let p be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
for a being read-write Int-Location
for I being Program of SCM+FSA st I is_closed_on s,p & I is_halting_on s,p & s . a = 0 holds
DataPart (Comput ((p +* (while=0 (a,I))),(Initialize s),((LifeSpan ((p +* I),(Initialize s))) + 3))) = DataPart (Comput ((p +* I),(Initialize s),(LifeSpan ((p +* I),(Initialize s)))))
let s be State of SCM+FSA; :: thesis: for a being read-write Int-Location
for I being Program of SCM+FSA st I is_closed_on s,p & I is_halting_on s,p & s . a = 0 holds
DataPart (Comput ((p +* (while=0 (a,I))),(Initialize s),((LifeSpan ((p +* I),(Initialize s))) + 3))) = DataPart (Comput ((p +* I),(Initialize s),(LifeSpan ((p +* I),(Initialize s)))))
let a be read-write Int-Location ; :: thesis: for I being Program of SCM+FSA st I is_closed_on s,p & I is_halting_on s,p & s . a = 0 holds
DataPart (Comput ((p +* (while=0 (a,I))),(Initialize s),((LifeSpan ((p +* I),(Initialize s))) + 3))) = DataPart (Comput ((p +* I),(Initialize s),(LifeSpan ((p +* I),(Initialize s)))))
let I be Program of SCM+FSA; :: thesis: ( I is_closed_on s,p & I is_halting_on s,p & s . a = 0 implies DataPart (Comput ((p +* (while=0 (a,I))),(Initialize s),((LifeSpan ((p +* I),(Initialize s))) + 3))) = DataPart (Comput ((p +* I),(Initialize s),(LifeSpan ((p +* I),(Initialize s))))) )
assume that
A1: ( I is_closed_on s,p & I is_halting_on s,p ) and
A2: s . a = 0 ; :: thesis: DataPart (Comput ((p +* (while=0 (a,I))),(Initialize s),((LifeSpan ((p +* I),(Initialize s))) + 3))) = DataPart (Comput ((p +* I),(Initialize s),(LifeSpan ((p +* I),(Initialize s)))))
set sI = Initialize s;
set pI = p +* I;
set s1 = Initialize s;
set p1 = p +* (while=0 (a,I));
A3: 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)) ) );

A4: now

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

now

A6: 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 A6, 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, A5, SCMFSA_9:19; :: thesis: verum

end;

hence S₁[k + 1] ; :: thesis: verum

end;

set i = a =0_goto 4;
set s2 = Comput ((p +* (while=0 (a,I))),(Initialize s),1);
set p2 = p +* (while=0 (a,I));
B7: IC in dom (Start-At (0,SCM+FSA)) by MEMSTR_0:15;
A8: IC (Initialize s) = IC (Start-At (0,SCM+FSA)) by B7, FUNCT_4:13
.= 0 by FUNCOP_1:72 ;
not a in dom (Start-At (0,SCM+FSA)) by SCMFSA_2:102;
then A9: (Initialize s) . a = s . a by FUNCT_4:11;
set loc4 = (card I) + 4;
A10: (p +* (while=0 (a,I))) /. (IC (Initialize s)) = (p +* (while=0 (a,I))) . (IC (Initialize s)) by PBOOLE:143;
0 in dom (while=0 (a,I)) by SCMFSA_9:10;
then (p +* (while=0 (a,I))) . 0 = (while=0 (a,I)) . 0 by FUNCT_4:13
.= a =0_goto 4 by SCMFSA_9:11 ;
then A11: CurInstr ((p +* (while=0 (a,I))),(Initialize s)) = a =0_goto 4 by A8, A10;
A12: 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 A11 ;
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 A13: DataPart (Comput ((p +* (while=0 (a,I))),(Initialize s),1)) = DataPart (Initialize s) by SCMFSA6A:7
.= DataPart (Initialize s) ;
A14: IC (Comput ((p +* (while=0 (a,I))),(Initialize s),1)) = 4 by A2, A12, A9, SCMFSA_2:70;
A15: 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)) )
B16: IC in dom (Start-At (0,SCM+FSA)) by MEMSTR_0:15;
IC (Comput ((p +* I),(Initialize s),0)) = IC (Initialize s) by EXTPRO_1:2
.= IC (Start-At (0,SCM+FSA)) by B16, 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 A14, A13, EXTPRO_1:2; :: thesis: verum

