let P be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
for I being Program of
for a being read-write Int-Location st I is_closed_on s,P & I is_halting_on s,P & s . a > 0 holds
( IC (Comput ((P +* (while>0 (a,I))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 3))) = 0 & ( for k being Element of NAT st k <= (LifeSpan ((P +* I),(Initialize s))) + 3 holds
IC (Comput ((P +* (while>0 (a,I))),(Initialize s),k)) in dom (while>0 (a,I)) ) )
let s be State of SCM+FSA; :: thesis: for I being Program of
for a being read-write Int-Location st I is_closed_on s,P & I is_halting_on s,P & s . a > 0 holds
( IC (Comput ((P +* (while>0 (a,I))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 3))) = 0 & ( for k being Element of NAT st k <= (LifeSpan ((P +* I),(Initialize s))) + 3 holds
IC (Comput ((P +* (while>0 (a,I))),(Initialize s),k)) in dom (while>0 (a,I)) ) )
set D = Int-Locations \/ FinSeq-Locations;
let I be Program of ; :: thesis: for a being read-write Int-Location st I is_closed_on s,P & I is_halting_on s,P & s . a > 0 holds
( IC (Comput ((P +* (while>0 (a,I))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 3))) = 0 & ( for k being Element of NAT st k <= (LifeSpan ((P +* I),(Initialize s))) + 3 holds
IC (Comput ((P +* (while>0 (a,I))),(Initialize s),k)) in dom (while>0 (a,I)) ) )
let a be read-write Int-Location ; :: thesis: ( I is_closed_on s,P & I is_halting_on s,P & s . a > 0 implies ( IC (Comput ((P +* (while>0 (a,I))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 3))) = 0 & ( for k being Element of NAT st k <= (LifeSpan ((P +* I),(Initialize s))) + 3 holds
IC (Comput ((P +* (while>0 (a,I))),(Initialize s),k)) in dom (while>0 (a,I)) ) ) )
assume A1: I is_closed_on s,P ; :: thesis: ( not I is_halting_on s,P or not s . a > 0 or ( IC (Comput ((P +* (while>0 (a,I))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 3))) = 0 & ( for k being Element of NAT st k <= (LifeSpan ((P +* I),(Initialize s))) + 3 holds
IC (Comput ((P +* (while>0 (a,I))),(Initialize s),k)) in dom (while>0 (a,I)) ) ) )
set sI = Initialize s;
set PI = P +* I;
set s1 = Initialize s;
set P1 = P +* (while>0 (a,I));
A2: while>0 (a,I) c= P +* (while>0 (a,I)) by FUNCT_4:25;
defpred S₁[ Nat] means ( $1 <= LifeSpan ((P +* I),(Initialize s)) implies ( IC (Comput ((P +* (while>0 (a,I))),(Initialize s),(1 + $1))) = (IC (Comput ((P +* I),(Initialize s),$1))) + 4 & DataPart (Comput ((P +* (while>0 (a,I))),(Initialize s),(1 + $1))) = DataPart (Comput ((P +* I),(Initialize s),$1)) ) );
assume A4: I is_halting_on s,P ; :: thesis: ( not s . a > 0 or ( IC (Comput ((P +* (while>0 (a,I))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 3))) = 0 & ( for k being Element of NAT st k <= (LifeSpan ((P +* I),(Initialize s))) + 3 holds
IC (Comput ((P +* (while>0 (a,I))),(Initialize s),k)) in dom (while>0 (a,I)) ) ) )

A5: now

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A6: S₁[k] ; :: thesis: S₁[k + 1]

now

A7: k + 0 < k + 1 by XREAL_1:6;
assume k + 1 <= LifeSpan ((P +* I),(Initialize s)) ; :: thesis: ( IC (Comput ((P +* (while>0 (a,I))),(Initialize s),((1 + k) + 1))) = (IC (Comput ((P +* I),(Initialize s),(k + 1)))) + 4 & DataPart (Comput ((P +* (while>0 (a,I))),(Initialize s),((1 + k) + 1))) = DataPart (Comput ((P +* I),(Initialize s),(k + 1))) )
then k < LifeSpan ((P +* I),(Initialize s)) by A7, XXREAL_0:2;
hence ( IC (Comput ((P +* (while>0 (a,I))),(Initialize s),((1 + k) + 1))) = (IC (Comput ((P +* I),(Initialize s),(k + 1)))) + 4 & DataPart (Comput ((P +* (while>0 (a,I))),(Initialize s),((1 + k) + 1))) = DataPart (Comput ((P +* I),(Initialize s),(k + 1))) ) by A1, A4, A6, Th44; :: thesis: verum

