let I be InitHalting good Program of SCM+FSA; :: thesis: for a being read-write Int-Location st ( for s being State of SCM+FSA st s . a > 0 holds
(IExec (I,s)) . a < s . a ) holds
while>0 (a,I) is InitHalting
let a be read-write Int-Location ; :: thesis: ( ( for s being State of SCM+FSA st s . a > 0 holds
(IExec (I,s)) . a < s . a ) implies while>0 (a,I) is InitHalting )
set D = Int-Locations \/ FinSeq-Locations;
defpred S₁[ Element of NAT ] means for t being State of SCM+FSA st t . a <= $1 holds
while>0 (a,I) is_halting_onInit t;
assume A1: for s being State of SCM+FSA st s . a > 0 holds
(IExec (I,s)) . a < s . a ; :: thesis: while>0 (a,I) is InitHalting

A2: now

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: S₁[k] ; :: thesis: S₁[k + 1]

now

let t be State of SCM+FSA; :: thesis: ( t . a <= k + 1 implies while>0 (a,I) is_halting_onInit b₁ )
assume A4: t . a <= k + 1 ; :: thesis: while>0 (a,I) is_halting_onInit b₁

per cases ( t . a <> k + 1 or t . a = k + 1 ) ;

suppose t . a <> k + 1 ; :: thesis: while>0 (a,I) is_halting_onInit b₁

then t . a < k + 1 by A4, XXREAL_0:1;
hence while>0 (a,I) is_halting_onInit t by A3, INT_1:20; :: thesis: verum

end;

suppose A5: t . a = k + 1 ; :: thesis: while>0 (a,I) is_halting_onInit b₁

set l0 = intloc 0;
set Iwhile = Initialized (while>0 (a,I));
set tW0 = t +* (Initialized (while>0 (a,I)));
set t1 = t +* (Initialized I);
set Wt = Comput ((ProgramPart (t +* (Initialized (while>0 (a,I))))),(t +* (Initialized (while>0 (a,I)))),((LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) + 3));
set A = NAT ;
set It = Comput ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)),(LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))));
A6: I is_halting_onInit t by SCM_HALT:36;
then A7: ProgramPart (t +* (Initialized I)) halts_on t +* (Initialized I) by SCM_HALT:def 5;
not intloc 0 in NAT by SCMFSA_2:84;
then not intloc 0 in (dom t) /\ NAT by XBOOLE_0:def 4;
then A8: not intloc 0 in dom (t | NAT) by RELAT_1:90;
A9: I is_closed_onInit t by SCM_HALT:35;
then A10: DataPart (Comput ((ProgramPart (t +* (Initialized (while>0 (a,I))))),(t +* (Initialized (while>0 (a,I)))),((LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) + 3))) = DataPart (Comput ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)),(LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))))) by A5, A6, Th21;
then A11: (Comput ((ProgramPart (t +* (Initialized (while>0 (a,I))))),(t +* (Initialized (while>0 (a,I)))),((LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) + 3))) . (intloc 0) = (Comput ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)),(LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))))) . (intloc 0) by SCMFSA6A:38
.= (Result ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) . (intloc 0) by A7, EXTPRO_1:23
.= ((Result ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) +* (t | NAT)) . (intloc 0) by A8, FUNCT_4:12
.= (IExec (I,t)) . (intloc 0) by SCMFSA6B:def 1
.= 1 by SCM_HALT:17 ;
not a in NAT by SCMFSA_2:84;
then not a in (dom t) /\ NAT by XBOOLE_0:def 4;
then A12: not a in dom (t | NAT) by RELAT_1:90;
A13: while>0 (a,I) c= Initialized (while>0 (a,I)) by SCMFSA6A:26;
Initialized (while>0 (a,I)) c= t +* (Initialized (while>0 (a,I))) by FUNCT_4:26;
then while>0 (a,I) c= t +* (Initialized (while>0 (a,I))) by A13, XBOOLE_1:1;
then while>0 (a,I) c= Comput ((ProgramPart (t +* (Initialized (while>0 (a,I))))),(t +* (Initialized (while>0 (a,I)))),((LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) + 3)) by AMI_1:81;
then A14: (Comput ((ProgramPart (t +* (Initialized (while>0 (a,I))))),(t +* (Initialized (while>0 (a,I)))),((LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) + 3))) +* (while>0 (a,I)) = Comput ((ProgramPart (t +* (Initialized (while>0 (a,I))))),(t +* (Initialized (while>0 (a,I)))),((LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) + 3)) by FUNCT_4:79;
(Comput ((ProgramPart (t +* (Initialized (while>0 (a,I))))),(t +* (Initialized (while>0 (a,I)))),((LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) + 3))) . a = (Comput ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)),(LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))))) . a by A10, SCMFSA6A:38
.= (Result ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) . a by A7, EXTPRO_1:23
.= ((Result ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) +* (t | NAT)) . a by A12, FUNCT_4:12
.= (IExec (I,t)) . a by SCMFSA6B:def 1 ;
then (Comput ((ProgramPart (t +* (Initialized (while>0 (a,I))))),(t +* (Initialized (while>0 (a,I)))),((LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) + 3))) . a < k + 1 by A1, A5;
then while>0 (a,I) is_halting_onInit Comput ((ProgramPart (t +* (Initialized (while>0 (a,I))))),(t +* (Initialized (while>0 (a,I)))),((LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) + 3)) by A3, INT_1:20;
then A15: ProgramPart ((Comput ((ProgramPart (t +* (Initialized (while>0 (a,I))))),(t +* (Initialized (while>0 (a,I)))),((LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) + 3))) +* (Initialized (while>0 (a,I)))) halts_on (Comput ((ProgramPart (t +* (Initialized (while>0 (a,I))))),(t +* (Initialized (while>0 (a,I)))),((LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) + 3))) +* (Initialized (while>0 (a,I))) by SCM_HALT:def 5;
IC (Comput ((ProgramPart (t +* (Initialized (while>0 (a,I))))),(t +* (Initialized (while>0 (a,I)))),((LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) + 3))) = 0 by A5, A9, A6, Th21;
then A16: (Comput ((ProgramPart (t +* (Initialized (while>0 (a,I))))),(t +* (Initialized (while>0 (a,I)))),((LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) + 3))) +* (Initialized (while>0 (a,I))) = Comput ((ProgramPart (t +* (Initialized (while>0 (a,I))))),(t +* (Initialized (while>0 (a,I)))),((LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) + 3)) by A11, A14, Th6;

