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(*
* Vericert: Verified high-level synthesis.
* Copyright (C) 2020-2022 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 AssocMapExt.
Import AssocMap.
#[export] Hint Resolve elements_correct elements_complete elements_keys_norepet : assocmap.
#[export] 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.
Definition merge_atom (new : option A) (old : option A) : option A :=
match new, old with
| Some _, _ => new
| _, _ => old
end.
Definition merge (m1 m2 : t A) : t A :=
PTree.combine merge_atom m1 m2.
Lemma merge_base_1 :
forall am x,
(merge (empty A) am) ! x = am ! x.
Proof using.
unfold merge; intros.
rewrite PTree.gcombine; eauto.
Qed.
Lemma merge_base_2 :
forall am x,
(merge am (empty A)) ! x = am ! x.
Proof using.
unfold merge; intros.
rewrite PTree.gcombine; eauto.
cbv [merge_atom]. destruct_match; crush.
Qed.
Lemma merge_add_assoc :
forall r am am' v x,
(merge (set r v am) am') ! x = (set r v (merge am am')) ! x.
Proof using.
unfold merge. intros; rewrite PTree.gcombine by auto.
unfold merge_atom. destruct_match.
destruct (peq r x); subst.
rewrite PTree.gss in Heqo. inv Heqo. rewrite PTree.gss. auto.
rewrite PTree.gso in Heqo by auto. rewrite PTree.gso by auto.
rewrite PTree.gcombine by auto. rewrite Heqo. auto.
destruct (peq r x); subst.
rewrite PTree.gss in Heqo. inv Heqo.
rewrite PTree.gso in Heqo by auto. rewrite PTree.gso by auto.
rewrite PTree.gcombine by auto. rewrite Heqo. auto.
Qed.
Lemma merge_correct_1 :
forall am bm k v,
am ! k = Some v ->
(merge am bm) ! k = Some v.
Proof using.
unfold merge; intros. rewrite PTree.gcombine by auto.
rewrite H; auto.
Qed.
Lemma merge_correct_2 :
forall am bm k v,
am ! k = None ->
bm ! k = Some v ->
(merge am bm) ! k = Some v.
Proof using.
unfold merge; intros. rewrite PTree.gcombine by auto.
rewrite H. rewrite H0. auto.
Qed.
Lemma merge_correct_3 :
forall am bm k,
am ! k = None ->
bm ! k = None ->
(merge am bm) ! k = None.
Proof using.
unfold merge; intros. rewrite PTree.gcombine by auto.
rewrite H. rewrite H0. auto.
Qed.
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.
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.
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.
#[local] Hint Resolve merge_base_1 : core.
#[local] Hint Resolve merge_base_2 : core.
#[local] Hint Resolve merge_add_assoc : core.
#[local] Hint Resolve merge_correct_1 : core.
#[local] Hint Resolve merge_correct_2 : core.
#[local] Hint Resolve merge_correct_3 : core.
#[local] Hint Resolve add_assoc : core.
#[local] Hint Resolve not_in_assoc : core.
#[local] Hint Resolve elements_iff : core.
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. Qed.
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.
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.
#[local] Hint Resolve elements_correct' : core.
#[local] Hint Resolve elements_correct_none : core.
#[local] Hint Resolve merge_fold_add : core.
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. Qed.
Lemma merge_fold_base :
forall am,
merge_fold (empty A) am = am.
Proof. auto. Qed.
End Operations.
#[export] Hint Resolve merge_base_1 : assocmap.
#[export] Hint Resolve merge_base_2 : assocmap.
#[export] Hint Resolve merge_add_assoc : assocmap.
#[export] Hint Resolve merge_correct_1 : assocmap.
#[export] Hint Resolve merge_correct_2 : assocmap.
#[export] Hint Resolve merge_correct_3 : assocmap.
#[export] Hint Resolve add_assoc : assocmap.
#[export] Hint Resolve not_in_assoc : assocmap.
#[export] Hint Resolve elements_iff : assocmap.
#[export] Hint Resolve elements_correct' : assocmap.
#[export] Hint Resolve merge_fold_not_in : assocmap.
#[export] Hint Resolve merge_fold_base : assocmap.
#[export] Hint Resolve elements_correct_none : assocmap.
#[export] Hint Resolve merge_fold_add : assocmap.
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 merge_get_default :
forall ars ars' r x,
ars ! r = Some x ->
(AssocMapExt.merge _ ars ars') # r = x.
Proof.
unfold AssocMapExt.merge; intros.
unfold "#", AssocMapExt.get_default.
rewrite AssocMap.gcombine by auto.
unfold AssocMapExt.merge_atom.
now rewrite !H.
Qed.
Lemma merge_get_default2 :
forall ars ars' r,
ars ! r = None ->
(AssocMapExt.merge _ ars ars') # r = ars' # r.
Proof.
unfold AssocMapExt.merge; intros.
unfold "#", AssocMapExt.get_default.
rewrite AssocMap.gcombine by auto.
unfold AssocMapExt.merge_atom.
now rewrite !H.
Qed.
Lemma merge_get_default3 :
forall A ars ars' r,
ars ! r = None ->
(AssocMapExt.merge A ars ars') ! r = ars' ! r.
Proof.
unfold AssocMapExt.merge; intros.
unfold "#", AssocMapExt.get_default.
rewrite AssocMap.gcombine by auto.
unfold AssocMapExt.merge_atom.
now rewrite !H.
Qed.
Lemma merge_get_default4 :
forall A ars ars' r x,
ars ! r = Some x ->
(AssocMapExt.merge A ars ars') ! r = Some x.
Proof.
unfold AssocMapExt.merge; intros.
unfold "#", AssocMapExt.get_default.
rewrite AssocMap.gcombine by auto.
unfold AssocMapExt.merge_atom.
now rewrite !H.
Qed.
Lemma merge_left_gso :
forall A ars ars' d (v: A) r,
d <> r ->
(AssocMapExt.merge _ (ars # d <- v) ars') ! r = (AssocMapExt.merge _ ars ars') ! r.
Proof.
unfold AssocMapExt.merge; intros.
rewrite ! AssocMap.gcombine by auto.
now rewrite AssocMap.gso by auto.
Qed.
Lemma merge_right_gso :
forall A ars ars' d (v: A) r,
d <> r ->
(AssocMapExt.merge _ ars (ars' # d <- v)) ! r = (AssocMapExt.merge _ ars ars') ! r.
Proof.
unfold AssocMapExt.merge; intros.
rewrite ! AssocMap.gcombine by auto.
now rewrite AssocMap.gso by auto.
Qed.
|
