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

for I being really-closed Program of SCM+FSA st p +* I halts_on Initialized s holds

for J being Program of SCM+FSA

for k being Nat st k <= LifeSpan ((p +* I),(Initialized s)) holds

Comput ((p +* I),(Initialized s),k) = Comput ((p +* (I ";" J)),(Initialized s),k)

let p be Instruction-Sequence of SCM+FSA; :: thesis: for I being really-closed Program of SCM+FSA st p +* I halts_on Initialized s holds

for J being Program of SCM+FSA

for k being Nat st k <= LifeSpan ((p +* I),(Initialized s)) holds

Comput ((p +* I),(Initialized s),k) = Comput ((p +* (I ";" J)),(Initialized s),k)

let I be really-closed Program of SCM+FSA; :: thesis: ( p +* I halts_on Initialized s implies for J being Program of SCM+FSA

for k being Nat st k <= LifeSpan ((p +* I),(Initialized s)) holds

Comput ((p +* I),(Initialized s),k) = Comput ((p +* (I ";" J)),(Initialized s),k) )

assume A1: p +* I halts_on Initialized s ; :: thesis: for J being Program of SCM+FSA

for k being Nat st k <= LifeSpan ((p +* I),(Initialized s)) holds

Comput ((p +* I),(Initialized s),k) = Comput ((p +* (I ";" J)),(Initialized s),k)

set s1 = Initialized s;

set p1 = p +* I;

A2: I c= p +* I by FUNCT_4:25;

let J be Program of SCM+FSA; :: thesis: for k being Nat st k <= LifeSpan ((p +* I),(Initialized s)) holds

Comput ((p +* I),(Initialized s),k) = Comput ((p +* (I ";" J)),(Initialized s),k)

set s2 = Initialized s;

set p2 = p +* (I ";" J);

defpred S_{1}[ Nat] means ( $1 <= LifeSpan ((p +* I),(Initialized s)) implies Comput ((p +* I),(Initialized s),$1) = Comput ((p +* (I ";" J)),(Initialized s),$1) );

A3: for m being Nat st S_{1}[m] holds

S_{1}[m + 1]
_{1}[ 0 ]
;

thus for k being Nat holds S_{1}[k]
from NAT_1:sch 2(A15, A3); :: thesis: verum

for I being really-closed Program of SCM+FSA st p +* I halts_on Initialized s holds

for J being Program of SCM+FSA

for k being Nat st k <= LifeSpan ((p +* I),(Initialized s)) holds

Comput ((p +* I),(Initialized s),k) = Comput ((p +* (I ";" J)),(Initialized s),k)

let p be Instruction-Sequence of SCM+FSA; :: thesis: for I being really-closed Program of SCM+FSA st p +* I halts_on Initialized s holds

for J being Program of SCM+FSA

for k being Nat st k <= LifeSpan ((p +* I),(Initialized s)) holds

Comput ((p +* I),(Initialized s),k) = Comput ((p +* (I ";" J)),(Initialized s),k)

let I be really-closed Program of SCM+FSA; :: thesis: ( p +* I halts_on Initialized s implies for J being Program of SCM+FSA

for k being Nat st k <= LifeSpan ((p +* I),(Initialized s)) holds

Comput ((p +* I),(Initialized s),k) = Comput ((p +* (I ";" J)),(Initialized s),k) )

assume A1: p +* I halts_on Initialized s ; :: thesis: for J being Program of SCM+FSA

for k being Nat st k <= LifeSpan ((p +* I),(Initialized s)) holds

Comput ((p +* I),(Initialized s),k) = Comput ((p +* (I ";" J)),(Initialized s),k)

set s1 = Initialized s;

set p1 = p +* I;

A2: I c= p +* I by FUNCT_4:25;

let J be Program of SCM+FSA; :: thesis: for k being Nat st k <= LifeSpan ((p +* I),(Initialized s)) holds

Comput ((p +* I),(Initialized s),k) = Comput ((p +* (I ";" J)),(Initialized s),k)

set s2 = Initialized s;

set p2 = p +* (I ";" J);

defpred S

A3: for m being Nat st S

S

proof

A15:
S
dom (I ";" J) = (dom I) \/ (dom (Reloc (J,(card I))))
by SCMFSA6A:39;

then A4: dom I c= dom (I ";" J) by XBOOLE_1:7;

set sx = Initialized s;

set px = p +* (I ";" J);

A5: I ";" J c= p +* (I ";" J) by FUNCT_4:25;

let m be Nat; :: thesis: ( S_{1}[m] implies S_{1}[m + 1] )

assume A6: ( m <= LifeSpan ((p +* I),(Initialized s)) implies Comput ((p +* I),(Initialized s),m) = Comput ((p +* (I ";" J)),(Initialized s),m) ) ; :: thesis: S_{1}[m + 1]

assume A7: m + 1 <= LifeSpan ((p +* I),(Initialized s)) ; :: thesis: Comput ((p +* I),(Initialized s),(m + 1)) = Comput ((p +* (I ";" J)),(Initialized s),(m + 1))

A8: Comput ((p +* I),(Initialized s),(m + 1)) = Following ((p +* I),(Comput ((p +* I),(Initialized s),m))) by EXTPRO_1:3

.= Exec ((CurInstr ((p +* I),(Comput ((p +* I),(Initialized s),m)))),(Comput ((p +* I),(Initialized s),m))) ;

