Library RelationAlgebra.ugregex

ugregex: untyped generalised regular expressions

we define here the syntax for untyped generalised regular expressions which we will actually use in computations

Require Import kat lsyntax positives sups glang comparisons boolean.
Set Implicit Arguments.

Section l.
Variable Pred: nat.
Notation Sigma := positive.
Notation Atom := (ord (pow2 Pred)).
Notation uglang := (traces_monoid_ops Atom traces_tt traces_tt).


we declare strict iteration as primitive for efficiency reasons: Kleene star is derived from strict iteration in a linear way, while deriving strict iteration out of Kleene star requires duplication.
Inductive ugregex :=
| u_var(i: Sigma)
| u_prd(p: expr (ord Pred))
| u_pls(e f: ugregex)
| u_dot(e f: ugregex)
| u_itr(e: ugregex).

zer and one are also derived operations
Definition u_zer := u_prd e_bot.
Definition u_one := u_prd e_top.
Definition u_str e := u_pls u_one (u_itr e).


we define the untyped guarded string language of an expression algebraically (by induction on the expression) ; below we characterise it coalgebraically

Fixpoint lang (e: ugregex): uglang :=
  match e with
    | u_var atsingle a
    | u_prd ptinj (fun ieval (set.mem i) p)
    | u_pls e flang e + lang f
    | u_dot e flang e × lang f
    | u_itr elang e ^+

we get a KA structure, by interpretation into languages

Definition u_leq e f := lang e <== lang f.
Definition u_weq e f := lang e == lang f.

Canonical Structure ugregex_lattice_ops :=
  lattice.mk_ops _ u_leq u_weq u_pls u_pls id u_zer u_zer.

CoInductive ugregex_unit := ugregex_tt.

Canonical Structure ugregex_monoid_ops :=
  monoid.mk_ops ugregex_unit _
  (fun _ _ _u_dot) (fun _u_one) (fun _u_itr) (fun _u_str)
  (fun _ _ _u_zer) (fun _ _ _ _ _u_zer) (fun _ _ _ _ _u_zer).

Notation tt := ugregex_tt.
Notation ugregex' := (ugregex_monoid_ops tt tt).

Global Instance ugregex_monoid_laws: monoid.laws BKA ugregex_monoid_ops.
  apply (laws_of_faithful_functor (f:=fun _ _: ugregex_unitlang)).
  constructor; simpl ob. intros ? ?. constructor.
   now intros ? ? ?.
   now intros ? ? ?.
   now intros _ ? ?.
   intros _ [a|]; simpl; intuition; discriminate.
   intros ? [a|]; simpl; intuition.
   intros _ ? x. rewrite str_itr. simpl (lang _). apply cup_weq. 2: reflexivity.
   intros [|]; simpl. intuition. tauto.
  now intros ? ? ?.
  now intros ? ? ?.

Global Instance ugregex_lattice_laws: lattice.laws BKA ugregex_lattice_ops.
Proof. apply (@lattice_laws _ _ ugregex_monoid_laws tt tt). Qed.

Note that ugregex actually comes with a KAT structure, but we do not need it

folding expressions
Ltac fold_ugregex := ra_fold ugregex_monoid_ops tt.

Coalgebraic characterisation of the language recognised by an expression

epsilon (optimised since it's quite simple)

Fixpoint epsilon_pred a (e: expr_ops (ord Pred) BL) :=
  match e with
    | e_botfalse
    | e_toptrue
    | e_var ia i
    | e_cup e fepsilon_pred a e ||| epsilon_pred a f
    | e_cap e fepsilon_pred a e &&& epsilon_pred a f
    | e_neg enegb (epsilon_pred a e)

Fixpoint epsilon a (e: ugregex) :=
  match e with
    | u_var _false
    | u_prd pepsilon_pred a p
    | u_pls e fepsilon a e ||| epsilon a f
    | u_dot e fepsilon a e &&& epsilon a f
    | u_itr eepsilon a e

derivatives (specification, unoptimised)

