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Parsers.v
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Parsers.v
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Require Import Ascii.
Require Import Coq.Lists.List.
Set Universe Polymorphism.
Definition string := list ascii.
Definition env_ty := list Type.
Require Import Coq.Logic.Eqdep.
Require Import Coq.Program.Equality.
Import EqdepTheory.
Section Parsers.
Set Universe Polymorphism.
Variable (Vars: Set).
Variable (Tys: Vars -> Set).
Inductive parser: Set -> Type :=
| pFail : forall {A: Set}, parser A
| pEps : parser unit
| pChar : ascii -> parser ascii
| pAny : parser ascii
| pCat : forall {A B: Set}, parser A -> parser B -> parser (A * B)
| pMap : forall {A B: Set}, parser A -> (A -> B) -> parser B
| pStar: forall {A: Set}, parser A -> parser (list A)
| pAlt : forall {A: Set}, parser A -> parser A -> parser A
| pVar : forall (v: Vars), parser (Tys v)
| pBind : forall {A: Set} {F: A -> Set} (p: parser A) (f: forall a, parser (F a)),
parser {a & F a}.
Arguments pFail {_}.
Arguments pCat {_ _} _ _.
Arguments pMap {_ _} _ _.
Arguments pStar {_} _.
Arguments pAlt {_} _ _.
Arguments pBind {_ _} _ _.
Variable (Ps: forall (v: Vars), parser (Tys v)).
Inductive Parses : string -> forall {A: Set}, parser A -> A -> Prop :=
| Eps : Parses nil pEps tt
| Chr : forall c, Parses (c :: nil) (pChar c) c
| Any : forall c, Parses (c :: nil) pAny c
| Concat :
forall s {A B: Set} (p: parser A) v s' (p': parser B) v',
Parses s p v ->
Parses s' p' v' ->
Parses (s ++ s') (pCat p p') (v, v')
| Map:
forall s {A B: Set} (f: A -> B) p v,
Parses s p v ->
Parses s (pMap p f) (f v)
| AltL:
forall s {A: Set} (p: parser A) p' v,
Parses s p v ->
Parses s (pAlt p p') v
| AltR:
forall s {A: Set} (p: parser A) p' v,
Parses s p' v ->
Parses s (pAlt p p') v
| StarEmp:
forall {A: Set} (p: parser A),
Parses nil (pStar p) nil
| StarIter:
forall s s' {A: Set} (p: parser A) v vs,
Parses s p v ->
Parses s' (pStar p) vs ->
Parses (s ++ s') (pStar p) (v :: vs)
| Var:
forall s var (v: Tys var),
Parses s (Ps var) v ->
Parses s (pVar var) v
| Bind:
forall s s' {A: Set} (F: A -> Set) (p: parser A) (f: forall a, parser (F a)) a b,
Parses s p a ->
Parses s' (f a) b ->
Parses (s ++ s') (pBind p f) (existT _ a b).
Ltac my_simpl :=
repeat (unfold eq_rect in *; simpl in *; try subst;
try match goal with
| H: context[ match ?Eq with | eq_refl => _ end] |- _ =>
pose (e := UIP_refl _ _ Eq);
erewrite e in *;
clear e
end).
Ltac simpl_exist H :=
pose (e' := projT2_eq H);
my_simpl;
try clear e'.
Lemma inv_map :
forall {A B : Set} {s p} {f: A -> B} {v: B},
Parses s (pMap p f) v ->
exists v': A, Parses s p v' /\ f v' = v.
Proof.
intros.
inversion H.
subst.
simpl_exist H6.
clear H6.
simpl_exist H4.
clear H4.
pose (e'' := projT2_eq H5).
my_simpl.
simpl_exist e''.
clear e''.
clear H5.
exists v0.
split; auto.
Qed.
