set i1 = F₃(),F₄() <=0_goto ((card F₂()) + 3);
set J = F₂() ';' (AddTo F₃(),F₄(),(- F₅()));
set i3 = goto (- ((card F₂()) + 2));
set b = DataLoc (F₁() . F₃()),F₄();
set FOR = for-down F₃(),F₄(),F₅(),F₂();
set pFOR = stop (for-down F₃(),F₄(),F₅(),F₂());
set iFOR = Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()));
set pJ = stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())));
set IsJ = Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))));
defpred S₁[ Nat] means for t being State of SCMPDS st t . (DataLoc (F₁() . F₃()),F₄()) <= $1 & P₁[ Dstate t] & t . F₃() = F₁() . F₃() holds
( for-down F₃(),F₄(),F₅(),F₂() is_closed_on t & for-down F₃(),F₄(),F₅(),F₂() is_halting_on t );
A4: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A5: S₁[k] ; :: thesis: S₁[k + 1]

now

let t be State of SCMPDS ; :: thesis: ( t . (DataLoc (F₁() . F₃()),F₄()) <= k + 1 & P₁[ Dstate t] & t . F₃() = F₁() . F₃() implies ( for-down F₃(),F₄(),F₅(),F₂() is_closed_on b₁ & for-down F₃(),F₄(),F₅(),F₂() is_halting_on b₁ ) )
assume A6: t . (DataLoc (F₁() . F₃()),F₄()) <= k + 1 ; :: thesis: ( P₁[ Dstate t] & t . F₃() = F₁() . F₃() implies ( for-down F₃(),F₄(),F₅(),F₂() is_closed_on b₁ & for-down F₃(),F₄(),F₅(),F₂() is_halting_on b₁ ) )
assume A7: P₁[ Dstate t] ; :: thesis: ( t . F₃() = F₁() . F₃() implies ( for-down F₃(),F₄(),F₅(),F₂() is_closed_on b₁ & for-down F₃(),F₄(),F₅(),F₂() is_halting_on b₁ ) )
assume A8: t . F₃() = F₁() . F₃() ; :: thesis: ( for-down F₃(),F₄(),F₅(),F₂() is_closed_on b₁ & for-down F₃(),F₄(),F₅(),F₂() is_halting_on b₁ )

per cases ( t . (DataLoc (F₁() . F₃()),F₄()) <= 0 or t . (DataLoc (F₁() . F₃()),F₄()) > 0 ) ;

suppose t . (DataLoc (F₁() . F₃()),F₄()) <= 0 ; :: thesis: ( for-down F₃(),F₄(),F₅(),F₂() is_closed_on b₁ & for-down F₃(),F₄(),F₅(),F₂() is_halting_on b₁ )

hence ( for-down F₃(),F₄(),F₅(),F₂() is_closed_on t & for-down F₃(),F₄(),F₅(),F₂() is_halting_on t ) by A8, SCMPDS_7:63; :: thesis: verum

end;

suppose A9: t . (DataLoc (F₁() . F₃()),F₄()) > 0 ; :: thesis: ( for-down F₃(),F₄(),F₅(),F₂() is_closed_on b₁ & for-down F₃(),F₄(),F₅(),F₂() is_halting_on b₁ )