A9: Comput ((p +* (I ";" J)),(Initialized s),(m + 1)) = Following ((p +* (I ";" J)),(Comput ((p +* (I ";" J)),(Initialized s),m))) by EXTPRO_1:3

.= Exec ((CurInstr ((p +* (I ";" J)),(Comput ((p +* (I ";" J)),(Initialized s),m)))),(Comput ((p +* (I ";" J)),(Initialized s),m))) ;

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

then A10: IC (Initialized s) in dom I by AFINSQ_1:65;

A11: IC (Comput ((p +* I),(Initialized s),m)) in dom I by AMISTD_1:21, A2, A10;

A12: (p +* I) /. (IC (Comput ((p +* I),(Initialized s),m))) = (p +* I) . (IC (Comput ((p +* I),(Initialized s),m))) by PBOOLE:143;

A13: CurInstr ((p +* I),(Comput ((p +* I),(Initialized s),m))) = I . (IC (Comput ((p +* I),(Initialized s),m))) by A11, A12, A2, GRFUNC_1:2;

A14: (p +* (I ";" J)) /. (IC (Comput ((p +* (I ";" J)),(Initialized s),m))) = (p +* (I ";" J)) . (IC (Comput ((p +* (I ";" J)),(Initialized s),m))) by PBOOLE:143;

m < LifeSpan ((p +* I),(Initialized s)) by A7, NAT_1:13;

then I . (IC (Comput ((p +* I),(Initialized s),m))) <> halt SCM+FSA by A1, A13, EXTPRO_1:def 15;

then CurInstr ((p +* I),(Comput ((p +* I),(Initialized s),m))) = (I ";" J) . (IC (Comput ((p +* I),(Initialized s),m))) by A11, A13, SCMFSA6A:15

.= CurInstr ((p +* (I ";" J)),(Comput ((p +* (I ";" J)),(Initialized s),m))) by A14, A7, A11, A4, A5, A6, GRFUNC_1:2, NAT_1:13 ;

hence Comput ((p +* I),(Initialized s),(m + 1)) = Comput ((p +* (I ";" J)),(Initialized s),(m + 1)) by A6, A7, A8, A9, NAT_1:13; :: thesis: verum

end;then A4: dom I c= dom (I ";" J) by XBOOLE_1:7;

set sx = Initialized s;

set px = p +* (I ";" J);

A5: I ";" J c= p +* (I ";" J) by FUNCT_4:25;

let m be Nat; :: thesis: ( S

assume A6: ( m <= LifeSpan ((p +* I),(Initialized s)) implies Comput ((p +* I),(Initialized s),m) = Comput ((p +* (I ";" J)),(Initialized s),m) ) ; :: thesis: S

assume A7: m + 1 <= LifeSpan ((p +* I),(Initialized s)) ; :: thesis: Comput ((p +* I),(Initialized s),(m + 1)) = Comput ((p +* (I ";" J)),(Initialized s),(m + 1))

A8: Comput ((p +* I),(Initialized s),(m + 1)) = Following ((p +* I),(Comput ((p +* I),(Initialized s),m))) by EXTPRO_1:3

.= Exec ((CurInstr ((p +* I),(Comput ((p +* I),(Initialized s),m)))),(Comput ((p +* I),(Initialized s),m))) ;

A9: Comput ((p +* (I ";" J)),(Initialized s),(m + 1)) = Following ((p +* (I ";" J)),(Comput ((p +* (I ";" J)),(Initialized s),m))) by EXTPRO_1:3

.= Exec ((CurInstr ((p +* (I ";" J)),(Comput ((p +* (I ";" J)),(Initialized s),m)))),(Comput ((p +* (I ";" J)),(Initialized s),m))) ;

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

then A10: IC (Initialized s) in dom I by AFINSQ_1:65;

A11: IC (Comput ((p +* I),(Initialized s),m)) in dom I by AMISTD_1:21, A2, A10;

A12: (p +* I) /. (IC (Comput ((p +* I),(Initialized s),m))) = (p +* I) . (IC (Comput ((p +* I),(Initialized s),m))) by PBOOLE:143;

A13: CurInstr ((p +* I),(Comput ((p +* I),(Initialized s),m))) = I . (IC (Comput ((p +* I),(Initialized s),m))) by A11, A12, A2, GRFUNC_1:2;

A14: (p +* (I ";" J)) /. (IC (Comput ((p +* (I ";" J)),(Initialized s),m))) = (p +* (I ";" J)) . (IC (Comput ((p +* (I ";" J)),(Initialized s),m))) by PBOOLE:143;

m < LifeSpan ((p +* I),(Initialized s)) by A7, NAT_1:13;

then I . (IC (Comput ((p +* I),(Initialized s),m))) <> halt SCM+FSA by A1, A13, EXTPRO_1:def 15;

then CurInstr ((p +* I),(Comput ((p +* I),(Initialized s),m))) = (I ";" J) . (IC (Comput ((p +* I),(Initialized s),m))) by A11, A13, SCMFSA6A:15

.= CurInstr ((p +* (I ";" J)),(Comput ((p +* (I ";" J)),(Initialized s),m))) by A14, A7, A11, A4, A5, A6, GRFUNC_1:2, NAT_1:13 ;

hence Comput ((p +* I),(Initialized s),(m + 1)) = Comput ((p +* (I ";" J)),(Initialized s),(m + 1)) by A6, A7, A8, A9, NAT_1:13; :: thesis: verum

thus for k being Nat holds S