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(*
* Vericert: Verified high-level synthesis.
* Copyright (C) 2020 Yann Herklotz <yann@yannherklotz.com>
* 2020 James Pollard <j@mes.dev>
*
* 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/>.
*)
Require Import compcert.lib.Maps.
Require Import vericert.common.Vericertlib.
Require Import vericert.hls.ValueInt.
Definition reg := positive.
Module AssocMap := Maps.PTree.
Module AssocMap_Properties := Maps.PTree_Properties.
Module AssocMapExt.
Import AssocMap.
Hint Resolve elements_correct elements_complete
elements_keys_norepet : assocmap.
Hint Resolve gso gss : assocmap.
Section Operations.
Variable A : Type.
Definition get_default (a : A) (k : reg) (m : t A) : A :=
match get k m with
| None => a
| Some v => v
end.
Fixpoint merge (m1 m2 : t A) : t A :=
match m1, m2 with
| Node l1 (Some a) r1, Node l2 _ r2 => Node (merge l1 l2) (Some a) (merge r1 r2)
| Node l1 None r1, Node l2 o r2 => Node (merge l1 l2) o (merge r1 r2)
| Leaf, _ => m2
| _, _ => m1
end.
Lemma merge_base_1 :
forall am,
merge (empty A) am = am.
Proof. auto. Qed.
Hint Resolve merge_base_1 : assocmap.
Lemma merge_base_2 :
forall am,
merge am (empty A) = am.
Proof.
unfold merge.
destruct am; trivial.
destruct o; trivial.
Qed.
Hint Resolve merge_base_2 : assocmap.
Lemma merge_add_assoc :
forall r am am' v,
merge (set r v am) am' = set r v (merge am am').
Proof.
induction r; intros; destruct am; destruct am'; try (destruct o); simpl;
try rewrite IHr; try reflexivity.
Qed.
Hint Resolve merge_add_assoc : assocmap.
Lemma merge_correct_1 :
forall am bm k v,
get k am = Some v ->
get k (merge am bm) = Some v.
Proof.
induction am; intros; destruct k; destruct bm; try (destruct o); simpl;
try rewrite gempty in H; try discriminate; try assumption; auto.
Qed.
Hint Resolve merge_correct_1 : assocmap.
Lemma merge_correct_2 :
forall am bm k v,
get k am = None ->
get k bm = Some v ->
get k (merge am bm) = Some v.
Proof.
induction am; intros; destruct k; destruct bm; try (destruct o); simpl;
try rewrite gempty in H; try discriminate; try assumption; auto.
Qed.
Hint Resolve merge_correct_2 : assocmap.
Lemma merge_correct_3 :
forall am bm k,
get k am = None ->
get k bm = None ->
get k (merge am bm) = None.
Proof.
induction am; intros; destruct k; destruct bm; try (destruct o); simpl;
try rewrite gempty in H; try discriminate; try assumption; auto.
Qed.
Hint Resolve merge_correct_3 : assocmap.
Definition merge_fold (am bm : t A) : t A :=
fold_right (fun p a => set (fst p) (snd p) a) bm (elements am).
Lemma add_assoc :
forall (k : elt) (v : A) l bm,
List.In (k, v) l ->
list_norepet (List.map fst l) ->
@get A k (fold_right (fun p a => set (fst p) (snd p) a) bm l) = Some v.
Proof.
induction l; intros.
- contradiction.
- destruct a as [k' v'].
destruct (peq k k').
+ inversion H. inversion H1. inversion H0. subst.
simpl. auto with assocmap.
inversion H0; subst. apply in_map with (f:=fst) in H1. contradiction.
+ 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 elt A) l) ->
@get A k bm = Some v ->
get k (fold_right (fun p a => set (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, get k am = Some v) <->
List.In k (List.map (@fst _ A) (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, get k am = Some v) <->
~ List.In k (List.map (@fst _ A) (elements am)).
Proof. auto using not_iff_compat with assocmap. Qed.
Hint Resolve elements_correct' : assocmap.
Lemma elements_correct_none :
forall am k,
get k am = None ->
~ List.In k (List.map (@fst _ A) (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_fold_add :
forall k v am bm,
am ! k = Some v ->
(merge_fold am bm) ! k = Some v.
Proof. unfold merge_fold; auto with assocmap. Qed.
Hint Resolve merge_fold_add : assocmap.
Lemma merge_fold_not_in :
forall k v am bm,
get k am = None ->
get k bm = Some v ->
get k (merge_fold am bm) = Some v.
Proof. intros. apply not_in_assoc; auto with assocmap. Qed.
Hint Resolve merge_fold_not_in : assocmap.
Lemma merge_fold_base :
forall am,
merge_fold (empty A) am = am.
Proof. auto. Qed.
Hint Resolve merge_fold_base : assocmap.
End Operations.
End AssocMapExt.
Import AssocMapExt.
Definition assocmap := AssocMap.t value.
Definition find_assocmap (n : nat) : reg -> assocmap -> value :=
get_default value (ZToValue 0).
Definition empty_assocmap : assocmap := AssocMap.empty value.
Definition merge_assocmap : assocmap -> assocmap -> assocmap := merge value.
Ltac unfold_merge :=
unfold merge_assocmap; try (repeat (rewrite merge_add_assoc));
rewrite AssocMapExt.merge_base_1.
Declare Scope assocmap.
Notation "a ! b" := (AssocMap.get b a) (at level 1) : assocmap.
Notation "a # ( b , c )" := (find_assocmap c b a) (at level 1) : assocmap.
Notation "a # b" := (find_assocmap 32 b a) (at level 1) : assocmap.
Notation "a ## b" := (List.map (fun c => find_assocmap 32 c a) b) (at level 1) : assocmap.
Notation "a # b '<-' c" := (AssocMap.set b c a) (at level 1, b at next level) : assocmap.
Local Open Scope assocmap.
Lemma find_get_assocmap :
forall assoc r v,
assoc ! r = Some v ->
assoc # r = v.
Proof. intros. unfold find_assocmap, AssocMapExt.get_default. rewrite H. trivial. Qed.
Lemma fso : forall m v k1 k2, k1 <> k2 -> (m # k1 <- v) # k2 = m # k2.
Proof.
unfold "_ # _".
unfold AssocMapExt.get_default.
intros.
destruct_match; rewrite AssocMap.gso in Heqo by auto; rewrite Heqo; auto.
Qed.
Lemma fss : forall m v k, (m # k <- v) # k = v.
Proof.
unfold "_ # _".
unfold AssocMapExt.get_default.
intros.
destruct_match; rewrite AssocMap.gss in Heqo by auto; inv Heqo; crush.
Qed.
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