Commit 061604dd authored by Robbert Krebbers's avatar Robbert Krebbers
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Remove generality for COFE/CMRA maps.

parent 287afa0d
Require Export modures.cmra.
Require Import prelude.pmap prelude.natmap prelude.gmap modures.option.
Require Export modures.cmra prelude.gmap.
Require Import modures.option.
Local Obligation Tactic := idtac.
Section map.
Context `{FinMap K M}.
Context `{Countable K}.
(* COFE *)
Global Instance map_dist `{Dist A} : Dist (M A) := λ n m1 m2,
Global Instance map_dist `{Dist A} : Dist (gmap K A) := λ n m1 m2,
i, m1 !! i ={n}= m2 !! i.
Program Definition map_chain `{Dist A} (c : chain (M A))
Program Definition map_chain `{Dist A} (c : chain (gmap K A))
(k : K) : chain (option A) := {| chain_car n := c n !! k |}.
Next Obligation. by intros A ? c k n i ?; apply (chain_cauchy c). Qed.
Global Instance map_compl `{Cofe A} : Compl (M A) := λ c,
Global Instance map_compl `{Cofe A} : Compl (gmap K A) := λ c,
map_imap (λ i _, compl (map_chain c i)) (c 1).
Global Instance map_cofe `{Cofe A} : Cofe (M A).
Global Instance map_cofe `{Cofe A} : Cofe (gmap K A).
Proof.
split.
* intros m1 m2; split.
......@@ -30,33 +31,33 @@ Proof.
by rewrite conv_compl; simpl; apply reflexive_eq.
Qed.
Global Instance lookup_ne `{Dist A} n k :
Proper (dist n ==> dist n) (lookup k : M A option A).
Proper (dist n ==> dist n) (lookup k : gmap K A option A).
Proof. by intros m1 m2. Qed.
Global Instance insert_ne `{Dist A} (i : K) n :
Proper (dist n ==> dist n ==> dist n) (insert (M:=M A) i).
Proper (dist n ==> dist n ==> dist n) (insert (M:=gmap K A) i).
Proof.
intros x y ? m m' ? j; destruct (decide (i = j)); simplify_map_equality;
[by constructor|by apply lookup_ne].
Qed.
Global Instance delete_ne `{Dist A} (i : K) n :
Proper (dist n ==> dist n) (delete (M:=M A) i).
Proper (dist n ==> dist n) (delete (M:=gmap K A) i).
Proof.
intros m m' ? j; destruct (decide (i = j)); simplify_map_equality;
[by constructor|by apply lookup_ne].
Qed.
Global Instance map_empty_timeless `{Dist A, Equiv A} : Timeless ( : M A).
Global Instance map_empty_timeless `{Dist A, Equiv A} : Timeless ( : gmap K A).
Proof.
intros m Hm i; specialize (Hm i); rewrite lookup_empty in Hm |- *.
inversion_clear Hm; constructor.
Qed.
Global Instance map_lookup_timeless `{Cofe A} (m : M A) i :
Global Instance map_lookup_timeless `{Cofe A} (m : gmap K A) i :
Timeless m Timeless (m !! i).
Proof.
intros ? [x|] Hx; [|by symmetry; apply (timeless _)].
rewrite (timeless m (<[i:=x]>m)), lookup_insert; auto.
by symmetry in Hx; inversion Hx; cofe_subst; rewrite insert_id.
Qed.
Global Instance map_ra_insert_timeless `{Cofe A} (m : M A) i x :
Global Instance map_ra_insert_timeless `{Cofe A} (m : gmap K A) i x :
Timeless x Timeless m Timeless (<[i:=x]>m).
Proof.
intros ?? m' Hm j; destruct (decide (i = j)); simplify_map_equality.
......@@ -64,11 +65,12 @@ Proof.
by apply (timeless _); rewrite <-Hm, lookup_insert_ne by done.
Qed.
Global Instance map_ra_singleton_timeless `{Cofe A} (i : K) (x : A) :
Timeless x Timeless ({[ i, x ]} : M A) := _.
Timeless x Timeless ({[ i, x ]} : gmap K A) := _.
Instance map_fmap_ne `{Dist A, Dist B} (f : A B) n :
Proper (dist n ==> dist n) f Proper (dist n ==> dist n) (fmap (M:=M) f).
Proper (dist n ==> dist n) f
Proper (dist n ==> dist n) (fmap (M:=gmap K) f).
Proof. by intros ? m m' Hm k; rewrite !lookup_fmap; apply option_fmap_ne. Qed.
Definition mapC (A : cofeT) : cofeT := CofeT (M A).
Canonical Structure mapC (A : cofeT) : cofeT := CofeT (gmap K A).
Definition mapC_map {A B} (f: A -n> B) : mapC A -n> mapC B :=
CofeMor (fmap f : mapC A mapC B).
Global Instance mapC_map_ne {A B} n :
......@@ -79,19 +81,19 @@ Proof.
Qed.