A10: dom (t | NAT ) = NAT by SCMPDS_6:1;
A11: not F₃() in dom (t | NAT ) by A10, SCMPDS_2:53;
A12: not DataLoc (F₁() . F₃()),F₄() in dom (t | NAT ) by A10, SCMPDS_2:53;
A13: 0 in dom (stop (for-down F₃(),F₄(),F₅(),F₂())) by SCMPDS_4:75;
- (- F₅()) > 0 by A1;
then - F₅() < 0 ;
then - F₅() <= - 1 by INT_1:21;
then A14: (- F₅()) + (t . (DataLoc (F₁() . F₃()),F₄())) <= (- 1) + (t . (DataLoc (F₁() . F₃()),F₄())) by XREAL_1:8;
(t . (DataLoc (F₁() . F₃()),F₄())) - 1 <= k by A6, XREAL_1:22;
then A15: (- F₅()) + (t . (DataLoc (F₁() . F₃()),F₄())) <= k by A14, XXREAL_0:2;
set t2 = t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))));
set t3 = t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())));
set t4 = Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1;
set Jt = IExec (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))),t;
A16: Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))) c= t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))) by FUNCT_4:26;
for-down F₃(),F₄(),F₅(),F₂() = (F₃(),F₄() <=0_goto ((card F₂()) + 3)) ';' ((F₂() ';' (AddTo F₃(),F₄(),(- F₅()))) ';' (goto (- ((card F₂()) + 2)))) by Lm3;
then A17: CurInstr (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))) = F₃(),F₄() <=0_goto ((card F₂()) + 3) by SCMPDS_6:22;
A18: Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),(0 + 1) = Following (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),0 ) by AMI_1:14
.= Following (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))) by AMI_1:13
.= Exec (F₃(),F₄() <=0_goto ((card F₂()) + 3)),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))) by A17, AMI_1:def 18 ;
A19: DataPart (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))) = DataPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))) by SCMPDS_4:24, SCMPDS_4:36;

now

let x be Int_position ; :: thesis: (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))) . x = (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1) . x
thus (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))) . x = (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))) . x by A19, SCMPDS_4:23
.= (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1) . x by A18, SCMPDS_2:68 ; :: thesis: verum

end;

