set i1 = (F₄(),F₅()) <=0_goto ((card F₃()) + 3);
set J = F₃() ';' (AddTo (F₄(),F₅(),(- F₆())));
set i3 = goto (- ((card F₃()) + 2));
set b = DataLoc ((F₁() . F₄()),F₅());
set FOR = for-down (F₄(),F₅(),F₆(),F₃());
set pFOR = stop (for-down (F₄(),F₅(),F₆(),F₃()));
set pJ = stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆()))));
defpred S₁[ Nat] means for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . (DataLoc ((F₁() . F₄()),F₅())) <= $1 & P₁[t] & t . F₄() = F₁() . F₄() holds
( for-down (F₄(),F₅(),F₆(),F₃()) is_closed_on t,Q & for-down (F₄(),F₅(),F₆(),F₃()) is_halting_on t,Q );
A4: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

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

now :: thesis: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . (DataLoc ((F₁() . F₄()),F₅())) <= k + 1 & P₁[t] & t . F₄() = F₁() . F₄() holds
( for-down (F₄(),F₅(),F₆(),F₃()) is_closed_on t,Q & for-down (F₄(),F₅(),F₆(),F₃()) is_halting_on t,Q )

let t be 0 -started State of SCMPDS; :: thesis: for Q being Instruction-Sequence of SCMPDS st t . (DataLoc ((F₁() . F₄()),F₅())) <= k + 1 & P₁[t] & t . F₄() = F₁() . F₄() holds
( for-down (F₄(),F₅(),F₆(),F₃()) is_closed_on b₂,b₃ & for-down (F₄(),F₅(),F₆(),F₃()) is_halting_on b₂,b₃ )
let Q be Instruction-Sequence of SCMPDS; :: thesis: ( t . (DataLoc ((F₁() . F₄()),F₅())) <= k + 1 & P₁[t] & t . F₄() = F₁() . F₄() implies ( for-down (F₄(),F₅(),F₆(),F₃()) is_closed_on b₁,b₂ & for-down (F₄(),F₅(),F₆(),F₃()) is_halting_on b₁,b₂ ) )
A6: Initialize t = t by MEMSTR_0:44;
assume A7: t . (DataLoc ((F₁() . F₄()),F₅())) <= k + 1 ; :: thesis: ( P₁[t] & t . F₄() = F₁() . F₄() implies ( for-down (F₄(),F₅(),F₆(),F₃()) is_closed_on b₁,b₂ & for-down (F₄(),F₅(),F₆(),F₃()) is_halting_on b₁,b₂ ) )
assume A8: P₁[t] ; :: thesis: ( t . F₄() = F₁() . F₄() implies ( for-down (F₄(),F₅(),F₆(),F₃()) is_closed_on b₁,b₂ & for-down (F₄(),F₅(),F₆(),F₃()) is_halting_on b₁,b₂ ) )
assume A9: t . F₄() = F₁() . F₄() ; :: thesis: ( for-down (F₄(),F₅(),F₆(),F₃()) is_closed_on b₁,b₂ & for-down (F₄(),F₅(),F₆(),F₃()) is_halting_on b₁,b₂ )

per cases ( t . (DataLoc ((F₁() . F₄()),F₅())) <= 0 or t . (DataLoc ((F₁() . F₄()),F₅())) > 0 ) ;

suppose t . (DataLoc ((F₁() . F₄()),F₅())) <= 0 ; :: thesis: ( for-down (F₄(),F₅(),F₆(),F₃()) is_closed_on b₁,b₂ & for-down (F₄(),F₅(),F₆(),F₃()) is_halting_on b₁,b₂ )

hence ( for-down (F₄(),F₅(),F₆(),F₃()) is_closed_on t,Q & for-down (F₄(),F₅(),F₆(),F₃()) is_halting_on t,Q ) by A9, SCMPDS_7:44; :: thesis: verum

end;

suppose A10: t . (DataLoc ((F₁() . F₄()),F₅())) > 0 ; :: thesis: ( for-down (F₄(),F₅(),F₆(),F₃()) is_closed_on b₁,b₂ & for-down (F₄(),F₅(),F₆(),F₃()) is_halting_on b₁,b₂ )

