Library iris.algebra.functions
From stdpp Require Import finite.
From iris.algebra Require Export cmra.
From iris.algebra Require Import updates.
Definition discrete_fun_insert {SI : sidx} `{!EqDecision A} {B : A → ofe}
(x : A) (y : B x) (f : discrete_fun B) : discrete_fun B := λ x',
match decide (x = x') with left H ⇒ eq_rect _ B y _ H | right _ ⇒ f x' end.
Global Instance: Params (@discrete_fun_insert) 6 := {}.
Definition discrete_fun_singleton {SI : sidx} `{!EqDecision A} {B : A → ucmra}
(x : A) (y : B x) : discrete_fun B := discrete_fun_insert x y ε.
Global Instance: Params (@discrete_fun_singleton) 6 := {}.
Section ofe.
Context {SI : sidx} `{!EqDecision A} {B : A → ofe}.
Implicit Types x : A.
Implicit Types f g : discrete_fun B.
Global Instance discrete_funO_ofe_discrete :
(∀ i, OfeDiscrete (B i)) → OfeDiscrete (discrete_funO B).
Proof. intros HB f f' Heq i. apply HB, Heq. Qed.
From iris.algebra Require Export cmra.
From iris.algebra Require Import updates.
Definition discrete_fun_insert {SI : sidx} `{!EqDecision A} {B : A → ofe}
(x : A) (y : B x) (f : discrete_fun B) : discrete_fun B := λ x',
match decide (x = x') with left H ⇒ eq_rect _ B y _ H | right _ ⇒ f x' end.
Global Instance: Params (@discrete_fun_insert) 6 := {}.
Definition discrete_fun_singleton {SI : sidx} `{!EqDecision A} {B : A → ucmra}
(x : A) (y : B x) : discrete_fun B := discrete_fun_insert x y ε.
Global Instance: Params (@discrete_fun_singleton) 6 := {}.
Section ofe.
Context {SI : sidx} `{!EqDecision A} {B : A → ofe}.
Implicit Types x : A.
Implicit Types f g : discrete_fun B.
Global Instance discrete_funO_ofe_discrete :
(∀ i, OfeDiscrete (B i)) → OfeDiscrete (discrete_funO B).
Proof. intros HB f f' Heq i. apply HB, Heq. Qed.
Properties of discrete_fun_insert.
Global Instance discrete_fun_insert_ne x :
NonExpansive2 (discrete_fun_insert (B:=B) x).
Proof.
intros n y1 y2 ? f1 f2 ? x'; rewrite /discrete_fun_insert.
by destruct (decide _) as [[]|].
Qed.
Global Instance discrete_fun_insert_proper x :
Proper ((≡) ==> (≡) ==> (≡)) (discrete_fun_insert (B:=B) x) := ne_proper_2 _.
Lemma discrete_fun_lookup_insert f x y : (discrete_fun_insert x y f) x = y.
Proof.
rewrite /discrete_fun_insert; destruct (decide _) as [Hx|]; last done.
by rewrite (proof_irrel Hx eq_refl).
Qed.
Lemma discrete_fun_lookup_insert_ne f x x' y :
x ≠ x' → (discrete_fun_insert x y f) x' = f x'.
Proof. by rewrite /discrete_fun_insert; destruct (decide _). Qed.
Global Instance discrete_fun_insert_discrete f x y :
Discrete f → Discrete y → Discrete (discrete_fun_insert x y f).
Proof.
intros ?? g Heq x'; destruct (decide (x = x')) as [->|].
- rewrite discrete_fun_lookup_insert.
apply: discrete. by rewrite -(Heq x') discrete_fun_lookup_insert.
- rewrite discrete_fun_lookup_insert_ne //.
apply: discrete. by rewrite -(Heq x') discrete_fun_lookup_insert_ne.
Qed.
End ofe.
Section cmra.
Context {SI : sidx} `{!EqDecision A} {B : A → ucmra}.
Implicit Types x : A.
Implicit Types f g : discrete_fun B.
Global Instance discrete_funR_cmra_discrete:
(∀ i, CmraDiscrete (B i)) → CmraDiscrete (discrete_funR B).
Proof. intros HB. split; [apply _|]. intros x Hv i. apply HB, Hv. Qed.
Global Instance discrete_fun_singleton_ne x :
NonExpansive (discrete_fun_singleton x : B x → _).
Proof.
intros n y1 y2 ?; apply discrete_fun_insert_ne; [done|].
by apply equiv_dist.
