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Commit 9774ce9c authored by Robbert Krebbers's avatar Robbert Krebbers
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Use scheme - then + then * for bullets.

parent 2e9c3f77
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theories/bsets.v
View file @ 9774ce9c
......@@ -21,11 +21,11 @@ Instance bset_collection {A} `{∀ x y : A, Decision (x = y)} :
Collection A (bset A).
Proof.
split; [split| |].
* by intros x ?.
* by intros x y; rewrite <-(bool_decide_spec (x = y)).
* split. apply orb_prop_elim. apply orb_prop_intro.
* split. apply andb_prop_elim. apply andb_prop_intro.
* intros X Y x; unfold elem_of, bset_elem_of; simpl.
- by intros x ?.
- by intros x y; rewrite <-(bool_decide_spec (x = y)).
- split. apply orb_prop_elim. apply orb_prop_intro.
- split. apply andb_prop_elim. apply andb_prop_intro.
- intros X Y x; unfold elem_of, bset_elem_of; simpl.
destruct (bset_car X x), (bset_car Y x); simpl; tauto.
Qed.
Instance bset_elem_of_dec {A} x (X : bset A) : Decision (x ∈ X) := _.
......
theories/co_pset.v
View file @ 9774ce9c
......@@ -65,10 +65,10 @@ Lemma coPset_eq t1 t2 :
Proof.
revert t2.
induction t1 as [b1|b1 l1 IHl r1 IHr]; intros [b2|b2 l2 r2] Ht ??; simpl in *.
* f_equal; apply (Ht 1).
* by discriminate (coPLeaf_wf (coPNode b2 l2 r2) b1).
* by discriminate (coPLeaf_wf (coPNode b1 l1 r1) b2).
* f_equal; [apply (Ht 1)| |].
- f_equal; apply (Ht 1).
- by discriminate (coPLeaf_wf (coPNode b2 l2 r2) b1).
- by discriminate (coPLeaf_wf (coPNode b1 l1 r1) b2).
- f_equal; [apply (Ht 1)| |].
+ apply IHl; try apply (λ x, Ht (x~0)); eauto.
+ apply IHr; try apply (λ x, Ht (x~1)); eauto.
Qed.
......@@ -163,13 +163,13 @@ Instance coPset_elem_of_dec (p : positive) (X : coPset) : Decision (p ∈ X) :=
Instance coPset_collection : Collection positive coPset.
Proof.
split; [split| |].
* by intros ??.
* intros p q. apply elem_to_Pset_singleton.
* intros [t] [t'] p; unfold elem_of, coPset_elem_of, coPset_union; simpl.
- by intros ??.
- intros p q. apply elem_to_Pset_singleton.
- intros [t] [t'] p; unfold elem_of, coPset_elem_of, coPset_union; simpl.
by rewrite elem_to_Pset_union, orb_True.
* intros [t] [t'] p; unfold elem_of,coPset_elem_of,coPset_intersection; simpl.
- intros [t] [t'] p; unfold elem_of,coPset_elem_of,coPset_intersection; simpl.
by rewrite elem_to_Pset_intersection, andb_True.
* intros [t] [t'] p; unfold elem_of, coPset_elem_of, coPset_difference; simpl.
- intros [t] [t'] p; unfold elem_of, coPset_elem_of, coPset_difference; simpl.
by rewrite elem_to_Pset_intersection,
elem_to_Pset_opp, andb_True, negb_True.
Qed.
......@@ -192,7 +192,7 @@ Lemma coPset_finite_spec X : set_finite X ↔ coPset_finite (`X).
Proof.
destruct X as [t Ht].
unfold set_finite, elem_of at 1, coPset_elem_of; simpl; clear Ht; split.
* induction t as [b|b l IHl r IHr]; simpl.
- induction t as [b|b l IHl r IHr]; simpl.
{ destruct b; simpl; [intros [l Hl]|done].
by apply (is_fresh (of_list l : Pset)), elem_of_of_list, Hl. }
intros [ll Hll]; rewrite andb_True; split.