end;

hence S₁[k + 1] ; :: thesis: verum

end;

reconsider l = LifeSpan ((P +* I),(Initialize s)) as Element of NAT ;
set loc4 = (card I) + 4;
set i = a >0_goto 4;
set s2 = Comput ((P +* (while>0 (a,I))),(Initialize s),1);
IC in dom (Start-At (0,SCM+FSA)) by MEMSTR_0:15;
then A8: IC in dom (Start-At (0,SCM+FSA)) ;
A9: IC (Initialize s) = IC (Start-At (0,SCM+FSA)) by A8, FUNCT_4:13
.= 0 by FUNCOP_1:72 ;
not a in dom (Start-At (0,SCM+FSA)) by SCMFSA_2:102;
then A10: (Initialize s) . a = s . a by FUNCT_4:11;
assume A11: s . a > 0 ; :: thesis: ( IC (Comput ((P +* (while>0 (a,I))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 3))) = 0 & ( for k being Element of NAT st k <= (LifeSpan ((P +* I),(Initialize s))) + 3 holds
IC (Comput ((P +* (while>0 (a,I))),(Initialize s),k)) in dom (while>0 (a,I)) ) )
A12: 0 in dom (while>0 (a,I)) by Th10;
A13: (P +* (while>0 (a,I))) /. (IC (Initialize s)) = (P +* (while>0 (a,I))) . (IC (Initialize s)) by PBOOLE:143;
(P +* (while>0 (a,I))) . 0 = (while>0 (a,I)) . 0 by A12, FUNCT_4:13
.= a >0_goto 4 by Th11 ;
then A14: CurInstr ((P +* (while>0 (a,I))),(Initialize s)) = a >0_goto 4 by A9, A13;
A15: Comput ((P +* (while>0 (a,I))),(Initialize s),(0 + 1)) = Following ((P +* (while>0 (a,I))),(Comput ((P +* (while>0 (a,I))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (while>0 (a,I))),(Initialize s)) by EXTPRO_1:2
.= Exec ((a >0_goto 4),(Initialize s)) by A14 ;
then ( ( for c being Int-Location holds (Comput ((P +* (while>0 (a,I))),(Initialize s),1)) . c = (Initialize s) . c ) & ( for f being FinSeq-Location holds (Comput ((P +* (while>0 (a,I))),(Initialize s),1)) . f = (Initialize s) . f ) ) by SCMFSA_2:71;
then A16: DataPart (Comput ((P +* (while>0 (a,I))),(Initialize s),1)) = DataPart (Initialize s) by SCMFSA6A:7
.= DataPart (Initialize s) ;
A17: IC (Comput ((P +* (while>0 (a,I))),(Initialize s),1)) = 4 by A11, A15, A10, SCMFSA_2:71;
A18: S₁[ 0 ]

proof

assume 0 <= LifeSpan ((P +* I),(Initialize s)) ; :: thesis: ( IC (Comput ((P +* (while>0 (a,I))),(Initialize s),(1 + 0))) = (IC (Comput ((P +* I),(Initialize s),0))) + 4 & DataPart (Comput ((P +* (while>0 (a,I))),(Initialize s),(1 + 0))) = DataPart (Comput ((P +* I),(Initialize s),0)) )
IC in dom (Start-At (0,SCM+FSA)) by MEMSTR_0:15;
then A19: IC in dom (Start-At (0,SCM+FSA)) ;
IC (Comput ((P +* I),(Initialize s),0)) = IC (Initialize s) by EXTPRO_1:2
.= IC (Start-At (0,SCM+FSA)) by A19, FUNCT_4:13
.= 0 by FUNCOP_1:72 ;
hence ( IC (Comput ((P +* (while>0 (a,I))),(Initialize s),(1 + 0))) = (IC (Comput ((P +* I),(Initialize s),0))) + 4 & DataPart (Comput ((P +* (while>0 (a,I))),(Initialize s),(1 + 0))) = DataPart (Comput ((P +* I),(Initialize s),0)) ) by A17, A16, EXTPRO_1:2; :: thesis: verum

end;