Qed.
Global Instance discrete_fun_singleton_proper x :
Proper ((≡) ==> (≡)) (discrete_fun_singleton x) := ne_proper _.
Lemma discrete_fun_lookup_singleton x (y : B x) : (discrete_fun_singleton x y) x = y.
Proof. by rewrite /discrete_fun_singleton discrete_fun_lookup_insert. Qed.
Lemma discrete_fun_lookup_singleton_ne x x' (y : B x) :
x ≠ x' → (discrete_fun_singleton x y) x' = ε.
Proof. intros; by rewrite /discrete_fun_singleton discrete_fun_lookup_insert_ne. Qed.
Global Instance discrete_fun_singleton_discrete x (y : B x) :
(∀ i, Discrete (ε : B i)) → Discrete y → Discrete (discrete_fun_singleton x y).
Proof. apply _. Qed.
Lemma discrete_fun_singleton_validN n x (y : B x) : ✓{n} discrete_fun_singleton x y ↔ ✓{n} y.
Proof.
split; [by move=>/(_ x); rewrite discrete_fun_lookup_singleton|].
move⇒Hx x'; destruct (decide (x = x')) as [->|];
rewrite ?discrete_fun_lookup_singleton ?discrete_fun_lookup_singleton_ne //.
by apply ucmra_unit_validN.
Qed.
Lemma discrete_fun_singleton_valid x (y : B x) : ✓ discrete_fun_singleton x y ↔ ✓ y.
Proof.
by rewrite !cmra_valid_validN; setoid_rewrite discrete_fun_singleton_validN.
Qed.
Lemma discrete_fun_singleton_unit x : discrete_fun_singleton x (ε : B x) ≡ ε.
Proof.
move⇒x'; destruct (decide (x = x')) as [->|];
by rewrite ?discrete_fun_lookup_singleton ?discrete_fun_lookup_singleton_ne.
Qed.
Lemma discrete_fun_singleton_core x (y : B x) :
core (discrete_fun_singleton x y) ≡ discrete_fun_singleton x (core y).
Proof.
move⇒x'; destruct (decide (x = x')) as [->|];
by rewrite discrete_fun_lookup_core ?discrete_fun_lookup_singleton
?discrete_fun_lookup_singleton_ne // (core_id_core _).
Qed.
Global Instance discrete_fun_singleton_core_id x (y : B x) :
CoreId y → CoreId (discrete_fun_singleton x y).
Proof. by rewrite !core_id_total discrete_fun_singleton_core⇒ →. Qed.
Lemma discrete_fun_singleton_op (x : A) (y1 y2 : B x) :
discrete_fun_singleton x y1 ⋅ discrete_fun_singleton x y2 ≡ discrete_fun_singleton x (y1 ⋅ y2).
Proof.
intros x'; destruct (decide (x' = x)) as [->|].
- by rewrite discrete_fun_lookup_op !discrete_fun_lookup_singleton.
- by rewrite discrete_fun_lookup_op !discrete_fun_lookup_singleton_ne // left_id.
Qed.
Lemma discrete_fun_insert_updateP x (P : B x → Prop) (Q : discrete_fun B → Prop) g y1 :
y1 ~~>: P → (∀ y2, P y2 → Q (discrete_fun_insert x y2 g)) →
discrete_fun_insert x y1 g ~~>: Q.
Proof.
intros Hy1 HP; apply cmra_total_updateP.
intros n gf Hg. destruct (Hy1 n (Some (gf x))) as (y2&?&?).
{ move: (Hg x). by rewrite discrete_fun_lookup_op discrete_fun_lookup_insert. }
∃ (discrete_fun_insert x y2 g); split; [auto|].
intros x'; destruct (decide (x' = x)) as [->|];
rewrite discrete_fun_lookup_op ?discrete_fun_lookup_insert //; [].
move: (Hg x'). by rewrite discrete_fun_lookup_op !discrete_fun_lookup_insert_ne.
Qed.
Lemma discrete_fun_insert_updateP' x (P : B x → Prop) g y1 :
y1 ~~>: P →
discrete_fun_insert x y1 g ~~>: λ g', ∃ y2, g' = discrete_fun_insert x y2 g ∧ P y2.
Proof. eauto using discrete_fun_insert_updateP. Qed.
Lemma discrete_fun_insert_update g x y1 y2 :
y1 ~~> y2 → discrete_fun_insert x y1 g ~~> discrete_fun_insert x y2 g.