(* CMRA *)
Global Instance map_op `{Op A} : Op (M A) := merge op.
Global Instance map_unit `{Unit A} : Unit (M A) := fmap unit.
Global Instance map_valid `{Valid A} : Valid (M A) := λ m, i, (m !! i).
Global Instance map_validN `{ValidN A} : ValidN (M A) := λ n m,
Global Instance map_op `{Op A} : Op (gmap K A) := merge op.
Global Instance map_unit `{Unit A} : Unit (gmap K A) := fmap unit.
Global Instance map_valid `{Valid A} : Valid (gmap K A) := λ m, i, (m !! i).
Global Instance map_validN `{ValidN A} : ValidN (gmap K A) := λ n m,
i, {n} (m!!i).
Global Instance map_minus `{Minus A} : Minus (M A) := merge minus.
Global Instance map_minus `{Minus A} : Minus (gmap K A) := merge minus.
Lemma lookup_op `{Op A} m1 m2 i : (m1 m2) !! i = m1 !! i m2 !! i.
Proof. by apply lookup_merge. Qed.
Lemma lookup_minus `{Minus A} m1 m2 i : (m1 m2) !! i = m1 !! i m2 !! i.
Proof. by apply lookup_merge. Qed.
Lemma lookup_unit `{Unit A} m i : unit m !! i = unit (m !! i).
Proof. by apply lookup_fmap. Qed.
Lemma map_included_spec `{CMRA A} (m1 m2 : M A) :
Lemma map_included_spec `{CMRA A} (m1 m2 : gmap K A) :
m1 m2 i, m1 !! i m2 !! i.
Proof.
split.
......@@ -99,7 +101,7 @@ Proof.
* intros Hm; exists (m2 m1); intros i.
by rewrite lookup_op, lookup_minus, ra_op_minus.
Qed.
Lemma map_includedN_spec `{CMRA A} (m1 m2 : M A) n :
Lemma map_includedN_spec `{CMRA A} (m1 m2 : gmap K A) n :
m1 {n} m2 i, m1 !! i {n} m2 !! i.
Proof.
split.
......@@ -107,7 +109,7 @@ Proof.
* intros Hm; exists (m2 m1); intros i.
by rewrite lookup_op, lookup_minus, cmra_op_minus.
Qed.
Global Instance map_cmra `{CMRA A} : CMRA (M A).
Global Instance map_cmra `{CMRA A} : CMRA (gmap K A).
Proof.
split.
* apply _.
......@@ -131,13 +133,13 @@ Proof.
* intros x y n; rewrite map_includedN_spec; intros ? i.
by rewrite lookup_op, lookup_minus, cmra_op_minus by done.
Qed.
Global Instance map_ra_empty `{RA A} : RAIdentity (M A).
Global Instance map_ra_empty `{RA A} : RAIdentity (gmap K A).
Proof.
split.
* by intros ?; rewrite lookup_empty.
* by intros m i; simpl; rewrite lookup_op, lookup_empty; destruct (m !! i).
Qed.
Global Instance map_cmra_extend `{CMRA A, !CMRAExtend A} : CMRAExtend (M A).
Global Instance map_cmra_extend `{CMRA A, !CMRAExtend A} : CMRAExtend (gmap K A).
Proof.
intros n m m1 m2 Hm Hm12.
assert ( i, m !! i ={n}= m1 !! i m2 !! i) as Hm12'
......@@ -156,9 +158,10 @@ Proof.
by destruct (m1 !! i), (m2 !! i); inversion_clear Hm12''.
Qed.
Global Instance map_empty_valid_timeless `{Valid A, ValidN A} :
ValidTimeless ( : M A).
ValidTimeless ( : gmap K A).
Proof. by intros ??; rewrite lookup_empty. Qed.
Global Instance map_ra_insert_valid_timeless `{Valid A,ValidN A} (m: M A) i x:
Global Instance map_ra_insert_valid_timeless `{Valid A, ValidN A}
(m : gmap K A) i x:
ValidTimeless x ValidTimeless m m !! i = None
ValidTimeless (<[i:=x]>m).
Proof.
......@@ -169,20 +172,20 @@ Proof.
by specialize (Hm j); simplify_map_equality.
Qed.
Global Instance map_ra_singleton_valid_timeless `{Valid A, ValidN A} (i : K) x :
ValidTimeless x ValidTimeless ({[ i, x ]} : M A).
ValidTimeless x ValidTimeless ({[ i, x ]} : gmap K A).
Proof.
intros ?; apply (map_ra_insert_valid_timeless _ _ _ _ _); simpl.
by rewrite lookup_empty.
Qed.
Lemma map_insert_valid `{ValidN A} (m : M A) n i x :
Lemma map_insert_valid `{ValidN A} (m : gmap K A) n i x :
{n} x {n} m {n} (<[i:=x]>m).
Proof. by intros ?? j; destruct (decide (i = j)); simplify_map_equality. Qed.