A20: for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A18, A5);
set s4 = Comput ((P +* (while>0 (a,I))),(Initialize s),(((1 + (LifeSpan ((P +* I),(Initialize s)))) + 1) + 1));
set s3 = Comput ((P +* (while>0 (a,I))),(Initialize s),((1 + (LifeSpan ((P +* I),(Initialize s)))) + 1));
A21: (card I) + 4 in dom (while>0 (a,I)) by Th38;
set s2 = Comput ((P +* (while>0 (a,I))),(Initialize s),(1 + (LifeSpan ((P +* I),(Initialize s)))));
S₁[l] by A20;
then A22: CurInstr ((P +* (while>0 (a,I))),(Comput ((P +* (while>0 (a,I))),(Initialize s),(1 + (LifeSpan ((P +* I),(Initialize s))))))) = goto ((card I) + 4) by A1, A4, Th45;
A23: Comput ((P +* (while>0 (a,I))),(Initialize s),((1 + (LifeSpan ((P +* I),(Initialize s)))) + 1)) = Following ((P +* (while>0 (a,I))),(Comput ((P +* (while>0 (a,I))),(Initialize s),(1 + (LifeSpan ((P +* I),(Initialize s))))))) by EXTPRO_1:3
.= Exec ((goto ((card I) + 4)),(Comput ((P +* (while>0 (a,I))),(Initialize s),(1 + (LifeSpan ((P +* I),(Initialize s))))))) by A22 ;
A24: IC (Comput ((P +* (while>0 (a,I))),(Initialize s),((1 + (LifeSpan ((P +* I),(Initialize s)))) + 1))) = (card I) + 4 by A23, SCMFSA_2:69;
A25: (P +* (while>0 (a,I))) /. (IC (Comput ((P +* (while>0 (a,I))),(Initialize s),((1 + (LifeSpan ((P +* I),(Initialize s)))) + 1)))) = (P +* (while>0 (a,I))) . (IC (Comput ((P +* (while>0 (a,I))),(Initialize s),((1 + (LifeSpan ((P +* I),(Initialize s)))) + 1)))) by PBOOLE:143;
(P +* (while>0 (a,I))) . ((card I) + 4) = (while>0 (a,I)) . ((card I) + 4) by A21, A2, GRFUNC_1:2
.= goto 0 by Th46 ;
then A26: CurInstr ((P +* (while>0 (a,I))),(Comput ((P +* (while>0 (a,I))),(Initialize s),((1 + (LifeSpan ((P +* I),(Initialize s)))) + 1)))) = goto 0 by A24, A25;
A27: Comput ((P +* (while>0 (a,I))),(Initialize s),(((1 + (LifeSpan ((P +* I),(Initialize s)))) + 1) + 1)) = Following ((P +* (while>0 (a,I))),(Comput ((P +* (while>0 (a,I))),(Initialize s),((1 + (LifeSpan ((P +* I),(Initialize s)))) + 1)))) by EXTPRO_1:3
.= Exec ((goto 0),(Comput ((P +* (while>0 (a,I))),(Initialize s),((1 + (LifeSpan ((P +* I),(Initialize s)))) + 1)))) by A26 ;
A28: IC (Comput ((P +* (while>0 (a,I))),(Initialize s),(((1 + (LifeSpan ((P +* I),(Initialize s)))) + 1) + 1))) = 0 by A27, SCMFSA_2:69;
hence IC (Comput ((P +* (while>0 (a,I))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 3))) = 0 ; :: thesis: for k being Element of NAT st k <= (LifeSpan ((P +* I),(Initialize s))) + 3 holds
IC (Comput ((P +* (while>0 (a,I))),(Initialize s),k)) in dom (while>0 (a,I))
A29: (((LifeSpan ((P +* I),(Initialize s))) + 1) + 1) + 1 = (LifeSpan ((P +* I),(Initialize s))) + (2 + 1) ;