Proof.
rewrite !cmra_update_updateP; eauto using discrete_fun_insert_updateP with subst.
Qed.
Lemma discrete_fun_singleton_updateP x (P : B x → Prop) (Q : discrete_fun B → Prop) y1 :
y1 ~~>: P → (∀ y2, P y2 → Q (discrete_fun_singleton x y2)) →
discrete_fun_singleton x y1 ~~>: Q.
Proof. rewrite /discrete_fun_singleton; eauto using discrete_fun_insert_updateP. Qed.
Lemma discrete_fun_singleton_updateP' x (P : B x → Prop) y1 :
y1 ~~>: P →
discrete_fun_singleton x y1 ~~>: λ g, ∃ y2, g = discrete_fun_singleton x y2 ∧ P y2.
Proof. eauto using discrete_fun_singleton_updateP. Qed.
Lemma discrete_fun_singleton_update x (y1 y2 : B x) :
y1 ~~> y2 → discrete_fun_singleton x y1 ~~> discrete_fun_singleton x y2.
Proof. eauto using discrete_fun_insert_update. Qed.
Lemma discrete_fun_singleton_updateP_empty x (P : B x → Prop) (Q : discrete_fun B → Prop) :
ε ~~>: P → (∀ y2, P y2 → Q (discrete_fun_singleton x y2)) → ε ~~>: Q.
Proof.
intros Hx HQ; apply cmra_total_updateP.
intros n gf Hg. destruct (Hx n (Some (gf x))) as (y2&?&?); first apply Hg.
∃ (discrete_fun_singleton x y2); split; [by apply HQ|].
intros x'; destruct (decide (x' = x)) as [->|].
- by rewrite discrete_fun_lookup_op discrete_fun_lookup_singleton.
- rewrite discrete_fun_lookup_op discrete_fun_lookup_singleton_ne //; by apply Hg.
Qed.
Lemma discrete_fun_singleton_updateP_empty' x (P : B x → Prop) :
ε ~~>: P → ε ~~>: λ g, ∃ y2, g = discrete_fun_singleton x y2 ∧ P y2.
Proof. eauto using discrete_fun_singleton_updateP_empty. Qed.
Lemma discrete_fun_singleton_update_empty x (y : B x) :
ε ~~> y → ε ~~> discrete_fun_singleton x y.
Proof.
rewrite !cmra_update_updateP;
eauto using discrete_fun_singleton_updateP_empty with subst.
Qed.
Lemma discrete_fun_updateP `{!Finite A} f P Q :
(∀ a, f a ~~>: P a) → (∀ f', (∀ a, P a (f' a)) → Q f') → f ~~>: Q.
Proof.
repeat setoid_rewrite cmra_total_updateP. intros Hf HP n h Hh.
destruct (finite_choice
(λ a y, P a y ∧ ✓{n} (y ⋅ (h a)))) as [f' Hf']; first naive_solver.
naive_solver.
Qed.
Lemma discrete_fun_updateP' `{!Finite A} f P :
(∀ a, f a ~~>: P a) → f ~~>: λ f', ∀ a, P a (f' a).
Proof. eauto using discrete_fun_updateP. Qed.
Lemma discrete_fun_update f g :
(∀ a, f a ~~> g a) → f ~~> g.
Proof.
repeat setoid_rewrite cmra_total_update.
intros Hfg n h Hh a. apply (Hfg a), Hh.
Qed.
End cmra.
NonExpansive2 (discrete_fun_insert (B:=B) x).
Proof.
intros n y1 y2 ? f1 f2 ? x'; rewrite /discrete_fun_insert.
by destruct (decide _) as [[]|].
Qed.
Global Instance discrete_fun_insert_proper x :
Proper ((≡) ==> (≡) ==> (≡)) (discrete_fun_insert (B:=B) x) := ne_proper_2 _.
Lemma discrete_fun_lookup_insert f x y : (discrete_fun_insert x y f) x = y.
Proof.
rewrite /discrete_fun_insert; destruct (decide _) as [Hx|]; last done.
by rewrite (proof_irrel Hx eq_refl).
Qed.
Lemma discrete_fun_lookup_insert_ne f x x' y :
x ≠ x' → (discrete_fun_insert x y f) x' = f x'.
Proof. by rewrite /discrete_fun_insert; destruct (decide _). Qed.
Global Instance discrete_fun_insert_discrete f x y :
Discrete f → Discrete y → Discrete (discrete_fun_insert x y f).
