Lvc.Constr.MapLookupList
Require Export Setoid Coq.Classes.Morphisms.
Require Import EqDec Computable Util LengthEq AutoIndTac.
Require Export CSet Containers.SetDecide MoreList.
Require Export MapBasics MapLookup MapUpdate.
Set Implicit Arguments.
Section MapLookupList.
Variable X : Type.
Context `{OrderedType X}.
Variable Y : Type.
Open Scope fmap_scope.
Fixpoint lookup_list (m:X → Y) (L:list X) : list Y :=
match L with
| nil ⇒ nil
| x::L ⇒ m x::lookup_list m L
end.
Lemma update_with_list_app (A A' : list X) (B B' : list Y) E
: length A = length B
→ update_with_list (A++A') (B++B') E = update_with_list A B (update_with_list A' B' E).
Proof.
intros. eapply length_length_eq in H0. general induction H0; simpl; eauto.
rewrite IHlength_eq; eauto.
Qed.
Lemma lookup_list_length (m:X → Y) (L:list X)
: length (lookup_list m L) = length L.
Proof.
induction L; simpl; eauto.
Qed.
Lemma lookup_list_agree (m m':X → Y) (L:list X)
: agree_on eq (of_list L) m m'
→ lookup_list m L = lookup_list m' L.
Proof.
general induction L; simpl in × |- *; eauto.
f_equal. eapply H0; cset_tac; eauto.
eapply IHL; eapply agree_on_incl; eauto. cset_tac; eauto.
Qed.
Lemma of_list_lookup_list `{OrderedType Y} (m:X → Y) L
: Proper (_eq ==> _eq) m
→ of_list (lookup_list m L) [=] lookup_set m (of_list L).
Proof.
general induction L; simpl.
- intros x. cset_tac; firstorder.
- rewrite IHL; eauto. intros x. split; intros.
+ eapply lookup_set_spec; eauto. eapply add_iff in H2; destruct H2.
× eexists a; split; eauto. eapply add_1; eauto.
× eapply lookup_set_spec in H2; eauto. dcr. eexists x0; split; eauto.
eapply add_2; eauto.
+ eapply lookup_set_spec in H2; eauto. dcr.
eapply add_iff in H4; destruct H4.
× eapply add_1. rewrite H5. eapply H1. eapply H2.
× eapply add_2. eapply lookup_set_spec; eauto.
Qed.
End MapLookupList.
Lemma lookup_id X (l:list X)
: lookup_list (@id X) l = l.
Proof.
general induction l; simpl; eauto.
f_equal; eauto.
Qed.
Global Instance update_with_list_inst X `{OrderedType X} Y `{OrderedType Y} :
Proper (eq ==> eq ==> (@feq X Y _eq ) ==> (@feq _ _ _eq)) (@update_with_list X _ Y).
Proof.
unfold respectful, Proper; intros. subst.
general induction y; simpl; eauto.
destruct y0; eauto. hnf; intros.
specialize (IHy y2 x1 y1 H3).
eapply update_inst; eauto.
Qed.
Global Instance lookup_list_inst X `{OrderedType X} Y:
Proper ((@feq X Y eq) ==> eq ==> eq) (@lookup_list X Y).
Proof.
unfold respectful, Proper, update, feq; intros; subst.
general induction y0; eauto.
simpl. f_equal; eauto.
Qed.
Lemma update_with_list_lookup_list X `{OrderedType X} Y `{OrderedType Y} (E:X → Y)
`{Proper _ (_eq ==> _eq) E} (Z : list X)
: @feq _ _ _eq (update_with_list Z (lookup_list E Z) E) E.
Proof.
general induction Z; simpl.
+ reflexivity.
+ setoid_rewrite IHZ; eauto. rewrite update_id; eauto. reflexivity.
Qed.
Lemma lookup_list_app X Y (A A':list X) (E:X → Y)
: lookup_list E (A ++ A') = List.app (lookup_list E A) (lookup_list E A').
Proof.
general induction A; simpl; eauto.
rewrite IHA; eauto.
Qed.
