Talk at Proof Summit 2011 on 2011/09/25

1. Proof Summit 2011
Coq
@tmiya
September 25,2011
@tmiya : Coq , 1

2. @tmiya_ SIer
2007 LL Spirit Coq
• Coq
• Haskell Scala
2009 Agda
• @yoshihiro503 bool Prop
• =⇒ Coq
2010 2 @kencoba Formal
Methods Forum
•
• ProofCafe
: Coq
@tmiya : Coq , 2

3. Coq
User Contribution
@tmiya : Coq , 3

4. — @kinaba d. y. d.
@tmiya : Coq , 4

5. (regular expression)
∅
"a" "b" ...
L1 , L2 {xy |x ∈ L1 , y ∈ L2 }
L1 , L2 L1 ∪ L2
L 0
∪ {x|x ∈ L} ∪ {xx|x ∈ L} ∪ . . .
@tmiya : Coq , 5

6. ”Derivatives of Regular Expressions”, Janusz Brzozowski, Journal
of the ACM 1964.
R(s) : s R
{
ν(R) (s = ””)
R(s) =
(∂a R)(s ) (s = a :: s )
ν(R) = R
∂a R = R a
NFA R a
∂a R
”Yacc is Dead” (http://arxiv.org/abs/1010.5023)
2011 Brzozowski
@tmiya : Coq , 6

7. R ν(R) ∂a R
∅ false ∅
true { ∅
(c = a)
"c" false
{ ∅ (c = a)
(∂a R)S (ν(R) = false)
RS ν(R) ∧ ν(S)
(∂a R)S + (∂a S) (ν(R) = true)
R +S ν(R) ∨ ν(S) (∂a R) + (∂a S)
R∗ true (∂a R)R ∗
⇒ d(fg ) = f (dg ) + (df )g
@tmiya : Coq , 7

8. (1/4)
30
Inductive RegExp : Set := (* *)
| Empty : RegExp (* *)
| Eps : RegExp (* *)
| Char : ascii -> RegExp (* *)
| Cat : RegExp -> RegExp -> RegExp (* *)
| Or : RegExp -> RegExp -> RegExp (* *)
| Star : RegExp -> RegExp (* *)
Notation "a ++ b" := (Cat a b).
Notation "a || b" := (Or a b).
@tmiya : Coq , 8

9. (2/4)
Fixpoint nu(re:RegExp):bool :=
match re with
| Empty => false
| Eps => true
| Char c => false
| Cat r s => (nu r && nu s)%bool
| Or r s => (nu r || nu s)%bool
| Star r => true
end.
@tmiya : Coq , 9

10. (3/4)
Fixpoint derive(a:ascii)(re:RegExp):RegExp :=
match re with
| Empty => Empty
| Eps => Empty
| Char c => match (ascii_dec c a) with
| left _ => Eps
| right _ => Empty
end
| Cat r s => match (nu r) with
| true => ((derive a r) ++ s) || (derive a s)
| false => (derive a r) ++ s
end
| Or r s => (derive a r) || (derive a s)
| Star r => (derive a r) ++ (Star r)
end.
Notation "re / a" := (derive a re).
@tmiya : Coq , 10

11. (4/4)
Fixpoint matches (re:RegExp)(s:string) : bool :=
match s with
| EmptyString => nu re
| String a w => matches (re / a) w
end.
Notation "re ~= s" := (matches re s) (at level 60).
@tmiya : Coq , 11

12. Kleene
Kleene
”A Completeness Theorem for Kleene Algebras and the Algebra of
Regular Events,” D. Kozen (1994)
∅ 0 1
• x + (y + z) = (x + y ) + z, x(yz) = (xy )z :
• x +y =y +z :
• x(y + z) = xy + xz, (x + y )z = xz + yz :
• x + 0 = 0 + x = x, 1x = x1 = x :
• x0 = 0x = 0 :
x +x =x :
Kleene-star (x ≤ y ⇔ x + y = y )
• 1 + xx ∗ ≤ x ∗ , 1 + x ∗ x ≤ x ∗
• x + yz ≤ z ⇒ y ∗ x ≤ z
• x + yz ≤ y ⇒ xy ∗ ≤ z
Kleene
:
:
@tmiya : Coq , 12

13. Kleene
(1/3)
Brzozowski Kleene
Coq
• 1500
•
Setoid
=⇒ setoid_rewrite tactic
• Brzozowski Coq
•
Kleene
• ”A tactic for deciding Kleene algebras”
•
@tmiya : Coq , 13

14. Kleene
(2/3)
Coq
•
induction re.
• Or Cat, Star
• =⇒ induction s.
Lemma divide_Cat : forall s r’ r’’, (r’ ++ r’’) ~== s ->
{s’:string & {s’’:string | s = (s’ ++ s’’)%string /
r’ ~== s’ / r’’ ~== s’’ }}.
@tmiya : Coq , 14

15. Kleene
(3/3)
+ +rr ∗ = r ∗
+ +r ∗ r = r∗
• r∗ r =⇒ r ∗
Lemma Star_to_list : forall s r, (Star r) ~== s ->
{ss:list string |
forallb (fun s => r ~= s) ss = true /
concat_list_string ss = s /
forallb (fun s => bneq_empty_string s) ss = true }.
• s
• refine (induction_ltof2 string str_length _ _).
Setoid
@tmiya : Coq , 15

16. User Contribution
Coq User Contribution
INRIA The Coq User’s Contributions
1. Makefile
• Make
-R . RegExp
Char.v
...
RegExp.v
(Coqdoc )
• $ coq_makefile -f Make -o Makefile
• $ make clean all all-gal.pdf html
• $ tar -cf RegExp.tar Makefile *.v
2. tar upload
3. Coq user contributions submit
• Coq
LGPL
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17. Brzozowski ( )
Kleene
Coq
INRIA User contribution
@tmiya : Coq , 17