Lemma inv_cat :
forall {A B : Set} {s p p'} {v: A * B},
Parses s (pCat p p') v ->
exists s' s'' v' v'',
Parses s' p v' /\
Parses s'' p' v'' /\
s = s' ++ s'' /\
v = (v', v'').
Proof.
intros.
inversion H.
subst.
exists s0.
exists s'.
exists v0.
exists v'.
simpl_exist H8.
simpl_exist H6.
simpl_exist H7.
repeat split; auto.
Qed.
Lemma inv_bind :
forall {A B: Set} {s p} {f: A -> parser B} {v},
Parses s (pBind p f) v ->
exists s' s'' v' v'',
s = s' ++ s'' /\
Parses s' p v' /\
Parses s'' (f v') v'' /\
v = existT _ v' v''.
Proof.
Admitted.
Definition hoare_pc_parses {A: Set} (p: parser A) (Pre: string -> Prop) (Post: A -> string -> Prop) :=
forall s v,
Pre s ->
Parses s p v ->
Post v s.
Notation "{{ Pre }} p {{ Post }}" := (hoare_pc_parses p Pre Post) (at level 80).
Lemma hoare_chr P:
forall c,
{{ P c }} pChar c {{ P }}.
Proof.
unfold hoare_pc_parses.
intros.
inversion H0.
simpl_exist H3.
auto.
Qed.
Lemma hoare_eps P:
{{ P tt }} pEps {{ P }}.
Proof.
unfold hoare_pc_parses.
intros.
inversion H0.
simpl_exist H2.
auto.
Qed.
Lemma hoare_concat {V V': Set} P P' P'' (p: parser V) (p': parser V'):
forall v v',
{{ P v }} p {{ P }} ->
{{ P' v' }} p' {{ P' }} ->
{{ P'' (v, v') }} pCat p p' {{ P'' }}.
Admitted.
Lemma hoare_map {A B: Set} P (p: parser A) (f: A -> B):
forall v,
{{ P (f v) }} pMap p f {{ P }}.
Admitted.
Fixpoint any {A} (ps: list (parser A)) : parser A :=
match ps with
| nil => pFail
| p :: ps' => pAlt p (any ps')
end.
Definition pure {A: Set} (a: A) : parser A := pMap pEps (fun _ => a).
Lemma pure_spec {A: Set}:
forall (a: A) s v,
Parses s (pure a) v ->
s = nil /\ v = a.
Proof.
unfold pure.
intros.
pose proof (inv_map H).
destruct H0 as [x [? ?]].
subst.
inversion H0.
split; exact eq_refl.
Qed.
Fixpoint fold {A B: Set} (f: A -> B -> B) (acc: B) (ps: list (parser A)) :=
match ps with
| nil => pure acc
| p :: ps' =>
pMap (pCat p (fold f acc ps')) (fun '(a, b) => f a b)
end.
Definition filter {A: Set} (p: parser A) (check: A -> bool) : parser A :=
pMap (pBind p (fun a => if check a then pure a else pFail))
(fun x => projT2 x).
Ltac destruct_all H :=
try match H with
| _ /\ _ =>
let Hl := fresh "H" in
let Hr := fresh "H" in
destruct H as [Hl Hr];
destruct_all Hl;
destruct_all Hr
| exists _, _ =>
let x := fresh "x" in
let H := fresh "H" in
destruct H as [x H];
destruct_all H
end.
Lemma filter_spec {A: Set} f:
forall s (p: parser A) (v : A),
Parses s (filter p f) v ->
f v = true.
Proof.
intros.
unfold filter in H.
pose proof (inv_map H).
destruct H0 as [? [? ?]].
subst.
clear H.
pose proof (inv_bind H0).
do 4 destruct H.
destruct H as [? [? [? ?]]].
subst.
simpl.
destruct (f x2) eqn:?.
- pose proof inv_map H2.
destruct H as [? [? ?]].
subst.
auto.
- inversion H2.
Qed.
