let P be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
for I being really-closed MacroInstruction of SCM+FSA
for a being read-write Int-Location st I is_halting_on s,P & s . a = 0 holds
( IC (Comput ((P +* (while=0 (a,I))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 2))) = 0 & ( for k being Nat st k <= (LifeSpan ((P +* I),(Initialize s))) + 2 holds
IC (Comput ((P +* (while=0 (a,I))),(Initialize s),k)) in dom (while=0 (a,I)) ) )
set D = Int-Locations \/ FinSeq-Locations;
let s be State of SCM+FSA; :: thesis: for I being really-closed MacroInstruction of SCM+FSA
for a being read-write Int-Location st I is_halting_on s,P & s . a = 0 holds
( IC (Comput ((P +* (while=0 (a,I))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 2))) = 0 & ( for k being Nat st k <= (LifeSpan ((P +* I),(Initialize s))) + 2 holds
IC (Comput ((P +* (while=0 (a,I))),(Initialize s),k)) in dom (while=0 (a,I)) ) )
let I be really-closed MacroInstruction of SCM+FSA ; :: thesis: for a being read-write Int-Location st I is_halting_on s,P & s . a = 0 holds
( IC (Comput ((P +* (while=0 (a,I))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 2))) = 0 & ( for k being Nat st k <= (LifeSpan ((P +* I),(Initialize s))) + 2 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_halting_on s,P & s . a = 0 implies ( IC (Comput ((P +* (while=0 (a,I))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 2))) = 0 & ( for k being Nat st k <= (LifeSpan ((P +* I),(Initialize s))) + 2 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));
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))) + 3 & DataPart (Comput ((P +* (while=0 (a,I))),(Initialize s),(1 + $1))) = DataPart (Comput ((P +* I),(Initialize s),$1)) ) );
assume A1: 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))) + 2))) = 0 & ( for k being Nat st k <= (LifeSpan ((P +* I),(Initialize s))) + 2 holds
IC (Comput ((P +* (while=0 (a,I))),(Initialize s),k)) in dom (while=0 (a,I)) ) ) )

A2: now :: thesis: for k being Nat st S₁[k] holds
S₁[k + 1]

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

now :: thesis: ( k + 1 <= LifeSpan ((P +* I),(Initialize s)) implies ( IC (Comput ((P +* (while=0 (a,I))),(Initialize s),((1 + k) + 1))) = (IC (Comput ((P +* I),(Initialize s),(k + 1)))) + 3 & DataPart (Comput ((P +* (while=0 (a,I))),(Initialize s),((1 + k) + 1))) = DataPart (Comput ((P +* I),(Initialize s),(k + 1))) ) )

A4: 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)))) + 3 & 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 A4, XXREAL_0:2;
hence ( IC (Comput ((P +* (while=0 (a,I))),(Initialize s),((1 + k) + 1))) = (IC (Comput ((P +* I),(Initialize s),(k + 1)))) + 3 & DataPart (Comput ((P +* (while=0 (a,I))),(Initialize s),((1 + k) + 1))) = DataPart (Comput ((P +* I),(Initialize s),(k + 1))) ) by A1, A3, Th2; :: thesis: verum

end;

hence S₁[k + 1] ; :: thesis: verum

end;

reconsider l = LifeSpan ((P +* I),(Initialize s)) as Element of NAT by ORDINAL1:def 12;
set loc4 = (card I) + 3;
set i = a =0_goto 3;
set s2 = Comput ((P +* (while=0 (a,I))),(Initialize s),1);
IC in dom (Start-At (0,SCM+FSA)) by MEMSTR_0:15;
then A5: IC (Initialize s) = IC (Start-At (0,SCM+FSA)) by FUNCT_4:13
.= 0 by FUNCOP_1:72 ;
not a in dom (Start-At (0,SCM+FSA)) by SCMFSA_2:102;
then A6: (Initialize s) . a = s . a by FUNCT_4:11;
assume A7: s . a = 0 ; :: thesis: ( IC (Comput ((P +* (while=0 (a,I))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 2))) = 0 & ( for k being Nat st k <= (LifeSpan ((P +* I),(Initialize s))) + 2 holds
IC (Comput ((P +* (while=0 (a,I))),(Initialize s),k)) in dom (while=0 (a,I)) ) )
A8: 0 in dom (while=0 (a,I)) by AFINSQ_1:65;
A9: (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 A8, FUNCT_4:13
.= a =0_goto 3 by SCMFSA_X:10 ;
then A10: CurInstr ((P +* (while=0 (a,I))),(Initialize s)) = a =0_goto 3 by A5, A9;
A11: 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
.= Exec ((a =0_goto 3),(Initialize s)) by A10 ;
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_2:70;
then A12: DataPart (Comput ((P +* (while=0 (a,I))),(Initialize s),1)) = DataPart (Initialize s) by SCMFSA_M:2;
A13: IC (Comput ((P +* (while=0 (a,I))),(Initialize s),1)) = 3 by A7, A11, A6, SCMFSA_2:70;
A14: 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))) + 3 & DataPart (Comput ((P +* (while=0 (a,I))),(Initialize s),(1 + 0))) = DataPart (Comput ((P +* I),(Initialize s),0)) )
A15: IC in dom (Start-At (0,SCM+FSA)) by MEMSTR_0:15;
IC (Comput ((P +* I),(Initialize s),0)) = IC (Start-At (0,SCM+FSA)) by A15, 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))) + 3 & DataPart (Comput ((P +* (while=0 (a,I))),(Initialize s),(1 + 0))) = DataPart (Comput ((P +* I),(Initialize s),0)) ) by A13, A12; :: thesis: verum

end;

