let I be InitHalting good Program of SCM+FSA; :: thesis: for a being read-write Int-Location st ex f being Function of (product the Object-Kind of SCM+FSA),INT st
for s, t being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA holds
( ( f . s > 0 implies f . (IExec (I,P,s)) < f . s ) & ( DataPart s = DataPart t implies f . s = f . t ) & ( f . s <= 0 implies s . a <= 0 ) & ( s . a <= 0 implies f . s <= 0 ) ) holds
while>0 (a,I) is InitHalting
let a be read-write Int-Location ; :: thesis: ( ex f being Function of (product the Object-Kind of SCM+FSA),INT st
for s, t being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA holds
( ( f . s > 0 implies f . (IExec (I,P,s)) < f . s ) & ( DataPart s = DataPart t implies f . s = f . t ) & ( f . s <= 0 implies s . a <= 0 ) & ( s . a <= 0 implies f . s <= 0 ) ) implies while>0 (a,I) is InitHalting )
set D = Data-Locations ;
given f being Function of (product the Object-Kind of SCM+FSA),INT such that A1: for s, t being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA holds
( ( f . s > 0 implies f . (IExec (I,P,s)) < f . s ) & ( DataPart s = DataPart t implies f . s = f . t ) & ( f . s <= 0 implies s . a <= 0 ) & ( s . a <= 0 implies f . s <= 0 ) ) ; :: thesis: while>0 (a,I) is InitHalting
defpred S₁[ Element of NAT ] means for t being State of SCM+FSA
for Q being Instruction-Sequence of SCM+FSA st f . t <= $1 holds
while>0 (a,I) is_halting_onInit t,Q;

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: for Q being Instruction-Sequence of SCM+FSA st f . t <= k + 1 holds
while>0 (a,I) is_halting_onInit b₂,b₃
let Q be Instruction-Sequence of SCM+FSA; :: thesis: ( f . t <= k + 1 implies while>0 (a,I) is_halting_onInit b₁,b₂ )
assume A4: f . t <= k + 1 ; :: thesis: while>0 (a,I) is_halting_onInit b₁,b₂

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

suppose f . t <> k + 1 ; :: thesis: while>0 (a,I) is_halting_onInit b₁,b₂

then f . t < k + 1 by A4, XXREAL_0:1;
hence while>0 (a,I) is_halting_onInit t,Q by A3, INT_1:7; :: thesis: verum

end;

suppose A5: f . t = k + 1 ; :: thesis: while>0 (a,I) is_halting_onInit b₁,b₂

set l0 = intloc 0;
set tW0 = Initialized t;
set QW = Q +* (while>0 (a,I));
set t1 = Initialized t;
set Q1 = Q +* I;
set Wt = Comput ((Q +* (while>0 (a,I))),(Initialized t),((LifeSpan ((Q +* I),(Initialized t))) + 3));
set It = Comput ((Q +* I),(Initialized t),(LifeSpan ((Q +* I),(Initialized t))));
A6: I is_closed_onInit t,Q by SCM_HALT:25;
A10: I is_halting_onInit t,Q by SCM_HALT:26;
then A11: Q +* I halts_on Initialized t by SCM_HALT:def 5;
A12: not t . a <= 0 by A1, A5;
then A13: DataPart (Comput ((Q +* (while>0 (a,I))),(Initialized t),((LifeSpan ((Q +* I),(Initialized t))) + 3))) = DataPart (Comput ((Q +* I),(Initialized t),(LifeSpan ((Q +* I),(Initialized t))))) by A6, A10, Th21;
then A14: (Comput ((Q +* (while>0 (a,I))),(Initialized t),((LifeSpan ((Q +* I),(Initialized t))) + 3))) . (intloc 0) = (Comput ((Q +* I),(Initialized t),(LifeSpan ((Q +* I),(Initialized t))))) . (intloc 0) by SCMFSA6A:7
.= (Result ((Q +* I),(Initialized t))) . (intloc 0) by A11, EXTPRO_1:23
.= (Result ((Q +* I),(Initialized t))) . (intloc 0)
.= (IExec (I,Q,t)) . (intloc 0) by SCMFSA6B:def 1
.= 1 by SCM_HALT:9 ;
DataPart (Comput ((Q +* (while>0 (a,I))),(Initialized t),((LifeSpan ((Q +* I),(Initialized t))) + 3))) = DataPart (Result ((Q +* I),(Initialized t))) by A13, A11, EXTPRO_1:23
.= DataPart (Result ((Q +* I),(Initialized t)))
.= DataPart (IExec (I,Q,t)) by SCMFSA6B:def 1 ;
then f . (Comput ((Q +* (while>0 (a,I))),(Initialized t),((LifeSpan ((Q +* I),(Initialized t))) + 3))) = f . (IExec (I,Q,t)) by A1;
then f . (Comput ((Q +* (while>0 (a,I))),(Initialized t),((LifeSpan ((Q +* I),(Initialized t))) + 3))) < k + 1 by A1, A5;
then while>0 (a,I) is_halting_onInit Comput ((Q +* (while>0 (a,I))),(Initialized t),((LifeSpan ((Q +* I),(Initialized t))) + 3)),Q +* (while>0 (a,I)) by A3, INT_1:7;
then A15: (Q +* (while>0 (a,I))) +* (while>0 (a,I)) halts_on Initialized (Comput ((Q +* (while>0 (a,I))),(Initialized t),((LifeSpan ((Q +* I),(Initialized t))) + 3))) by SCM_HALT:def 5;
IC (Comput ((Q +* (while>0 (a,I))),(Initialized t),((LifeSpan ((Q +* I),(Initialized t))) + 3))) = 0 by A12, A6, A10, Th21;
then A16: Initialized (Comput ((Q +* (while>0 (a,I))),(Initialized t),((LifeSpan ((Q +* I),(Initialized t))) + 3))) = Comput ((Q +* (while>0 (a,I))),(Initialized t),((LifeSpan ((Q +* I),(Initialized t))) + 3)) by A14, Th6;
A17: (Q +* (while>0 (a,I))) +* (while>0 (a,I)) = Q +* (while>0 (a,I)) by FUNCT_4:93;

