let P be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
for I being parahalting Program of SCM+FSA st Initialize ((intloc 0) .--> 1) c= s & I c= P holds
for k being Element of NAT st k <= LifeSpan (P,s) holds
CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),s,k))) <> halt SCM+FSA
let s be State of SCM+FSA; :: thesis: for I being parahalting Program of SCM+FSA st Initialize ((intloc 0) .--> 1) c= s & I c= P holds
for k being Element of NAT st k <= LifeSpan (P,s) holds
CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),s,k))) <> halt SCM+FSA
let I be parahalting Program of SCM+FSA; :: thesis: ( Initialize ((intloc 0) .--> 1) c= s & I c= P implies for k being Element of NAT st k <= LifeSpan (P,s) holds
CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),s,k))) <> halt SCM+FSA )
set s2 = s;
set P2 = P +* (loop I);
assume A1: Initialize ((intloc 0) .--> 1) c= s ; :: thesis: ( not I c= P or for k being Element of NAT st k <= LifeSpan (P,s) holds
CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),s,k))) <> halt SCM+FSA )
Start-At (0,SCM+FSA) c= Initialize ((intloc 0) .--> 1) by FUNCT_4:25;
then Start-At (0,SCM+FSA) c= s by A1, XBOOLE_1:1;
then A2: s is 0 -started by MEMSTR_0:29;
then reconsider s1 = s as 0 -started State of SCM+FSA ;
assume A3: I c= P ; :: thesis: for k being Element of NAT st k <= LifeSpan (P,s) holds
CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),s,k))) <> halt SCM+FSA
then A4: P halts_on s by A2, AMISTD_1:def 11;
let k be Element of NAT ; :: thesis: ( k <= LifeSpan (P,s) implies CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),s,k))) <> halt SCM+FSA )
assume A5: k <= LifeSpan (P,s) ; :: thesis: CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),s,k))) <> halt SCM+FSA
set lk = IC (Comput (P,s,k));
assume A6: CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),s,k))) = halt SCM+FSA ; :: thesis: contradiction
A7: dom I = dom (loop I) by FUNCT_4:99;
A8: IC (Comput (P,s1,k)) in dom I by A3, AMISTD_1:def 10;
then A9: (loop I) . (IC (Comput (P,s,k))) in rng (loop I) by A7, FUNCT_1:def 3;
A10: (P +* (loop I)) /. (IC (Comput ((P +* (loop I)),s,k))) = (P +* (loop I)) . (IC (Comput ((P +* (loop I)),s,k))) by PBOOLE:143;
Comput (P,s1,k) = Comput ((P +* (loop I)),s1,k) by A4, A5, Th80, A3;
then CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),s,k))) = (P +* (loop I)) . (IC (Comput (P,s,k))) by A10
.= (loop I) . (IC (Comput (P,s,k))) by A8, A7, FUNCT_4:13 ;
hence contradiction by A6, A9, Th75; :: thesis: verum