let s be State of SCM+FSA ; :: thesis: for I being Program of SCM+FSA st Directed I is_pseudo-closed_on s holds
( I ';' (Stop SCM+FSA ) is_closed_on s & I ';' (Stop SCM+FSA ) is_halting_on s & LifeSpan (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))) = pseudo-LifeSpan s,(Directed I) & ( for n being Element of NAT st n < pseudo-LifeSpan s,(Directed I) holds
IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) ) & ( for n being Element of NAT st n <= pseudo-LifeSpan s,(Directed I) holds
DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) ) )
set D = Int-Locations \/ FinSeq-Locations ;
set A = NAT ;
let I be Program of SCM+FSA ; :: thesis: ( Directed I is_pseudo-closed_on s implies ( I ';' (Stop SCM+FSA ) is_closed_on s & I ';' (Stop SCM+FSA ) is_halting_on s & LifeSpan (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))) = pseudo-LifeSpan s,(Directed I) & ( for n being Element of NAT st n < pseudo-LifeSpan s,(Directed I) holds
IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) ) & ( for n being Element of NAT st n <= pseudo-LifeSpan s,(Directed I) holds
DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) ) ) )
set I0 = Directed I;
set I1 = I ';' (Stop SCM+FSA );
set s00 = s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ));
set s10 = s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ));
reconsider k = pseudo-LifeSpan (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),(Directed I) as Element of NAT ;
(Stop SCM+FSA ) . 0 = halt SCM+FSA by AFINSQ_1:38;
then A1: halt SCM+FSA = (Stop SCM+FSA ) . ((card I) -' (card I)) by XREAL_1:234;
A2: DataPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))) = DataPart s by SCMFSA8A:11
.= DataPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))) by SCMFSA8A:11 ;
assume A3: Directed I is_pseudo-closed_on s ; :: thesis: ( I ';' (Stop SCM+FSA ) is_closed_on s & I ';' (Stop SCM+FSA ) is_halting_on s & LifeSpan (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))) = pseudo-LifeSpan s,(Directed I) & ( for n being Element of NAT st n < pseudo-LifeSpan s,(Directed I) holds
IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) ) & ( for n being Element of NAT st n <= pseudo-LifeSpan s,(Directed I) holds
DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) ) )
then A4: Directed I is_pseudo-closed_on s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )) by Th50;
defpred S1[ Nat] means ( k <= $1 implies ( IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),$1) = card I & CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),$1)),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),$1) = halt SCM+FSA ) );
A5: (I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ) c= s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:26;
I ';' (Stop SCM+FSA ) c= (I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ) by SCMFSA8A:9;
then A6: I ';' (Stop SCM+FSA ) c= s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )) by A5, XBOOLE_1:1;
A7: Directed I c= I ';' (Stop SCM+FSA ) by SCMFSA6A:55;
then A8: dom (Directed I) c= dom (I ';' (Stop SCM+FSA )) by GRFUNC_1:8;
ProgramPart (Relocated (Directed I),0 ) c= I ';' (Stop SCM+FSA ) by A7, Th9;
then A9: ProgramPart (Relocated (Directed I),0 ) c= s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )) by A6, XBOOLE_1:1;
s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )) = (s +* (I ';' (Stop SCM+FSA ))) +* (Start-At 0 ,SCM+FSA ) by FUNCT_4:15;
then A10: IC (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))) = 0 by AMI_1:111;
A11: (Directed I) +* (Start-At 0 ,SCM+FSA ) c= s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:26;
A12: now
let n be Element of NAT ; :: thesis: ( n <= pseudo-LifeSpan (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),(Directed I) implies ( IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) & DataPart (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) ) )
assume A13: n <= pseudo-LifeSpan (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),(Directed I) ; :: thesis: ( IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) & DataPart (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) )
then (IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n)) + 0 = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) by A11, A4, A9, A10, A2, Th51;
hence IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) ; :: thesis: DataPart (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)
thus DataPart (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) by A11, A4, A9, A10, A2, A13, Th51; :: thesis: verum
end;
A14: k = pseudo-LifeSpan s,(Directed I) by A3, Th50;
A15: (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))) +* ((Directed I) +* (Start-At 0 ,SCM+FSA )) = s +* (((Directed I) +* (Start-At 0 ,SCM+FSA )) +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))) by FUNCT_4:15
.= s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )) ;
A16: now
let n be Element of NAT ; :: thesis: ( n < pseudo-LifeSpan (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),(Directed I) implies ( CurInstr (ProgramPart (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) = CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) & IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) in dom (Directed I) & CurInstr (ProgramPart (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) <> halt SCM+FSA ) )
assume A17: n < pseudo-LifeSpan (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),(Directed I) ; :: thesis: ( CurInstr (ProgramPart (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) = CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) & IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) in dom (Directed I) & CurInstr (ProgramPart (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) <> halt SCM+FSA )
