Commit 1078c169 authored by Robbert Krebbers's avatar Robbert Krebbers

Stronger "generic proper" lemmas for `big_op{L,M,S,MS}`.

- Renamed them from `_forall` into `_gen_proper`, to avoid confusion
  with `big_sep{L,M,S,MS}_forall`, which are actually about `∀`.
- For lists and maps there now two variants, `_gen_proper_2`, in case
  the maps or lists on both sides are different, and `_gen_proper`,
  in case the maps or lists on both sides are the same.
parent 547bbb5a
......@@ -101,19 +101,36 @@ Section list.
Lemma big_opL_unit l : ([^o list] ky l, monoid_unit) (monoid_unit : M).
Proof. induction l; rewrite /= ?left_id //. Qed.
Lemma big_opL_forall R f g l :
Lemma big_opL_gen_proper_2 (R : relation M) f g l1 l2 :
R monoid_unit monoid_unit
Proper (R ==> R ==> R) o
( k,
match l1 !! k, l2 !! k with
| Some y1, Some y2 => R (f k y1) (g k y2)
| None, None => True
| _, _ => False
end)
R ([^o list] k y l1, f k y) ([^o list] k y l2, g k y).
Proof.
intros ??. revert l2 f g. induction l1 as [|x1 l1 IH]=> -[|x2 l2] //= f g Hfg.
- by specialize (Hfg 0).
- by specialize (Hfg 0).
- f_equiv; [apply (Hfg 0)|]. apply IH. intros k. apply (Hfg (S k)).
Qed.
Lemma big_opL_gen_proper R f g l :
Reflexive R
Proper (R ==> R ==> R) o
( k y, l !! k = Some y R (f k y) (g k y))
R ([^o list] k y l, f k y) ([^o list] k y l, g k y).
Proof.
intros ??. revert f g. induction l as [|x l IH]=> f g ? //=; f_equiv; eauto.
intros. apply big_opL_gen_proper_2; [done..|].
intros k. destruct (l !! k) eqn:?; auto.
Qed.
Lemma big_opL_ext f g l :
( k y, l !! k = Some y f k y = g k y)
([^o list] k y l, f k y) = [^o list] k y l, g k y.
Proof. apply big_opL_forall; apply _. Qed.
Proof. apply big_opL_gen_proper; apply _. Qed.
Lemma big_opL_permutation (f : A M) l1 l2 :
l1 l2 ([^o list] x l1, f x) ([^o list] x l2, f x).
......@@ -132,11 +149,11 @@ Section list.
Lemma big_opL_ne f g l n :
( k y, l !! k = Some y f k y {n} g k y)
([^o list] k y l, f k y) {n} ([^o list] k y l, g k y).
Proof. apply big_opL_forall; apply _. Qed.
Proof. apply big_opL_gen_proper; apply _. Qed.
Lemma big_opL_proper f g l :
( k y, l !! k = Some y f k y g k y)
([^o list] k y l, f k y) ([^o list] k y l, g k y).
Proof. apply big_opL_forall; apply _. Qed.
Proof. apply big_opL_gen_proper; apply _. Qed.
Global Instance big_opL_ne' n :
Proper (pointwise_relation _ (pointwise_relation _ (dist n)) ==> (=) ==> dist n)
......@@ -179,30 +196,72 @@ Section gmap.
Implicit Types m : gmap K A.
Implicit Types f g : K A M.
Lemma big_opM_forall R f g m :
Reflexive R Proper (R ==> R ==> R) o
Lemma big_opM_empty f : ([^o map] kx , f k x) = monoid_unit.
Proof. by rewrite big_opM_eq /big_opM_def map_to_list_empty. Qed.
Lemma big_opM_insert f m i x :
m !! i = None
([^o map] ky <[i:=x]> m, f k y) f i x `o` [^o map] ky m, f k y.
Proof. intros ?. by rewrite big_opM_eq /big_opM_def map_to_list_insert. Qed.