Lemma map_insert_op `{Op A} (m1 m2 : M A) i x :
Lemma map_insert_op `{Op A} (m1 m2 : gmap K A) i x :
m2 !! i = None <[i:=x]>(m1 m2) = <[i:=x]>m1 m2.
Proof. by intros Hi; apply (insert_merge_l _); rewrite Hi. Qed.
Definition mapRA (A : cmraT) : cmraT := CMRAT (M A).
Canonical Structure mapRA (A : cmraT) : cmraT := CMRAT (gmap K A).
Global Instance map_fmap_cmra_monotone `{CMRA A, CMRA B} (f : A B)
`{!CMRAMonotone f} : CMRAMonotone (fmap f : M A M B).
`{!CMRAMonotone f} : CMRAMonotone (fmap f : gmap K A gmap K B).
Proof.
split.
* intros m1 m2 n; rewrite !map_includedN_spec; intros Hm i.
......@@ -195,47 +198,27 @@ Global Instance mapRA_map_ne {A B} n :
Proper (dist n ==> dist n) (@mapRA_map A B) := mapC_map_ne n.
Global Instance mapRA_map_monotone {A B : cmraT} (f : A -n> B)
`{!CMRAMonotone f} : CMRAMonotone (mapRA_map f) := _.
End map.
Section map_dom.
Context `{FinMapDom K M D, Fresh K D, !FreshSpec K D}.
Lemma map_dom_op `{Op A} (m1 m2: M A) : dom D (m1 m2) dom D m1 dom D m2.
Proof.
apply elem_of_equiv; intros i; rewrite elem_of_union, !elem_of_dom.
unfold is_Some; setoid_rewrite lookup_op.
destruct (m1 !! i), (m2 !! i); naive_solver.
Qed.
Lemma map_update_alloc `{CMRA A} (m : M A) x :
x m : λ m', i, m' = <[i:=x]>m m !! i = None.
Proof.
intros ? mf n Hm. set (i := fresh (dom D (m mf))).
assert (i dom D m i dom D mf) as [??].
{ rewrite <-not_elem_of_union, <-map_dom_op; apply is_fresh. }
exists (<[i:=x]>m); split; [exists i; split; [done|]|].
* by apply not_elem_of_dom.
* rewrite <-map_insert_op by (by apply not_elem_of_dom).
by apply map_insert_valid; [apply cmra_valid_validN|].
Qed.
End map_dom.
Arguments mapC {_} _ {_ _ _ _ _ _ _ _ _} _.
Arguments mapRA {_} _ {_ _ _ _ _ _ _ _ _} _.
Context `{Fresh K (gset K), !FreshSpec K (gset K)}.
Canonical Structure natmapC := mapC natmap.
Definition natmapC_map {A B}
(f : A -n> B) : natmapC A -n> natmapC B := mapC_map f.
Canonical Structure PmapC := mapC Pmap.
Definition PmapC_map {A B} (f : A -n> B) : PmapC A -n> PmapC B := mapC_map f.
Canonical Structure gmapC K `{Countable K} := mapC (gmap K).
Definition gmapC_map `{Countable K} {A B}
(f : A -n> B) : gmapC K A -n> gmapC K B := mapC_map f.
Canonical Structure natmapRA := mapRA natmap.
Definition natmapRA_map {A B : cmraT}
(f : A -n> B) : natmapRA A -n> natmapRA B := mapRA_map f.
Canonical Structure PmapRA := mapRA Pmap.
Definition PmapRA_map {A B : cmraT}
(f : A -n> B) : PmapRA A -n> PmapRA B := mapRA_map f.
Canonical Structure gmapRA K `{Countable K} := mapRA (gmap K).
Definition gmapRA_map `{Countable K} {A B : cmraT}
(f : A -n> B) : gmapRA K A -n> gmapRA K B := mapRA_map f.
Lemma map_dom_op `{Op A} (m1 m2 : gmap K A) :
dom (gset K) (m1 m2) dom _ m1 dom _ m2.
Proof.
apply elem_of_equiv; intros i; rewrite elem_of_union, !elem_of_dom.
unfold is_Some; setoid_rewrite lookup_op.
destruct (m1 !! i), (m2 !! i); naive_solver.
Qed.
Lemma map_update_alloc `{CMRA A} (m : gmap K A) x :
x m : λ m', i, m' = <[i:=x]>m m !! i = None.
Proof.
intros ? mf n Hm. set (i := fresh (dom (gset K) (m mf))).
assert (i dom (gset K) m i dom (gset K) mf) as [??].
{ rewrite <-not_elem_of_union, <-map_dom_op; apply is_fresh. }
exists (<[i:=x]>m); split; [exists i; split; [done|]|].
* by apply not_elem_of_dom.
* rewrite <-map_insert_op by (by apply not_elem_of_dom).
by apply map_insert_valid; [apply cmra_valid_validN|].
Qed.
End map.
Arguments mapC _ {_ _} _.
Arguments mapRA _ {_ _} _.
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