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Commit ebcb867c authored by Robbert Krebbers's avatar Robbert Krebbers Committed by Ralf Jung
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theories/base.v
View file @ ebcb867c
......@@ -23,6 +23,7 @@ Arguments compose _ _ _ _ _ _ /.
Arguments flip _ _ _ _ _ _ /.
Arguments const _ _ _ _ /.
Typeclasses Transparent id compose flip const.
Instance: Params (@pair) 2.
(** Change [True] and [False] into notations in order to enable overloading.
We will use this in the file [assertions] to give [True] and [False] a
......@@ -792,6 +793,11 @@ Instance pointwise_transitive {A} `{R : relation B} :
Transitive R → Transitive (pointwise_relation A R) | 9.
Proof. firstorder. Qed.
(** ** Unit *)
Instance unit_equiv : Equiv unit := λ _ _, True.
Instance unit_equivalence : Equivalence (@equiv unit _).
Proof. repeat split. Qed.
(** ** Products *)
Instance prod_map_injective {A A' B B'} (f : A → A') (g : B → B') :
Injective (=) (=) f → Injective (=) (=) g →
......@@ -825,6 +831,15 @@ Section prod_relation.
Proof. firstorder eauto. Qed.
End prod_relation.
Instance prod_equiv `{Equiv A,Equiv B} : Equiv (A * B) := prod_relation (≡) (≡).
Instance pair_proper `{Equiv A, Equiv B} :
Proper ((≡) ==> (≡) ==> (≡)) (@pair A B) | 0 := _.
Instance fst_proper `{Equiv A, Equiv B} :
Proper ((≡) ==> (≡)) (@fst A B) | 0 := _.
Instance snd_proper `{Equiv A, Equiv B} :
Proper ((≡) ==> (≡)) (@snd A B) | 0 := _.
Typeclasses Opaque prod_equiv.
(** ** Other *)
Lemma or_l P Q : ¬Q → P ∨ Q ↔ P.
Proof. tauto. Qed.
......
theories/collections.v
View file @ ebcb867c
......@@ -249,7 +249,7 @@ Ltac unfold_elem_of :=
For goals that do not involve [≡], [⊆], [map], or quantifiers this tactic is
generally powerful enough. This tactic either fails or proves the goal. *)
Tactic Notation "solve_elem_of" tactic3(tac) :=
simpl in *;
setoid_subst;
decompose_empty;
unfold_elem_of;
solve [intuition (simplify_equality; tac)].
......@@ -261,7 +261,7 @@ fails or loops on very small goals generated by [solve_elem_of] already. We
use the [naive_solver] tactic as a substitute. This tactic either fails or
proves the goal. *)
Tactic Notation "esolve_elem_of" tactic3(tac) :=
simpl in *;
setoid_subst;
decompose_empty;
unfold_elem_of;
naive_solver tac.
......@@ -273,6 +273,11 @@ Section collection.
Global Instance: Lattice C.
Proof. split. apply _. firstorder auto. solve_elem_of. Qed.
Global Instance difference_proper : Proper ((≡) ==> (≡) ==> (≡)) (∖).
Proof.
intros X1 X2 HX Y1 Y2 HY; apply elem_of_equiv; intros x.
by rewrite !elem_of_difference, HX, HY.
Qed.
Lemma intersection_singletons x : {[x]} ∩ {[x]} ≡ {[x]}.
Proof. esolve_elem_of. Qed.
Lemma difference_twice X Y : (X ∖ Y) ∖ Y ≡ X ∖ Y.
......@@ -283,6 +288,8 @@ Section collection.
Proof. esolve_elem_of. Qed.
Lemma difference_union_distr_l X Y Z : (X ∪ Y) ∖ Z ≡ X ∖ Z ∪ Y ∖ Z.
Proof. esolve_elem_of. Qed.
Lemma difference_union_distr_r X Y Z : Z ∖ (X ∪ Y) ≡ (Z ∖ X) ∩ (Z ∖ Y).
Proof. esolve_elem_of. Qed.
Lemma difference_intersection_distr_l X Y Z : (X ∩ Y) ∖ Z ≡ X ∖ Z ∩ Y ∖ Z.
Proof. esolve_elem_of. Qed.
......@@ -298,6 +305,8 @@ Section collection.
Proof. unfold_leibniz. apply difference_diag. Qed.
Lemma difference_union_distr_l_L X Y Z : (X ∪ Y) ∖ Z = X ∖ Z ∪ Y ∖ Z.
Proof. unfold_leibniz. apply difference_union_distr_l. Qed.
