = s . f )

( f <> intloc 0 & f <> IC ) by SCMFSA_2:57, SCMFSA_2:58;

then A1: not f in dom (Initialize ((intloc 0) .--> 1)) by SCMFSA_M:11, TARSKI:def 2;

A2: (IExec (I,p,s)) . f = (Result ((p +* I),(Initialized s))) . f by SCMFSA6B:def 1;

A3: Initialize ((intloc 0) .--> 1) c= Initialized s by FUNCT_4:25;

A4: I c= p +* I by FUNCT_4:25;

then p +* I halts_on Initialized s by Def1, A3;

then consider n being Nat such that

A5: Result ((p +* I),(Initialized s)) = Comput ((p +* I),(Initialized s),n) and

CurInstr ((p +* I),(Result ((p +* I),(Initialized s)))) = halt SCM+FSA by EXTPRO_1:def 9;

IC (Initialized s) = 0 by MEMSTR_0:def 11;

then IC (Initialized s) in dom I by AFINSQ_1:65;

then A6: for m being Nat st m < n holds

IC (Comput ((p +* I),(Initialized s),m)) in dom I by AMISTD_1:21, A4;

assume not f in UsedI*Loc I ; :: thesis: (IExec (I,p,s)) . f = s . f

hence (IExec (I,p,s)) . f = (Initialized s) . f by A2, A5, A6, FUNCT_4:25, SF_MASTR:63

.= s . f by A1, FUNCT_4:11 ;

:: thesis: verum

for I being really-closed InitHalting Program of SCM+FSA

for f being FinSeq-Location st not f in UsedI*Loc I holds

(IExec (I,p,s)) . f = s . f

let p be Instruction-Sequence of SCM+FSA; :: thesis: for I being really-closed InitHalting Program of SCM+FSA

for f being FinSeq-Location st not f in UsedI*Loc I holds

(IExec (I,p,s)) . f = s . f

let I be really-closed InitHalting Program of SCM+FSA; :: thesis: for f being FinSeq-Location st not f in UsedI*Loc I holds

(IExec (I,p,s)) . f = s . f

let f be FinSeq-Location ; :: thesis: ( not f in UsedI*Loc I implies (IExec (I,p,s)) . f = s . f )

( f <> intloc 0 & f <> IC ) by SCMFSA_2:57, SCMFSA_2:58;

then A1: not f in dom (Initialize ((intloc 0) .--> 1)) by SCMFSA_M:11, TARSKI:def 2;

A2: (IExec (I,p,s)) . f = (Result ((p +* I),(Initialized s))) . f by SCMFSA6B:def 1;

A3: Initialize ((intloc 0) .--> 1) c= Initialized s by FUNCT_4:25;

A4: I c= p +* I by FUNCT_4:25;

then p +* I halts_on Initialized s by Def1, A3;

then consider n being Nat such that

A5: Result ((p +* I),(Initialized s)) = Comput ((p +* I),(Initialized s),n) and

CurInstr ((p +* I),(Result ((p +* I),(Initialized s)))) = halt SCM+FSA by EXTPRO_1:def 9;

IC (Initialized s) = 0 by MEMSTR_0:def 11;

then IC (Initialized s) in dom I by AFINSQ_1:65;

then A6: for m being Nat st m < n holds

IC (Comput ((p +* I),(Initialized s),m)) in dom I by AMISTD_1:21, A4;

assume not f in UsedI*Loc I ; :: thesis: (IExec (I,p,s)) . f = s . f

hence (IExec (I,p,s)) . f = (Initialized s) . f by A2, A5, A6, FUNCT_4:25, SF_MASTR:63

.= s . f by A1, FUNCT_4:11 ;

:: thesis: verum