A11: 0 in dom (stop (for-down (F₄(),F₅(),F₆(),F₃()))) by COMPOS_1:36;
- (- F₆()) > 0 by A1;
then - F₆() < 0 ;
then - F₆() <= - 1 by INT_1:8;
then A12: (- F₆()) + (t . (DataLoc ((F₁() . F₄()),F₅()))) <= (- 1) + (t . (DataLoc ((F₁() . F₄()),F₅()))) by XREAL_1:6;
(t . (DataLoc ((F₁() . F₄()),F₅()))) - 1 <= k by A7, XREAL_1:20;
then A13: (- F₆()) + (t . (DataLoc ((F₁() . F₄()),F₅()))) <= k by A12, XXREAL_0:2;
set Q2 = Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))));
set Q3 = Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())));
set t4 = Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,1);
set Q4 = Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())));
set Jt = IExec ((F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))),Q,t);
A14: stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))) c= Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆()))))) by FUNCT_4:25;
A15: for-down (F₄(),F₅(),F₆(),F₃()) = ((F₄(),F₅()) <=0_goto ((card F₃()) + 3)) ';' ((F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))) ';' (goto (- ((card F₃()) + 2)))) by Lm2;
A16: Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,(0 + 1)) = Following ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),(Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,0))) by EXTPRO_1:3
.= Following ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t) by EXTPRO_1:2
.= Exec (((F₄(),F₅()) <=0_goto ((card F₃()) + 3)),t) by A15, A6, SCMPDS_6:11 ;
for x being Int_position holds t . x = (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,1)) . x by A16, SCMPDS_2:56;
then A17: DataPart t = DataPart (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,1)) by SCMPDS_4:8;
A18: (IExec ((F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))),Q,t)) . (DataLoc ((F₁() . F₄()),F₅())) = (t . (DataLoc ((F₁() . F₄()),F₅()))) - F₆() by A3, A8, A9, A10;
A19: (IExec ((F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))),Q,t)) . F₄() = t . F₄() by A3, A8, A9, A10;
set m2 = LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t);
set t5 = Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),(Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,1)),(LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)));
set Q5 = Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())));
set l1 = (card (F₃() ';' (AddTo (F₄(),F₅(),(- F₆()))))) + 1;
A20: card (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))) = (card F₃()) + 1 by SCMP_GCD:4;
A21: t . (DataLoc ((t . F₄()),F₅())) = t . (DataLoc ((F₁() . F₄()),F₅())) by A9;
A22: IC t = 0 by A6, MEMSTR_0:47;
A23: ( F₃() is_closed_on t,Q & F₃() is_halting_on t,Q ) by A3, A8, A9, A10;
then A24: F₃() ';' (AddTo (F₄(),F₅(),(- F₆()))) is_closed_on t,Q by Th5;
then A25: F₃() ';' (AddTo (F₄(),F₅(),(- F₆()))) is_closed_on t,Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆()))))) by A6, SCMPDS_6:24;
(card F₃()) + 2 < (card F₃()) + 3 by XREAL_1:6;
then A26: (card (F₃() ';' (AddTo (F₄(),F₅(),(- F₆()))))) + 1 in dom (for-down (F₄(),F₅(),F₆(),F₃())) by A20, SCMPDS_7:42;
set m3 = (LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1;
set t6 = Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1));
set Q6 = Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())));
A27: Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1)) = Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),(Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,1)),(LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t))) by EXTPRO_1:4;
A28: F₃() ';' (AddTo (F₄(),F₅(),(- F₆()))) is_halting_on t,Q by A23, Th5;
then A29: Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆()))))) halts_on t by A6, SCMPDS_6:def 3;
(Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))) +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆()))))) halts_on t by A28, A6, SCMPDS_6:def 3;
then A30: F₃() ';' (AddTo (F₄(),F₅(),(- F₆()))) is_halting_on t,Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆()))))) by A6, SCMPDS_6:def 3;
set m4 = ((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1) + 1;
set t7 = Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,(((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1) + 1));
set Q7 = Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())));
A31: stop (for-down (F₄(),F₅(),F₆(),F₃())) c= Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃()))) by FUNCT_4:25;
A32: for-down (F₄(),F₅(),F₆(),F₃()) c= stop (for-down (F₄(),F₅(),F₆(),F₃())) by AFINSQ_1:74;
then A33: for-down (F₄(),F₅(),F₆(),F₃()) c= Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃()))) by A31, XBOOLE_1:1;
Shift ((F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))),1) c= for-down (F₄(),F₅(),F₆(),F₃()) by Lm3;
then Shift ((F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))),1) c= stop (for-down (F₄(),F₅(),F₆(),F₃())) by A32, XBOOLE_1:1;
then A34: Shift ((F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))),1) c= Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃()))) by A31, XBOOLE_1:1;