......@@ -200,7 +200,7 @@ Proof.
rewrite elem_of_list_omap; intros; exists (i~0); auto.
+ apply IHr; exists (omap (maybe (~1)) ll); intros i.
rewrite elem_of_list_omap; intros; exists (i~1); auto.
* induction t as [b|b l IHl r IHr]; simpl; [by exists []; destruct b|].
- induction t as [b|b l IHl r IHr]; simpl; [by exists []; destruct b|].
rewrite andb_True; intros [??]; destruct IHl as [ll ?], IHr as [rl ?]; auto.
exists ([1] ++ ((~0) <$> ll) ++ ((~1) <$> rl))%list; intros [i|i|]; simpl;
rewrite elem_of_cons, elem_of_app, !elem_of_list_fmap; naive_solver.
......@@ -237,8 +237,8 @@ Qed.
Lemma coPpick_elem_of X : ¬set_finite X → coPpick X ∈ X.
Proof.
destruct X as [t ?]; unfold coPpick; destruct (coPpick_raw _) as [j|] eqn:?.
* by intros; apply coPpick_raw_elem_of.
* by intros []; apply coPset_finite_spec, coPpick_raw_None.
- by intros; apply coPpick_raw_elem_of.
- by intros []; apply coPset_finite_spec, coPpick_raw_None.
Qed.
(** * Conversion to psets *)
......@@ -270,8 +270,8 @@ Fixpoint of_Pset_raw (t : Pmap_raw ()) : coPset_raw :=
Lemma of_Pset_wf t : Pmap_wf t → coPset_wf (of_Pset_raw t).
Proof.
induction t as [|[] l IHl r IHr]; simpl; rewrite ?andb_True; auto.
* intros [??]; destruct l as [|[]], r as [|[]]; simpl in *; auto.
* destruct l as [|[]], r as [|[]]; simpl in *; rewrite ?andb_true_r;
- intros [??]; destruct l as [|[]], r as [|[]]; simpl in *; auto.
- destruct l as [|[]], r as [|[]]; simpl in *; rewrite ?andb_true_r;
rewrite ?andb_True; rewrite ?andb_True in IHl, IHr; intuition.
Qed.
Lemma elem_of_of_Pset_raw i t : e_of i (of_Pset_raw t) ↔ t !! i = Some ().
......@@ -327,9 +327,9 @@ Definition coPset_suffixes (p : positive) : coPset :=
Lemma elem_coPset_suffixes p q : p ∈ coPset_suffixes q ↔ ∃ q', p = q' ++ q.
Proof.
unfold elem_of, coPset_elem_of; simpl; split.
* revert p; induction q; intros [?|?|]; simpl;
- revert p; induction q; intros [?|?|]; simpl;
rewrite ?coPset_elem_of_node; naive_solver.
* by intros [q' ->]; induction q; simpl; rewrite ?coPset_elem_of_node.
- by intros [q' ->]; induction q; simpl; rewrite ?coPset_elem_of_node.
Qed.
Lemma coPset_suffixes_infinite p : ¬set_finite (coPset_suffixes p).
Proof.
......
theories/collections.v
View file @ 9774ce9c
......@@ -48,8 +48,8 @@ Section simple_collection.
Lemma elem_of_subseteq_singleton x X : x ∈ X ↔ {[ x ]} ⊆ X.
Proof.
split.
* intros ??. rewrite elem_of_singleton. by intros ->.
* intros Ex. by apply (Ex x), elem_of_singleton.
- intros ??. rewrite elem_of_singleton. by intros ->.
- intros Ex. by apply (Ex x), elem_of_singleton.
Qed.
Global Instance singleton_proper : Proper ((=) ==> (≡)) (singleton (B:=C)).
Proof. by repeat intro; subst. Qed.
......@@ -59,9 +59,9 @@ Section simple_collection.
Lemma elem_of_union_list Xs x : x ∈ ⋃ Xs ↔ ∃ X, X ∈ Xs ∧ x ∈ X.
Proof.
split.