Lemma lookup_list_nodup X `{OrderedType X} Y (Z:list X) (Z':list Y) f
: length Z = length Z'
→ NoDupA _eq Z
→ lookup_list (f [Z <-- Z']) Z = Z'.
Proof.
intros. length_equify. general induction H0; simpl in *; dcr; eauto.
- f_equal.
+ lud; intuition.
+ erewrite lookup_list_agree; eauto.
eapply agree_on_update_dead; eauto. reflexivity.
Qed.
Hint Resolve lookup_list_agree : cset.
Lemma lookup_list_map X Y (f:X→Y) L
: lookup_list f L = f ⊝ L.
Proof.
induction L; simpl; f_equal; eauto.
Qed.
Smpl Add match goal with
| [ |- context [ ❬@lookup_list ?X ?Y ?f ?L❭ ] ] ⇒
rewrite (@lookup_list_length X Y f L)
end : len.
Lemma map_app X `{OrderedType X} Y `{OrderedType Y} (f:X→Y)
`{Proper _ (_eq ==> _eq) f} s t
: map f (s ∪ t) [=] map f s ∪ map f t.
Proof.
cset_tac.
Qed.
Lemma map_add X `{OrderedType X} Y `{OrderedType Y} (f:X→Y)
`{Proper _ (_eq ==> _eq) f} x t
: map f ({x; t}) [=] {f x; map f t}.
Proof.
cset_tac.
Qed.
Lemma map_empty X `{OrderedType X} Y `{OrderedType Y} (f:X→Y)
`{Proper _ (_eq ==> _eq) f}
: map f ∅ [=] ∅.
Proof.
cset_tac.
Qed.
Instance map_Proper X `{OrderedType X} Y `{OrderedType Y}
: Proper (@fpeq X Y _eq _ _ ==> _eq ==> _eq) map.
Proof.
unfold Proper, respectful; intros. inv H1; dcr.
hnf; intros. cset_tac'.
- eexists x1. rewrite <- H2, H9. split; eauto. eapply H3.
- eexists x1. rewrite H2, H9. split; eauto. symmetry. eapply H3.
Qed.
Lemma map_single {X} `{OrderedType X} Y `{OrderedType Y} (f:X→Y)
`{Proper _ (_eq ==> _eq) f} x
: map f {{x}} [=] {{f x}}.
Proof.
hnf; intros. rewrite map_iff; eauto.
split; intros.
- destruct H2; dcr. cset_tac'. rewrite H3; eauto.
- cset_tac.
Qed.
Lemma of_list_map X `{OrderedType X} Y `{OrderedType Y}
(f:X→Y) `{Proper _ (_eq ==> _eq) f} L
: of_list (f ⊝ L) [=] map f (of_list L).
Proof.
general induction L; simpl; eauto.
- rewrite map_add; eauto.
rewrite IHL; eauto; reflexivity.
Qed.
Lemma map_union X `{OrderedType X} Y `{OrderedType Y} (f:X→Y) `{Proper _ (_eq ==> _eq) f} s t
: map f (s ∪ t) [=] map f s ∪ map f t.
Proof.
cset_tac; eauto.
Qed.
Lemma agree_on_update_list X `{OrderedType X} Y (L:list X) (L':list Y) (V:X→Y)
`{Proper _ (_eq ==> eq) V} V' D (Len:❬L❭= ❬L'❭)
: agree_on eq (D \ of_list L) V V'
→ lookup_list V L = L'
→ agree_on eq D V (V'[L <-- L']).
Proof.
intros. hnf; intros.
decide (x ∈ of_list L).
- edestruct update_with_list_lookup_in_list; try eapply i; dcr.
Focus 2. rewrite H7.
rewrite lookup_list_map in H2. subst. inv_get.
eapply H0; eauto. eauto.
- rewrite lookup_set_update_not_in_Z; eauto.
eapply H1; cset_tac.
Qed.
Lemma map_singleton X `{OrderedType X}
(f : X → X) `{Proper _ (_eq ==> _eq) f} (x : X)
: map f (singleton x) [=] singleton (f x)
.
Proof.
apply set_incl.