A35: IC (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,1)) = (IC t) + 1 by A10, A16, A21, SCMPDS_2:56
.= 0 + 1 by A22 ;
then A36: DataPart (Comput ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t,(LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)))) = DataPart (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),(Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,1)),(LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)))) by A14, A30, A25, A17, A34, SCMPDS_7:18;
then A37: DataPart (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),(Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,1)),(LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)))) = DataPart (Result ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) by A29, EXTPRO_1:23
.= DataPart (IExec ((F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))),Q,t)) by SCMPDS_4:def 5 ;
A38: IC (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),(Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,1)),(LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)))) = (card (F₃() ';' (AddTo (F₄(),F₅(),(- F₆()))))) + 1 by A14, A30, A25, A35, A17, A34, SCMPDS_7:18;
then A39: CurInstr ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),(Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1)))) = (Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))) . ((card (F₃() ';' (AddTo (F₄(),F₅(),(- F₆()))))) + 1) by A27, PBOOLE:143
.= (for-down (F₄(),F₅(),F₆(),F₃())) . ((card (F₃() ';' (AddTo (F₄(),F₅(),(- F₆()))))) + 1) by A26, A33, GRFUNC_1:2
.= goto (- ((card F₃()) + 2)) by A20, SCMPDS_7:43 ;
A40: Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,(((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1) + 1)) = Following ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),(Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1)))) by EXTPRO_1:3
.= Exec ((goto (- ((card F₃()) + 2))),(Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1)))) by A39 ;
IC (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,(((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1) + 1))) = ICplusConst ((Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1))),(0 - ((card F₃()) + 2))) by A40, SCMPDS_2:54
.= 0 by A20, A38, A27, SCMPDS_7:1 ;
then A41: Initialize (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,(((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1) + 1))) = Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,(((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1) + 1)) by MEMSTR_0:46;
InsCode (goto (- ((card F₃()) + 2))) = 14 by SCMPDS_2:12;
then InsCode (goto (- ((card F₃()) + 2))) in {0,4,5,6,14} by ENUMSET1:def 3;
then Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,(((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1) + 1)) = Initialize (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1))) by A40, A41, SCMPDS_8:3
.= Initialize (IExec ((F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))),Q,t)) by A37, A27, MEMSTR_0:80 ;
then A42: P₁[ Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,(((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1) + 1))] by A3, A8, A9, A10;
A43: (Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))) +* (stop (for-down (F₄(),F₅(),F₆(),F₃()))) = Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃()))) ;
(Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),(Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,1)),(LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)))) . (DataLoc ((F₁() . F₄()),F₅())) = (Comput ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t,(LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)))) . (DataLoc ((F₁() . F₄()),F₅())) by A36, SCMPDS_4:8
.= (Result ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) . (DataLoc ((F₁() . F₄()),F₅())) by A29, EXTPRO_1:23
.= (t . (DataLoc ((F₁() . F₄()),F₅()))) - F₆() by A18, SCMPDS_4:def 5 ;
then A44: (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,(((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1) + 1))) . (DataLoc ((F₁() . F₄()),F₅())) = (- F₆()) + (t . (DataLoc ((F₁() . F₄()),F₅()))) by A27, A40, SCMPDS_2:54;
(Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),(Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,1)),(LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)))) . F₄() = (Comput ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t,(LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)))) . F₄() by A36, SCMPDS_4:8
.= (Result ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) . F₄() by A29, EXTPRO_1:23
.= F₁() . F₄() by A9, A19, SCMPDS_4:def 5 ;
then A45: (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,(((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1) + 1))) . F₄() = F₁() . F₄() by A27, A40, SCMPDS_2:54;
then A46: for-down (F₄(),F₅(),F₆(),F₃()) is_closed_on Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,(((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1) + 1)),Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃()))) by A5, A42, A44, A13, A41;