now

consider m being Element of NAT such that
A17: CurInstr ((ProgramPart (Comput ((ProgramPart (t +* (Initialized (while>0 (a,I))))),(t +* (Initialized (while>0 (a,I)))),((LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) + 3)))),(Comput ((ProgramPart (Comput ((ProgramPart (t +* (Initialized (while>0 (a,I))))),(t +* (Initialized (while>0 (a,I)))),((LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) + 3)))),(Comput ((ProgramPart (t +* (Initialized (while>0 (a,I))))),(t +* (Initialized (while>0 (a,I)))),((LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) + 3))),m))) = halt SCM+FSA by A16, A15, EXTPRO_1:30;
take mm = ((LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) + 3) + m; :: thesis: CurInstr ((ProgramPart (t +* (Initialized (while>0 (a,I))))),(Comput ((ProgramPart (t +* (Initialized (while>0 (a,I))))),(t +* (Initialized (while>0 (a,I)))),mm))) = halt SCM+FSA
T: ProgramPart (t +* (Initialized (while>0 (a,I)))) = ProgramPart (Comput ((ProgramPart (t +* (Initialized (while>0 (a,I))))),(t +* (Initialized (while>0 (a,I)))),((LifeSpan ((ProgramPart (t +* (Initialized I))),(t +* (Initialized I)))) + 3))) by AMI_1:123;
thus CurInstr ((ProgramPart (t +* (Initialized (while>0 (a,I))))),(Comput ((ProgramPart (t +* (Initialized (while>0 (a,I))))),(t +* (Initialized (while>0 (a,I)))),mm))) = halt SCM+FSA by A17, T, EXTPRO_1:5; :: thesis: verum

end;

then ProgramPart (t +* (Initialized (while>0 (a,I)))) halts_on t +* (Initialized (while>0 (a,I))) by EXTPRO_1:30;
hence while>0 (a,I) is_halting_onInit t by SCM_HALT:def 5; :: thesis: verum

end;

end;

end;

hence S₁[k + 1] ; :: thesis: verum

end;

A18: S₁[ 0 ] by Th16;
A19: for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A18, A2);

now

let t be State of SCM+FSA; :: thesis: while>0 (a,I) is_halting_onInit b₁

per cases ( t . a <= 0 or t . a > 0 ) ;

suppose t . a <= 0 ; :: thesis: while>0 (a,I) is_halting_onInit b₁

hence while>0 (a,I) is_halting_onInit t by Th16; :: thesis: verum

end;

suppose t . a > 0 ; :: thesis: while>0 (a,I) is_halting_onInit b₁

then reconsider n = t . a as Element of NAT by INT_1:16;
S₁[n] by A19;
hence while>0 (a,I) is_halting_onInit t ; :: thesis: verum

end;

end;

end;

hence while>0 (a,I) is InitHalting by SCM_HALT:36; :: thesis: verum