Fixpoint deriv a i (e: ugregex'): ugregex' :=
  match e with
    | u_prd _ ⇒ 0
    | u_var jofbool (eqb_pos i j)
    | u_pls e fderiv a i e + deriv a i f
    | u_dot e fderiv a i e × f + ofbool (epsilon (set.mem a) e) × deriv a i f
    | u_itr ederiv a i e × (e: ugregex')^*

corresponding coalgebraic notion of language

Fixpoint lang' (e: ugregex') (w: trace Atom) :=
  match w with
    | tnil ais_true (epsilon (set.mem a) e)
    | tcons a i wlang' (deriv a i e) w

characterisation of epsilon through languages

Lemma epsilon_iff_lang_nil a e: epsilon (set.mem a) e (lang e) (tnil a).
 induction e; simpl.
  intuition discriminate.
   apply eq_bool_iff. simpl epsilon. induction p; simpl.
    rewrite <-Bool.orb_lazy_alt. congruence.
    rewrite <-Bool.andb_lazy_alt. congruence.
  setoid_rewrite Bool.orb_true_iff. now apply cup_weq.
  setoid_rewrite Bool.andb_true_iff. setoid_rewrite traces_dot_nil. now apply cap_weq.
  split. O. simpl. tauto.
  intros [i Hi]. rewrite IHe. clear IHe. induction i. assumption.
  apply IHi. setoid_rewrite traces_dot_nil in Hi. apply Hi.

Lemma lang_0: lang u_zer == 0.
Proof. intros [?|???]; simpl; intuition; discriminate. Qed.

Lemma lang_1: lang u_one == 1.
Proof. intros [a|???]; simpl. intuition. reflexivity. Qed.

Lemma lang_ofbool b: lang (ofbool b: ugregex') == ofbool b.
Proof. case b. apply lang_1. apply lang_0. Qed.

Global Instance lang_leq: Proper (leq ==> leq) lang.
Proof. now intros ? ?. Qed.
Global Instance lang_weq: Proper (weq ==> weq) lang.
Proof. now intros ? ?. Qed.

Lemma lang_sup J: lang (sup id J) == sup lang J.
Proof. apply f_sup_weq. apply lang_0. reflexivity. Qed.

Lemma deriv_sup a i J: deriv a i (sup id J) = sup (deriv a i) J.
Proof. apply f_sup_eq; now f_equal. Qed.

characterisation of derivatives through languages

Lemma deriv_traces a i e: lang (deriv a i e) == traces_deriv a i (lang e).
  symmetry. induction e; simpl deriv. simpl lang.
   rewrite lang_ofbool. apply traces_deriv_single.
   rewrite lang_0. apply traces_deriv_inj.
   setoid_rewrite traces_deriv_pls. now apply cup_weq.
   generalize (epsilon_iff_lang_nil a e1). case epsilon; intro He1.
    fold_ugregex. setoid_rewrite dot1x. simpl lang.
    setoid_rewrite traces_deriv_dot_1. now rewrite IHe1, IHe2.
     now rewrite <-He1.
    fold_ugregex. setoid_rewrite dot0x. rewrite cupxb. simpl lang.
    setoid_rewrite traces_deriv_dot_2. now rewrite IHe1.
     rewrite <-He1. discriminate.
   simpl lang.
    setoid_rewrite traces_deriv_itr. now rewrite IHe, str_itr, <-inj_top.

the two definitions of languages (algebraic and coalgebraic) coincide, by unicity of the coalgebra morphism from expressions to languages

Theorem lang_lang' e: lang e == lang' e.
  symmetry. intro w. revert e. induction w as [a|a i w IH]; simpl lang'; intro e.
  - apply epsilon_iff_lang_nil.
  - rewrite IH. apply deriv_traces.

as a consequence, lang' preserves equality
Corollary lang'_weq: Proper (weq ==> weq) lang'.
Proof. intros ? ? H. apply lang_weq in H. now rewrite 2 lang_lang' in H. Qed.

Comparing expressions

Fixpoint ugregex_compare (x y: ugregex) :=
  match x,y with
    | u_prd a, u_prd b
    | u_var a, u_var bcmp a b
    | u_pls x x', u_pls y y'
    | u_dot x x', u_dot y y'lex (ugregex_compare x y) (ugregex_compare x' y')
    | u_itr x, u_itr yugregex_compare x y
    | u_var _, _Lt
    | _, u_var _Gt
    | u_prd _, _Lt
    | _, u_prd _Gt
    | u_itr _ , _Lt
    | _, u_itr _Gt
    | u_pls _ _ , _Lt
    | _, u_pls _ _Gt

Lemma ugregex_compare_spec x y: compare_spec (x=y) (ugregex_compare x y).
  revert y; induction x; destruct y; try (constructor; congruence); simpl.
  case cmp_spec; constructor; congruence.
  case cmp_spec; constructor; congruence.
  eapply lex_spec; eauto. intuition congruence.
  eapply lex_spec; eauto. intuition congruence.
  case IHx; constructor; congruence.

Canonical Structure ugregex_cmp := mk_simple_cmp _ ugregex_compare_spec.

End l.