Lemma difference_union_distr_r_L X Y Z : Z ∖ (X ∪ Y) = (Z ∖ X) ∩ (Z ∖ Y).
Proof. unfold_leibniz. apply difference_union_distr_r. Qed.
Lemma difference_intersection_distr_l_L X Y Z :
(X ∩ Y) ∖ Z = X ∖ Z ∩ Y ∖ Z.
Proof. unfold_leibniz. apply difference_intersection_distr_l. Qed.
......@@ -518,16 +527,13 @@ Section collection_monad.
Global Instance collection_fmap_proper {A B} (f : A → B) :
Proper ((≡) ==> (≡)) (fmap f).
Proof. intros X Y E. esolve_elem_of. Qed.
Global Instance collection_ret_proper {A} :
Proper ((=) ==> (≡)) (@mret M _ A).
Proof. intros X Y E. esolve_elem_of. Qed.
Proof. intros X Y [??]; split; esolve_elem_of. Qed.
Global Instance collection_bind_proper {A B} (f : A → M B) :
Proper ((≡) ==> (≡)) (mbind f).
Proof. intros X Y E. esolve_elem_of. Qed.
Proof. intros X Y [??]; split; esolve_elem_of. Qed.
Global Instance collection_join_proper {A} :
Proper ((≡) ==> (≡)) (@mjoin M _ A).
Proof. intros X Y E. esolve_elem_of. Qed.
Proof. intros X Y [??]; split; esolve_elem_of. Qed.
Lemma collection_bind_singleton {A B} (f : A → M B) x : {[ x ]} ≫= f ≡ f x.
Proof. esolve_elem_of. Qed.
......
theories/fin_collections.v
View file @ ebcb867c
......@@ -3,7 +3,7 @@
(** This file collects definitions and theorems on finite collections. Most
importantly, it implements a fold and size function and some useful induction
principles on finite collections . *)
Require Import Permutation ars listset.
Require Import Permutation relations listset.
Require Export numbers collections.
Instance collection_size `{Elements A C} : Size C := length ∘ elements.
......
theories/fin_map_dom.v
View file @ ebcb867c
......@@ -77,15 +77,15 @@ Proof. rewrite not_elem_of_dom. apply delete_partial_alter. Qed.
Lemma delete_insert_dom {A} (m : M A) i x :
i ∉ dom D m → delete i (<[i:=x]>m) = m.
Proof. rewrite not_elem_of_dom. apply delete_insert. Qed.
Lemma map_disjoint_dom {A} (m1 m2 : M A) : m1 ⊥ m2 ↔ dom D m1 ∩ dom D m2 ≡ ∅.
Lemma map_disjoint_dom {A} (m1 m2 : M A) : m1 ⊥ₘ m2 ↔ dom D m1 ∩ dom D m2 ≡ ∅.
Proof.
rewrite map_disjoint_spec, elem_of_equiv_empty.
setoid_rewrite elem_of_intersection.
setoid_rewrite elem_of_dom. unfold is_Some. naive_solver.
Qed.
Lemma map_disjoint_dom_1 {A} (m1 m2 : M A) : m1 ⊥ m2 → dom D m1 ∩ dom D m2 ≡ ∅.
Lemma map_disjoint_dom_1 {A} (m1 m2 : M A) : m1 ⊥ₘ m2 → dom D m1 ∩ dom D m2 ≡ ∅.
Proof. apply map_disjoint_dom. Qed.
Lemma map_disjoint_dom_2 {A} (m1 m2 : M A) : dom D m1 ∩ dom D m2 ≡ ∅ → m1 ⊥ m2.
Lemma map_disjoint_dom_2 {A} (m1 m2 : M A) : dom D m1 ∩ dom D m2 ≡ ∅ → m1 ⊥ₘ m2.
Proof. apply map_disjoint_dom. Qed.
Lemma dom_union {A} (m1 m2 : M A) : dom D (m1 ∪ m2) ≡ dom D m1 ∪ dom D m2.
Proof.
......
theories/fin_maps.v
View file @ ebcb867c
......@@ -5,7 +5,7 @@ finite maps and collects some theory on it. Most importantly, it proves useful
induction principles for finite maps and implements the tactic
[simplify_map_equality] to simplify goals involving finite maps. *)
Require Import Permutation.
Require Export ars vector orders.
Require Export relations vector orders.
(** * Axiomatization of finite maps *)
(** We require Leibniz equality to be extensional on finite maps. This of
......@@ -71,25 +71,26 @@ Instance map_intersection_with `{Merge M} {A} : IntersectionWith A (M A) :=
Instance map_difference_with `{Merge M} {A} : DifferenceWith A (M A) :=
λ f, merge (difference_with f).