then IncAddr (CurInstr (ProgramPart (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n)),0 = CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) by A11, A4, A9, A10, A2, Th51;
hence CurInstr (ProgramPart (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) = CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) by Th8; :: thesis: ( IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) in dom (Directed I) & CurInstr (ProgramPart (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) <> halt SCM+FSA )
thus IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) in dom (Directed I) by A15, A4, A17, SCMFSA8A:31; :: thesis: CurInstr (ProgramPart (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) <> halt SCM+FSA
thus CurInstr (ProgramPart (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) <> halt SCM+FSA by A15, A4, A17, SCMFSA8A:31; :: thesis: verum
end;
A18: now
let n be Element of NAT ; :: thesis: ( CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) = halt SCM+FSA implies not k > n )
assume A19: CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) = halt SCM+FSA ; :: thesis: not k > n
reconsider l = IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) as Element of NAT ;
Z: (ProgramPart (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n)) /. (IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n)) = (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) . (IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n)) by AMI_1:150;
assume A20: k > n ; :: thesis: contradiction
then A21: l in dom (Directed I) by A3, A14, SCMFSA8A:def 5;
CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) = CurInstr (ProgramPart (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) by A16, A20
.= (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))) . l by Z, AMI_1:54
.= (Directed I) . l by A21, Th26 ;
then halt SCM+FSA in rng (Directed I) by A19, A21, FUNCT_1:def 5;
hence contradiction by AMI_1:def 53; :: thesis: verum
end;
z: card (Stop SCM+FSA ) = 1 by SCMNORM:3;
then card (I ';' (Stop SCM+FSA )) = (card I) + 1 by SCMFSA6A:61;
then card I < card (I ';' (Stop SCM+FSA )) by NAT_1:13;
then A22: card I in dom (I ';' (Stop SCM+FSA )) by SCMFSA6A:15;
card I < (card I) + (card (Stop SCM+FSA )) by NAT_1:13, z;
then (I ';' (Stop SCM+FSA )) . (card I) = IncAddr (halt SCM+FSA ),(card I) by A1, Th13
.= halt SCM+FSA by SCMFSA_4:8 ;
then A23: (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))) . (card I) = halt SCM+FSA by A22, Th26;
A24: for n being Element of NAT st S1[n] holds
S1[n + 1]
proof
let n be Element of NAT ; :: thesis: ( S1[n] implies S1[n + 1] )
assume A25: S1[n] ; :: thesis: S1[n + 1]
Z: (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1))) /. (IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1))) = (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1)) . (IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1))) by AMI_1:150;
assume A26: k <= n + 1 ; :: thesis: ( IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1)) = card I & CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1))),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1)) = halt SCM+FSA )
hereby :: thesis: CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1))),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1)) = halt SCM+FSA
per cases ( k = n + 1 or k <= n ) by A26, NAT_1:8;
suppose k = n + 1 ; :: thesis: IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1)) = card I
hence IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1)) = IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),k) by A12
.= card (ProgramPart (Directed I)) by A3, A14, SCMFSA8A:def 5
.= card (Directed I) by AMI_1:105
.= card I by SCMFSA8A:34 ;
:: thesis: verum
end;
suppose A27: k <= n ; :: thesis: IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1)) = card I
T: ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))) = ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) by AMI_1:144;
Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1) = Following (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) by T ;
hence IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1)) = card I by A25, A27, AMI_1:def 8; :: thesis: verum
end;
end;
end;
hence CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1))),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1)) = halt SCM+FSA by A23, Z, AMI_1:54; :: thesis: verum
end;
Y: (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),0 )) /. (IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),0 )) = (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),0 ) . (IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),0 )) by AMI_1:150;
A28: S1[ 0 ]
proof
assume k <= 0 ; :: thesis: ( IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),0 ) = card I & CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),0 )),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),0 ) = halt SCM+FSA )
then k = 0 ;
hence IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),0 ) = IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),k) by A12
.= card (ProgramPart (Directed I)) by A3, A14, SCMFSA8A:def 5
.= card (Directed I) by AMI_1:105
.= card I by SCMFSA8A:34 ;
:: thesis: CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),0 )),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),0 ) = halt SCM+FSA
hence CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),0 )),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),0 ) = halt SCM+FSA by A23, Y, AMI_1:13; :: thesis: verum