then A20: DataPart (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))) = DataPart (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1) by SCMPDS_4:23;
A21: (IExec (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))),t) . (DataLoc (F₁() . F₃()),F₄()) = (t . (DataLoc (F₁() . F₃()),F₄())) - F₅() by A3, A7, A8, A9;
A22: (IExec (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))),t) . F₃() = t . F₃() by A3, A7, A8, A9;
set m2 = LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))));
set t5 = Comput (ProgramPart (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1)),(Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1),(LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))));
set l1 = (card (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))) + 1;
A23: card (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))) = (card F₂()) + 1 by SCMP_GCD:8;
A24: dom (t | NAT ) = NAT by SCMPDS_6:1;
A25: not DataLoc (F₁() . F₃()),F₄() in dom (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))) by SCMPDS_4:31;
not F₃() in dom (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))) by SCMPDS_4:31;
then A26: (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))) . (DataLoc ((t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))) . F₃()),F₄()) = (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))) . (DataLoc (F₁() . F₃()),F₄()) by A8, FUNCT_4:12
.= t . (DataLoc (F₁() . F₃()),F₄()) by A25, FUNCT_4:12 ;
A27: IC (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))) = 0 by SCMPDS_6:21;
A28: ( F₂() is_closed_on t & F₂() is_halting_on t ) by A3, A7, A8, A9;
then A29: F₂() ';' (AddTo F₃(),F₄(),(- F₅())) is_closed_on t by Th6;
then A30: F₂() ';' (AddTo F₃(),F₄(),(- F₅())) is_closed_on t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))) by SCMPDS_6:38;
(card F₂()) + 2 < (card F₂()) + 3 by XREAL_1:8;
then A31: (card (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))) + 1 in dom (for-down F₃(),F₄(),F₅(),F₂()) by A23, SCMPDS_7:61;
set m3 = (LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1;
set t6 = Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1);
ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))) = ProgramPart (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1) by AMI_1:144;
then A32: Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) = Comput (ProgramPart (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1)),(Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1),(LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) by AMI_1:51;
F₂() ';' (AddTo F₃(),F₄(),(- F₅())) is_halting_on t by A28, Th6;
then A33: ProgramPart (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))) halts_on t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))) by SCMPDS_6:def 3;
t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))) = (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))) +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))) by A16, FUNCT_4:79;
then ProgramPart ((t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))) +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))) halts_on (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))) +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))) by A33;
then A34: F₂() ';' (AddTo F₃(),F₄(),(- F₅())) is_halting_on t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))) by SCMPDS_6:def 3;
set m4 = ((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1;
set t7 = Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),(((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1);
A35: IExec (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))),t = (Result (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) +* (t | NAT ) by SCMPDS_4:def 8;
( for-down F₃(),F₄(),F₅(),F₂() c= Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())) & Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())) c= t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))) ) by FUNCT_4:26, SCMPDS_6:17;
then A36: for-down F₃(),F₄(),F₅(),F₂() c= t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))) by XBOOLE_1:1;
Shift (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))),1 c= for-down F₃(),F₄(),F₅(),F₂() by Lm4;
then Shift (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))),1 c= t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))) by A36, XBOOLE_1:1;
then A37: Shift (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))),1 c= Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1 by AMI_1:81;
A38: IC (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1) = (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1) . (IC SCMPDS ) by AMI_1:def 15
.= succ (IC (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))) by A9, A18, A26, SCMPDS_2:68
.= 0 + 1 by A27 ;
then A39: DataPart (Comput (ProgramPart (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))),(t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))),(LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))))) = DataPart (Comput (ProgramPart (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1)),(Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1),(LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))))) by A23, A16, A34, A30, A20, A37, SCMPDS_7:36;
then A40: DataPart (Comput (ProgramPart (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1)),(Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1),(LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))))) = DataPart (Result (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) by A33, AMI_1:122
.= DataPart ((Result (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) +* (t | NAT )) by A24, AMI_2:29, FUNCT_4:76, SCMPDS_2:100
.= DataPart (IExec (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))),t) by SCMPDS_4:def 8 ;
Y: (ProgramPart (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1))) /. (IC (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1))) = (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1)) . (IC (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1))) by AMI_1:150;
A41: IC (Comput (ProgramPart (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1)),(Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1),(LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))))) = (card (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))) + 1 by A23, A16, A34, A30, A38, A20, A37, SCMPDS_7:36;
then A42: CurInstr (ProgramPart (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1))),(Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1)) = (Comput (ProgramPart (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1)),(Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1),(LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))))) . ((card (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))) + 1) by A32, AMI_1:def 16, Y
.= (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1) . ((card (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))) + 1) by AMI_1:54
.= (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))) . ((card (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))) + 1) by AMI_1:54
.= (for-down F₃(),F₄(),F₅(),F₂()) . ((card (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))) + 1) by A31, A36, GRFUNC_1:8
.= goto (- ((card F₂()) + 2)) by A23, SCMPDS_7:62 ;
T: ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))) = ProgramPart (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1)) by AMI_1:144;
A43: Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),(((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1) = Following (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1)) by AMI_1:14
.= Exec (goto (- ((card F₂()) + 2))),(Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1)) by A42, AMI_1:def 18, T ;
InsCode (goto (- ((card F₂()) + 2))) = 0 by SCMPDS_2:21;
then InsCode (goto (- ((card F₂()) + 2))) in {0 ,4,5,6} by ENUMSET1:def 2;
then Dstate (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),(((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1)) = Dstate (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1)) by A43, SCMPDS_8:3
.= Dstate (IExec (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))),t) by A40, A32, SCMPDS_8:2 ;
then A44: P₁[ Dstate (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),(((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1))] by A3, A7, A8, A9;
IC (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),(((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1)) = (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),(((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1)) . (IC SCMPDS ) by AMI_1:def 15
.= ICplusConst (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1)),(0 - ((card F₂()) + 2)) by A43, SCMPDS_2:66
.= 0 by A23, A41, A32, SCMPDS_7:1 ;
then A45: (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),(((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1)) +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))) = Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),(((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1) by SCMPDS_7:37;
(Comput (ProgramPart (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1)),(Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1),(LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))))) . (DataLoc (F₁() . F₃()),F₄()) = (Comput (ProgramPart (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))),(t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))),(LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))))) . (DataLoc (F₁() . F₃()),F₄()) by A39, SCMPDS_4:23
.= (Result (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) . (DataLoc (F₁() . F₃()),F₄()) by A33, AMI_1:122
.= (t . (DataLoc (F₁() . F₃()),F₄())) - F₅() by A21, A35, A12, FUNCT_4:12 ;
then A46: (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),(((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1)) . (DataLoc (F₁() . F₃()),F₄()) = (- F₅()) + (t . (DataLoc (F₁() . F₃()),F₄())) by A32, A43, SCMPDS_2:66;
(Comput (ProgramPart (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1)),(Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1),(LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))))) . F₃() = (Comput (ProgramPart (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))),(t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))),(LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))))) . F₃() by A39, SCMPDS_4:23
.= (Result (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) . F₃() by A33, AMI_1:122
.= F₁() . F₃() by A8, A22, A35, A11, FUNCT_4:12 ;
then A47: (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),(((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1)) . F₃() = F₁() . F₃() by A32, A43, SCMPDS_2:66;
then A48: for-down F₃(),F₄(),F₅(),F₂() is_closed_on Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),(((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1) by A5, A44, A46, A15;