Instance map_equiv `{∀ A, Lookup K A (M A), Equiv A} : Equiv (M A) | 1 := λ m1 m2,
∀ i, m1 !! i ≡ m2 !! i.
(** The relation [intersection_forall R] on finite maps describes that the
relation [R] holds for each pair in the intersection. *)
Definition map_Forall `{Lookup K A M} (P : K → A → Prop) : M → Prop :=
λ m, ∀ i x, m !! i = Some x → P i x.
Definition map_Forall2 `{∀ A, Lookup K A (M A)} {A B}
(R : A → B → Prop) (P : A → Prop) (Q : B → Prop)
(m1 : M A) (m2 : M B) : Prop := ∀ i,
match m1 !! i, m2 !! i with
| Some x, Some y => R x y
| Some x, None => P x
| None, Some y => Q y
| None, None => True
end.
Definition map_relation `{∀ A, Lookup K A (M A)} {A B} (R : A → B → Prop)
(P : A → Prop) (Q : B → Prop) (m1 : M A) (m2 : M B) : Prop := ∀ i,
option_relation R P Q (m1 !! i) (m2 !! i).
Definition map_included `{∀ A, Lookup K A (M A)} {A}
(R : relation A) : relation (M A) := map_Forall2 R (λ _, False) (λ _, True).
Instance map_disjoint `{∀ A, Lookup K A (M A)} {A} : Disjoint (M A) :=
map_Forall2 (λ _ _, False) (λ _, True) (λ _, True).
(R : relation A) : relation (M A) := map_relation R (λ _, False) (λ _, True).
Definition map_disjoint `{∀ A, Lookup K A (M A)} {A} : relation (M A) :=
map_relation (λ _ _, False) (λ _, True) (λ _, True).
Infix "⊥ₘ" := map_disjoint (at level 70) : C_scope.
Hint Extern 0 (_ ⊥ₘ _) => symmetry; eassumption.
Notation "( m ⊥ₘ.)" := (map_disjoint m) (only parsing) : C_scope.
Notation "(.⊥ₘ m )" := (λ m2, m2 ⊥ₘ m) (only parsing) : C_scope.
Instance map_subseteq `{∀ A, Lookup K A (M A)} {A} : SubsetEq (M A) :=
map_Forall2 (=) (λ _, False) (λ _, True).
map_included (=).
(** The union of two finite maps only has a meaningful definition for maps
that are disjoint. However, as working with partial functions is inconvenient
......@@ -114,12 +115,67 @@ Definition map_imap `{∀ A, Insert K A (M A), ∀ A, Empty (M A),
Section theorems.
Context `{FinMap K M}.
(** ** Setoids *)
Section setoid.
Context `{Equiv A}.
Global Instance map_equivalence `{!Equivalence ((≡) : relation A)} :
Equivalence ((≡) : relation (M A)).
Proof.
split.
* by intros m i.
* by intros m1 m2 ? i.
* by intros m1 m2 m3 ?? i; transitivity (m2 !! i).
Qed.
Global Instance lookup_proper (i : K) :
Proper ((≡) ==> (≡)) (lookup (M:=M A) i).
Proof. by intros m1 m2 Hm. Qed.
Global Instance partial_alter_proper :
Proper (((≡) ==> (≡)) ==> (=) ==> (≡) ==> (≡)) partial_alter.
Proof.
by intros f1 f2 Hf i ? <- m1 m2 Hm j; destruct (decide (i = j)) as [->|];
rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne by done;
try apply Hf; apply lookup_proper.
Qed.
Global Instance insert_proper (i : K) :
Proper ((≡) ==> (≡) ==> (≡)) (insert (M:=M A) i).
Proof. by intros ???; apply partial_alter_proper; [constructor|]. Qed.
Global Instance delete_proper (i : K) :
Proper ((≡) ==> (≡)) (delete (M:=M A) i).
Proof. by apply partial_alter_proper; [constructor|]. Qed.
Global Instance alter_proper :
Proper (((≡) ==> (≡)) ==> (=) ==> (≡) ==> (≡)) (alter (A:=A) (M:=M A)).
Proof.
intros ?? Hf; apply partial_alter_proper.
by destruct 1; constructor; apply Hf.
Qed.
Lemma merge_ext f g
`{!PropHolds (f None None = None), !PropHolds (g None None = None)} :
((≡) ==> (≡) ==> (≡))%signature f g →
((≡) ==> (≡) ==> (≡))%signature (merge f) (merge g).