Fixpoint repeat {A} (n: nat) (p: parser A) : parser (list A) :=
match n with
| 0 => pure nil
| S n' => pMap (pCat p (repeat n' p)) (fun '(x, xs) => x :: xs)
end.
Lemma repeat_spec {A: Set} n p:
forall s (vs: list A),
Parses s (repeat n p) vs ->
length vs = n.
Proof.
induction n; intros; simpl.
- unfold repeat in H.
pose proof (pure_spec _ _ _ H).
destruct H0.
subst.
exact eq_refl.
- unfold repeat in H.
fold (@repeat A) in H.
pose proof inv_map H.
destruct H0 as [? [? ?]].
subst.
pose proof inv_cat H0.
destruct H1 as [? [? [? [? [? [? [?]]]]]]].
subst.
simpl.
erewrite IHn; [exact eq_refl|].
eauto.
Qed.
Lemma repeat_len_spec {A: Set} n:
forall s p (v: list A) n',
(forall s' v', Parses s' p v' -> length s' = n') ->
Parses s (repeat n p) v ->
length s = n * n'.
Proof.
intros.
generalize H.
induction n; intros; simpl.
- unfold repeat in H.
pose proof (@pure_spec (list A)).
specialize (H2 nil s v H0).
destruct H2.
subst.
exact eq_refl.
-
unfold repeat in H.
fold (@repeat A) in H.
admit.
Admitted.
End Parsers.
Arguments parser {_ _} _.
Arguments Parses {_ _ _} _ _ _.
Arguments pFail {_ _ _}.
Arguments pEps {_ _}.
Arguments pAny {_ _}.
Arguments pChar {_ _} _.
Arguments pCat {_ _ _ _} _ _.
Arguments pMap {_ _ _ _} _ _.
Arguments pStar {_ _ _} _.
Arguments pAlt {_ _ _} _ _.
Arguments pVar {_ _} _.
Arguments pBind {_ _ _ _} _ _.
Arguments any {_ _ _} _.
Arguments pure {_ _ _} _.
Arguments filter {_ _ _} _ _.
Arguments repeat {_ _ _} _ _.
Arguments fold {_ _ _ _} _ _ _.
Infix "$" := (pCat) (at level 80).
Infix "@" := (pMap) (at level 80).
Infix "||" := (pAlt) (at level 50, left associativity).
Infix ">>=" := (pBind) (at level 80).
Notation "p1 $> p2" := (pCat p1 p2 @ fun '(_,x) => x) (at level 50).
Notation "p1 <$ p2" := (pCat p1 p2 @ fun '(x,_) => x) (at level 50).
Local Open Scope char_scope.
Definition check_prefix (ss : string * string) :=
match fst ss with
| nil => true
| _ => false
end.
Ltac obl_tac := Tactics.program_simpl.
Program Fixpoint add_prefix {s} (c: ascii) (ss: list {ss' | s = fst ss' ++ snd ss'}) : list {ss' | c :: s = fst ss' ++ snd ss'} :=
match ss with
| nil => nil
| pr :: ss' =>
let '(x, y) := pr in
match x with
| nil => (nil, c :: y) :: (c :: nil, y) :: add_prefix c ss'
| _ => (c :: x, y) :: add_prefix c ss'
end
end.
Program Fixpoint splits (s: list ascii) : list {ss' | s = fst ss' ++ snd ss'} :=
match s with
| nil => (nil, nil) :: nil
| c :: s' =>
let ss := splits s' in
add_prefix c ss
end.
Ltac cat_all ss :=
match ss with
| ?x :: ?xs =>
idtac "catting with:";
idtac x;
(cat_all xs) +
(eapply Concat with (s := fst x) (s' := snd x))
end.
Ltac pose_all ss :=
match ss with
| ?s :: ?ss' =>
pose s; pose_all ss'
| nil => idtac
end.
Definition mp_ty (A: Set) := A -> list ascii -> bool.
