let s1, s2 be State of SCM+FSA; :: thesis: for p1, p2 being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being InitHalting Program of SCM+FSA st Initialize ((intloc 0) .--> 1) c= s1 & Initialize ((intloc 0) .--> 1) c= s2 & I c= p1 & I c= p2 & NPP s1 = NPP s2 holds
for k being Element of NAT holds
( NPP (Comput (p1,s1,k)) = NPP (Comput (p2,s2,k)) & CurInstr (p1,(Comput (p1,s1,k))) = CurInstr (p2,(Comput (p2,s2,k))) )
let p1, p2 be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for I being InitHalting Program of SCM+FSA st Initialize ((intloc 0) .--> 1) c= s1 & Initialize ((intloc 0) .--> 1) c= s2 & I c= p1 & I c= p2 & NPP s1 = NPP s2 holds
for k being Element of NAT holds
( NPP (Comput (p1,s1,k)) = NPP (Comput (p2,s2,k)) & CurInstr (p1,(Comput (p1,s1,k))) = CurInstr (p2,(Comput (p2,s2,k))) )
let I be InitHalting Program of SCM+FSA; :: thesis: ( Initialize ((intloc 0) .--> 1) c= s1 & Initialize ((intloc 0) .--> 1) c= s2 & I c= p1 & I c= p2 & NPP s1 = NPP s2 implies for k being Element of NAT holds
( NPP (Comput (p1,s1,k)) = NPP (Comput (p2,s2,k)) & CurInstr (p1,(Comput (p1,s1,k))) = CurInstr (p2,(Comput (p2,s2,k))) ) )
assume that
A1: Initialize ((intloc 0) .--> 1) c= s1 and
A2: Initialize ((intloc 0) .--> 1) c= s2 and
A3: I c= p1 and
A4: I c= p2 and
A5: NPP s1 = NPP s2 ; :: thesis: for k being Element of NAT holds
( NPP (Comput (p1,s1,k)) = NPP (Comput (p2,s2,k)) & CurInstr (p1,(Comput (p1,s1,k))) = CurInstr (p2,(Comput (p2,s2,k))) )
let k be Element of NAT ; :: thesis: ( NPP (Comput (p1,s1,k)) = NPP (Comput (p2,s2,k)) & CurInstr (p1,(Comput (p1,s1,k))) = CurInstr (p2,(Comput (p2,s2,k))) )
A6: IC (Comput (p1,s1,k)) in dom I by A1, Def1, A3;
A7: IC (Comput (p2,s2,k)) in dom I by A2, Def1, A4;
for m being Element of NAT st m < k holds
IC (Comput (p2,s2,m)) in dom I by A2, Def1, A4;
hence NPP (Comput (p1,s1,k)) = NPP (Comput (p2,s2,k)) by A5, A3, A4, AMISTD_2:66; :: thesis: CurInstr (p1,(Comput (p1,s1,k))) = CurInstr (p2,(Comput (p2,s2,k)))
then A8: IC (Comput (p1,s1,k)) = IC (Comput (p2,s2,k)) by COMPOS_1:230;
thus CurInstr (p2,(Comput (p2,s2,k))) = p2 . (IC (Comput (p2,s2,k))) by PBOOLE:158
.= I . (IC (Comput (p2,s2,k))) by A7, A4, GRFUNC_1:8
.= p1 . (IC (Comput (p1,s1,k))) by A8, A6, A3, GRFUNC_1:8
.= CurInstr (p1,(Comput (p1,s1,k))) by PBOOLE:158 ; :: thesis: verum