Proof.
intros ?? g Heq x'; destruct (decide (x = x')) as [->|].
- rewrite discrete_fun_lookup_insert.
apply: discrete. by rewrite -(Heq x') discrete_fun_lookup_insert.
- rewrite discrete_fun_lookup_insert_ne //.
apply: discrete. by rewrite -(Heq x') discrete_fun_lookup_insert_ne.
Qed.
End ofe.
Section cmra.
Context {SI : sidx} `{!EqDecision A} {B : A → ucmra}.
Implicit Types x : A.
Implicit Types f g : discrete_fun B.
Global Instance discrete_funR_cmra_discrete:
(∀ i, CmraDiscrete (B i)) → CmraDiscrete (discrete_funR B).
Proof. intros HB. split; [apply _|]. intros x Hv i. apply HB, Hv. Qed.
Global Instance discrete_fun_singleton_ne x :
NonExpansive (discrete_fun_singleton x : B x → _).
Proof.
intros n y1 y2 ?; apply discrete_fun_insert_ne; [done|].
by apply equiv_dist.
Qed.
Global Instance discrete_fun_singleton_proper x :
Proper ((≡) ==> (≡)) (discrete_fun_singleton x) := ne_proper _.
Lemma discrete_fun_lookup_singleton x (y : B x) : (discrete_fun_singleton x y) x = y.
Proof. by rewrite /discrete_fun_singleton discrete_fun_lookup_insert. Qed.
Lemma discrete_fun_lookup_singleton_ne x x' (y : B x) :
x ≠ x' → (discrete_fun_singleton x y) x' = ε.
Proof. intros; by rewrite /discrete_fun_singleton discrete_fun_lookup_insert_ne. Qed.
Global Instance discrete_fun_singleton_discrete x (y : B x) :
(∀ i, Discrete (ε : B i)) → Discrete y → Discrete (discrete_fun_singleton x y).
Proof. apply _. Qed.
Lemma discrete_fun_singleton_validN n x (y : B x) : ✓{n} discrete_fun_singleton x y ↔ ✓{n} y.
Proof.
split; [by move=>/(_ x); rewrite discrete_fun_lookup_singleton|].
move⇒Hx x'; destruct (decide (x = x')) as [->|];
rewrite ?discrete_fun_lookup_singleton ?discrete_fun_lookup_singleton_ne //.
by apply ucmra_unit_validN.
Qed.
Lemma discrete_fun_singleton_valid x (y : B x) : ✓ discrete_fun_singleton x y ↔ ✓ y.
Proof.
by rewrite !cmra_valid_validN; setoid_rewrite discrete_fun_singleton_validN.
Qed.
Lemma discrete_fun_singleton_unit x : discrete_fun_singleton x (ε : B x) ≡ ε.
Proof.
move⇒x'; destruct (decide (x = x')) as [->|];
by rewrite ?discrete_fun_lookup_singleton ?discrete_fun_lookup_singleton_ne.
Qed.
Lemma discrete_fun_singleton_core x (y : B x) :
core (discrete_fun_singleton x y) ≡ discrete_fun_singleton x (core y).
Proof.
move⇒x'; destruct (decide (x = x')) as [->|];
by rewrite discrete_fun_lookup_core ?discrete_fun_lookup_singleton
?discrete_fun_lookup_singleton_ne // (core_id_core _).
Qed.
Global Instance discrete_fun_singleton_core_id x (y : B x) :
CoreId y → CoreId (discrete_fun_singleton x y).
Proof. by rewrite !core_id_total discrete_fun_singleton_core⇒ →. Qed.
Lemma discrete_fun_singleton_op (x : A) (y1 y2 : B x) :
discrete_fun_singleton x y1 ⋅ discrete_fun_singleton x y2 ≡ discrete_fun_singleton x (y1 ⋅ y2).
Proof.
intros x'; destruct (decide (x' = x)) as [->|].
- by rewrite discrete_fun_lookup_op !discrete_fun_lookup_singleton.
- by rewrite discrete_fun_lookup_op !discrete_fun_lookup_singleton_ne // left_id.
Qed.
Lemma discrete_fun_insert_updateP x (P : B x → Prop) (Q : discrete_fun B → Prop) g y1 :
y1 ~~>: P → (∀ y2, P y2 → Q (discrete_fun_insert x y2 g)) →
discrete_fun_insert x y1 g ~~>: Q.