end;
A29: for n being Element of NAT holds S1[n] from NAT_1:sch 1(A28, A24);
now
let n be Element of NAT ; :: thesis: IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),b1) in dom (I ';' (Stop SCM+FSA ))
per cases ( n < k or k <= n ) ;
suppose A30: n < k ; :: thesis: IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),b1) in dom (I ';' (Stop SCM+FSA ))
then IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) by A12;
then IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) in dom (Directed I) by A3, A14, A30, SCMFSA8A:def 5;
hence IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) in dom (I ';' (Stop SCM+FSA )) by A8; :: thesis: verum
end;
end;
end;
hence I ';' (Stop SCM+FSA ) is_closed_on s by SCMFSA7B:def 7; :: thesis: ( I ';' (Stop SCM+FSA ) is_halting_on s & LifeSpan (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))) = pseudo-LifeSpan s,(Directed I) & ( for n being Element of NAT st n < pseudo-LifeSpan s,(Directed I) holds
IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) ) & ( for n being Element of NAT st n <= pseudo-LifeSpan s,(Directed I) holds
DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) ) )
set s1 = s +* (I +* (Start-At 0 ,SCM+FSA ));
A31: card (ProgramPart (Directed I)) = card (Directed I) by AMI_1:105
.= card I by SCMFSA8A:34 ;
S1[k] by A29;
then A32: ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))) halts_on s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )) by AMI_1:146;
hence I ';' (Stop SCM+FSA ) is_halting_on s by SCMFSA7B:def 8; :: thesis: ( LifeSpan (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))) = pseudo-LifeSpan s,(Directed I) & ( for n being Element of NAT st n < pseudo-LifeSpan s,(Directed I) holds
IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) ) & ( for n being Element of NAT st n <= pseudo-LifeSpan s,(Directed I) holds
DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) ) )
CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),k)),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),k) = halt SCM+FSA by A29;
then A33: LifeSpan (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))) = k by A32, A18, AMI_1:def 46;
defpred S2[ Nat] means ( $1 < pseudo-LifeSpan s,(Directed I) implies ( IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),$1) in dom I & IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),$1) = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),$1) & DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),$1) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),$1) ) );
A34: for n being Element of NAT st S2[n] holds
S2[n + 1]
proof
let n be Element of NAT ; :: thesis: ( S2[n] implies S2[n + 1] )
set l = IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n);
set l0 = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n);
assume A35: S2[n] ; :: thesis: S2[n + 1]
assume A36: n + 1 < pseudo-LifeSpan s,(Directed I) ; :: thesis: ( IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(n + 1)) in dom I & IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(n + 1)) = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1)) & DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(n + 1)) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1)) )
then A37: IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) in dom (Directed I) by A35, FUNCT_4:105, NAT_1:12;
A38: for f being FinSeq-Location holds (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) . f = (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) . f by A35, A36, NAT_1:12, SCMFSA6A:38;
for a being Int-Location holds (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) . a = (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) . a by A35, A36, NAT_1:12, SCMFSA6A:38;
then A39: Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n, Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n equal_outside NAT by A35, A36, A38, NAT_1:12, SCMFSA6A:28;
A40: now
A41: dom I = dom (Directed I) by FUNCT_4:105;
assume A42: I . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n)) = halt SCM+FSA ; :: thesis: contradiction
Y: (ProgramPart (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n)) /. (IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n)) = (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) . (IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n)) by AMI_1:150;
n < k by A14, A36, NAT_1:12;
then A43: CurInstr (ProgramPart (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) = (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) . (IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)) by A12, Y
.= (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))) . (IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)) by AMI_1:54
.= (Directed I) . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n)) by A35, A36, A41, Th26, NAT_1:12
.= goto (card I) by A35, A36, A42, NAT_1:12, SCMFSA8A:30 ;
T: ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))) = ProgramPart (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) by AMI_1:144;
A44: IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),(n + 1)) = IC (Following (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n)) by AMI_1:14
.= card I by A43, T, SCMFSA_2:95
.= card (Directed I) by SCMFSA8A:34 ;
IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),(n + 1)) in dom (Directed I) by A3, A36, SCMFSA8A:31;
hence contradiction by A44, SCMFSA6A:15; :: thesis: verum
end;
Y: (ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n)) /. (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n)) = (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n)) by AMI_1:150;