now

let k be Element of NAT ; :: thesis: IC (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),b₁) in dom (stop (for-down F₃(),F₄(),F₅(),F₂()))

per cases ( k < ((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1 or k >= ((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1 ) ;

suppose k < ((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1 ; :: thesis: IC (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),b₁) in dom (stop (for-down F₃(),F₄(),F₅(),F₂()))

then A49: k <= (LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1 by INT_1:20;

hereby :: thesis: verum

per cases ( k <= LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))) or k = (LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1 ) by A49, NAT_1:8;

suppose A50: k <= LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))) ; :: thesis: IC (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),k) in dom (stop (for-down F₃(),F₄(),F₅(),F₂()))

hereby :: thesis: verum

per cases ( k = 0 or k <> 0 ) ;

suppose k = 0 ; :: thesis: IC (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),k) in dom (stop (for-down F₃(),F₄(),F₅(),F₂()))

hence IC (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),k) in dom (stop (for-down F₃(),F₄(),F₅(),F₂())) by A13, A27, AMI_1:13; :: thesis: verum

end;

suppose k <> 0 ; :: thesis: IC (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),k) in dom (stop (for-down F₃(),F₄(),F₅(),F₂()))

then consider kn being Nat such that
A51: k = kn + 1 by NAT_1:6;
reconsider kn = kn as Element of NAT by ORDINAL1:def 13;
reconsider lm = IC (Comput (ProgramPart (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))),(t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))),kn) as Element of NAT ;
t: ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))) = ProgramPart (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1) by AMI_1:144;
kn < k by A51, XREAL_1:31;
then kn < LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))) by A50, XXREAL_0:2;
then (IC (Comput (ProgramPart (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))),(t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))),kn)) + 1 = IC (Comput (ProgramPart (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1)),(Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),1),kn) by A23, A16, A34, A30, A38, A20, A37, SCMPDS_7:34;
then A52: IC (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),k) = lm + 1 by A51, AMI_1:51, t;
IC (Comput (ProgramPart (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))),(t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))))),kn) in dom (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))) by A29, SCMPDS_6:def 2;
then lm < card (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))) by SCMPDS_4:1;
then lm < (card (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))) + 1 by SCMPDS_5:7;
then A53: lm + 1 <= (card (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))) + 1 by INT_1:20;
(card F₂()) + 2 < (card F₂()) + 4 by XREAL_1:8;
then lm + 1 < (card F₂()) + 4 by A23, A53, XXREAL_0:2;
then lm + 1 < card (stop (for-down F₃(),F₄(),F₅(),F₂())) by Lm2;
hence IC (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),k) in dom (stop (for-down F₃(),F₄(),F₅(),F₂())) by A52, SCMPDS_4:1; :: thesis: verum

end;

end;

end;

end;

suppose A54: k = (LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1 ; :: thesis: IC (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),k) in dom (stop (for-down F₃(),F₄(),F₅(),F₂()))