Lemma big_opM_delete f m i x :
m !! i = Some x
([^o map] ky m, f k y) f i x `o` [^o map] ky delete i m, f k y.
Proof.
intros. rewrite -big_opM_insert ?lookup_delete //.
by rewrite insert_delete insert_id.
Qed.
Lemma big_opM_gen_proper_2 (R : relation M) f g m1 m2 :
subrelation () R Equivalence R
Proper (R ==> R ==> R) o
( k,
match m1 !! k, m2 !! k with
| Some y1, Some y2 => R (f k y1) (g k y2)
| None, None => True
| _, _ => False
end)
R ([^o map] k x m1, f k x) ([^o map] k x m2, g k x).
Proof.
intros HR ??. revert m2 f g.
induction m1 as [|k x1 m1 Hm1k IH] using map_ind=> m2 f g Hfg.
{ destruct m2 as [|k x2 m2 _ _] using map_ind.
{ rewrite !big_opM_empty. by apply HR. }
generalize (Hfg k). by rewrite lookup_empty lookup_insert. }
generalize (Hfg k). rewrite lookup_insert.
destruct (m2 !! k) as [x2|] eqn:Hm2k; [intros Hk|done].
etrans; [by apply HR, big_opM_insert|].
etrans; [|by symmetry; apply HR, big_opM_delete].
f_equiv; [done|]. apply IH=> k'. destruct (decide (k = k')) as [->|?].
- by rewrite lookup_delete Hm1k.
- generalize (Hfg k'). rewrite lookup_insert_ne // lookup_delete_ne //.
Qed.
Lemma big_opM_gen_proper R f g m :
Reflexive R
Proper (R ==> R ==> R) o
( k x, m !! k = Some x R (f k x) (g k x))
R ([^o map] k x m, f k x) ([^o map] k x m, g k x).
Proof.
intros ?? Hf. rewrite big_opM_eq. apply (big_opL_forall R); auto.
intros ?? Hf. rewrite big_opM_eq. apply (big_opL_gen_proper R); auto.
intros k [i x] ?%elem_of_list_lookup_2. by apply Hf, elem_of_map_to_list.
Qed.
Lemma big_opM_ext f g m :
( k x, m !! k = Some x f k x = g k x)
([^o map] k x m, f k x) = ([^o map] k x m, g k x).
Proof. apply big_opM_forall; apply _. Qed.
Proof. apply big_opM_gen_proper; apply _. Qed.
(** The lemmas [big_opM_ne] and [big_opM_proper] are more generic than the
instances as they also give [m !! k = Some y] in the premise. *)
Lemma big_opM_ne f g m n :
( k x, m !! k = Some x f k x {n} g k x)
([^o map] k x m, f k x) {n} ([^o map] k x m, g k x).
Proof. apply big_opM_forall; apply _. Qed.
Proof. apply big_opM_gen_proper; apply _. Qed.
Lemma big_opM_proper f g m :
( k x, m !! k = Some x f k x g k x)
([^o map] k x m, f k x) ([^o map] k x m, g k x).
Proof. apply big_opM_forall; apply _. Qed.
Proof. apply big_opM_gen_proper; apply _. Qed.
Global Instance big_opM_ne' n :
Proper (pointwise_relation _ (pointwise_relation _ (dist n)) ==> (=) ==> dist n)
......@@ -213,22 +272,6 @@ Section gmap.
(big_opM o (K:=K) (A:=A)).
Proof. intros f g Hf m ? <-. apply big_opM_proper; intros; apply Hf. Qed.
Lemma big_opM_empty f : ([^o map] kx , f k x) = monoid_unit.
Proof. by rewrite big_opM_eq /big_opM_def map_to_list_empty. Qed.
Lemma big_opM_insert f m i x :
m !! i = None
([^o map] ky <[i:=x]> m, f k y) f i x `o` [^o map] ky m, f k y.