* induction Xs; simpl; intros HXs; [by apply elem_of_empty in HXs|].
- induction Xs; simpl; intros HXs; [by apply elem_of_empty in HXs|].
setoid_rewrite elem_of_cons. apply elem_of_union in HXs. naive_solver.
* intros [X []]. induction 1; simpl; [by apply elem_of_union_l |].
- intros [X []]. induction 1; simpl; [by apply elem_of_union_l |].
intros. apply elem_of_union_r; auto.
Qed.
Lemma non_empty_singleton x : ({[ x ]} : C) ≢ ∅.
......@@ -113,9 +113,9 @@ Section of_option_list.
Lemma elem_of_of_list (x : A) l : x ∈ of_list l ↔ x ∈ l.
Proof.
split.
* induction l; simpl; [by rewrite elem_of_empty|].
- induction l; simpl; [by rewrite elem_of_empty|].
rewrite elem_of_union,elem_of_singleton; intros [->|?]; constructor; auto.
* induction 1; simpl; rewrite elem_of_union, elem_of_singleton; auto.
- induction 1; simpl; rewrite elem_of_union, elem_of_singleton; auto.
Qed.
End of_option_list.
......@@ -356,11 +356,11 @@ Section collection_ops.
Forall2 (∈) xs Xs ∧ y ∈ Y ∧ foldr (λ x, (≫= f x)) (Some y) xs = Some x.
Proof.
split.
* revert x. induction Xs; simpl; intros x HXs; [eexists [], x; intuition|].
- revert x. induction Xs; simpl; intros x HXs; [eexists [], x; intuition|].
rewrite elem_of_intersection_with in HXs; destruct HXs as (x1&x2&?&?&?).
destruct (IHXs x2) as (xs & y & hy & ? & ?); trivial.
eexists (x1 :: xs), y. intuition (simplify_option_equality; auto).
* intros (xs & y & Hxs & ? & Hx). revert x Hx.
- intros (xs & y & Hxs & ? & Hx). revert x Hx.
induction Hxs; intros; simplify_option_equality; [done |].
rewrite elem_of_intersection_with. naive_solver.
Qed.
......@@ -389,8 +389,8 @@ Section NoDup.
Global Instance: Proper (R ==> (≡) ==> iff) elem_of_upto.
Proof.
intros ?? E1 ?? E2. split; intros [z [??]]; exists z.
* rewrite <-E1, <-E2; intuition.
* rewrite E1, E2; intuition.
- rewrite <-E1, <-E2; intuition.
- rewrite E1, E2; intuition.
Qed.
Global Instance: Proper ((≡) ==> iff) set_NoDup.
Proof. firstorder. Qed.
......@@ -575,8 +575,8 @@ Section collection_monad.
l ∈ mapM f k ↔ Forall2 (λ x y, x ∈ f y) l k.
Proof.
split.
* revert l. induction k; solve_elem_of.
* induction 1; solve_elem_of.
- revert l. induction k; solve_elem_of.
- induction 1; solve_elem_of.
Qed.
Lemma collection_mapM_length {A B} (f : A → M B) l k :
l ∈ mapM f k → length l = length k.
......
theories/fin_collections.v
View file @ 9774ce9c
......@@ -20,9 +20,9 @@ Proof. by exists (elements X); intros; rewrite elem_of_elements. Qed.
Global Instance elements_proper: Proper ((≡) ==> (≡ₚ)) (elements (C:=C)).
Proof.
intros ?? E. apply NoDup_Permutation.
* apply NoDup_elements.
* apply NoDup_elements.
* intros. by rewrite !elem_of_elements, E.
- apply NoDup_elements.
- apply NoDup_elements.
- intros. by rewrite !elem_of_elements, E.
Qed.
Global Instance collection_size_proper: Proper ((≡) ==> (=)) (@size C _).
Proof. intros ?? E. apply Permutation_length. by rewrite E. Qed.
......@@ -45,9 +45,9 @@ Lemma size_singleton (x : A) : size {[ x ]} = 1.
