let s be State of SCM+FSA; :: thesis: for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being InitHalting 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 +* (Directed I)),(Comput ((p +* (Directed I)),s,k))) <> halt SCM+FSA
let p 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= s & I c= p holds
for k being Element of NAT st k <= LifeSpan (p,s) holds
CurInstr ((p +* (Directed I)),(Comput ((p +* (Directed I)),s,k))) <> halt SCM+FSA
set A = NAT ;
let I be InitHalting 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 +* (Directed I)),(Comput ((p +* (Directed I)),s,k))) <> halt SCM+FSA )
set s2 = s +* EP;
set p2 = p +* (Directed I);
set m = LifeSpan (p,s);
E2: s +* EP = s by FUNCT_4:22;
assume A2: 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 +* (Directed I)),(Comput ((p +* (Directed I)),s,k))) <> halt SCM+FSA )
assume A3: I c= p ; :: thesis: for k being Element of NAT st k <= LifeSpan (p,s) holds
CurInstr ((p +* (Directed I)),(Comput ((p +* (Directed I)),s,k))) <> halt SCM+FSA
then A4: p halts_on s by A2, Def2;
A5: now
set s1 = s +* EP;
set p1 = p +* (I ';' I);
E1: s = s +* EP by FUNCT_4:22;
let k be Element of NAT ; :: thesis: ( k <= LifeSpan (p,s) implies NPP (Comput (p,s,k)) = NPP (Comput ((p +* (Directed I)),(s +* EP),k)) )
defpred S1[ Nat] means ( $1 <= k implies NPP (Comput ((p +* (I ';' I)),(s +* EP),$1)) = NPP (Comput ((p +* (Directed I)),(s +* EP),$1)) );
assume A6: k <= LifeSpan (p,s) ; :: thesis: NPP (Comput (p,s,k)) = NPP (Comput ((p +* (Directed I)),(s +* EP),k))
A7: for n being Element of NAT st S1[n] holds
S1[n + 1]
proof
A8: Directed I c= I ';' I by SCMFSA6A:55;
let n be Element of NAT ; :: thesis: ( S1[n] implies S1[n + 1] )
A9: dom I c= dom (I ';' I) by SCMFSA6A:56;
assume A10: ( n <= k implies NPP (Comput ((p +* (I ';' I)),(s +* EP),n)) = NPP (Comput ((p +* (Directed I)),(s +* EP),n)) ) ; :: thesis: S1[n + 1]
A11: Comput ((p +* (Directed I)),(s +* EP),(n + 1)) = Following ((p +* (Directed I)),(Comput ((p +* (Directed I)),(s +* EP),n))) by EXTPRO_1:4
.= Exec ((CurInstr ((p +* (Directed I)),(Comput ((p +* (Directed I)),(s +* EP),n)))),(Comput ((p +* (Directed I)),(s +* EP),n))) ;
A12: Comput ((p +* (I ';' I)),(s +* EP),(n + 1)) = Following ((p +* (I ';' I)),(Comput ((p +* (I ';' I)),(s +* EP),n))) by EXTPRO_1:4
.= Exec ((CurInstr ((p +* (I ';' I)),(Comput ((p +* (I ';' I)),(s +* EP),n)))),(Comput ((p +* (I ';' I)),(s +* EP),n))) ;
A13: n <= n + 1 by NAT_1:12;
assume A14: n + 1 <= k ; :: thesis: NPP (Comput ((p +* (I ';' I)),(s +* EP),(n + 1))) = NPP (Comput ((p +* (Directed I)),(s +* EP),(n + 1)))
then A15: IC (Comput ((p +* (I ';' I)),(s +* EP),n)) = IC (Comput ((p +* (Directed I)),(s +* EP),n)) by A10, A13, COMPOS_1:230, XXREAL_0:2;
n <= k by A14, A13, XXREAL_0:2;
then n <= LifeSpan (p,s) by A6, XXREAL_0:2;