now :: thesis: for k being Nat holds IC (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,k)) in dom (stop (for-down (F₄(),F₅(),F₆(),F₃())))

let k be Nat; :: thesis: IC (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,b₁)) in dom (stop (for-down (F₄(),F₅(),F₆(),F₃())))

per cases ( k < ((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1) + 1 or k >= ((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1) + 1 ) ;

suppose k < ((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1) + 1 ; :: thesis: IC (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,b₁)) in dom (stop (for-down (F₄(),F₅(),F₆(),F₃())))

then A47: k <= (LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1 by INT_1:7;

hereby :: thesis: verum

per cases ( k <= LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t) or k = (LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1 ) by A47, NAT_1:8;

suppose A48: k <= LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t) ; :: thesis: IC (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,k)) in dom (stop (for-down (F₄(),F₅(),F₆(),F₃())))

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

suppose k = 0 ; :: thesis: IC (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,k)) in dom (stop (for-down (F₄(),F₅(),F₆(),F₃())))

hence IC (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,k)) in dom (stop (for-down (F₄(),F₅(),F₆(),F₃()))) by A11, A22, EXTPRO_1:2; :: thesis: verum

end;

suppose k <> 0 ; :: thesis: IC (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,k)) in dom (stop (for-down (F₄(),F₅(),F₆(),F₃())))

then consider kn being Nat such that
A49: k = kn + 1 by NAT_1:6;
reconsider kn = kn as Nat ;
reconsider lm = IC (Comput ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t,kn)) as Element of NAT ;
kn < k by A49, XREAL_1:29;
then kn < LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t) by A48, XXREAL_0:2;
then (IC (Comput ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t,kn))) + 1 = IC (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),(Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,1)),kn)) by A14, A30, A25, A35, A17, A34, SCMPDS_7:16;
then A50: IC (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,k)) = lm + 1 by A49, EXTPRO_1:4;
IC (Comput ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t,kn)) in dom (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆()))))) by A24, A6, SCMPDS_6:def 2;
then lm < card (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆()))))) by AFINSQ_1:66;
then lm < (card (F₃() ';' (AddTo (F₄(),F₅(),(- F₆()))))) + 1 by COMPOS_1:55;
then A51: lm + 1 <= (card (F₃() ';' (AddTo (F₄(),F₅(),(- F₆()))))) + 1 by INT_1:7;
(card F₃()) + 2 < (card F₃()) + 4 by XREAL_1:6;
then lm + 1 < (card F₃()) + 4 by A20, A51, XXREAL_0:2;
then lm + 1 < card (stop (for-down (F₄(),F₅(),F₆(),F₃()))) by Lm1;
hence IC (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,k)) in dom (stop (for-down (F₄(),F₅(),F₆(),F₃()))) by A50, AFINSQ_1:66; :: thesis: verum

end;

end;

end;

suppose A52: k = (LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1 ; :: thesis: IC (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,k)) in dom (stop (for-down (F₄(),F₅(),F₆(),F₃())))