Proof.
by intros Hf ?? Hm1 ?? Hm2 i; rewrite !lookup_merge by done; apply Hf.
Qed.
Global Instance union_with_proper :
Proper (((≡) ==> (≡) ==> (≡)) ==> (≡) ==> (≡) ==> (≡)) union_with.
Proof.
intros ?? Hf ?? Hm1 ?? Hm2 i; apply (merge_ext _ _); auto.
by do 2 destruct 1; first [apply Hf | constructor].
Qed.
Global Instance map_leibniz `{!LeibnizEquiv A} : LeibnizEquiv (M A).
Proof.
intros m1 m2; split.
* by intros Hm; apply map_eq; intros i; unfold_leibniz; apply lookup_proper.
* by intros <-; intros i; fold_leibniz.
Qed.
End setoid.
(** ** General properties *)
Lemma map_eq_iff {A} (m1 m2 : M A) : m1 = m2 ↔ ∀ i, m1 !! i = m2 !! i.
Proof. split. by intros ->. apply map_eq. Qed.
Lemma map_subseteq_spec {A} (m1 m2 : M A) :
m1 ⊆ m2 ↔ ∀ i x, m1 !! i = Some x → m2 !! i = Some x.
Proof.
unfold subseteq, map_subseteq, map_Forall2. split; intros Hm i;
unfold subseteq, map_subseteq, map_relation. split; intros Hm i;
specialize (Hm i); destruct (m1 !! i), (m2 !! i); naive_solver.
Qed.
Global Instance: EmptySpec (M A).
......@@ -129,9 +185,10 @@ Proof.
Qed.
Global Instance: ∀ {A} (R : relation A), PreOrder R → PreOrder (map_included R).
Proof.
split; [intros m i; by destruct (m !! i)|].
split; [intros m i; by destruct (m !! i); simpl|].
intros m1 m2 m3 Hm12 Hm23 i; specialize (Hm12 i); specialize (Hm23 i).
destruct (m1 !! i), (m2 !! i), (m3 !! i); try done; etransitivity; eauto.
destruct (m1 !! i), (m2 !! i), (m3 !! i); simplify_equality';
done || etransitivity; eauto.
Qed.
Global Instance: PartialOrder ((⊆) : relation (M A)).
Proof.
......@@ -357,8 +414,8 @@ Lemma insert_included {A} R `{!Reflexive R} (m : M A) i x :
(∀ y, m !! i = Some y → R y x) → map_included R m (<[i:=x]>m).
Proof.
intros ? j; destruct (decide (i = j)) as [->|].
* rewrite lookup_insert. destruct (m !! j); eauto.
* rewrite lookup_insert_ne by done. by destruct (m !! j).
* rewrite lookup_insert. destruct (m !! j); simpl; eauto.
* rewrite lookup_insert_ne by done. by destruct (m !! j); simpl.
Qed.
Lemma insert_subseteq {A} (m : M A) i x : m !! i = None → m ⊆ <[i:=x]>m.
Proof. apply partial_alter_subseteq. Qed.
......@@ -465,6 +522,17 @@ Proof.
* by rewrite lookup_omap, !lookup_singleton.
* by rewrite lookup_omap, !lookup_singleton_ne.
Qed.
Lemma map_fmap_id {A} (m : M A) : id <$> m = m.
Proof. apply map_eq; intros i; by rewrite lookup_fmap, option_fmap_id. Qed.
Lemma map_fmap_compose {A B C} (f : A → B) (g : B → C) (m : M A) :
g ∘ f <$> m = g <$> f <$> m.
Proof. apply map_eq; intros i; by rewrite !lookup_fmap,option_fmap_compose. Qed.
Lemma map_fmap_ext {A B} (f1 f2 : A → B) m :
(∀ i x, m !! i = Some x → f1 x = f2 x) → f1 <$> m = f2 <$> m.
Proof.
intros Hi; apply map_eq; intros i; rewrite !lookup_fmap.
by destruct (m !! i) eqn:?; simpl; erewrite ?Hi by eauto.
Qed.
(** ** Properties of conversion to lists *)
Lemma map_to_list_unique {A} (m : M A) i x y :
......@@ -807,7 +875,7 @@ Lemma insert_merge_r m1 m2 i x z :
Proof. by intros; apply partial_alter_merge_r. Qed.
End more_merge.
(** ** Properties on the [map_Forall2] relation *)
(** ** Properties on the [map_relation] relation *)
Section Forall2.
Context {A B} (R : A → B → Prop) (P : A → Prop) (Q : B → Prop).