Proof.
change (length (elements {[ x ]}) = length [x]).
apply Permutation_length, NoDup_Permutation.
* apply NoDup_elements.
* apply NoDup_singleton.
* intros y.
- apply NoDup_elements.
- apply NoDup_singleton.
- intros y.
by rewrite elem_of_elements, elem_of_singleton, elem_of_list_singleton.
Qed.
Lemma size_singleton_inv X x y : size X = 1 → x ∈ X → y ∈ X → x = y.
......@@ -59,8 +59,8 @@ Qed.
Lemma collection_choose_or_empty X : (∃ x, x ∈ X) ∨ X ≡ ∅.
Proof.
destruct (elements X) as [|x l] eqn:HX; [right|left].
* apply equiv_empty; intros x. by rewrite <-elem_of_elements, HX, elem_of_nil.
* exists x. rewrite <-elem_of_elements, HX. by left.
- apply equiv_empty; intros x. by rewrite <-elem_of_elements, HX, elem_of_nil.
- exists x. rewrite <-elem_of_elements, HX. by left.
Qed.
Lemma collection_choose X : X ≢ ∅ → ∃ x, x ∈ X.
Proof. intros. by destruct (collection_choose_or_empty X). Qed.
......@@ -75,17 +75,17 @@ Lemma size_1_elem_of X : size X = 1 → ∃ x, X ≡ {[ x ]}.
Proof.
intros E. destruct (size_pos_elem_of X); auto with lia.
exists x. apply elem_of_equiv. split.
* rewrite elem_of_singleton. eauto using size_singleton_inv.
* solve_elem_of.
- rewrite elem_of_singleton. eauto using size_singleton_inv.
- solve_elem_of.
Qed.
Lemma size_union X Y : X ∩ Y ≡ ∅ → size (X ∪ Y) = size X + size Y.
Proof.
intros [E _]. unfold size, collection_size. simpl. rewrite <-app_length.
apply Permutation_length, NoDup_Permutation.
* apply NoDup_elements.
* apply NoDup_app; repeat split; try apply NoDup_elements.
- apply NoDup_elements.
- apply NoDup_app; repeat split; try apply NoDup_elements.
intros x; rewrite !elem_of_elements; solve_elem_of.
* intros. by rewrite elem_of_app, !elem_of_elements, elem_of_union.
- intros. by rewrite elem_of_app, !elem_of_elements, elem_of_union.
Qed.
Instance elem_of_dec_slow (x : A) (X : C) : Decision (x ∈ X) | 100.
Proof.
......@@ -129,9 +129,9 @@ Proof.
intros ? Hemp Hadd. apply well_founded_induction with (⊂).
{ apply collection_wf. }
intros X IH. destruct (collection_choose_or_empty X) as [[x ?]|HX].
* rewrite (union_difference {[ x ]} X) by solve_elem_of.
- rewrite (union_difference {[ x ]} X) by solve_elem_of.
apply Hadd. solve_elem_of. apply IH; solve_elem_of.
* by rewrite HX.
- by rewrite HX.
Qed.
Lemma collection_fold_ind {B} (P : B → C → Prop) (f : A → B → B) (b : B) :
Proper ((=) ==> (≡) ==> iff) P →
......@@ -143,9 +143,9 @@ Proof.
{ intros help ?. apply help; [apply NoDup_elements|].
symmetry. apply elem_of_elements. }
induction 1 as [|x l ?? IH]; simpl.
* intros X HX. setoid_rewrite elem_of_nil in HX.
- intros X HX. setoid_rewrite elem_of_nil in HX.
rewrite equiv_empty. done. solve_elem_of.
* intros X HX. setoid_rewrite elem_of_cons in HX.
- intros X HX. setoid_rewrite elem_of_cons in HX.
rewrite (union_difference {[ x ]} X) by solve_elem_of.
apply Hadd. solve_elem_of. apply IH. solve_elem_of.
Qed.
......
theories/fin_maps.v
View file @ 9774ce9c
......@@ -121,9 +121,9 @@ Section setoid.