(card (F₂() ';' (AddTo F₃(),F₄(),(- F₅())))) + 1 in dom (stop (for-down F₃(),F₄(),F₅(),F₂())) by A31, SCMPDS_6:18;
hence IC (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),k) in dom (stop (for-down F₃(),F₄(),F₅(),F₂())) by A23, A16, A34, A30, A38, A20, A37, A32, A54, SCMPDS_7:36; :: thesis: verum

end;

end;

end;

end;

suppose k >= ((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1 ; :: thesis: IC (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),b₁) in dom (stop (for-down F₃(),F₄(),F₅(),F₂()))

then consider nn being Nat such that
A55: k = (((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1) + nn by NAT_1:10;
reconsider nn = nn as Element of NAT by ORDINAL1:def 13;
ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))) = ProgramPart (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),(((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1)) by AMI_1:144;
then Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),k = Comput (ProgramPart ((Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),(((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1)) +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),((Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),(((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1)) +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),nn by A45, A55, AMI_1:51;
hence IC (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),k) in dom (stop (for-down F₃(),F₄(),F₅(),F₂())) by A48, SCMPDS_6:def 2; :: thesis: verum

end;

end;

end;

hence for-down F₃(),F₄(),F₅(),F₂() is_closed_on t by SCMPDS_6:def 2; :: thesis: for-down F₃(),F₄(),F₅(),F₂() is_halting_on t
for-down F₃(),F₄(),F₅(),F₂() is_halting_on Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),(((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1) by A5, A47, A44, A46, A15;
then ProgramPart (Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),(((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1)) halts_on Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),(((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1) by A45, SCMPDS_6:def 3;
then ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))) halts_on Comput (ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))))),(t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))),(((LifeSpan (t +* (Initialized (stop (F₂() ';' (AddTo F₃(),F₄(),(- F₅()))))))) + 1) + 1) by AMI_1:144;
then ProgramPart (t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂())))) halts_on t +* (Initialized (stop (for-down F₃(),F₄(),F₅(),F₂()))) by AMI_1:93;
hence for-down F₃(),F₄(),F₅(),F₂() is_halting_on t by SCMPDS_6:def 3; :: thesis: verum

end;

end;

end;

hence S₁[k + 1] ; :: thesis: verum

end;

thus ( P₁[F₁()] or not P₁[F₁()] ) ; :: thesis: ( for-down F₃(),F₄(),F₅(),F₂() is_closed_on F₁() & for-down F₃(),F₄(),F₅(),F₂() is_halting_on F₁() )
A56: S₁[ 0 ] by SCMPDS_7:63;
A57: for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A56, A4);

per cases ( F₁() . (DataLoc (F₁() . F₃()),F₄()) <= 0 or F₁() . (DataLoc (F₁() . F₃()),F₄()) > 0 ) ;

suppose F₁() . (DataLoc (F₁() . F₃()),F₄()) <= 0 ; :: thesis: ( for-down F₃(),F₄(),F₅(),F₂() is_closed_on F₁() & for-down F₃(),F₄(),F₅(),F₂() is_halting_on F₁() )

hence ( for-down F₃(),F₄(),F₅(),F₂() is_closed_on F₁() & for-down F₃(),F₄(),F₅(),F₂() is_halting_on F₁() ) by SCMPDS_7:63; :: thesis: verum

end;

suppose F₁() . (DataLoc (F₁() . F₃()),F₄()) > 0 ; :: thesis: ( for-down F₃(),F₄(),F₅(),F₂() is_closed_on F₁() & for-down F₃(),F₄(),F₅(),F₂() is_halting_on F₁() )

then reconsider m = F₁() . (DataLoc (F₁() . F₃()),F₄()) as Element of NAT by INT_1:16;
S₁[m] by A57;
hence ( for-down F₃(),F₄(),F₅(),F₂() is_closed_on F₁() & for-down F₃(),F₄(),F₅(),F₂() is_halting_on F₁() ) by A2; :: thesis: verum

end;

end;