Context `{∀ x y, Decision (R x y), ∀ x, Decision (P x), ∀ y, Decision (Q y)}.
......@@ -819,8 +887,8 @@ Let f (mx : option A) (my : option B) : option bool :=
| None, Some y => Some (bool_decide (Q y))
| None, None => None
end.
Lemma map_Forall2_alt (m1 : M A) (m2 : M B) :
map_Forall2 R P Q m1 m2 ↔ map_Forall (λ _, Is_true) (merge f m1 m2).
Lemma map_relation_alt (m1 : M A) (m2 : M B) :
map_relation R P Q m1 m2 ↔ map_Forall (λ _, Is_true) (merge f m1 m2).
Proof.
split.
* intros Hm i P'; rewrite lookup_merge by done; intros.
......@@ -830,71 +898,72 @@ Proof.
destruct (m1 !! i), (m2 !! i); simplify_equality'; auto;
by eapply bool_decide_unpack, Hm.
Qed.
Global Instance map_Forall2_dec `{∀ x y, Decision (R x y), ∀ x, Decision (P x),
∀ y, Decision (Q y)} m1 m2 : Decision (map_Forall2 R P Q m1 m2).
Global Instance map_relation_dec `{∀ x y, Decision (R x y), ∀ x, Decision (P x),
∀ y, Decision (Q y)} m1 m2 : Decision (map_relation R P Q m1 m2).
Proof.
refine (cast_if (decide (map_Forall (λ _, Is_true) (merge f m1 m2))));
abstract by rewrite map_Forall2_alt.
abstract by rewrite map_relation_alt.
Defined.
(** Due to the finiteness of finite maps, we can extract a witness if the
relation does not hold. *)
Lemma map_not_Forall2 (m1 : M A) (m2 : M B) :
¬map_Forall2 R P Q m1 m2 ↔ ∃ i,
¬map_relation R P Q m1 m2 ↔ ∃ i,
(∃ x y, m1 !! i = Some x ∧ m2 !! i = Some y ∧ ¬R x y)
∨ (∃ x, m1 !! i = Some x ∧ m2 !! i = None ∧ ¬P x)
∨ (∃ y, m1 !! i = None ∧ m2 !! i = Some y ∧ ¬Q y).
Proof.
split.
* rewrite map_Forall2_alt, (map_not_Forall _). intros (i&?&Hm&?); exists i.
* rewrite map_relation_alt, (map_not_Forall _). intros (i&?&Hm&?); exists i.
rewrite lookup_merge in Hm by done.
destruct (m1 !! i), (m2 !! i); naive_solver auto 2 using bool_decide_pack.
* by intros [i[(x&y&?&?&?)|[(x&?&?&?)|(y&?&?&?)]]] Hm;
* unfold map_relation, option_relation.
by intros [i[(x&y&?&?&?)|[(x&?&?&?)|(y&?&?&?)]]] Hm;
specialize (Hm i); simplify_option_equality.
Qed.
End Forall2.
(** ** Properties on the disjoint maps *)
Lemma map_disjoint_spec {A} (m1 m2 : M A) :
m1 ⊥ m2 ↔ ∀ i x y, m1 !! i = Some x → m2 !! i = Some y → False.
m1 ⊥ₘ m2 ↔ ∀ i x y, m1 !! i = Some x → m2 !! i = Some y → False.
Proof.
split; intros Hm i; specialize (Hm i);
destruct (m1 !! i), (m2 !! i); naive_solver.
Qed.
Lemma map_disjoint_alt {A} (m1 m2 : M A) :
m1 ⊥ m2 ↔ ∀ i, m1 !! i = None ∨ m2 !! i = None.
m1 ⊥ₘ m2 ↔ ∀ i, m1 !! i = None ∨ m2 !! i = None.
Proof.
split; intros Hm1m2 i; specialize (Hm1m2 i);
destruct (m1 !! i), (m2 !! i); naive_solver.
Qed.
Lemma map_not_disjoint {A} (m1 m2 : M A) :
¬m1 ⊥ m2 ↔ ∃ i x1 x2, m1 !! i = Some x1 ∧ m2 !! i = Some x2.
¬m1 ⊥ₘ m2 ↔ ∃ i x1 x2, m1 !! i = Some x1 ∧ m2 !! i = Some x2.
Proof.
unfold disjoint, map_disjoint. rewrite map_not_Forall2 by solve_decision.
split; [|naive_solver].
intros [i[(x&y&?&?&?)|[(x&?&?&[])|(y&?&?&[])]]]; naive_solver.