Global Instance map_equivalence : Equivalence ((≡) : relation (M A)).
Proof.
split.
* by intros m i.
* by intros m1 m2 ? i.
* by intros m1 m2 m3 ?? i; transitivity (m2 !! i).
- 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).
......@@ -258,9 +258,9 @@ Proof.
{ by rewrite lookup_partial_alter_ne,
!lookup_partial_alter, lookup_partial_alter_ne. }
destruct (decide (jj = i)) as [->|?].
* by rewrite lookup_partial_alter,
- by rewrite lookup_partial_alter,
!lookup_partial_alter_ne, lookup_partial_alter by congruence.
* by rewrite !lookup_partial_alter_ne by congruence.
- by rewrite !lookup_partial_alter_ne by congruence.
Qed.
Lemma partial_alter_self_alt {A} (m : M A) i x :
x = m !! i → partial_alter (λ _, x) i m = m.
......@@ -307,8 +307,8 @@ Lemma lookup_alter_Some {A} (f : A → A) m i j y :
(i = j ∧ ∃ x, m !! j = Some x ∧ y = f x) ∨ (i ≠ j ∧ m !! j = Some y).
Proof.
destruct (decide (i = j)) as [->|?].
* rewrite lookup_alter. naive_solver (simplify_option_equality; eauto).
* rewrite lookup_alter_ne by done. naive_solver.
- rewrite lookup_alter. naive_solver (simplify_option_equality; eauto).
- rewrite lookup_alter_ne by done. naive_solver.
Qed.
Lemma lookup_alter_None {A} (f : A → A) m i j :
alter f i m !! j = None ↔ m !! j = None.
......@@ -333,9 +333,9 @@ Lemma lookup_delete_Some {A} (m : M A) i j y :
delete i m !! j = Some y ↔ i ≠ j ∧ m !! j = Some y.
Proof.
split.
* destruct (decide (i = j)) as [->|?];
- destruct (decide (i = j)) as [->|?];
rewrite ?lookup_delete, ?lookup_delete_ne; intuition congruence.
* intros [??]. by rewrite lookup_delete_ne.
- intros [??]. by rewrite lookup_delete_ne.
Qed.
Lemma lookup_delete_is_Some {A} (m : M A) i j :
is_Some (delete i m !! j) ↔ i ≠ j ∧ is_Some (m !! j).
......@@ -412,9 +412,9 @@ Lemma lookup_insert_Some {A} (m : M A) i j x y :
<[i:=x]>m !! j = Some y ↔ (i = j ∧ x = y) ∨ (i ≠ j ∧ m !! j = Some y).
Proof.
split.
* destruct (decide (i = j)) as [->|?];
- destruct (decide (i = j)) as [->|?];
rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
* intros [[-> ->]|[??]]; [apply lookup_insert|]. by rewrite lookup_insert_ne.
- intros [[-> ->]|[??]]; [apply lookup_insert|]. by rewrite lookup_insert_ne.
Qed.
Lemma lookup_insert_is_Some {A} (m : M A) i j x :
is_Some (<[i:=x]>m !! j) ↔ i = j ∨ i ≠ j ∧ is_Some (m !! j).
......@@ -435,8 +435,8 @@ 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); simpl; eauto.
* rewrite lookup_insert_ne by done. by destruct (m !! j); simpl.
- 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.
......@@ -462,8 +462,8 @@ Lemma delete_insert_subseteq {A} (m1 m2 : M A) i x :
Proof.
rewrite !map_subseteq_spec.
intros Hix Hi j y Hj. destruct (decide (i = j)) as [->|?].
* rewrite lookup_insert. congruence.
* rewrite lookup_insert_ne by done. apply Hi. by rewrite lookup_delete_ne.
- rewrite lookup_insert. congruence.
- rewrite lookup_insert_ne by done. apply Hi. by rewrite lookup_delete_ne.
Qed.
Lemma insert_delete_subset {A} (m1 m2 : M A) i x :
m1 !! i = None → <[i:=x]> m1 ⊂ m2 → m1 ⊂ delete i m2.