U: (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)) /. (IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)) = (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) . (IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)) by AMI_1:150;
A45: CurInstr (ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = (s +* (I +* (Start-At 0 ,SCM+FSA ))) . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n)) by Y, AMI_1:54
.= I . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n)) by A35, A36, Th26, NAT_1:12
.= (Directed I) . (IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)) by A35, A36, A40, NAT_1:12, SCMFSA8A:30
.= (I ';' (Stop SCM+FSA )) . (IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)) by A7, A37, GRFUNC_1:8
.= (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))) . (IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)) by A8, A37, Th26
.= CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) by U, AMI_1:54 ;
T: ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))) = ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) by AMI_1:144;
A46: Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1) = Following (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) by A45, T ;
pseudo-LifeSpan s,(Directed I) = k by A3, Th50;
then A47: IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),(n + 1)) = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1)) by A12, A36;
A48: dom (Directed I) = dom I by FUNCT_4:105;
T: ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA ))) = ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) by AMI_1:144;
A49: Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(n + 1) = Following (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n)),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) by T ;
then A50: for f being FinSeq-Location holds (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(n + 1)) . f = (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1)) . f by A46, A39, SCMFSA6A:31, SCMFSA6A:32;
IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),(n + 1)) in dom (Directed I) by A3, A36, SCMFSA8A:31;
hence IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(n + 1)) in dom I by A47, A49, A46, A39, A48, AMI_1:121, SCMFSA6A:32; :: thesis: ( IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(n + 1)) = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1)) & DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(n + 1)) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1)) )
thus IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(n + 1)) = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1)) by A49, A46, A39, AMI_1:121, SCMFSA6A:32; :: thesis: DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(n + 1)) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1))
for a being Int-Location holds (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(n + 1)) . a = (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1)) . a by A49, A46, A39, SCMFSA6A:30, SCMFSA6A:32;
hence DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(n + 1)) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),(n + 1)) by A50, SCMFSA6A:38; :: thesis: verum
end;
IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),k) = card I by A29;
then A51: IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),(LifeSpan (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))))) = card I by A12, A33;
for n being Element of NAT st not IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),n) in dom (Directed I) holds
LifeSpan (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))) <= n by A16, A33;
hence LifeSpan (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))) = pseudo-LifeSpan s,(Directed I) by A3, A51, A31, SCMFSA8A:def 5; :: thesis: ( ( for n being Element of NAT st n < pseudo-LifeSpan s,(Directed I) holds
IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) ) & ( for n being Element of NAT st n <= pseudo-LifeSpan s,(Directed I) holds
DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) ) )
A52: S2[ 0 ]
proof
A53: IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),0 ) = IC (s +* (I +* (Start-At 0 ,SCM+FSA ))) by AMI_1:13
.= IC ((s +* I) +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:15
.= 0 by AMI_1:111 ;
assume 0 < pseudo-LifeSpan s,(Directed I) ; :: thesis: ( IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),0 ) in dom I & IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),0 ) = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),0 ) & DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),0 ) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),0 ) )
then IC (Comput (ProgramPart (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))),0 ) in dom (Directed I) by A3, SCMFSA8A:31;
then IC (s +* ((Directed I) +* (Start-At 0 ,SCM+FSA ))) in dom (Directed I) by AMI_1:13;
then 0 in dom (Directed I) by Th31;
hence IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),0 ) in dom I by A53, FUNCT_4:105; :: thesis: ( IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),0 ) = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),0 ) & DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),0 ) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),0 ) )
thus IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),0 ) = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),0 ) by A10, A53, AMI_1:13; :: thesis: DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),0 ) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),0 )
thus DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),0 ) = DataPart (s +* (I +* (Start-At 0 ,SCM+FSA ))) by AMI_1:13