Qed.
Global Instance: Symmetric (@disjoint (M A) _).
Global Instance: Symmetric (map_disjoint : relation (M A)).
Proof. intros A m1 m2. rewrite !map_disjoint_spec. naive_solver. Qed.
Lemma map_disjoint_empty_l {A} (m : M A) : ∅ ⊥ m.
Lemma map_disjoint_empty_l {A} (m : M A) : ∅ ⊥ₘ m.
Proof. rewrite !map_disjoint_spec. intros i x y. by rewrite lookup_empty. Qed.
Lemma map_disjoint_empty_r {A} (m : M A) : m ⊥ ∅.
Lemma map_disjoint_empty_r {A} (m : M A) : m ⊥ₘ ∅.
Proof. rewrite !map_disjoint_spec. intros i x y. by rewrite lookup_empty. Qed.
Lemma map_disjoint_weaken {A} (m1 m1' m2 m2' : M A) :
m1' ⊥ m2' → m1 ⊆ m1' → m2 ⊆ m2' → m1 ⊥ m2.
m1' ⊥ₘ m2' → m1 ⊆ m1' → m2 ⊆ m2' → m1 ⊥ₘ m2.
Proof. rewrite !map_subseteq_spec, !map_disjoint_spec. eauto. Qed.
Lemma map_disjoint_weaken_l {A} (m1 m1' m2 : M A) :
m1' ⊥ m2 → m1 ⊆ m1' → m1 ⊥ m2.
m1' ⊥ₘ m2 → m1 ⊆ m1' → m1 ⊥ₘ m2.
Proof. eauto using map_disjoint_weaken. Qed.
Lemma map_disjoint_weaken_r {A} (m1 m2 m2' : M A) :
m1 ⊥ m2' → m2 ⊆ m2' → m1 ⊥ m2.
m1 ⊥ₘ m2' → m2 ⊆ m2' → m1 ⊥ₘ m2.
Proof. eauto using map_disjoint_weaken. Qed.
Lemma map_disjoint_Some_l {A} (m1 m2 : M A) i x:
m1 ⊥ m2 → m1 !! i = Some x → m2 !! i = None.
m1 ⊥ₘ m2 → m1 !! i = Some x → m2 !! i = None.
Proof. rewrite map_disjoint_spec, eq_None_not_Some. intros ?? [??]; eauto. Qed.
Lemma map_disjoint_Some_r {A} (m1 m2 : M A) i x:
m1 ⊥ m2 → m2 !! i = Some x → m1 !! i = None.
Proof. rewrite (symmetry_iff (⊥)). apply map_disjoint_Some_l. Qed.
Lemma map_disjoint_singleton_l {A} (m : M A) i x : {[i, x]} ⊥ m ↔ m !! i = None.
m1 ⊥ₘ m2 → m2 !! i = Some x → m1 !! i = None.
Proof. rewrite (symmetry_iff map_disjoint). apply map_disjoint_Some_l. Qed.
Lemma map_disjoint_singleton_l {A} (m: M A) i x : {[i, x]} ⊥ₘ m ↔ m !! i = None.
Proof.
split; [|rewrite !map_disjoint_spec].
* intro. apply (map_disjoint_Some_l {[i, x]} _ _ x);
......@@ -904,20 +973,20 @@ Proof.
+ by rewrite lookup_singleton_ne.
Qed.
Lemma map_disjoint_singleton_r {A} (m : M A) i x :
m ⊥ {[i, x]} ↔ m !! i = None.
Proof. by rewrite (symmetry_iff (⊥)), map_disjoint_singleton_l. Qed.
m ⊥ₘ {[i, x]} ↔ m !! i = None.
Proof. by rewrite (symmetry_iff map_disjoint), map_disjoint_singleton_l. Qed.
Lemma map_disjoint_singleton_l_2 {A} (m : M A) i x :
m !! i = None → {[i, x]} ⊥ m.
m !! i = None → {[i, x]} ⊥ₘ m.
Proof. by rewrite map_disjoint_singleton_l. Qed.
Lemma map_disjoint_singleton_r_2 {A} (m : M A) i x :
m !! i = None → m ⊥ {[i, x]}.
m !! i = None → m ⊥ₘ {[i, x]}.
Proof. by rewrite map_disjoint_singleton_r. Qed.
Lemma map_disjoint_delete_l {A} (m1 m2 : M A) i : m1 ⊥ m2 → delete i m1 ⊥ m2.
Lemma map_disjoint_delete_l {A} (m1 m2 : M A) i : m1 ⊥ₘ m2 → delete i m1 ⊥ₘ m2.
