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 pJ = 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 (ProgramPart t) = NAT by COMPOS_1:34;
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 COMPOS_1:135;
- (- 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 = (Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))));
set t3 = (Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())));
set t4 = Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1);
set Jt = IExec ((F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))),t);
I1: (Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))) = t +* (Initialize (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))) by COMPOS_1:125;
I2: (Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))) = t +* (Initialize (stop (for-down (F₃(),F₄(),F₅(),F₂())))) by COMPOS_1:125;
A16: Initialize (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))) c= (Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))) by I1, FUNCT_4:26;
B17: for-down (F₃(),F₄(),F₅(),F₂()) = ((F₃(),F₄()) <=0_goto ((card F₂()) + 3)) ';' ((F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))) ';' (goto (- ((card F₂()) + 2)))) by Lm3;
A18: Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),(0 + 1)) = Following ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),0))) by EXTPRO_1:4
.= Following ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))) by EXTPRO_1:3
.= Exec (((F₃(),F₄()) <=0_goto ((card F₂()) + 3)),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))) by B17, I2, SCMPDS_6:22 ;
A19: DataPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))) = DataPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))) by COMPOS_1:138, FUNCT_7:134;

now

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

end;

then A20: DataPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))) = DataPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (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 ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))));
set t5 = Comput ((ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1)),(LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (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 (ProgramPart t) = NAT by COMPOS_1:34;
A25: not DataLoc ((F₁() . F₃()),F₄()) in dom (Initialize (stop (for-down (F₃(),F₄(),F₅(),F₂())))) by SCMPDS_4:31;
not F₃() in dom (Initialize (stop (for-down (F₃(),F₄(),F₅(),F₂())))) by SCMPDS_4:31;
then A26: ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))) . (DataLoc ((((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))) . F₃()),F₄())) = ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))) . (DataLoc ((F₁() . F₃()),F₄())) by A8, I2, FUNCT_4:12
.= t . (DataLoc ((F₁() . F₃()),F₄())) by A25, I2, FUNCT_4:12 ;
A27: IC ((Initialize t) +* (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 (Initialize t) +* (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 ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1;
set t6 = Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1));
ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))) = ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1)) by AMI_1:123;
then A32: Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1)) = Comput ((ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1)),(LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))))) by EXTPRO_1:5;
I3: (Initialize ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))) = ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))) +* (Initialize (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))) by COMPOS_1:125;
B33: F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))) is_halting_on t by A28, Th6;
then A33: ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))) halts_on (Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))) by SCMPDS_6:def 3;
(Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))) = (Initialize ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))) by A16, I3, FUNCT_4:79;
then ProgramPart ((Initialize ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))) halts_on (Initialize ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))) by B33, SCMPDS_6:def 3;
then A34: F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))) is_halting_on (Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))) by SCMPDS_6:def 3;
set m4 = ((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1) + 1;
set t7 = Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1) + 1));
A35: IExec ((F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))),t) = (Result ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) +* (t | NAT) by SCMPDS_4:def 8;
( for-down (F₃(),F₄(),F₅(),F₂()) c= Initialize (stop (for-down (F₃(),F₄(),F₅(),F₂()))) & Initialize (stop (for-down (F₃(),F₄(),F₅(),F₂()))) c= (Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))) ) by I2, FUNCT_4:26, SCMPDS_6:17;
then A36: for-down (F₃(),F₄(),F₅(),F₂()) c= (Initialize t) +* (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= (Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))) by A36, XBOOLE_1:1;
then A37: Shift ((F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))),1) c= Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1) by AMI_1:81;
A38: IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1)) = (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1)) . (IC SCMPDS)
.= succ (IC ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))) by A9, A18, A26, SCMPDS_2:68
.= 0 + 1 by A27 ;
then A39: DataPart (Comput ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))),(LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))))) = DataPart (Comput ((ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1)),(LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))))) by A23, A16, A34, A30, A20, A37, SCMPDS_7:36;
then A40: DataPart (Comput ((ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1)),(LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))))) = DataPart (Result ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) by A33, EXTPRO_1:23
.= DataPart ((Result ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (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 ;
A41: IC (Comput ((ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1)),(LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))))) = (card (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))) + 1 by A16, A34, A30, A38, A20, A37, SCMPDS_7:36;
then A42: CurInstr ((ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1)))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1)))) = (Comput ((ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1)),(LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))))) . ((card (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))) + 1) by A32, COMPOS_1:38
.= (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1)) . ((card (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))) + 1) by AMI_1:54
.= ((Initialize t) +* (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 ;
A43: Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1) + 1)) = Following ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1)))) by EXTPRO_1:4
.= Exec ((goto (- ((card F₂()) + 2))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1)))) by A42, AMI_1:123 ;
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 ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1) + 1))) = Dstate (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (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 ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1) + 1)))] by A3, A7, A8, A9;
IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1) + 1))) = (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1) + 1))) . (IC SCMPDS)
.= ICplusConst ((Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (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: (Initialize (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1) + 1)))) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))) = Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1) + 1)) by SCMPDS_7:37;
(Comput ((ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1)),(LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))))) . (DataLoc ((F₁() . F₃()),F₄())) = (Comput ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))),(LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))))) . (DataLoc ((F₁() . F₃()),F₄())) by A39, SCMPDS_4:23
.= (Result ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) . (DataLoc ((F₁() . F₃()),F₄())) by A33, EXTPRO_1:23
.= (t . (DataLoc ((F₁() . F₃()),F₄()))) - F₅() by A21, A35, A12, FUNCT_4:12 ;
then A46: (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (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 ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1)),(LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))))) . F₃() = (Comput ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))),(LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))))) . F₃() by A39, SCMPDS_4:23
.= (Result ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) . F₃() by A33, EXTPRO_1:23
.= F₁() . F₃() by A8, A22, A35, A11, FUNCT_4:12 ;
then A47: (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (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 ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1) + 1)) by A5, A44, A46, A15;

