let P be Instruction-Sequence of SCMPDS; :: thesis: for s being 0 -started State of SCMPDS
for I being parahalting Program of
for k being Nat st I c= P & k <= LifeSpan ((P +* (stop I)),s) holds
Comput (P,s,k) = Comput ((P +* (stop I)),s,k)
let s be 0 -started State of SCMPDS; :: thesis: for I being parahalting Program of
for k being Nat st I c= P & k <= LifeSpan ((P +* (stop I)),s) holds
Comput (P,s,k) = Comput ((P +* (stop I)),s,k)
let I be parahalting Program of ; :: thesis: for k being Nat st I c= P & k <= LifeSpan ((P +* (stop I)),s) holds
Comput (P,s,k) = Comput ((P +* (stop I)),s,k)
let k be Nat; :: thesis: ( I c= P & k <= LifeSpan ((P +* (stop I)),s) implies Comput (P,s,k) = Comput ((P +* (stop I)),s,k) )
set m = LifeSpan ((P +* (stop I)),s);
assume that
A1: I c= P and
A2: k <= LifeSpan ((P +* (stop I)),s) ; :: thesis: Comput (P,s,k) = Comput ((P +* (stop I)),s,k)
set s2 = s;
set P2 = P +* (stop I);
defpred S₁[ Nat] means ( $1 <= LifeSpan ((P +* (stop I)),s) implies Comput (P,s,$1) = Comput ((P +* (stop I)),s,$1) );
A3: P = P +* I by A1, FUNCT_4:98;

A4: now :: thesis: for k being Nat st S₁[k] holds
S₁[k + 1]

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A5: S₁[k] ; :: thesis: S₁[k + 1]

now :: thesis: ( k + 1 <= LifeSpan ((P +* (stop I)),s) implies Comput (P,s,(k + 1)) = Comput ((P +* (stop I)),s,(k + 1)) )

A6: Comput ((P +* (stop I)),s,(k + 1)) = Following ((P +* (stop I)),(Comput ((P +* (stop I)),s,k))) by EXTPRO_1:3;
A7: Comput (P,s,(k + 1)) = Following (P,(Comput (P,s,k))) by EXTPRO_1:3;
A8: k < k + 1 by XREAL_1:29;
assume A9: k + 1 <= LifeSpan ((P +* (stop I)),s) ; :: thesis: Comput (P,s,(k + 1)) = Comput ((P +* (stop I)),s,(k + 1))
then A10: k < LifeSpan ((P +* (stop I)),s) by A8, XXREAL_0:2;
then IC (Comput ((P +* (stop I)),s,k)) in dom I by Th12;
then A11: IC (Comput ((P +* (stop I)),s,k)) in dom (stop I) by FUNCT_4:12;
A12: IC (Comput ((P +* (stop I)),s,k)) in dom I by A10, Th12;
CurInstr (P,(Comput (P,s,k))) = P . (IC (Comput ((P +* (stop I)),s,k))) by A5, A9, A8, PBOOLE:143, XXREAL_0:2
.= I . (IC (Comput ((P +* (stop I)),s,k))) by A3, A10, Th12, FUNCT_4:13
.= (stop I) . (IC (Comput ((P +* (stop I)),s,k))) by A12, AFINSQ_1:def 3
.= (P +* (stop I)) . (IC (Comput ((P +* (stop I)),s,k))) by A11, FUNCT_4:13
.= CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),s,k))) by PBOOLE:143 ;
hence Comput (P,s,(k + 1)) = Comput ((P +* (stop I)),s,(k + 1)) by A5, A9, A8, A7, A6, XXREAL_0:2; :: thesis: verum

end;

hence S₁[k + 1] ; :: thesis: verum

end;

A13: S₁[ 0 ] ;
for k being Nat holds S₁[k] from NAT_1:sch 2(A13, A4);
hence Comput (P,s,k) = Comput ((P +* (stop I)),s,k) by A2; :: thesis: verum