Definition find {A: Set} (a: A) (s: list ascii) (mp: mp_ty A) := mp a s.
Scheme Equality for list.
Definition mp_empty {A: Set} : mp_ty A := fun _ _ => false.
Search (ascii -> ascii -> bool).
Definition mp_assoc {A: Set} {v_eqb: A -> A -> bool} (v: A) (s: list ascii) (mp: mp_ty A) : mp_ty A :=
fun v' s' => if v_eqb v v' then list_beq ascii eqb s s' else find v s mp.
Ltac parse Vs Tys Ps Mp V_eqb :=
compute;
match goal with
| |- Parses ?s _ ?p _ =>
idtac "current s:";
idtac s;
idtac "current parser:";
idtac p;
idtac "current map:";
idtac Mp
end;
match goal with
| |- Parses ?s _ (_ @ _) _ => eapply (Map Vs Tys Ps _); parse Vs Tys Ps Mp V_eqb
| |- Parses ?s _ (pChar _) _ => eapply Chr
| |- Parses ?s _ pAny _ => eapply Any
| |- Parses ?s _ (pVar ?v) _ =>
assert (find v s Mp = false) by exact eq_refl;
let Mp' := fresh "Mp" in
pose (Mp' := mp_assoc (v_eqb := V_eqb) v s Mp);
subst Mp;
eapply (Var Vs Tys Ps _ v); parse Vs Tys Ps Mp' V_eqb
| |- Parses ?s _ (?pl || ?pr) _ =>
(eapply AltL + eapply AltR); parse Vs Tys Ps Mp V_eqb
| |- Parses nil _ pEps _ => eapply Eps
| |- Parses nil _ (pStar _) _ => eapply StarEmp
| |- Parses ?s _ (_ $ _ ) _ =>
set (tmp := (splits s));
vm_compute in tmp;
match goal with
| _ := ?v |- _ =>
pose_all v
end;
clear tmp;
multimatch goal with
| sn := ?v |- Parses ?gs _ _ _ =>
let H := fresh "H" in
assert (H: fst (proj1_sig sn) ++ snd (proj1_sig sn) = gs) by exact eq_refl;
subst sn;
simpl in H;
erewrite <- H;
clear H;
eapply Concat;
parse Vs Tys Ps Mp V_eqb
end
| |- Parses ?s _ (pStar _ ) _ =>
set (tmp := (splits s));
vm_compute in tmp;
match goal with
| _ := ?v |- _ =>
pose_all v
end;
clear tmp;
multimatch goal with
| sn := ?v |- Parses ?gs _ _ _ =>
let H := fresh "H" in
assert (H: fst (proj1_sig sn) ++ snd (proj1_sig sn) = gs) by exact eq_refl;
subst sn;
simpl in H;
erewrite <- H;
clear H;
eapply StarIter;
parse Vs Tys Ps Mp V_eqb
end
| |- Parses ?s _ (_ >>= _ ) _ =>
set (tmp := (splits s));
vm_compute in tmp;
match goal with
| _ := ?v |- _ =>
pose_all v
end;
clear tmp;
multimatch goal with
| sn := ?v |- Parses ?gs _ _ _ =>
let H := fresh "H" in
assert (H: fst (proj1_sig sn) ++ snd (proj1_sig sn) = gs) by exact eq_refl;
subst sn;
simpl in H;
erewrite <- H;
clear H;
eapply Bind;
parse Vs Tys Ps Mp V_eqb
end
end.
(*
Ltac parse_one :=
match goal with
| |- Parses ?s _ (_ @ _) _ => eapply Map
| |- Parses ?s _ (pChar _) _ => eapply Chr
| |- Parses nil _ _ pEps => eapply Eps
end.