Proof.
intros Hy1 HP; apply cmra_total_updateP.
intros n gf Hg. destruct (Hy1 n (Some (gf x))) as (y2&?&?).
{ move: (Hg x). by rewrite discrete_fun_lookup_op discrete_fun_lookup_insert. }
∃ (discrete_fun_insert x y2 g); split; [auto|].
intros x'; destruct (decide (x' = x)) as [->|];
rewrite discrete_fun_lookup_op ?discrete_fun_lookup_insert //; [].
move: (Hg x'). by rewrite discrete_fun_lookup_op !discrete_fun_lookup_insert_ne.
Qed.
Lemma discrete_fun_insert_updateP' x (P : B x → Prop) g y1 :
y1 ~~>: P →
discrete_fun_insert x y1 g ~~>: λ g', ∃ y2, g' = discrete_fun_insert x y2 g ∧ P y2.
Proof. eauto using discrete_fun_insert_updateP. Qed.
Lemma discrete_fun_insert_update g x y1 y2 :
y1 ~~> y2 → discrete_fun_insert x y1 g ~~> discrete_fun_insert x y2 g.
Proof.
rewrite !cmra_update_updateP; eauto using discrete_fun_insert_updateP with subst.
Qed.
Lemma discrete_fun_singleton_updateP x (P : B x → Prop) (Q : discrete_fun B → Prop) y1 :
y1 ~~>: P → (∀ y2, P y2 → Q (discrete_fun_singleton x y2)) →
discrete_fun_singleton x y1 ~~>: Q.
Proof. rewrite /discrete_fun_singleton; eauto using discrete_fun_insert_updateP. Qed.
Lemma discrete_fun_singleton_updateP' x (P : B x → Prop) y1 :
y1 ~~>: P →
discrete_fun_singleton x y1 ~~>: λ g, ∃ y2, g = discrete_fun_singleton x y2 ∧ P y2.
Proof. eauto using discrete_fun_singleton_updateP. Qed.
Lemma discrete_fun_singleton_update x (y1 y2 : B x) :
y1 ~~> y2 → discrete_fun_singleton x y1 ~~> discrete_fun_singleton x y2.
Proof. eauto using discrete_fun_insert_update. Qed.
Lemma discrete_fun_singleton_updateP_empty x (P : B x → Prop) (Q : discrete_fun B → Prop) :
ε ~~>: P → (∀ y2, P y2 → Q (discrete_fun_singleton x y2)) → ε ~~>: Q.
Proof.
intros Hx HQ; apply cmra_total_updateP.
intros n gf Hg. destruct (Hx n (Some (gf x))) as (y2&?&?); first apply Hg.
∃ (discrete_fun_singleton x y2); split; [by apply HQ|].
intros x'; destruct (decide (x' = x)) as [->|].
- by rewrite discrete_fun_lookup_op discrete_fun_lookup_singleton.
- rewrite discrete_fun_lookup_op discrete_fun_lookup_singleton_ne //; by apply Hg.
Qed.
Lemma discrete_fun_singleton_updateP_empty' x (P : B x → Prop) :
ε ~~>: P → ε ~~>: λ g, ∃ y2, g = discrete_fun_singleton x y2 ∧ P y2.
Proof. eauto using discrete_fun_singleton_updateP_empty. Qed.
Lemma discrete_fun_singleton_update_empty x (y : B x) :
ε ~~> y → ε ~~> discrete_fun_singleton x y.
Proof.
rewrite !cmra_update_updateP;
eauto using discrete_fun_singleton_updateP_empty with subst.
Qed.
Lemma discrete_fun_updateP `{!Finite A} f P Q :
(∀ a, f a ~~>: P a) → (∀ f', (∀ a, P a (f' a)) → Q f') → f ~~>: Q.
Proof.
repeat setoid_rewrite cmra_total_updateP. intros Hf HP n h Hh.
destruct (finite_choice
(λ a y, P a y ∧ ✓{n} (y ⋅ (h a)))) as [f' Hf']; first naive_solver.
naive_solver.
Qed.
Lemma discrete_fun_updateP' `{!Finite A} f P :
(∀ a, f a ~~>: P a) → f ~~>: λ f', ∀ a, P a (f' a).
Proof. eauto using discrete_fun_updateP. Qed.
Lemma discrete_fun_update f g :
(∀ a, f a ~~> g a) → f ~~> g.
Proof.
repeat setoid_rewrite cmra_total_update.
intros Hfg n h Hh a. apply (Hfg a), Hh.
Qed.
End cmra.