......@@ -477,10 +477,10 @@ Lemma insert_subset_inv {A} (m1 m2 : M A) i x :
∃ m2', m2 = <[i:=x]>m2' ∧ m1 ⊂ m2' ∧ m2' !! i = None.
Proof.
intros Hi Hm1m2. exists (delete i m2). split_ands.
* rewrite insert_delete. done. eapply lookup_weaken, strict_include; eauto.
- rewrite insert_delete. done. eapply lookup_weaken, strict_include; eauto.
by rewrite lookup_insert.
* eauto using insert_delete_subset.
* by rewrite lookup_delete.
- eauto using insert_delete_subset.
- by rewrite lookup_delete.
Qed.
Lemma insert_empty {A} i (x : A) : <[i:=x]>∅ = {[i := x]}.
Proof. done. Qed.
......@@ -510,8 +510,8 @@ Qed.
Lemma alter_singleton {A} (f : A → A) i x : alter f i {[i := x]} = {[i := f x]}.
Proof.
intros. apply map_eq. intros i'. destruct (decide (i = i')) as [->|?].
* by rewrite lookup_alter, !lookup_singleton.
* by rewrite lookup_alter_ne, !lookup_singleton_ne.
- by rewrite lookup_alter, !lookup_singleton.
- by rewrite lookup_alter_ne, !lookup_singleton_ne.
Qed.
Lemma alter_singleton_ne {A} (f : A → A) i j x :
i ≠ j → alter f i {[j := x]} = {[j := x]}.
......@@ -528,15 +528,15 @@ Proof. apply map_empty; intros i. by rewrite lookup_omap, lookup_empty. Qed.
Lemma fmap_insert {A B} (f: A → B) m i x: f <$> <[i:=x]>m = <[i:=f x]>(f <$> m).
Proof.
apply map_eq; intros i'; destruct (decide (i' = i)) as [->|].
* by rewrite lookup_fmap, !lookup_insert.
* by rewrite lookup_fmap, !lookup_insert_ne, lookup_fmap by done.
- by rewrite lookup_fmap, !lookup_insert.
- by rewrite lookup_fmap, !lookup_insert_ne, lookup_fmap by done.
Qed.
Lemma omap_insert {A B} (f : A → option B) m i x y :
f x = Some y → omap f (<[i:=x]>m) = <[i:=y]>(omap f m).
Proof.
intros; apply map_eq; intros i'; destruct (decide (i' = i)) as [->|].
* by rewrite lookup_omap, !lookup_insert.
* by rewrite lookup_omap, !lookup_insert_ne, lookup_omap by done.
- by rewrite lookup_omap, !lookup_insert.
- by rewrite lookup_omap, !lookup_insert_ne, lookup_omap by done.
Qed.
Lemma map_fmap_singleton {A B} (f : A → B) i x : f <$> {[i := x]} = {[i := f x]}.
Proof.
......@@ -585,8 +585,8 @@ Proof.
setoid_rewrite elem_of_cons.
intros [?|?] Hdup; simplify_equality; [by rewrite lookup_insert|].
destruct (decide (i = j)) as [->|].
* rewrite lookup_insert; f_equal; eauto.
* rewrite lookup_insert_ne by done; eauto.
- rewrite lookup_insert; f_equal; eauto.
- rewrite lookup_insert_ne by done; eauto.
Qed.
Lemma elem_of_map_of_list_1 {A} (l : list (K * A)) i x :
NoDup (l.*1) → (i,x) ∈ l → map_of_list l !! i = Some x.
......@@ -617,8 +617,8 @@ Lemma not_elem_of_map_of_list_2 {A} (l : list (K * A)) i :
Proof.
induction l as [|[j y] l IH]; csimpl; [rewrite elem_of_nil; tauto|].
rewrite elem_of_cons. destruct (decide (i = j)); simplify_equality.
* by rewrite lookup_insert.
* by rewrite lookup_insert_ne; intuition.
- by rewrite lookup_insert.
- by rewrite lookup_insert_ne; intuition.
