let s be State of SCM+FSA; :: thesis: for p being Instruction-Sequence of SCM+FSA
for a being read-write Int-Location
for J being good really-closed MacroInstruction of SCM+FSA st not J destroys a & ProperTimesBody a,J,s,p holds
for k being Nat st k <= s . a holds
(((StepTimes (a,J,p,s)) . k) . a) + k = s . a
let p be Instruction-Sequence of SCM+FSA; :: thesis: for a being read-write Int-Location
for J being good really-closed MacroInstruction of SCM+FSA st not J destroys a & ProperTimesBody a,J,s,p holds
for k being Nat st k <= s . a holds
(((StepTimes (a,J,p,s)) . k) . a) + k = s . a
let a be read-write Int-Location; :: thesis: for J being good really-closed MacroInstruction of SCM+FSA st not J destroys a & ProperTimesBody a,J,s,p holds
for k being Nat st k <= s . a holds
(((StepTimes (a,J,p,s)) . k) . a) + k = s . a
let J be good really-closed MacroInstruction of SCM+FSA ; :: thesis: ( not J destroys a & ProperTimesBody a,J,s,p implies for k being Nat st k <= s . a holds
(((StepTimes (a,J,p,s)) . k) . a) + k = s . a )
set I = J;
assume that
A1: not J destroys a and
A2: ProperTimesBody a,J,s,p ; :: thesis: for k being Nat st k <= s . a holds
(((StepTimes (a,J,p,s)) . k) . a) + k = s . a
set Is = Initialized s;
set ST = StepTimes (a,J,p,s);
set SW = StepWhile>0 (a,(J ";" (SubFrom (a,(intloc 0)))),p,(Initialized s));
defpred S₁[ Nat] means ( $1 <= s . a implies (((StepTimes (a,J,p,s)) . $1) . a) + $1 = s . a );
A3: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume that
A4: ( k <= s . a implies (((StepTimes (a,J,p,s)) . k) . a) + k = s . a ) and
A5: k + 1 <= s . a ; :: thesis: (((StepTimes (a,J,p,s)) . (k + 1)) . a) + (k + 1) = s . a
reconsider sa = s . a as Element of NAT by A5, INT_1:3;
A6: k < sa by A5, NAT_1:13;
then A7: ((StepTimes (a,J,p,s)) . k) . (intloc 0) = 1 by A1, A2, Th51;

A8: now :: thesis: not ((StepWhile>0 (a,(J ";" (SubFrom (a,(intloc 0)))),p,(Initialized s))) . k) . a <= 0

assume ((StepWhile>0 (a,(J ";" (SubFrom (a,(intloc 0)))),p,(Initialized s))) . k) . a <= 0 ; :: thesis: contradiction
then (((StepWhile>0 (a,(J ";" (SubFrom (a,(intloc 0)))),p,(Initialized s))) . k) . a) + k < (s . a) + 0 by A6, XREAL_1:8;
hence contradiction by A4, A6; :: thesis: verum

end;

J is_halting_on (StepTimes (a,J,p,s)) . k,p +* (Times (a,J)) by A2, A6;
then A9: J is_halting_on Initialized ((StepTimes (a,J,p,s)) . k),p +* (Times (a,J)) by A7, SCMFSA8B:42;
Macro (SubFrom (a,(intloc 0))) is_halting_on IExec (J,(p +* (Times (a,J))),((StepTimes (a,J,p,s)) . k)),p +* (Times (a,J)) by SCMFSA7B:19;
then J ";" (SubFrom (a,(intloc 0))) is_halting_on Initialized ((StepTimes (a,J,p,s)) . k),p +* (Times (a,J)) by A9, SFMASTR1:3;
then DataPart ((StepWhile>0 (a,(J ";" (SubFrom (a,(intloc 0)))),p,(Initialized s))) . (k + 1)) = DataPart (IExec ((J ";" (SubFrom (a,(intloc 0)))),(p +* (Times (a,J))),((StepTimes (a,J,p,s)) . k))) by A7, A8, Th32;
then ((StepTimes (a,J,p,s)) . (k + 1)) . a = (IExec ((J ";" (SubFrom (a,(intloc 0)))),(p +* (Times (a,J))),((StepTimes (a,J,p,s)) . k))) . a by SCMFSA_M:2
.= (Exec ((SubFrom (a,(intloc 0))),(IExec (J,(p +* (Times (a,J))),((StepTimes (a,J,p,s)) . k))))) . a by A9, SFMASTR1:11
.= ((IExec (J,(p +* (Times (a,J))),((StepTimes (a,J,p,s)) . k))) . a) - ((IExec (J,(p +* (Times (a,J))),((StepTimes (a,J,p,s)) . k))) . (intloc 0)) by SCMFSA_2:65
.= ((IExec (J,(p +* (Times (a,J))),((StepTimes (a,J,p,s)) . k))) . a) - 1 by A9, SCMFSA8C:67
.= ((Initialized ((StepTimes (a,J,p,s)) . k)) . a) - 1 by A9, A1, SCMFSA8C:95
.= (((StepTimes (a,J,p,s)) . k) . a) - 1 by SCMFSA_M:37 ;
hence (((StepTimes (a,J,p,s)) . (k + 1)) . a) + (k + 1) = s . a by A4, A6; :: thesis: verum

end;

A10: S₁[ 0 ]

proof

assume 0 <= s . a ; :: thesis: (((StepTimes (a,J,p,s)) . 0) . a) + 0 = s . a
thus (((StepTimes (a,J,p,s)) . 0) . a) + 0 = (Initialized s) . a by SCMFSA_9:def 5
.= s . a by SCMFSA_M:37 ; :: thesis: verum

end;

thus for k being Nat holds S₁[k] from NAT_1:sch 2(A10, A3); :: thesis: verum