- cset_tac.
- cset_tac.
Qed.
Require Import EqDec Computable Util LengthEq AutoIndTac.
Require Export CSet Containers.SetDecide MoreList.
Require Export MapBasics MapLookup MapUpdate.
Set Implicit Arguments.
Section MapLookupList.
Variable X : Type.
Context `{OrderedType X}.
Variable Y : Type.
Open Scope fmap_scope.
Fixpoint lookup_list (m:X → Y) (L:list X) : list Y :=
match L with
| nil ⇒ nil
| x::L ⇒ m x::lookup_list m L
end.
Lemma update_with_list_app (A A' : list X) (B B' : list Y) E
: length A = length B
→ update_with_list (A++A') (B++B') E = update_with_list A B (update_with_list A' B' E).
Proof.
intros. eapply length_length_eq in H0. general induction H0; simpl; eauto.
rewrite IHlength_eq; eauto.
Qed.
Lemma lookup_list_length (m:X → Y) (L:list X)
: length (lookup_list m L) = length L.
Proof.
induction L; simpl; eauto.
Qed.
Lemma lookup_list_agree (m m':X → Y) (L:list X)
: agree_on eq (of_list L) m m'
→ lookup_list m L = lookup_list m' L.
Proof.
general induction L; simpl in × |- *; eauto.
f_equal. eapply H0; cset_tac; eauto.
eapply IHL; eapply agree_on_incl; eauto. cset_tac; eauto.
Qed.
Lemma of_list_lookup_list `{OrderedType Y} (m:X → Y) L
: Proper (_eq ==> _eq) m
→ of_list (lookup_list m L) [=] lookup_set m (of_list L).
Proof.
general induction L; simpl.
- intros x. cset_tac; firstorder.
- rewrite IHL; eauto. intros x. split; intros.
+ eapply lookup_set_spec; eauto. eapply add_iff in H2; destruct H2.
× eexists a; split; eauto. eapply add_1; eauto.
× eapply lookup_set_spec in H2; eauto. dcr. eexists x0; split; eauto.
eapply add_2; eauto.
+ eapply lookup_set_spec in H2; eauto. dcr.
eapply add_iff in H4; destruct H4.
× eapply add_1. rewrite H5. eapply H1. eapply H2.
× eapply add_2. eapply lookup_set_spec; eauto.
Qed.
End MapLookupList.
Lemma lookup_id X (l:list X)
: lookup_list (@id X) l = l.
Proof.
general induction l; simpl; eauto.
f_equal; eauto.
Qed.
Global Instance update_with_list_inst X `{OrderedType X} Y `{OrderedType Y} :
Proper (eq ==> eq ==> (@feq X Y _eq ) ==> (@feq _ _ _eq)) (@update_with_list X _ Y).
Proof.
unfold respectful, Proper; intros. subst.
general induction y; simpl; eauto.
destruct y0; eauto. hnf; intros.
specialize (IHy y2 x1 y1 H3).
eapply update_inst; eauto.
Qed.
Global Instance lookup_list_inst X `{OrderedType X} Y:
Proper ((@feq X Y eq) ==> eq ==> eq) (@lookup_list X Y).
Proof.
unfold respectful, Proper, update, feq; intros; subst.
general induction y0; eauto.
simpl. f_equal; eauto.
Qed.
Lemma update_with_list_lookup_list X `{OrderedType X} Y `{OrderedType Y} (E:X → Y)
`{Proper _ (_eq ==> _eq) E} (Z : list X)
: @feq _ _ _eq (update_with_list Z (lookup_list E Z) E) E.
Proof.
general induction Z; simpl.
+ reflexivity.
+ setoid_rewrite IHZ; eauto. rewrite update_id; eauto. reflexivity.
Qed.
Lemma lookup_list_app X Y (A A':list X) (E:X → Y)
: lookup_list E (A ++ A') = List.app (lookup_list E A) (lookup_list E A').
Proof.
general induction A; simpl; eauto.
rewrite IHA; eauto.
Qed.
