Commit aead381e authored by Jacques-Henri Jourdan's avatar Jacques-Henri Jourdan

Csum: sum of two CMRAs

parent 93df90de
Pipeline #1195 passed with stage
......@@ -55,6 +55,7 @@ algebra/upred_tactics.v
algebra/upred_big_op.v
algebra/frac.v
algebra/one_shot.v
algebra/csum.v
algebra/list.v
program_logic/model.v
program_logic/adequacy.v
......
From iris.algebra Require Export cmra.
From iris.algebra Require Import upred.
Local Arguments pcore _ _ !_ /.
Local Arguments cmra_pcore _ !_ /.
Local Arguments validN _ _ _ !_ /.
Local Arguments valid _ _ !_ /.
Local Arguments cmra_validN _ _ !_ /.
Local Arguments cmra_valid _ !_ /.
Inductive csum (A B : Type) :=
| Cinl : A -> csum A B
| Cinr : B -> csum A B
| CsumBot : csum A B.
Arguments Cinl {_ _} _.
Arguments Cinr {_ _} _.
Arguments CsumBot {_ _}.
Section cofe.
Context {A B : cofeT}.
Implicit Types a : A.
Implicit Types b : B.
(* Cofe *)
Inductive csum_equiv : Equiv (csum A B) :=
| Cinl_equiv a a' : a a' -> Cinl a Cinl a'
| Cinlr_equiv b b' : b b' -> Cinr b Cinr b'
| CsumBot_equiv : CsumBot CsumBot.
Existing Instance csum_equiv.
Inductive csum_dist : Dist (csum A B) :=
| Cinl_dist n a a' : a {n} a' -> Cinl a {n} Cinl a'
| Cinlr_dist n b b' : b {n} b' -> Cinr b {n} Cinr b'
| CsumBot_dist n : CsumBot {n} CsumBot.
Existing Instance csum_dist.
Global Instance Cinl_ne n : Proper (dist n ==> dist n) (@Cinl A B).
Proof. by constructor. Qed.
Global Instance Cinl_proper : Proper (() ==> ()) (@Cinl A B).
Proof. by constructor. Qed.
Global Instance Cinl_inj : Inj () () (@Cinl A B).
Proof. by inversion_clear 1. Qed.
Global Instance Cinl_inj_dist n : Inj (dist n) (dist n) (@Cinl A B).
Proof. by inversion_clear 1. Qed.
Global Instance Cinr_ne n : Proper (dist n ==> dist n) (@Cinr A B).
Proof. by constructor. Qed.
Global Instance Cinr_proper : Proper (() ==> ()) (@Cinr A B).
Proof. by constructor. Qed.
Global Instance Cinr_inj : Inj () () (@Cinr A B).
Proof. by inversion_clear 1. Qed.
Global Instance Cinr_inj_dist n : Inj (dist n) (dist n) (@Cinr A B).
Proof. by inversion_clear 1. Qed.
Program Definition csum_chain_l (c : chain (csum A B)) (a : A) : chain A :=
{| chain_car n := match c n return _ with Cinl a' => a' | _ => a end |}.
Next Obligation. intros c a n i ?; simpl. by destruct (chain_cauchy c n i). Qed.
Program Definition csum_chain_r (c : chain (csum A B)) (b : B) : chain B :=
{| chain_car n := match c n return _ with Cinr b' => b' | _ => b end |}.
Next Obligation. intros c b n i ?; simpl. by destruct (chain_cauchy c n i). Qed.
Instance csum_compl : Compl (csum A B) := λ c,
match c 0 with
| Cinl a => Cinl (compl (csum_chain_l c a))
| Cinr b => Cinr (compl (csum_chain_r c b))
| CsumBot => CsumBot
end.
Definition csum_cofe_mixin : CofeMixin (csum A B).
Proof.
split.
- intros mx my; split.
+ by destruct 1; constructor; try apply equiv_dist.
+ intros Hxy; feed inversion (Hxy 0); subst; constructor; try done;
apply equiv_dist=> n; by feed inversion (Hxy n).
- intros n; split.
+ by intros [|a|]; constructor.
+ by destruct 1; constructor.
+ destruct 1; inversion_clear 1; constructor; etrans; eauto.
- by inversion_clear 1; constructor; apply dist_S.
- intros n c; rewrite /compl /csum_compl.
feed inversion (chain_cauchy c 0 n); first auto with lia; constructor.