Qed.
Lemma not_elem_of_map_of_list {A} (l : list (K * A)) i :
i ∉ l.*1 ↔ map_of_list l !! i = None.
......@@ -665,11 +665,11 @@ Lemma map_to_list_insert {A} (m : M A) i x :
m !! i = None → map_to_list (<[i:=x]>m) ≡ₚ (i,x) :: map_to_list m.
Proof.
intros. apply map_of_list_inj; csimpl.
* apply NoDup_fst_map_to_list.
* constructor; auto using NoDup_fst_map_to_list.
- apply NoDup_fst_map_to_list.
- constructor; auto using NoDup_fst_map_to_list.
rewrite elem_of_list_fmap. intros [[??] [? Hlookup]]; subst; simpl in *.
rewrite elem_of_map_to_list in Hlookup. congruence.
* by rewrite !map_of_to_list.
- by rewrite !map_of_to_list.
Qed.
Lemma map_of_list_nil {A} : map_of_list (@nil (K * A)) = ∅.
Proof. done. Qed.
......@@ -702,13 +702,13 @@ Lemma lookup_imap {A B} (f : K → A → option B) m i :
map_imap f m !! i = m !! i ≫= f i.
Proof.
unfold map_imap; destruct (m !! i ≫= f i) as [y|] eqn:Hi; simpl.
* destruct (m !! i) as [x|] eqn:?; simplify_equality'.
- destruct (m !! i) as [x|] eqn:?; simplify_equality'.
apply elem_of_map_of_list_1_help.
{ apply elem_of_list_omap; exists (i,x); split;
[by apply elem_of_map_to_list|by simplify_option_equality]. }
intros y'; rewrite elem_of_list_omap; intros ([i' x']&Hi'&?).
by rewrite elem_of_map_to_list in Hi'; simplify_option_equality.
* apply not_elem_of_map_of_list; rewrite elem_of_list_fmap.
- apply not_elem_of_map_of_list; rewrite elem_of_list_fmap.
intros ([i' x]&->&Hi'); simplify_equality'.
rewrite elem_of_list_omap in Hi'; destruct Hi' as ([j y]&Hj&?).
rewrite elem_of_map_to_list in Hj; simplify_option_equality.
......@@ -726,8 +726,8 @@ Proof.
by intros (?&?&?&?&?); simplify_option_equality. }
unfold map_of_collection; rewrite <-elem_of_map_of_list by done.
rewrite elem_of_list_omap. setoid_rewrite elem_of_elements; split.
* intros (?&?&?); simplify_option_equality; eauto.
* intros [??]; exists i; simplify_option_equality; eauto.
- intros (?&?&?); simplify_option_equality; eauto.
- intros [??]; exists i; simplify_option_equality; eauto.
Qed.
(** ** Induction principles *)
......@@ -741,8 +741,8 @@ Proof.
{ apply map_to_list_empty_inv_alt in Hml. by subst. }
inversion_clear Hnodup.
apply map_to_list_insert_inv in Hml; subst m. apply Hins.
* by apply not_elem_of_map_of_list_1.
* apply IH; auto using map_to_of_list.
- by apply not_elem_of_map_of_list_1.
- apply IH; auto using map_to_of_list.
Qed.
Lemma map_to_list_length {A} (m1 m2 : M A) :
m1 ⊂ m2 → length (map_to_list m1) < length (map_to_list m2).
......@@ -759,8 +759,8 @@ Qed.
Lemma map_wf {A} : wf (strict (@subseteq (M A) _)).
Proof.
apply (wf_projected (<) (length ∘ map_to_list)).
* by apply map_to_list_length.
* by apply lt_wf.
- by apply map_to_list_length.
- by apply lt_wf.
Qed.
(** ** Properties of the [map_Forall] predicate *)
......@@ -770,8 +770,8 @@ Context {A} (P : K → A → Prop).
Lemma map_Forall_to_list m : map_Forall P m ↔ Forall (curry P) (map_to_list m).
Proof.
rewrite Forall_forall. split.
