let s be State of SCMPDS; :: thesis: for P being the Instructions of SCMPDS -valued ManySortedSet of NAT
for I, J being Program of SCMPDS
for k being Element of NAT st k <= LifeSpan ((P +* (stop I)),(Initialize s)) & I c= J & I is_closed_on s,P & I is_halting_on s,P holds
IC (Comput ((P +* J),(Initialize s),k)) in dom (stop I)
let P be the Instructions of SCMPDS -valued ManySortedSet of NAT ; :: thesis: for I, J being Program of SCMPDS
for k being Element of NAT st k <= LifeSpan ((P +* (stop I)),(Initialize s)) & I c= J & I is_closed_on s,P & I is_halting_on s,P holds
IC (Comput ((P +* J),(Initialize s),k)) in dom (stop I)
let I, J be Program of SCMPDS; :: thesis: for k being Element of NAT st k <= LifeSpan ((P +* (stop I)),(Initialize s)) & I c= J & I is_closed_on s,P & I is_halting_on s,P holds
IC (Comput ((P +* J),(Initialize s),k)) in dom (stop I)
let k be Element of NAT ; :: thesis: ( k <= LifeSpan ((P +* (stop I)),(Initialize s)) & I c= J & I is_closed_on s,P & I is_halting_on s,P implies IC (Comput ((P +* J),(Initialize s),k)) in dom (stop I) )
set ss = Initialize s;
set PP = P +* (stop I);
set s1 = Comput ((P +* J),(Initialize s),k);
set s2 = Comput ((P +* (stop I)),(Initialize s),k);
assume that
A1: k <= LifeSpan ((P +* (stop I)),(Initialize s)) and
A2: I c= J and
A3: I is_closed_on s,P and
A4: I is_halting_on s,P ; :: thesis: IC (Comput ((P +* J),(Initialize s),k)) in dom (stop I)
NPP (Comput ((P +* J),(Initialize s),k)) = NPP (Comput ((P +* (stop I)),(Initialize s),k)) by A1, A2, A3, A4, Th39;
then IC (Comput ((P +* J),(Initialize s),k)) = IC (Comput ((P +* (stop I)),(Initialize s),k)) by COMPOS_1:230;
hence IC (Comput ((P +* J),(Initialize s),k)) in dom (stop I) by A3, SCMPDS_6:def 2; :: thesis: verum