+ rewrite (conv_compl n (csum_chain_l c a')) /=. destruct (c n); naive_solver.
+ rewrite (conv_compl n (csum_chain_r c b')) /=. destruct (c n); naive_solver.
Qed.
Canonical Structure csumC : cofeT := CofeT (csum A B) csum_cofe_mixin.
Global Instance csum_discrete : Discrete A Discrete B Discrete csumC.
Proof. by inversion_clear 3; constructor; apply (timeless _). Qed.
Global Instance csum_leibniz :
LeibnizEquiv A LeibnizEquiv B LeibnizEquiv (csumC A B).
Proof. by destruct 3; f_equal; apply leibniz_equiv. Qed.
Global Instance Cinl_timeless a : Timeless a Timeless (Cinl a).
Proof. by inversion_clear 2; constructor; apply (timeless _). Qed.
Global Instance Cinr_timeless b : Timeless b Timeless (Cinr b).
Proof. by inversion_clear 2; constructor; apply (timeless _). Qed.
End cofe.
Arguments csumC : clear implicits.
(* Functor on COFEs *)
Definition csum_map {A A' B B'} (fA : A A') (fB : B B')
(x : csum A B) : csum A' B' :=
match x with
| Cinl a => Cinl (fA a)
| Cinr b => Cinr (fB b)
| CsumBot => CsumBot
end.
Instance: Params (@csum_map) 4.
Lemma csum_map_id {A B} (x : csum A B) : csum_map id id x = x.
Proof. by destruct x. Qed.
Lemma csum_map_compose {A A' A'' B B' B''} (f : A A') (f' : A' A'')
(g : B B') (g' : B' B'') (x : csum A B) :
csum_map (f' f) (g' g) x = csum_map f' g' (csum_map f g x).
Proof. by destruct x. Qed.
Lemma csum_map_ext {A A' B B' : cofeT} (f f' : A A') (g g' : B B') x :
( x, f x f' x) ( x, g x g' x) csum_map f g x csum_map f' g' x.
Proof. by destruct x; constructor. Qed.
Instance csum_map_cmra_ne {A A' B B' : cofeT} n :
Proper ((dist n ==> dist n) ==> (dist n ==> dist n) ==> dist n ==> dist n)
(@csum_map A A' B B').
Proof. intros f f' Hf g g' Hg []; destruct 1; constructor; by apply Hf || apply Hg. Qed.
Definition csumC_map {A A' B B'} (f : A -n> A') (g : B -n> B') :
csumC A B -n> csumC A' B' :=
CofeMor (csum_map f g).
Instance csumC_map_ne A A' B B' n :
Proper (dist n ==> dist n ==> dist n) (@csumC_map A A' B B').
Proof. by intros f f' Hf g g' Hg []; constructor. Qed.
Section cmra.
Context {A B : cmraT}.
Implicit Types a : A.
Implicit Types b : B.
(* CMRA *)
Instance csum_valid : Valid (csum A B) := λ x,
match x with
| Cinl a => a
| Cinr b => b
| CsumBot => False
end.
Instance csum_validN : ValidN (csum A B) := λ n x,
match x with
| Cinl a => {n} a
| Cinr b => {n} b
| CsumBot => False
end.
Instance csum_pcore : PCore (csum A B) := λ x,
match x with
| Cinl a => Cinl <$> pcore a
| Cinr b => Cinr <$> pcore b
| CsumBot => Some CsumBot
end.
Instance csum_op : Op (csum A B) := λ x y,
match x, y with
| Cinl a, Cinl a' => Cinl (a a')
| Cinr b, Cinr b' => Cinr (b b')
| _, _ => CsumBot
end.
Lemma Cinl_op a a' : Cinl a Cinl a' = Cinl (a a').
Proof. done. Qed.
Lemma Cinr_op b b' : Cinr b Cinr b' = Cinr (b b').
Proof. done. Qed.
Lemma csum_included x y :
x y y = CsumBot ( a a', x = Cinl a y = Cinl a' a a')
( b b', x = Cinr b y = Cinr b' b b').
Proof.
split.
- intros [z Hy]; destruct x as [a|b|], z as [a'|b'|]; inversion_clear Hy; auto.
+ right; left; eexists _, _; split_and!; eauto. eexists; eauto.
+ right; right; eexists _, _; split_and!; eauto. eexists; eauto.