Proof.
rewrite !map_disjoint_alt. intros Hdisjoint j. destruct (Hdisjoint j); auto.
rewrite lookup_delete_None. tauto.
Qed.
Lemma map_disjoint_delete_r {A} (m1 m2 : M A) i : m1 ⊥ m2 → m1 ⊥ delete i m2.
Lemma map_disjoint_delete_r {A} (m1 m2 : M A) i : m1 ⊥ₘ m2 → m1 ⊥ₘ delete i m2.
Proof. symmetry. by apply map_disjoint_delete_l. Qed.
(** ** Properties of the [union_with] operation *)
......@@ -1036,7 +1105,7 @@ Qed.
Lemma map_positive_l_alt {A} (m1 m2 : M A) : m1 ≠ ∅ → m1 ∪ m2 ≠ ∅.
Proof. eauto using map_positive_l. Qed.
Lemma lookup_union_Some {A} (m1 m2 : M A) i x :
m1 ⊥ m2 → (m1 ∪ m2) !! i = Some x ↔ m1 !! i = Some x ∨ m2 !! i = Some x.
m1 ⊥ₘ m2 → (m1 ∪ m2) !! i = Some x ↔ m1 !! i = Some x ∨ m2 !! i = Some x.
Proof.
intros Hdisjoint. rewrite lookup_union_Some_raw.
intuition eauto using map_disjoint_Some_r.
......@@ -1045,9 +1114,9 @@ Lemma lookup_union_Some_l {A} (m1 m2 : M A) i x :
m1 !! i = Some x → (m1 ∪ m2) !! i = Some x.
Proof. intro. rewrite lookup_union_Some_raw; intuition. Qed.
Lemma lookup_union_Some_r {A} (m1 m2 : M A) i x :
m1 ⊥ m2 → m2 !! i = Some x → (m1 ∪ m2) !! i = Some x.
m1 ⊥ₘ m2 → m2 !! i = Some x → (m1 ∪ m2) !! i = Some x.
Proof. intro. rewrite lookup_union_Some; intuition. Qed.
Lemma map_union_commutative {A} (m1 m2 : M A) : m1 ⊥ m2 → m1 ∪ m2 = m2 ∪ m1.
Lemma map_union_commutative {A} (m1 m2 : M A) : m1 ⊥ₘ m2 → m1 ∪ m2 = m2 ∪ m1.
Proof.
intros Hdisjoint. apply (merge_commutative (union_with (λ x _, Some x))).
intros i. specialize (Hdisjoint i).
......@@ -1065,14 +1134,14 @@ Lemma map_union_subseteq_l {A} (m1 m2 : M A) : m1 ⊆ m1 ∪ m2.
Proof.
rewrite map_subseteq_spec. intros ? i x. rewrite lookup_union_Some_raw. tauto.
Qed.
Lemma map_union_subseteq_r {A} (m1 m2 : M A) : m1 ⊥ m2 → m2 ⊆ m1 ∪ m2.
Lemma map_union_subseteq_r {A} (m1 m2 : M A) : m1 ⊥ₘ m2 → m2 ⊆ m1 ∪ m2.
Proof.
intros. rewrite map_union_commutative by done. by apply map_union_subseteq_l.
Qed.
Lemma map_union_subseteq_l_alt {A} (m1 m2 m3 : M A) : m1 ⊆ m2 → m1 ⊆ m2 ∪ m3.
Proof. intros. transitivity m2; auto using map_union_subseteq_l. Qed.
Lemma map_union_subseteq_r_alt {A} (m1 m2 m3 : M A) :
m2 ⊥ m3 → m1 ⊆ m3 → m1 ⊆ m2 ∪ m3.
m2 ⊥ₘ m3 → m1 ⊆ m3 → m1 ⊆ m2 ∪ m3.
Proof. intros. transitivity m3; auto using map_union_subseteq_r. Qed.
Lemma map_union_preserving_l {A} (m1 m2 m3 : M A) : m1 ⊆ m2 → m3 ∪ m1 ⊆ m3 ∪ m2.
Proof.
......@@ -1080,52 +1149,52 @@ Proof.
rewrite !lookup_union_Some_raw. naive_solver.
Qed.
Lemma map_union_preserving_r {A} (m1 m2 m3 : M A) :
m2 ⊥ m3 → m1 ⊆ m2 → m1 ∪ m3 ⊆ m2 ∪ m3.
m2 ⊥ₘ m3 → m1 ⊆ m2 → m1 ∪ m3 ⊆ m2 ∪ m3.