Proof. intros ?. by rewrite big_opM_eq /big_opM_def map_to_list_insert. Qed.
Lemma big_opM_delete f m i x :
m !! i = Some x
([^o map] ky m, f k y) f i x `o` [^o map] ky delete i m, f k y.
Proof.
intros. rewrite -big_opM_insert ?lookup_delete //.
by rewrite insert_delete insert_id.
Qed.
Lemma big_opM_singleton f i x : ([^o map] ky {[i:=x]}, f k y) f i x.
Proof.
rewrite -insert_empty big_opM_insert/=; last auto using lookup_empty.
......@@ -292,30 +335,30 @@ Section gset.
Implicit Types X : gset A.
Implicit Types f : A M.
Lemma big_opS_forall R f g X :
Lemma big_opS_gen_proper R f g X :
Reflexive R Proper (R ==> R ==> R) o
( x, x X R (f x) (g x))
R ([^o set] x X, f x) ([^o set] x X, g x).
Proof.
rewrite big_opS_eq. intros ?? Hf. apply (big_opL_forall R); auto.
rewrite big_opS_eq. intros ?? Hf. apply (big_opL_gen_proper R); auto.
intros k x ?%elem_of_list_lookup_2. by apply Hf, elem_of_elements.
Qed.
Lemma big_opS_ext f g X :
( x, x X f x = g x)
([^o set] x X, f x) = ([^o set] x X, g x).
Proof. apply big_opS_forall; apply _. Qed.
Proof. apply big_opS_gen_proper; apply _. Qed.
(** The lemmas [big_opS_ne] and [big_opS_proper] are more generic than the
instances as they also give [x ∈ X] in the premise. *)
Lemma big_opS_ne f g X n :
( x, x X f x {n} g x)
([^o set] x X, f x) {n} ([^o set] x X, g x).
Proof. apply big_opS_forall; apply _. Qed.
Proof. apply big_opS_gen_proper; apply _. Qed.
Lemma big_opS_proper f g X :
( x, x X f x g x)
([^o set] x X, f x) ([^o set] x X, g x).
Proof. apply big_opS_forall; apply _. Qed.
Proof. apply big_opS_gen_proper; apply _. Qed.
Global Instance big_opS_ne' n :
Proper (pointwise_relation _ (dist n) ==> (=) ==> dist n) (big_opS o (A:=A)).
......@@ -386,30 +429,30 @@ Section gmultiset.
Implicit Types X : gmultiset A.
Implicit Types f : A M.
Lemma big_opMS_forall R f g X :
Lemma big_opMS_gen_proper R f g X :
Reflexive R Proper (R ==> R ==> R) o
( x, x X R (f x) (g x))
R ([^o mset] x X, f x) ([^o mset] x X, g x).
Proof.
rewrite big_opMS_eq. intros ?? Hf. apply (big_opL_forall R); auto.
rewrite big_opMS_eq. intros ?? Hf. apply (big_opL_gen_proper R); auto.
intros k x ?%elem_of_list_lookup_2. by apply Hf, gmultiset_elem_of_elements.
Qed.
Lemma big_opMS_ext f g X :
( x, x X f x = g x)
([^o mset] x X, f x) = ([^o mset] x X, g x).
Proof. apply big_opMS_forall; apply _. Qed.
Proof. apply big_opMS_gen_proper; apply _. Qed.
(** The lemmas [big_opMS_ne] and [big_opMS_proper] are more generic than the
instances as they also give [x ∈ X] in the premise. *)
Lemma big_opMS_ne f g X n :
( x, x X f x {n} g x)
([^o mset] x X, f x) {n} ([^o mset] x X, g x).
Proof. apply big_opMS_forall; apply _. Qed.
Proof. apply big_opMS_gen_proper; apply _. Qed.
Lemma big_opMS_proper f g X :
( x, x X f x g x)
([^o mset] x X, f x) ([^o mset] x X, g x).