- intros [->|[(a&a'&->&->&c&?)|(b&b'&->&->&c&?)]].
+ destruct x; exists CsumBot; constructor.
+ exists (Cinl c); by constructor.
+ exists (Cinr c); by constructor.
Qed.
Lemma csum_cmra_mixin : CMRAMixin (csum A B).
Proof.
split.
- intros n []; destruct 1; constructor; by cofe_subst.
- intros ???? [n a a' Ha|n b b' Hb|n] [=]; subst; eauto.
+ destruct (pcore a) as [ca|] eqn:?; simplify_option_eq.
destruct (cmra_pcore_ne n a a' ca) as (ca'&->&?); auto.
exists (Cinl ca'); by repeat constructor.
+ destruct (pcore b) as [cb|] eqn:?; simplify_option_eq.
destruct (cmra_pcore_ne n b b' cb) as (cb'&->&?); auto.
exists (Cinr cb'); by repeat constructor.
- intros ? [a|b|] [a'|b'|] H; inversion_clear H; cofe_subst; done.
- intros [a|b|]; rewrite /= ?cmra_valid_validN; naive_solver eauto using O.
- intros n [a|b|]; simpl; auto using cmra_validN_S.
- intros [a1|b1|] [a2|b2|] [a3|b3|]; constructor; by rewrite ?assoc.
- intros [a1|b1|] [a2|b2|]; constructor; by rewrite 1?comm.
- intros [a|b|] ? [=]; subst; auto.
+ destruct (pcore a) as [ca|] eqn:?; simplify_option_eq.
constructor; eauto using cmra_pcore_l.
+ destruct (pcore b) as [cb|] eqn:?; simplify_option_eq.
constructor; eauto using cmra_pcore_l.
- intros [a|b|] ? [=]; subst; auto.
+ destruct (pcore a) as [ca|] eqn:?; simplify_option_eq.
feed inversion (cmra_pcore_idemp a ca); repeat constructor; auto.
+ destruct (pcore b) as [cb|] eqn:?; simplify_option_eq.
feed inversion (cmra_pcore_idemp b cb); repeat constructor; auto.
- intros x y ? [->|[(a&a'&->&->&?)|(b&b'&->&->&?)]]%csum_included [=].
+ exists CsumBot. rewrite csum_included; eauto.
+ destruct (pcore a) as [ca|] eqn:?; simplify_option_eq.
destruct (cmra_pcore_preserving a a' ca) as (ca'&->&?); auto.
exists (Cinl ca'). rewrite csum_included; eauto 10.
+ destruct (pcore b) as [cb|] eqn:?; simplify_option_eq.
destruct (cmra_pcore_preserving b b' cb) as (cb'&->&?); auto.
exists (Cinr cb'). rewrite csum_included; eauto 10.
- intros n [a1|b1|] [a2|b2|]; simpl; eauto using cmra_validN_op_l; done.
- intros n [a|b|] y1 y2 Hx Hx'.
+ destruct y1 as [a1|b1|], y2 as [a2|b2|]; try (exfalso; by inversion_clear Hx').
apply (inj Cinl) in Hx'.
destruct (cmra_extend n a a1 a2) as ([z1 z2]&?&?&?); auto.
exists (Cinl z1, Cinl z2). by repeat constructor.
+ destruct y1 as [a1|b1|], y2 as [a2|b2|]; try (exfalso; by inversion_clear Hx').
apply (inj Cinr) in Hx'.
destruct (cmra_extend n b b1 b2) as ([z1 z2]&?&?&?); auto.
exists (Cinr z1, Cinr z2). by repeat constructor.
+ by exists (CsumBot, CsumBot); destruct y1, y2; inversion_clear Hx'.
Qed.
Canonical Structure csumR :=
CMRAT (csum A B) csum_cofe_mixin csum_cmra_mixin.
Global Instance csum_cmra_discrete :
CMRADiscrete A CMRADiscrete B CMRADiscrete csumR.
Proof.
split; first apply _.
by move=>[a|b|] HH /=; try apply cmra_discrete_valid.
Qed.
Global Instance Cinl_persistent a : Persistent a Persistent (Cinl a).
Proof. rewrite /Persistent /=. inversion_clear 1; by repeat constructor. Qed.
Global Instance Cinr_persistent b : Persistent b Persistent (Cinr b).
Proof. rewrite /Persistent /=. inversion_clear 1; by repeat constructor. Qed.