A16: for k being Nat holds S₁[k] from NAT_1:sch 2(A14, A2);
set s4 = Comput ((P +* (while=0 (a,I))),(Initialize s),((1 + (LifeSpan ((P +* I),(Initialize s)))) + 1));
set s3 = Comput ((P +* (while=0 (a,I))),(Initialize s),(1 + (LifeSpan ((P +* I),(Initialize s)))));
set s2 = Comput ((P +* (while=0 (a,I))),(Initialize s),(1 + (LifeSpan ((P +* I),(Initialize s)))));
S₁[l] by A16;
then A17: CurInstr ((P +* (while=0 (a,I))),(Comput ((P +* (while=0 (a,I))),(Initialize s),(1 + (LifeSpan ((P +* I),(Initialize s))))))) = goto 0 by A1, Th3;
A18: CurInstr ((P +* (while=0 (a,I))),(Comput ((P +* (while=0 (a,I))),(Initialize s),(1 + (LifeSpan ((P +* I),(Initialize s))))))) = goto 0 by A17;
A19: 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 0),(Comput ((P +* (while=0 (a,I))),(Initialize s),(1 + (LifeSpan ((P +* I),(Initialize s))))))) by A18 ;
A20: IC (Comput ((P +* (while=0 (a,I))),(Initialize s),((1 + (LifeSpan ((P +* I),(Initialize s)))) + 1))) = 0 by A19, SCMFSA_2:69;
hence IC (Comput ((P +* (while=0 (a,I))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 2))) = 0 ; :: thesis: for k being Nat st k <= (LifeSpan ((P +* I),(Initialize s))) + 2 holds
IC (Comput ((P +* (while=0 (a,I))),(Initialize s),k)) in dom (while=0 (a,I))

A21: now :: thesis: for k being Nat st k <= (LifeSpan ((P +* I),(Initialize s))) + 2 & k <> 0 holds
IC (Comput ((P +* (while=0 (a,I))),(Initialize s),k)) in dom (while=0 (a,I))

let k be Nat; :: thesis: ( k <= (LifeSpan ((P +* I),(Initialize s))) + 2 & k <> 0 implies IC (Comput ((P +* (while=0 (a,I))),(Initialize s),b₁)) in dom (while=0 (a,I)) )
assume A22: k <= (LifeSpan ((P +* I),(Initialize s))) + 2 ; :: 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
A23: 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 A24: ( k <= (LifeSpan ((P +* I),(Initialize s))) + 1 or k = (LifeSpan ((P +* I),(Initialize s))) + 2 ) by A22, 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))) + 2 ) by A24;

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 A23, XREAL_1:6;
then A25: IC (Comput ((P +* (while=0 (a,I))),(Initialize s),(1 + n))) = (IC (Comput ((P +* I),(Initialize s),n))) + 3 by A16;
reconsider m = IC (Comput ((P +* I),(Initialize s),n)) as Element of NAT ;
A26: I c= P +* I by FUNCT_4:25;
IC (Initialize s) = 0 by MEMSTR_0:def 11;
then IC (Initialize s) in dom I by AFINSQ_1:65;
then m in dom I by AMISTD_1:21, A26;
then m < card I by AFINSQ_1:66;
then A27: m + 3 < (card I) + 5 by XREAL_1:8;
card (while=0 (a,I)) = (card I) + 5 by SCMFSA_X:3;
hence IC (Comput ((P +* (while=0 (a,I))),(Initialize s),k)) in dom (while=0 (a,I)) by A23, A25, A27, AFINSQ_1:66; :: thesis: verum

end;

suppose k >= (LifeSpan ((P +* I),(Initialize s))) + 2 ; :: thesis: IC (Comput ((P +* (while=0 (a,I))),(Initialize s),b₁)) in dom (while=0 (a,I))

then k = (LifeSpan ((P +* I),(Initialize s))) + 2 by A22, XXREAL_0:1;
hence IC (Comput ((P +* (while=0 (a,I))),(Initialize s),k)) in dom (while=0 (a,I)) by A20, AFINSQ_1:65; :: thesis: verum

end;

end;

end;

now :: thesis: for k being Nat st k <= (LifeSpan ((P +* I),(Initialize s))) + 2 holds
IC (Comput ((P +* (while=0 (a,I))),(Initialize s),k)) in dom (while=0 (a,I))

let k be Nat; :: thesis: ( k <= (LifeSpan ((P +* I),(Initialize s))) + 2 implies IC (Comput ((P +* (while=0 (a,I))),(Initialize s),b₁)) in dom (while=0 (a,I)) )
assume A28: k <= (LifeSpan ((P +* I),(Initialize s))) + 2 ; :: 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 A8, A5; :: 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 A21, A28; :: thesis: verum

end;

end;

end;

hence for k being Nat st k <= (LifeSpan ((P +* I),(Initialize s))) + 2 holds
IC (Comput ((P +* (while=0 (a,I))),(Initialize s),k)) in dom (while=0 (a,I)) ; :: thesis: verum