now

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

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

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

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

hereby :: thesis: verum

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

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

hence IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),k)) in dom (stop (for-down (F₃(),F₄(),F₅(),F₂()))) by A13, A27, EXTPRO_1:3; :: thesis: verum

end;

suppose k <> 0 ; :: thesis: IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (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 ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))),kn)) as Element of NAT ;
t: ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))) = ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1)) by AMI_1:123;
kn < k by A51, XREAL_1:31;
then kn < LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))) by A50, XXREAL_0:2;
then (IC (Comput ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))),kn))) + 1 = IC (Comput ((ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),1))),(Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (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 ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),k)) = lm + 1 by A51, t, EXTPRO_1:5;
IC (Comput ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (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 AFINSQ_1:70;
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 ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),k)) in dom (stop (for-down (F₃(),F₄(),F₅(),F₂()))) by A52, AFINSQ_1:70; :: thesis: verum

end;

end;

end;

end;

suppose A54: k = (LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1 ; :: thesis: IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (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 ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),k)) in dom (stop (for-down (F₃(),F₄(),F₅(),F₂()))) by A16, A34, A30, A38, A20, A37, A32, A54, SCMPDS_7:36; :: thesis: verum

end;

end;

end;

end;

suppose k >= ((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1) + 1 ; :: thesis: IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (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 ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (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 ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))) = ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1) + 1))) by AMI_1:123;
then Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),k) = Comput ((ProgramPart ((Initialize (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1) + 1)))) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1) + 1)))) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),nn) by A45, A55, EXTPRO_1:5;
hence IC (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (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 ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1) + 1)) by A5, A47, A44, A46, A15;
then ProgramPart (Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1) + 1))) halts_on Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1) + 1)) by A45, SCMPDS_6:def 3;
then ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))) halts_on Comput ((ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))))),((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))),(((LifeSpan ((ProgramPart ((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅()))))))),((Initialize t) +* (stop (F₂() ';' (AddTo (F₃(),F₄(),(- F₅())))))))) + 1) + 1)) by AMI_1:123;
then ProgramPart ((Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂())))) halts_on (Initialize t) +* (stop (for-down (F₃(),F₄(),F₅(),F₂()))) by EXTPRO_1:22;
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;