(** Internalized properties *)
Lemma csum_equivI {M} (x y : csum A B) :
(x y) (match x, y with
| Cinl a, Cinl a' => a a'
| Cinr b, Cinr b' => b b'
| CsumBot, CsumBot => True
| _, _ => False
end : uPred M).
Proof.
uPred.unseal; do 2 split; first by destruct 1.
by destruct x, y; try destruct 1; try constructor.
Qed.
Lemma csum_validI {M} (x : csum A B) :
( x) (match x with
| Cinl a => a
| Cinr b => b
| CsumBot => False
end : uPred M).
Proof. uPred.unseal. by destruct x. Qed.
(** Updates *)
Lemma csum_update_l (a1 a2 : A) : a1 ~~> a2 Cinl a1 ~~> Cinl a2.
Proof.
intros Ha n [[a|b|]|] ?; simpl in *; auto.
- by apply (Ha n (Some a)).
- by apply (Ha n None).
Qed.
Lemma csum_update_r (b1 b2 : B) : b1 ~~> b2 Cinr b1 ~~> Cinr b2.
Proof.
intros Hb n [[a|b|]|] ?; simpl in *; auto.
- by apply (Hb n (Some b)).
- by apply (Hb n None).
Qed.
Lemma csum_updateP_l (P : A Prop) (Q : csum A B Prop) a :
a ~~>: P ( a', P a' Q (Cinl a')) Cinl a ~~>: Q.
Proof.
intros Hx HP n mf Hm. destruct mf as [[a'|b'|]|]; try by destruct Hm.
- destruct (Hx n (Some a')) as (c&?&?); naive_solver.
- destruct (Hx n None) as (c&?&?); naive_solver eauto using cmra_validN_op_l.
Qed.
Lemma csum_updateP_r (P : B Prop) (Q : csum A B Prop) b :
b ~~>: P ( b', P b' Q (Cinr b')) Cinr b ~~>: Q.
Proof.
intros Hx HP n mf Hm. destruct mf as [[a'|b'|]|]; try by destruct Hm.
- destruct (Hx n (Some b')) as (c&?&?); naive_solver.
- destruct (Hx n None) as (c&?&?); naive_solver eauto using cmra_validN_op_l.
Qed.
Lemma csum_updateP'_l (P : A Prop) a :
a ~~>: P Cinl a ~~>: λ m', a', m' = Cinl a' P a'.
Proof. eauto using csum_updateP_l. Qed.
Lemma csum_updateP'_r (P : B Prop) b :
b ~~>: P Cinr b ~~>: λ m', b', m' = Cinr b' P b'.
Proof. eauto using csum_updateP_r. Qed.
End cmra.
Arguments csumR : clear implicits.
(* Functor *)
Instance csum_map_cmra_monotone {A A' B B' : cmraT} (f : A A') (g : B B') :
CMRAMonotone f CMRAMonotone g CMRAMonotone (csum_map f g).
Proof.
split; try apply _.
- intros n [a|b|]; simpl; auto using validN_preserving.
- intros x y; rewrite !csum_included.
intros [->|[(a&a'&->&->&?)|(b&b'&->&->&?)]]; simpl;
eauto 10 using included_preserving.
Qed.
Program Definition csumRF (Fa Fb : rFunctor) : rFunctor := {|
rFunctor_car A B := csumR (rFunctor_car Fa A B) (rFunctor_car Fb A B);
rFunctor_map A1 A2 B1 B2 fg := csumC_map (rFunctor_map Fa fg) (rFunctor_map Fb fg)
|}.
Next Obligation.
by intros Fa Fb A1 A2 B1 B2 n f g Hfg; apply csumC_map_ne; try apply rFunctor_ne.
Qed.
Next Obligation.
intros Fa Fb A B x. rewrite /= -{2}(csum_map_id x).
apply csum_map_ext=>y; apply rFunctor_id.
Qed.
Next Obligation.
intros Fa Fb A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -csum_map_compose.
apply csum_map_ext=>y; apply rFunctor_compose.
Qed.
Instance csumRF_contractive Fa Fb :
rFunctorContractive Fa rFunctorContractive Fb
rFunctorContractive (csumRF Fa Fb).
Proof.
by intros ?? A1 A2 B1 B2 n f g Hfg; apply csumC_map_ne; try apply rFunctor_contractive.
Qed.
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