Ltac parse' := repeat parse_one. *)
(* a numeric parser *)
Definition dig {v env} : @parser v env nat :=
any (
(pChar "0" @ fun _ => 0) ::
(pChar "1" @ fun _ => 1) ::
(pChar "2" @ fun _ => 2) ::
(pChar "3" @ fun _ => 3) ::
(pChar "4" @ fun _ => 4) ::
(pChar "5" @ fun _ => 5) ::
(pChar "6" @ fun _ => 6) ::
(pChar "7" @ fun _ => 7) ::
(pChar "8" @ fun _ => 8) ::
(pChar "9" @ fun _ => 9) ::
nil
).
Definition pref_fun (x: nat * list nat) := fst x :: snd x.
Definition num {v env} := (@dig v env $ pStar dig) @ pref_fun.
Lemma length_cons:
forall A (x: A) xs,
length (x :: xs) = 1 + length xs.
Proof.
intros.
unfold length.
fold (length xs).
exact eq_refl.
Qed.
Require Import Coq.micromega.Lia.
Lemma pref_incr: forall x, 0 < length (pref_fun x).
Proof.
intros.
destruct x;
unfold pref_fun.
erewrite length_cons.
Lia.lia.
Qed.
Lemma num_nonempty :
forall V env_t env_p s v,
@Parses V env_t env_p s _ num v -> length v > 0.
Proof.
Admitted.
Definition emp_ty (v: Empty_set) : Set := match v with end.
Definition emp_p (v: Empty_set) : @parser Empty_set emp_ty (emp_ty v) := match v with end.
Example parsing_alt : {v & @Parses Empty_set emp_ty emp_p ("1" :: nil) _ (pChar "0" || pChar "1") v}.
Proof.
econstructor.
parse Empty_set emp_ty emp_p tt tt.
Defined.
Definition ab_parser : @parser Empty_set emp_ty (nat * nat) :=
((pChar "a" @ fun _ => 1) || ((pChar "a" $ pChar "a") @ fun _ => 2))
$
(((pChar "b") @ fun _ => 1) || ((pChar "b" $ pChar "b") @ fun _ => 2)).
Example parsing_cat : {v & @Parses Empty_set emp_ty emp_p ("a" :: "a" :: "b" :: nil) _ ab_parser v}.
Proof.
econstructor.
set (v := ab_parser).
vm_compute in v.
subst v.
Opaque "++".
parse Empty_set emp_ty emp_p tt tt.
Defined.
Section NetString.
Inductive V : Set := | ns.
Definition Tys (v: V) : Set := list ascii.
Fixpoint digs_2_num (digs: list nat) : nat :=
match digs with
| nil => 0
| d :: ds => d + digs_2_num ds * 10
end.
Definition num' : @parser V Tys (nat * nat) := num @ fun ds => (digs_2_num ds, length ds).
Definition net_string : @parser V Tys (list ascii * nat) :=
(num' >>= fun '(n, len) =>
(pChar ":" $ (repeat n pAny) $ pChar ",") @ (fun '(x, y, z) => (x :: y ++ z :: nil, len + n + 2)))
@
fun (x : {a & (list ascii * nat)%type}) => projT2 x.
Inductive fmt :=
| base
| nest: list fmt -> fmt.
Fixpoint fmt2parser (f: fmt) : @parser V Tys (list ascii * nat) :=
match f with
| base => net_string
| nest fms =>
(num' >>= fun '(n, len) =>
let f := fun '(cs, len) '(cs', len') => (cs ++ cs', len + len') in
let inner := fold f (nil, 0) (List.map fmt2parser fms) in
(pChar ":" $ filter inner (fun '(_, n') => Nat.eqb n n') $ pChar ",")
@
fun '(x, y) => (fst x :: fst (snd x) ++ y :: nil, snd (snd x) + 1 + len))
@
fun (x : {a & (list ascii * nat)%type}) => projT2 x
end.
Definition fmt_2 := nest (base :: base :: nil).
Definition Ps (v: V) : @parser V Tys (Tys v) := pFail.
Definition hello := "5" :: ":" :: "h" :: "e" :: "l" :: "l" :: "o" :: "," :: nil.