Lemma lookup_list_nodup X `{OrderedType X} Y (Z:list X) (Z':list Y) f
: length Z = length Z'
→ NoDupA _eq Z
→ lookup_list (f [Z <-- Z']) Z = Z'.
Proof.
intros. length_equify. general induction H0; simpl in *; dcr; eauto.
- f_equal.
+ lud; intuition.
+ erewrite lookup_list_agree; eauto.
eapply agree_on_update_dead; eauto. reflexivity.
Qed.
Hint Resolve lookup_list_agree : cset.
Lemma lookup_list_map X Y (f:X→Y) L
: lookup_list f L = f ⊝ L.
Proof.
induction L; simpl; f_equal; eauto.
Qed.
Smpl Add match goal with
| [ |- context [ ❬@lookup_list ?X ?Y ?f ?L❭ ] ] ⇒
rewrite (@lookup_list_length X Y f L)
end : len.
Lemma map_app X `{OrderedType X} Y `{OrderedType Y} (f:X→Y)
`{Proper _ (_eq ==> _eq) f} s t
: map f (s ∪ t) [=] map f s ∪ map f t.
Proof.
cset_tac.
Qed.
Lemma map_add X `{OrderedType X} Y `{OrderedType Y} (f:X→Y)
`{Proper _ (_eq ==> _eq) f} x t
: map f ({x; t}) [=] {f x; map f t}.
Proof.
cset_tac.
Qed.
Lemma map_empty X `{OrderedType X} Y `{OrderedType Y} (f:X→Y)
`{Proper _ (_eq ==> _eq) f}
: map f ∅ [=] ∅.
Proof.
cset_tac.
Qed.
Instance map_Proper X `{OrderedType X} Y `{OrderedType Y}
: Proper (@fpeq X Y _eq _ _ ==> _eq ==> _eq) map.
Proof.
unfold Proper, respectful; intros. inv H1; dcr.
hnf; intros. cset_tac'.
- eexists x1. rewrite <- H2, H9. split; eauto. eapply H3.
- eexists x1. rewrite H2, H9. split; eauto. symmetry. eapply H3.
Qed.
Lemma map_single {X} `{OrderedType X} Y `{OrderedType Y} (f:X→Y)
`{Proper _ (_eq ==> _eq) f} x
: map f {{x}} [=] {{f x}}.
Proof.
hnf; intros. rewrite map_iff; eauto.
split; intros.
- destruct H2; dcr. cset_tac'. rewrite H3; eauto.
- cset_tac.
Qed.
Lemma of_list_map X `{OrderedType X} Y `{OrderedType Y}
(f:X→Y) `{Proper _ (_eq ==> _eq) f} L
: of_list (f ⊝ L) [=] map f (of_list L).
Proof.
general induction L; simpl; eauto.
- rewrite map_add; eauto.
rewrite IHL; eauto; reflexivity.
Qed.
Lemma map_union X `{OrderedType X} Y `{OrderedType Y} (f:X→Y) `{Proper _ (_eq ==> _eq) f} s t
: map f (s ∪ t) [=] map f s ∪ map f t.
Proof.
cset_tac; eauto.
Qed.
Lemma agree_on_update_list X `{OrderedType X} Y (L:list X) (L':list Y) (V:X→Y)
`{Proper _ (_eq ==> eq) V} V' D (Len:❬L❭= ❬L'❭)
: agree_on eq (D \ of_list L) V V'
→ lookup_list V L = L'
→ agree_on eq D V (V'[L <-- L']).
Proof.
intros. hnf; intros.
decide (x ∈ of_list L).
- edestruct update_with_list_lookup_in_list; try eapply i; dcr.
Focus 2. rewrite H7.
rewrite lookup_list_map in H2. subst. inv_get.
eapply H0; eauto. eauto.
- rewrite lookup_set_update_not_in_Z; eauto.
eapply H1; cset_tac.
Qed.
Lemma map_singleton X `{OrderedType X}
(f : X → X) `{Proper _ (_eq ==> _eq) f} (x : X)
: map f (singleton x) [=] singleton (f x)
.
Proof.
apply set_incl.
- cset_tac.
- cset_tac.
Qed.