A30: now

let k be Element of NAT ; :: thesis: ( k <= (LifeSpan ((P +* I),(Initialize s))) + 3 & k <> 0 implies IC (Comput ((P +* (while>0 (a,I))),(Initialize s),b₁)) in dom (while>0 (a,I)) )
assume A31: k <= (LifeSpan ((P +* I),(Initialize s))) + 3 ; :: thesis: ( k <> 0 implies IC (Comput ((P +* (while>0 (a,I))),(Initialize s),b₁)) in dom (while>0 (a,I)) )
assume k <> 0 ; :: thesis: IC (Comput ((P +* (while>0 (a,I))),(Initialize s),b₁)) in dom (while>0 (a,I))
then consider n being Nat such that
A32: k = n + 1 by NAT_1:6;
( k <= (LifeSpan ((P +* I),(Initialize s))) + 1 or k >= ((LifeSpan ((P +* I),(Initialize s))) + 1) + 1 ) by NAT_1:13;
then A33: ( k <= (LifeSpan ((P +* I),(Initialize s))) + 1 or k = ((LifeSpan ((P +* I),(Initialize s))) + 1) + 1 or k > ((LifeSpan ((P +* I),(Initialize s))) + 1) + 1 ) by XXREAL_0:1;
reconsider n = n as Element of NAT by ORDINAL1:def 12;

per cases ( k <= (LifeSpan ((P +* I),(Initialize s))) + 1 or k = ((LifeSpan ((P +* I),(Initialize s))) + 1) + 1 or k >= (LifeSpan ((P +* I),(Initialize s))) + 3 ) by A29, A33, NAT_1:13;

suppose k <= (LifeSpan ((P +* I),(Initialize s))) + 1 ; :: thesis: IC (Comput ((P +* (while>0 (a,I))),(Initialize s),b₁)) in dom (while>0 (a,I))

then n <= LifeSpan ((P +* I),(Initialize s)) by A32, XREAL_1:6;
then A34: IC (Comput ((P +* (while>0 (a,I))),(Initialize s),(1 + n))) = (IC (Comput ((P +* I),(Initialize s),n))) + 4 by A20;
reconsider m = IC (Comput ((P +* I),(Initialize s),n)) as Element of NAT ;
m in dom I by A1, SCMFSA7B:def 6;
then m < card I by AFINSQ_1:66;
then A35: m + 4 < (card I) + 6 by XREAL_1:8;
card (while>0 (a,I)) = (card I) + 6 by Th5;
hence IC (Comput ((P +* (while>0 (a,I))),(Initialize s),k)) in dom (while>0 (a,I)) by A32, A34, A35, AFINSQ_1:66; :: thesis: verum

end;

suppose k = ((LifeSpan ((P +* I),(Initialize s))) + 1) + 1 ; :: thesis: IC (Comput ((P +* (while>0 (a,I))),(Initialize s),b₁)) in dom (while>0 (a,I))

hence IC (Comput ((P +* (while>0 (a,I))),(Initialize s),k)) in dom (while>0 (a,I)) by A24, Th38; :: thesis: verum

end;

suppose k >= (LifeSpan ((P +* I),(Initialize s))) + 3 ; :: thesis: IC (Comput ((P +* (while>0 (a,I))),(Initialize s),b₁)) in dom (while>0 (a,I))

then k = (LifeSpan ((P +* I),(Initialize s))) + 3 by A31, XXREAL_0:1;
hence IC (Comput ((P +* (while>0 (a,I))),(Initialize s),k)) in dom (while>0 (a,I)) by A28, Th10; :: thesis: verum

end;

end;

end;

now

let k be Element of NAT ; :: thesis: ( k <= (LifeSpan ((P +* I),(Initialize s))) + 3 implies IC (Comput ((P +* (while>0 (a,I))),(Initialize s),b₁)) in dom (while>0 (a,I)) )
assume A36: k <= (LifeSpan ((P +* I),(Initialize s))) + 3 ; :: thesis: IC (Comput ((P +* (while>0 (a,I))),(Initialize s),b₁)) in dom (while>0 (a,I))

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

suppose k = 0 ; :: thesis: IC (Comput ((P +* (while>0 (a,I))),(Initialize s),b₁)) in dom (while>0 (a,I))

hence IC (Comput ((P +* (while>0 (a,I))),(Initialize s),k)) in dom (while>0 (a,I)) by A12, A9, EXTPRO_1:2; :: thesis: verum

end;

suppose k <> 0 ; :: thesis: IC (Comput ((P +* (while>0 (a,I))),(Initialize s),b₁)) in dom (while>0 (a,I))

hence IC (Comput ((P +* (while>0 (a,I))),(Initialize s),k)) in dom (while>0 (a,I)) by A30, A36; :: thesis: verum

end;

end;

end;

hence for k being Element of NAT st k <= (LifeSpan ((P +* I),(Initialize s))) + 3 holds
IC (Comput ((P +* (while>0 (a,I))),(Initialize s),k)) in dom (while>0 (a,I)) ; :: thesis: verum