Definition world := "6" :: "w" :: "o" :: "r" :: "l" :: "d" :: "!" :: "," :: nil.
Definition hello_world :=
"1" :: "7" :: ":" :: hello ++ world ++ "," :: nil.
Example parse_net_num: {v & @Parses V Tys Ps ("5" :: nil) _ num' v}.
Proof.
econstructor.
parse V Tys Ps tt tt.
Defined.
Eval vm_compute in parse_net_num.
Example parse_net_hello : {v & @Parses V Tys Ps hello _ net_string v}.
Proof.
econstructor.
(* parse Vs Tys Ps tt tt. *)
eapply Map.
match goal with
| |- Parses ?s _ (_ >>= _ ) _ =>
set (tmp := (splits s));
vm_compute in tmp;
match goal with
| _ := ?v |- _ =>
pose_all v
end;
clear tmp
end.
clear s.
clear s1.
clear s2.
clear s3.
clear s4.
clear s5.
clear s6.
clear s7.
multimatch goal with
| sn := ?v |- Parses ?gs _ _ _ =>
let H := fresh "H" in
assert (H: fst (proj1_sig sn) ++ snd (proj1_sig sn) = gs) by exact eq_refl;
subst sn;
simpl in H;
erewrite <- H;
eapply Bind;
parse V Tys Ps tt tt
end.
Defined.
Example parse_fmt_2 : {v & @Parses V Tys Ps hello_world _ (fmt2parser fmt_2) v}.
Proof.
econstructor.
set (foo := fmt2parser fmt_2).
compute in foo.
subst.
Fail parse V Tys Ps tt tt.
Admitted.
End NetString.
Section Expr.
Inductive ExprV : Set := | e.
Scheme Equality for ExprV.
Definition ExprTys (x: ExprV) : Set :=
match x with
| e => nat
end.
(* e ::= num | (e) | e + e | e * e *)
Definition e_base : @parser ExprV ExprTys nat :=
num @ digs_2_num.
Definition e_parens : @parser ExprV ExprTys nat :=
pChar "(" $> pVar e <$ pChar ")".
Definition e_plus : @parser ExprV ExprTys nat :=
pVar e $ (pChar "+" $> pVar e) @ fun '(x, y) => x + y.
Definition e_times : @parser ExprV ExprTys nat :=
pVar e $ (pChar "*" $> pVar e) @ fun '(x, y) => x * y.
Definition expr_p : parser nat :=
e_base || e_parens || e_plus || e_times.
Definition ExprPs (v: ExprV) : @parser ExprV ExprTys (ExprTys v) :=
match v with | e => expr_p end.
Example expr_simpl : {v & @Parses ExprV ExprTys ExprPs ("4" :: "2" :: nil) _ expr_p v}.
Proof.
econstructor.
pose (me := @mp_empty ExprV).
parse ExprV ExprTys ExprPs me ExprV_beq.
Defined.
Example expr_var : {v & @Parses ExprV ExprTys ExprPs ("4" :: "2" :: nil) _ (pVar e) v}.
Proof.
econstructor.
pose (me := @mp_empty ExprV).
idtac "starting".
parse ExprV ExprTys ExprPs me ExprV_beq.
Defined.
Example expr_parens : {v & @Parses ExprV ExprTys ExprPs ( "(" :: "3" :: ")" :: nil) _ expr_p v}.
Proof.
econstructor.
pose (me := @mp_empty ExprV).
idtac "starting".
parse ExprV ExprTys ExprPs me ExprV_beq.
Defined.
Example expr_plus_times : {v & @Parses ExprV ExprTys ExprPs ( "3" :: "+" :: "4" :: "*" :: "5" :: nil) _ expr_p v}.
Proof.
econstructor.
pose (me := @mp_empty ExprV).
idtac "starting".
parse ExprV ExprTys ExprPs me ExprV_beq.
Defined.
End Expr.