Commit bd222fbf authored by Robbert Krebbers's avatar Robbert Krebbers

Fold operation on finite maps.

parent f9bc9466
......@@ -112,6 +112,11 @@ Definition map_imap `{∀ A, Insert K A (M A), ∀ A, Empty (M A),
A, FinMapToList K A (M A)} {A B} (f : K A option B) (m : M A) : M B :=
map_of_list (omap (λ ix, (fst ix,) <$> curry f ix) (map_to_list m)).
(* Folds a function [f] over a map. The order in which the function is called
is unspecified. *)
Definition map_fold `{FinMapToList K A M} {B}
(f : K A B B) (b : B) : M B := foldr (curry f) b map_to_list.
(** * Theorems *)
Section theorems.
Context `{FinMap K M}.
......@@ -814,6 +819,47 @@ Proof.
- by apply lt_wf.
(** ** The fold operation *)
Lemma map_fold_empty {A B} (f : K A B B) (b : B) :
map_fold f b = b.
Proof. unfold map_fold; simpl. by rewrite map_to_list_empty. Qed.
Lemma map_fold_insert {A B} (R : relation B) `{!PreOrder R}
(f : K A B B) (b : B) (i : K) (x : A) (m : M A) :
( j z, Proper (R ==> R) (f j z))
( j1 j2 z1 z2 y, R (f j1 z1 (f j2 z2 y)) (f j2 z2 (f j1 z1 y)))
m !! i = None
R (map_fold f b (<[i:=x]> m)) (f i x (map_fold f b m)).
intros. unfold map_fold; simpl.
assert ( kz, Proper (R ==> R) (curry f kz)) by (intros []; apply _).
trans (foldr (curry f) b ((i, x) :: map_to_list m)); [|done].
eapply (foldr_permutation R (curry f) b), map_to_list_insert; auto.
intros [] []; simpl; eauto.
Lemma map_fold_ind {A B} (P : B M A Prop) (f : K A B B) (b : B) :
P b
( i x m r, m !! i = None P r m P (f i x r) (<[i:=x]> m))
m, P (map_fold f b m) m.
intros Hemp Hinsert.
cut ( l, NoDup l
m, ( i x, m !! i = Some x (i,x) l) P (foldr (curry f) b l) m).
{ intros help ?. apply help; [apply NoDup_map_to_list|].
intros i x. by rewrite elem_of_map_to_list. }
induction 1 as [|[i x] l ?? IH]; simpl.
{ intros m Hm. cut (m = ); [by intros ->|]. apply map_empty; intros i.
apply eq_None_not_Some; intros [x []%Hm%elem_of_nil]. }
intros m Hm. assert (m !! i = Some x) by (apply Hm; by left).
rewrite <-(insert_id m i x), <-insert_delete by done.
apply Hinsert; auto using lookup_delete.
apply IH. intros j y. rewrite lookup_delete_Some, Hm. split.
- by intros [? [[= ??]|?]%elem_of_cons].
- intros ?; split; [intros ->|by right].
assert (m !! j = Some y) by (apply Hm; by right). naive_solver.
(** ** Properties of the [map_Forall] predicate *)
Section map_Forall.
Context {A} (P : K A Prop).
......@@ -3210,8 +3210,8 @@ Definition foldr_app := @fold_right_app.
Lemma foldl_app {A B} (f : A B A) (l k : list B) (a : A) :
foldl f a (l ++ k) = foldl f (foldl f a l) k.
Proof. revert a. induction l; simpl; auto. Qed.
Lemma foldr_permutation {A B} (R : relation B) `{!Equivalence R}
(f : A B B) (b : B) `{!Proper ((=) ==> R ==> R) f}
Lemma foldr_permutation {A B} (R : relation B) `{!PreOrder R}
(f : A B B) (b : B) `{! x, Proper (R ==> R) (f x)}
(Hf : a1 a2 b, R (f a1 (f a2 b)) (f a2 (f a1 b))) :
Proper (() ==> R) (foldr f b).
Proof. induction 1; simpl; [done|by f_equiv|apply Hf|etrans; eauto]. Qed.
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