.= DataPart s by SCMFSA8A:11
.= DataPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))) by SCMFSA8A:11
.= DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),0 ) by AMI_1:13 ; :: thesis: verum
end;
A54: for n being Element of NAT holds S2[n] from NAT_1:sch 1(A52, A34);
 hence for n being Element of NAT st n < pseudo-LifeSpan s,(Directed I) holds
IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) ; :: thesis: for n being Element of NAT st n <= pseudo-LifeSpan s,(Directed I) holds
DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)
let n be Element of NAT ; :: thesis: ( n <= pseudo-LifeSpan s,(Directed I) implies DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) )
assume A55: n <= pseudo-LifeSpan s,(Directed I) ; :: thesis: DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)
per cases ( n < pseudo-LifeSpan s,(Directed I) or n = pseudo-LifeSpan s,(Directed I) ) by A55, XXREAL_0:1;
suppose n < pseudo-LifeSpan s,(Directed I) ; :: thesis: DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)
hence DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) by A54; :: thesis: verum
end;
suppose A56: n = pseudo-LifeSpan s,(Directed I) ; :: thesis: DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)
hereby :: thesis: verum
per cases ( n = 0 or ex m being Nat st n = m + 1 ) by NAT_1:6;
suppose ex m being Nat st n = m + 1 ; :: thesis: DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)
then consider m being Nat such that
A58: n = m + 1 ;
reconsider m = m as Element of NAT by ORDINAL1:def 13;
T: ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))) = ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m) by AMI_1:144;
A59: Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n = Following (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m) by A58, AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m) by T ;
set i = CurInstr (ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m);
T: ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA ))) = ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) by AMI_1:144;
A60: Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n = Following (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) by A58, AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) by T ;
set l0 = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m);
set l = IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m);
A61: m + 0 < pseudo-LifeSpan s,(Directed I) by A56, A58, XREAL_1:8;
then A62: IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) = IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m) by A54;
A63: IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) in dom I by A54, A61;
then A64: IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m) in dom (Directed I) by A62, FUNCT_4:105;
Y: (ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)) /. (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)) = (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)) by AMI_1:150;
Z: (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m)) /. (IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m)) = (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m) . (IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m)) by AMI_1:150;
A65: CurInstr (ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) = (s +* (I +* (Start-At 0 ,SCM+FSA ))) . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)) by Y, AMI_1:54
.= I . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)) by A54, A61, Th26 ;
Directed I c= I ';' (Stop SCM+FSA ) by SCMFSA6A:55;
then A66: (Directed I) . (IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m)) = (I ';' (Stop SCM+FSA )) . (IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m)) by A64, GRFUNC_1:8
.= (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))) . (IC (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m)) by A8, A64, Th26
.= CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m) by Z, AMI_1:54 ;
A67: DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m) by A54, A61;
hereby :: thesis: verum
per cases ( CurInstr (ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) = halt SCM+FSA or CurInstr (ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) <> halt SCM+FSA ) ;
suppose A68: CurInstr (ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) = halt SCM+FSA ; :: thesis: DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)
then CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m) = goto (card I) by A63, A62, A65, A66, SCMFSA8A:30;
then InsCode (CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m)) = 6 by SCMFSA_2:47;
then A69: InsCode (CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m)) in {0 ,6,7,8} by ENUMSET1:def 2;
thus DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) by A60, A68, AMI_1:def 8
.= DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m) by A54, A61
.= DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) by A59, A69, Th32 ; :: thesis: verum
end;
suppose CurInstr (ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) <> halt SCM+FSA ; :: thesis: DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n)
then CurInstr (ProgramPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),m) = CurInstr (ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m)),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),m) by A63, A62, A65, A66, SCMFSA8A:30;
hence DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),n) = DataPart (Comput (ProgramPart (s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA ))),n) by A60, A59, A67, SCMFSA6C:5; :: thesis: verum
end;
end;
end;
end;
end;
end;
end;
end;