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(*
* CoqUp: Verified high-level synthesis.
* Copyright (C) 2020 Yann Herklotz <yann@yannherklotz.com>
*
* This program is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program. If not, see <https://www.gnu.org/licenses/>.
*)
From coqup Require Import Coquplib Value.
From Coq Require Import FSets.FMapPositive.
From compcert Require Import Maps.
Search Maps.PTree.get not List.In.
Definition reg := positive.
Module AssocMap := PositiveMap.
Module AssocMapExt.
Import AssocMap.
Hint Resolve elements_3w : assocmap.
Hint Resolve elements_correct : assocmap.
Hint Resolve gss : assocmap.
Hint Resolve gso : assocmap.
Section Operations.
Variable elt : Type.
Definition find_default (a : elt) (k : reg) (m : t elt) : elt :=
match find k m with
| None => a
| Some v => v
end.
Definition merge (am bm : t elt) : t elt :=
fold_right (fun p a => add (fst p) (snd p) a) bm (elements am).
Lemma add_assoc :
forall k v l bm,
List.In (k, v) l ->
SetoidList.NoDupA (@eq_key elt) l ->
@find elt k (fold_right (fun p a => add (fst p) (snd p) a) bm l) = Some v.
Proof.
Hint Resolve SetoidList.InA_alt : assocmap.
Hint Extern 1 (exists _, _) => apply list_in_map_inv : assocmap.
induction l; intros.
- contradiction.
- destruct a as [k' v'].
destruct (peq k k').
+ inversion H. inversion H1. inversion H0. subst.
simpl. auto with assocmap.
subst. inversion H0. subst. apply in_map with (f := fst) in H1. simpl in *.
unfold not in H4. exfalso. apply H4. apply SetoidList.InA_alt.
auto with assocmap.
+ inversion H. inversion H1. inversion H0. subst. simpl. rewrite gso; try assumption.
apply IHl. contradiction. contradiction.
simpl. rewrite gso; try assumption. apply IHl. assumption. inversion H0. subst. assumption.
Qed.
Hint Resolve add_assoc : assocmap.
Lemma not_in_assoc :
forall k v l bm,
~ List.In k (List.map (@fst key elt) l) ->
@find elt k bm = Some v ->
@find elt k (fold_right (fun p a => add (fst p) (snd p) a) bm l) = Some v.
Proof.
induction l; intros.
- assumption.
- destruct a as [k' v'].
destruct (peq k k'); subst;
simpl in *; apply Decidable.not_or in H; destruct H. contradiction.
rewrite AssocMap.gso; auto.
Qed.
Hint Resolve not_in_assoc : assocmap.
Lemma elements_iff :
forall am k,
(exists v, find k am = Some v) <->
List.In k (List.map (@fst _ elt) (elements am)).
Proof.
split; intros.
destruct H. apply elements_correct in H. apply in_map with (f := fst) in H. apply H.
apply list_in_map_inv in H. destruct H. destruct H. subst.
exists (snd x). apply elements_complete. assert (x = (fst x, snd x)) by apply surjective_pairing.
rewrite H in H0; assumption.
Qed.
Hint Resolve elements_iff : assocmap.
Lemma elements_correct' :
forall am k,
~ (exists v, find k am = Some v) <->
~ List.In k (List.map (@fst _ elt) (elements am)).
Proof. auto using not_iff_compat with assocmap. Qed.
Hint Resolve elements_correct' : assocmap.
Lemma elements_correct_none :
forall am k,
find k am = None ->
~ List.In k (List.map (@fst _ elt) (elements am)).
Proof.
intros. apply elements_correct'. unfold not. intros.
destruct H0. rewrite H in H0. discriminate.
Qed.
Hint Resolve elements_correct_none : assocmap.
Lemma merge_add :
forall k v am bm,
find k am = Some v ->
find k (merge am bm) = Some v.
Proof. unfold merge. auto with assocmap. Qed.
Hint Resolve merge_add : assocmap.
Lemma merge_not_in :
forall k v am bm,
find k am = None ->
find k bm = Some v ->
find k (merge am bm) = Some v.
Proof. intros. apply not_in_assoc; auto with assocmap. Qed.
Hint Resolve merge_not_in : assocmap.
Lemma merge_base :
forall am,
merge (empty elt) am = am.
Proof. auto. Qed.
Hint Resolve merge_base : assocmap.
End Operations.
End AssocMapExt.
Import AssocMapExt.
Definition assocmap := AssocMap.t value.
Definition find_assocmap (n : nat) : reg -> assocmap -> value :=
find_default value (ZToValue n 0).
Definition empty_assocmap : assocmap := AssocMap.empty value.
Definition merge_assocmap : assocmap -> assocmap -> assocmap := merge value.
Module AssocMapNotation.
Notation "a ! b" := (AssocMap.find b a) (at level 1).
Notation "a # ( b , c )" := (find_assocmap c b a) (at level 1).
Notation "a # b" := (find_assocmap 32 b a) (at level 1).
Notation "a ## b" := (List.map (fun c => find_assocmap 32 c a) b) (at level 1).
End AssocMapNotation.
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