then NPP (Comput (p,s,n)) = NPP (Comput ((p +* (I ';' I)),(s +* EP),n)) by A2, A4, Th18, A3, E1;
then IC (Comput (p,s,n)) = IC (Comput ((p +* (I ';' I)),(s +* EP),n)) by COMPOS_1:230;
then A16: IC (Comput ((p +* (I ';' I)),(s +* EP),n)) in dom I by A2, Def1, A3;
then A17: IC (Comput ((p +* (Directed I)),(s +* EP),n)) in dom (Directed I) by A15, FUNCT_4:105;
A18: CurInstr ((p +* (Directed I)),(Comput ((p +* (Directed I)),(s +* EP),n))) = (p +* (Directed I)) . (IC (Comput ((p +* (Directed I)),(s +* EP),n))) by PBOOLE:158
.= (Directed I) . (IC (Comput ((p +* (Directed I)),(s +* EP),n))) by A17, FUNCT_4:14 ;
CurInstr ((p +* (I ';' I)),(Comput ((p +* (I ';' I)),(s +* EP),n))) = (p +* (I ';' I)) . (IC (Comput ((p +* (I ';' I)),(s +* EP),n))) by PBOOLE:158
.= (I ';' I) . (IC (Comput ((p +* (I ';' I)),(s +* EP),n))) by A9, A16, FUNCT_4:14
.= (Directed I) . (IC (Comput ((p +* (I ';' I)),(s +* EP),n))) by A8, A15, A17, GRFUNC_1:8 ;
hence NPP (Comput ((p +* (I ';' I)),(s +* EP),(n + 1))) = NPP (Comput ((p +* (Directed I)),(s +* EP),(n + 1))) by A10, A14, A13, A15, A18, A12, A11, AMISTD_2:def 20, XXREAL_0:2; :: thesis: verum
end;
( Comput ((p +* (I ';' I)),(s +* EP),0) = s +* EP & Comput ((p +* (Directed I)),(s +* EP),0) = s +* EP ) by EXTPRO_1:3;
then NPP (Comput ((p +* (Directed I)),(s +* EP),0)) = NPP (Comput ((p +* (I ';' I)),(s +* EP),0)) ;
then A19: S1[ 0 ] ;
for n being Element of NAT holds S1[n] from NAT_1:sch 1(A19, A7);
then A20: NPP (Comput ((p +* (I ';' I)),(s +* EP),k)) = NPP (Comput ((p +* (Directed I)),(s +* EP),k)) ;
NPP (Comput (p,s,k)) = NPP (Comput ((p +* (I ';' I)),(s +* EP),k)) by A2, A4, A6, Th18, A3, E1;
hence NPP (Comput (p,s,k)) = NPP (Comput ((p +* (Directed I)),(s +* EP),k)) by A20; :: thesis: verum
end;
let k be Element of NAT ; :: thesis: ( k <= LifeSpan (p,s) implies CurInstr ((p +* (Directed I)),(Comput ((p +* (Directed I)),s,k))) <> halt SCM+FSA )
set lk = IC (Comput (p,s,k));
A21: ( IC (Comput (p,s,k)) in dom I & dom I = dom (Directed I) ) by A2, A3, Def1, FUNCT_4:105;
then A22: (Directed I) . (IC (Comput (p,s,k))) in rng (Directed I) by FUNCT_1:def 5;
assume k <= LifeSpan (p,s) ; :: thesis: CurInstr ((p +* (Directed I)),(Comput ((p +* (Directed I)),s,k))) <> halt SCM+FSA
then NPP (Comput (p,s,k)) = NPP (Comput ((p +* (Directed I)),(s +* EP),k)) by A5;
then IC (Comput (p,s,k)) = IC (Comput ((p +* (Directed I)),(s +* EP),k)) by COMPOS_1:230;
then A23: CurInstr ((p +* (Directed I)),(Comput ((p +* (Directed I)),(s +* EP),k))) = (p +* (Directed I)) . (IC (Comput (p,s,k))) by PBOOLE:158
.= (Directed I) . (IC (Comput (p,s,k))) by A21, FUNCT_4:14 ;
assume CurInstr ((p +* (Directed I)),(Comput ((p +* (Directed I)),s,k))) = halt SCM+FSA ; :: thesis: contradiction
hence contradiction by A23, A22, E2, SCMFSA6A:18; :: thesis: verum