end;

for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A15, A4);
then A17: S₁[ LifeSpan ((p +* I),(Initialize s))] ;
set s4 = Comput ((p +* (while=0 (a,I))),(Initialize s),(((1 + (LifeSpan ((p +* I),(Initialize s)))) + 1) + 1));
set p4 = p +* (while=0 (a,I));
set s3 = Comput ((p +* (while=0 (a,I))),(Initialize s),((1 + (LifeSpan ((p +* I),(Initialize s)))) + 1));
set p3 = p +* (while=0 (a,I));
A18: (card I) + 4 in dom (while=0 (a,I)) by SCMFSA_9:13;
set s2 = Comput ((p +* (while=0 (a,I))),(Initialize s),(1 + (LifeSpan ((p +* I),(Initialize s)))));
A19: 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, A17, SCMFSA_9:20;
A20: 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 A19 ;
then A21: ( ( for c being Int-Location holds (Comput ((p +* (while=0 (a,I))),(Initialize s),((1 + (LifeSpan ((p +* I),(Initialize s)))) + 1))) . c = (Comput ((p +* (while=0 (a,I))),(Initialize s),(1 + (LifeSpan ((p +* I),(Initialize s)))))) . c ) & ( for f being FinSeq-Location holds (Comput ((p +* (while=0 (a,I))),(Initialize s),((1 + (LifeSpan ((p +* I),(Initialize s)))) + 1))) . f = (Comput ((p +* (while=0 (a,I))),(Initialize s),(1 + (LifeSpan ((p +* I),(Initialize s)))))) . f ) ) by SCMFSA_2:69;
A22: (p +* (while=0 (a,I))) . ((card I) + 4) = (while=0 (a,I)) . ((card I) + 4) by A18, A3, GRFUNC_1:2
.= goto 0 by SCMFSA_9:21 ;
A23: (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;
IC (Comput ((p +* (while=0 (a,I))),(Initialize s),((1 + (LifeSpan ((p +* I),(Initialize s)))) + 1))) = (card I) + 4 by A20, SCMFSA_2:69;
then A24: CurInstr ((p +* (while=0 (a,I))),(Comput ((p +* (while=0 (a,I))),(Initialize s),((1 + (LifeSpan ((p +* I),(Initialize s)))) + 1)))) = goto 0 by A22, A23;
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 A24 ;
then ( ( for c being Int-Location holds (Comput ((p +* (while=0 (a,I))),(Initialize s),(((1 + (LifeSpan ((p +* I),(Initialize s)))) + 1) + 1))) . c = (Comput ((p +* (while=0 (a,I))),(Initialize s),((1 + (LifeSpan ((p +* I),(Initialize s)))) + 1))) . c ) & ( for f being FinSeq-Location holds (Comput ((p +* (while=0 (a,I))),(Initialize s),(((1 + (LifeSpan ((p +* I),(Initialize s)))) + 1) + 1))) . f = (Comput ((p +* (while=0 (a,I))),(Initialize s),((1 + (LifeSpan ((p +* I),(Initialize s)))) + 1))) . f ) ) by SCMFSA_2:69;
hence DataPart (Comput ((p +* (while=0 (a,I))),(Initialize s),((LifeSpan ((p +* I),(Initialize s))) + 3))) = DataPart (Comput ((p +* (while=0 (a,I))),(Initialize s),((1 + (LifeSpan ((p +* I),(Initialize s)))) + 1))) by SCMFSA6A:7
.= DataPart (Comput ((p +* I),(Initialize s),(LifeSpan ((p +* I),(Initialize s))))) by A17, A21, SCMFSA6A:7 ;
:: thesis: verum