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Sophie Quinton
rtproofs
Commits
9cc1d641
Commit
9cc1d641
authored
Aug 04, 2016
by
Felipe Cerqueira
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Add new function for fixedpoint iteration
parent
af370ec8
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util/fixedpoint.v
util/fixedpoint.v
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util/fixedpoint.v
View file @
9cc1d641
...
...
@@ 70,4 +70,47 @@ Section FixedPoint.
by
rewrite
addnS
;
apply
fun_mon_iter_mon_helper
.
Qed
.
End
FixedPoint
.
\ No newline at end of file
End
FixedPoint
.
(* In this section we define a fixedpoint iteration function
that stops as soon as it finds the solution. If no solution
is found, the function returns None. *)
Section
Iteration
.
Variable
T
:
eqType
.
Variable
f
:
T
>
T
.
Fixpoint
iter_fixpoint
max_steps
(
x
:
T
)
:
=
if
max_steps
is
step
.+
1
then
let
x'
:
=
f
x
in
if
x
==
x'
then
Some
x
else
iter_fixpoint
step
x'
else
None
.
Section
Lemmas
.
(* We prove that iter_fixpoint either returns either None
or Some y, where y is a fixed point. *)
Lemma
iter_fixpoint_cases
:
forall
max_steps
x0
,
iter_fixpoint
max_steps
x0
=
None
\/
exists
y
,
iter_fixpoint
max_steps
x0
=
Some
y
/\
y
=
f
y
.
Proof
.
induction
max_steps
.
{
by
ins
;
simpl
;
destruct
(
x0
==
f
x0
)
;
left
.
}
{
intros
x0
;
simpl
.
destruct
(
x0
==
f
x0
)
eqn
:
EQ1
;
first
by
right
;
exists
x0
;
split
;
last
by
apply
/
eqP
.
by
destruct
(
IHmax_steps
(
f
x0
))
as
[
NONE

FOUND
].
}
Qed
.
End
Lemmas
.
End
Iteration
.
\ No newline at end of file
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