70 lines
1.8 KiB
Coq
70 lines
1.8 KiB
Coq
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Require Import List.
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Import ListNotations.
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(* Syntax of re *)
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Inductive re {A:Type}: Type :=
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| ε | Null: re
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| Char: A -> re
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| Cat: re -> re -> re
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| Alt: re -> re -> re
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| Star: re -> re.
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Arguments re: clear implicits.
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Module ReNotations.
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Notation "∅" := Null.
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Notation "'↑' a" := (Char a) (at level 10).
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Notation "r '∗'" := (Star r) (at level 20).
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Infix ";" := Cat (at level 60, right associativity).
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Infix "│" := Alt (at level 65, right associativity).
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End ReNotations.
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(* Small-step operational semantics for re *)
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Inductive reDe {A:Type}: list A -> re A -> Prop :=
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| εDe: reDe [] ε
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| CharDe: forall (a:A), reDe [a] (Char a)
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| CatDe: forall (r1 r2:re A) (l1 l2: list A),
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reDe l1 r1 -> reDe l2 r2 -> reDe (l1++l2) (Cat r1 r2)
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| AltDeL: forall (r1 r2:re A) (l: list A),
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reDe l r1 -> reDe l (Alt r1 r2)
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| AltDeR: forall (r1 r2:re A) (l: list A),
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reDe l r2 -> reDe l (Alt r1 r2)
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| StarDeMore: forall (r:re A) (l1 l2:list A),
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reDe l1 r -> reDe l2 (Star r) -> reDe (l1++l2) (Star r)
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| StarDeDone: forall r:re A, reDe [] r.
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Notation "w '⊨' r" := (reDe w r) (at level 50, only parsing).
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Section Deriv.
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Context {A:Type}.
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Variable A_eqb: A -> A -> bool.
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(* Brzozowski derivative *)
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Fixpoint δ (a:A) (r: re A): re A :=
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match r with
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| ε | Null => Null
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| Char c =>
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if A_eqb c a then ε
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else Null
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| Cat r1 r2 => Cat (δ a r1) r2
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| Alt r1 r2 => Alt (δ a r1) (δ a r2)
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| Star r' => δ a r' (* ε case won't be possible in derivative *)
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end.
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End Deriv.
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Theorem foo (A:Type) (A_eqb: A -> A -> bool) (r:re A) (a:A) (w:list A):
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(a::w) ⊨ r -> w ⊨ (δ A_eqb a r).
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Proof.
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induction r; simpl.
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- intro H; inversion H.
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- intro H; inversion H.
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- intro H.
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pv_opt.
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repeat in_inv.
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induction w.
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+ constructor.
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+ induction w.
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Abort.
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