(card (F₃() ';' (AddTo (F₄(),F₅(),(- F₆()))))) + 1 in dom (stop (for-down (F₄(),F₅(),F₆(),F₃()))) by A26, COMPOS_1:62;
hence IC (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,k)) in dom (stop (for-down (F₄(),F₅(),F₆(),F₃()))) by A14, A30, A25, A35, A17, A34, A27, A52, SCMPDS_7:18; :: thesis: verum

end;

end;

end;

end;

suppose k >= ((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1) + 1 ; :: thesis: IC (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,b₁)) in dom (stop (for-down (F₄(),F₅(),F₆(),F₃())))

then consider nn being Nat such that
A53: k = (((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1) + 1) + nn by NAT_1:10;
reconsider nn = nn as Nat ;
Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,k) = Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),(Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,(((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1) + 1))),nn) by A53, EXTPRO_1:4
.= Comput (((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))) +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),(Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,(((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1) + 1))),nn) ;
hence IC (Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,k)) in dom (stop (for-down (F₄(),F₅(),F₆(),F₃()))) by A46, A41, SCMPDS_6:def 2; :: thesis: verum

end;

end;

end;

hence for-down (F₄(),F₅(),F₆(),F₃()) is_closed_on t,Q by A6, SCMPDS_6:def 2; :: thesis: for-down (F₄(),F₅(),F₆(),F₃()) is_halting_on t,Q
for-down (F₄(),F₅(),F₆(),F₃()) is_halting_on Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,(((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1) + 1)),Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃()))) by A5, A45, A42, A44, A13, A41;
then Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃()))) halts_on Comput ((Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃())))),t,(((LifeSpan ((Q +* (stop (F₃() ';' (AddTo (F₄(),F₅(),(- F₆())))))),t)) + 1) + 1)) by A43, A41, SCMPDS_6:def 3;
then Q +* (stop (for-down (F₄(),F₅(),F₆(),F₃()))) halts_on t by EXTPRO_1:22;
hence for-down (F₄(),F₅(),F₆(),F₃()) is_halting_on t,Q by A6, SCMPDS_6:def 3; :: thesis: verum

end;

end;

end;

hence S₁[k + 1] ; :: thesis: verum

end;

A54: S₁[ 0 ] by SCMPDS_7:44;
A55: for k being Nat holds S₁[k] from NAT_1:sch 2(A54, A4);

per cases ( F₁() . (DataLoc ((F₁() . F₄()),F₅())) <= 0 or F₁() . (DataLoc ((F₁() . F₄()),F₅())) > 0 ) ;

suppose F₁() . (DataLoc ((F₁() . F₄()),F₅())) <= 0 ; :: thesis: ( for-down (F₄(),F₅(),F₆(),F₃()) is_closed_on F₁(),F₂() & for-down (F₄(),F₅(),F₆(),F₃()) is_halting_on F₁(),F₂() )

hence ( for-down (F₄(),F₅(),F₆(),F₃()) is_closed_on F₁(),F₂() & for-down (F₄(),F₅(),F₆(),F₃()) is_halting_on F₁(),F₂() ) by SCMPDS_7:44; :: thesis: verum

end;

suppose F₁() . (DataLoc ((F₁() . F₄()),F₅())) > 0 ; :: thesis: ( for-down (F₄(),F₅(),F₆(),F₃()) is_closed_on F₁(),F₂() & for-down (F₄(),F₅(),F₆(),F₃()) is_halting_on F₁(),F₂() )

then reconsider m = F₁() . (DataLoc ((F₁() . F₄()),F₅())) as Element of NAT by INT_1:3;
S₁[m] by A55;
hence ( for-down (F₄(),F₅(),F₆(),F₃()) is_closed_on F₁(),F₂() & for-down (F₄(),F₅(),F₆(),F₃()) is_halting_on F₁(),F₂() ) by A2; :: thesis: verum

end;

end;