now

consider m being Element of NAT such that
A18: CurInstr ((Q +* (while>0 (a,I))),(Comput ((Q +* (while>0 (a,I))),(Comput ((Q +* (while>0 (a,I))),(Initialized t),((LifeSpan ((Q +* I),(Initialized t))) + 3))),m))) = halt SCM+FSA by A16, A15, A17, EXTPRO_1:29;
take mm = ((LifeSpan ((Q +* I),(Initialized t))) + 3) + m; :: thesis: CurInstr ((Q +* (while>0 (a,I))),(Comput ((Q +* (while>0 (a,I))),(Initialized t),mm))) = halt SCM+FSA
thus CurInstr ((Q +* (while>0 (a,I))),(Comput ((Q +* (while>0 (a,I))),(Initialized t),mm))) = halt SCM+FSA by A18, EXTPRO_1:4; :: thesis: verum

end;

then Q +* (while>0 (a,I)) halts_on Initialized t by EXTPRO_1:29;
hence while>0 (a,I) is_halting_onInit t,Q by SCM_HALT:def 5; :: thesis: verum

end;

end;

end;

hence S₁[k + 1] ; :: thesis: verum

end;

A19: S₁[ 0 ]

proof

let t be State of SCM+FSA; :: thesis: for Q being Instruction-Sequence of SCM+FSA st f . t <= 0 holds
while>0 (a,I) is_halting_onInit t,Q
let Q be Instruction-Sequence of SCM+FSA; :: thesis: ( f . t <= 0 implies while>0 (a,I) is_halting_onInit t,Q )
assume f . t <= 0 ; :: thesis: while>0 (a,I) is_halting_onInit t,Q
then t . a <= 0 by A1;
hence while>0 (a,I) is_halting_onInit t,Q by Th16; :: thesis: verum

end;

A20: for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A19, A2);

now

let t be State of SCM+FSA; :: thesis: for Q being Instruction-Sequence of SCM+FSA holds while>0 (a,I) is_halting_onInit Q,b₂

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

suppose f . t <= 0 ; :: thesis: for Q being Instruction-Sequence of SCM+FSA holds while>0 (a,I) is_halting_onInit Q,b₂

then t . a <= 0 by A1;
hence for Q being Instruction-Sequence of SCM+FSA holds while>0 (a,I) is_halting_onInit t,Q by Th16; :: thesis: verum

end;

suppose f . t > 0 ; :: thesis: for Q being Instruction-Sequence of SCM+FSA holds while>0 (a,I) is_halting_onInit Q,b₂

then reconsider n = f . t as Element of NAT by INT_1:3;
S₁[n] by A20;
hence for Q being Instruction-Sequence of SCM+FSA holds while>0 (a,I) is_halting_onInit t,Q ; :: thesis: verum

end;

end;

end;

hence while>0 (a,I) is InitHalting by SCM_HALT:26; :: thesis: verum