Proof. apply big_opMS_forall; apply _. Qed.
Proof. apply big_opMS_gen_proper; apply _. Qed.
Global Instance big_opMS_ne' n :
Proper (pointwise_relation _ (dist n) ==> (=) ==> dist n) (big_opMS o (A:=A)).
......
......@@ -102,7 +102,7 @@ Section sep_list.
Lemma big_sepL_mono Φ Ψ l :
( k y, l !! k = Some y Φ k y Ψ k y)
([ list] k y l, Φ k y) [ list] k y l, Ψ k y.
Proof. apply big_opL_forall; apply _. Qed.
Proof. apply big_opL_gen_proper; apply _. Qed.
Lemma big_sepL_ne Φ Ψ l n :
( k y, l !! k = Some y Φ k y {n} Ψ k y)
([ list] k y l, Φ k y)%I {n} ([ list] k y l, Ψ k y)%I.
......@@ -525,7 +525,7 @@ Section and_list.
Lemma big_andL_mono Φ Ψ l :
( k y, l !! k = Some y Φ k y Ψ k y)
([ list] k y l, Φ k y) [ list] k y l, Ψ k y.
Proof. apply big_opL_forall; apply _. Qed.
Proof. apply big_opL_gen_proper; apply _. Qed.
Lemma big_andL_ne Φ Ψ l n :
( k y, l !! k = Some y Φ k y {n} Ψ k y)
([ list] k y l, Φ k y)%I {n} ([ list] k y l, Ψ k y)%I.
......@@ -624,7 +624,7 @@ Section or_list.
Lemma big_orL_mono Φ Ψ l :
( k y, l !! k = Some y Φ k y Ψ k y)
([ list] k y l, Φ k y) [ list] k y l, Ψ k y.
Proof. apply big_opL_forall; apply _. Qed.
Proof. apply big_opL_gen_proper; apply _. Qed.
Lemma big_orL_ne Φ Ψ l n :
( k y, l !! k = Some y Φ k y {n} Ψ k y)
([ list] k y l, Φ k y)%I {n} ([ list] k y l, Ψ k y)%I.
......@@ -722,7 +722,7 @@ Section map.
Lemma big_sepM_mono Φ Ψ m :
( k x, m !! k = Some x Φ k x Ψ k x)
([ map] k x m, Φ k x) [ map] k x m, Ψ k x.
Proof. apply big_opM_forall; apply _ || auto. Qed.
Proof. apply big_opM_gen_proper; apply _ || auto. Qed.
Lemma big_sepM_ne Φ Ψ m n :
( k x, m !! k = Some x Φ k x {n} Ψ k x)
([ map] k x m, Φ k x)%I {n} ([ map] k x m, Ψ k x)%I.
......@@ -1243,7 +1243,7 @@ Section gset.
Lemma big_sepS_mono Φ Ψ X :
( x, x X Φ x Ψ x)
([ set] x X, Φ x) [ set] x X, Ψ x.
Proof. intros. apply big_opS_forall; apply _ || auto. Qed.
Proof. intros. apply big_opS_gen_proper; apply _ || auto. Qed.
Lemma big_sepS_ne Φ Ψ X n :
( x, x X Φ x {n} Ψ x)
([ set] x X, Φ x)%I {n} ([ set] x X, Ψ x)%I.
......@@ -1414,7 +1414,7 @@ Section gmultiset.
Lemma big_sepMS_mono Φ Ψ X :
( x, x X Φ x Ψ x)
([ mset] x X, Φ x) [ mset] x X, Ψ x.
Proof. intros. apply big_opMS_forall; apply _ || auto. Qed.
Proof. intros. apply big_opMS_gen_proper; apply _ || auto. Qed.
Lemma big_sepMS_ne Φ Ψ X n :
( x, x X Φ x {n} Ψ x)
([ mset] x X, Φ x)%I {n} ([ mset] x X, Ψ x)%I.
......
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