* intros Hforall [i x]. rewrite elem_of_map_to_list. by apply (Hforall i x).
* intros Hforall i x. rewrite <-elem_of_map_to_list. by apply (Hforall (i,x)).
- intros Hforall [i x]. rewrite elem_of_map_to_list. by apply (Hforall i x).
- intros Hforall i x. rewrite <-elem_of_map_to_list. by apply (Hforall (i,x)).
Qed.
Lemma map_Forall_empty : map_Forall P ∅.
Proof. intros i x. by rewrite lookup_empty. Qed.
......@@ -874,24 +874,24 @@ Lemma partial_alter_merge g g1 g2 m1 m2 i :
merge f (partial_alter g1 i m1) (partial_alter g2 i m2).
Proof.
intro. apply map_eq. intros j. destruct (decide (i = j)); subst.
* by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _).
* by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _).
- by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _).
- by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _).
Qed.
Lemma partial_alter_merge_l g g1 m1 m2 i :
g (f (m1 !! i) (m2 !! i)) = f (g1 (m1 !! i)) (m2 !! i) →
partial_alter g i (merge f m1 m2) = merge f (partial_alter g1 i m1) m2.
Proof.
intro. apply map_eq. intros j. destruct (decide (i = j)); subst.
* by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _).
* by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _).
- by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _).
- by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _).
Qed.
Lemma partial_alter_merge_r g g2 m1 m2 i :
g (f (m1 !! i) (m2 !! i)) = f (m1 !! i) (g2 (m2 !! i)) →
partial_alter g i (merge f m1 m2) = merge f m1 (partial_alter g2 i m2).
Proof.
intro. apply map_eq. intros j. destruct (decide (i = j)); subst.
* by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _).
* by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _).
- by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _).
- by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _).
Qed.
Lemma insert_merge m1 m2 i x y z :
f (Some y) (Some z) = Some x →
......@@ -928,10 +928,10 @@ 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.
- intros Hm i P'; rewrite lookup_merge by done; intros.
specialize (Hm i). destruct (m1 !! i), (m2 !! i);
simplify_equality'; auto using bool_decide_pack.
* intros Hm i. specialize (Hm i). rewrite lookup_merge in Hm by done.
- intros Hm i. specialize (Hm i). rewrite lookup_merge in Hm by done.
destruct (m1 !! i), (m2 !! i); simplify_equality'; auto;
by eapply bool_decide_unpack, Hm.
Qed.
......@@ -950,10 +950,10 @@ Lemma map_not_Forall2 (m1 : M A) (m2 : M B) :
∨ (∃ y, m1 !! i = None ∧ m2 !! i = Some y ∧ ¬Q y).
Proof.
split.
* rewrite map_relation_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.
* unfold map_relation, option_relation.
- unfold map_relation, option_relation.
by intros [i[(x&y&?&?&?)|[(x&?&?&?)|(y&?&?&?)]]] Hm;
specialize (Hm i); simplify_option_equality.
Qed.
......@@ -1003,9 +1003,9 @@ 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);
- intro. apply (map_disjoint_Some_l {[i := x]} _ _ x);
auto using lookup_singleton.
* intros ? j y1 y2. destruct (decide (i = j)) as [->|].
- intros ? j y1 y2. destruct (decide (i = j)) as [->|].
+ rewrite lookup_singleton. intuition congruence.
+ by rewrite lookup_singleton_ne.
Qed.
......@@ -1238,8 +1238,8 @@ Proof.
apply map_eq. intros j. apply option_eq. intros y.
rewrite lookup_union_Some_raw.
destruct (decide (i = j)); subst.
* rewrite !lookup_singleton, lookup_insert. intuition congruence.
* rewrite !lookup_singleton_ne, lookup_insert_ne; intuition congruence.
- rewrite !lookup_singleton, lookup_insert. intuition congruence.
- rewrite !lookup_singleton_ne, lookup_insert_ne; intuition congruence.
Qed.
Lemma insert_union_singleton_r {A} (m : M A) i x :
m !! i = None →