Proof.
intros. rewrite !(map_union_commutative _ m3)
by eauto using map_disjoint_weaken_l.
by apply map_union_preserving_l.
Qed.
Lemma map_union_reflecting_l {A} (m1 m2 m3 : M A) :
m3 ⊥ m1 → m3 ⊥ m2 → m3 ∪ m1 ⊆ m3 ∪ m2 → m1 ⊆ m2.
m3 ⊥ₘ m1 → m3 ⊥ₘ m2 → m3 ∪ m1 ⊆ m3 ∪ m2 → m1 ⊆ m2.
Proof.
rewrite !map_subseteq_spec. intros Hm31 Hm32 Hm i x ?. specialize (Hm i x).
rewrite !lookup_union_Some in Hm by done. destruct Hm; auto.
by rewrite map_disjoint_spec in Hm31; destruct (Hm31 i x x).
Qed.
Lemma map_union_reflecting_r {A} (m1 m2 m3 : M A) :
m1 ⊥ m3 → m2 ⊥ m3 → m1 ∪ m3 ⊆ m2 ∪ m3 → m1 ⊆ m2.
m1 ⊥ₘ m3 → m2 ⊥ₘ m3 → m1 ∪ m3 ⊆ m2 ∪ m3 → m1 ⊆ m2.
Proof.
intros ??. rewrite !(map_union_commutative _ m3) by done.
by apply map_union_reflecting_l.
Qed.
Lemma map_union_cancel_l {A} (m1 m2 m3 : M A) :
m1 ⊥ m3 → m2 ⊥ m3 → m3 ∪ m1 = m3 ∪ m2 → m1 = m2.
m1 ⊥ₘ m3 → m2 ⊥ₘ m3 → m3 ∪ m1 = m3 ∪ m2 → m1 = m2.
Proof.
intros. apply (anti_symmetric (⊆));
apply map_union_reflecting_l with m3; auto using (reflexive_eq (R:=(⊆))).
Qed.
Lemma map_union_cancel_r {A} (m1 m2 m3 : M A) :
m1 ⊥ m3 → m2 ⊥ m3 → m1 ∪ m3 = m2 ∪ m3 → m1 = m2.
m1 ⊥ₘ m3 → m2 ⊥ₘ m3 → m1 ∪ m3 = m2 ∪ m3 → m1 = m2.
Proof.
intros. apply (anti_symmetric (⊆));
apply map_union_reflecting_r with m3; auto using (reflexive_eq (R:=(⊆))).
Qed.
Lemma map_disjoint_union_l {A} (m1 m2 m3 : M A) :
m1 ∪ m2 ⊥ m3 ↔ m1 ⊥ m3 ∧ m2 ⊥ m3.
m1 ∪ m2 ⊥ₘ m3 ↔ m1 ⊥ₘ m3 ∧ m2 ⊥ₘ m3.
Proof.
rewrite !map_disjoint_alt. setoid_rewrite lookup_union_None. naive_solver.
Qed.
Lemma map_disjoint_union_r {A} (m1 m2 m3 : M A) :
m1 ⊥ m2 ∪ m3 ↔ m1 ⊥ m2 ∧ m1 ⊥ m3.
m1 ⊥ₘ m2 ∪ m3 ↔ m1 ⊥ₘ m2 ∧ m1 ⊥ₘ m3.
Proof.
rewrite !map_disjoint_alt. setoid_rewrite lookup_union_None. naive_solver.
Qed.
Lemma map_disjoint_union_l_2 {A} (m1 m2 m3 : M A) :
m1 ⊥ m3 → m2 ⊥ m3 → m1 ∪ m2 ⊥ m3.
m1 ⊥ₘ m3 → m2 ⊥ₘ m3 → m1 ∪ m2 ⊥ₘ m3.
Proof. by rewrite map_disjoint_union_l. Qed.
Lemma map_disjoint_union_r_2 {A} (m1 m2 m3 : M A) :
m1 ⊥ m2 → m1 ⊥ m3 → m1 ⊥ m2 ∪ m3.
m1 ⊥ₘ m2 → m1 ⊥ₘ m3 → m1 ⊥ₘ m2 ∪ m3.
Proof. by rewrite map_disjoint_union_r. Qed.
Lemma insert_union_singleton_l {A} (m : M A) i x : <[i:=x]>m = {[i,x]} ∪ m.
Proof.
......@@ -1142,22 +1211,22 @@ Proof.
by apply map_disjoint_singleton_l.
Qed.
Lemma map_disjoint_insert_l {A} (m1 m2 :