Commit f6909092 authored by Ralf Jung's avatar Ralf Jung
Browse files

change notation of step-indexed equality to ≡{n}≡

parent fbedbd17
...@@ -16,16 +16,16 @@ Section agree. ...@@ -16,16 +16,16 @@ Section agree.
Context {A : cofeT}. Context {A : cofeT}.
Instance agree_validN : ValidN (agree A) := λ n x, Instance agree_validN : ValidN (agree A) := λ n x,
agree_is_valid x n n', n' n x n' ={n'}= x n. agree_is_valid x n n', n' n x n' {n'} x n.
Lemma agree_valid_le (x : agree A) n n' : Lemma agree_valid_le (x : agree A) n n' :
agree_is_valid x n n' n agree_is_valid x n'. agree_is_valid x n n' n agree_is_valid x n'.
Proof. induction 2; eauto using agree_valid_S. Qed. Proof. induction 2; eauto using agree_valid_S. Qed.
Instance agree_equiv : Equiv (agree A) := λ x y, Instance agree_equiv : Equiv (agree A) := λ x y,
( n, agree_is_valid x n agree_is_valid y n) ( n, agree_is_valid x n agree_is_valid y n)
( n, agree_is_valid x n x n ={n}= y n). ( n, agree_is_valid x n x n {n} y n).
Instance agree_dist : Dist (agree A) := λ n x y, Instance agree_dist : Dist (agree A) := λ n x y,
( n', n' n agree_is_valid x n' agree_is_valid y n') ( n', n' n agree_is_valid x n' agree_is_valid y n')
( n', n' n agree_is_valid x n' x n' ={n'}= y n'). ( n', n' n agree_is_valid x n' x n' {n'} y n').
Program Instance agree_compl : Compl (agree A) := λ c, Program Instance agree_compl : Compl (agree A) := λ c,
{| agree_car n := c n n; agree_is_valid n := agree_is_valid (c n) n |}. {| agree_car n := c n n; agree_is_valid n := agree_is_valid (c n) n |}.
Next Obligation. intros; apply agree_valid_0. Qed. Next Obligation. intros; apply agree_valid_0. Qed.
...@@ -51,14 +51,14 @@ Proof. ...@@ -51,14 +51,14 @@ Proof.
Qed. Qed.
Canonical Structure agreeC := CofeT agree_cofe_mixin. Canonical Structure agreeC := CofeT agree_cofe_mixin.
Lemma agree_car_ne (x y : agree A) n : {n} x x ={n}= y x n ={n}= y n. Lemma agree_car_ne (x y : agree A) n : {n} x x {n} y x n {n} y n.
Proof. by intros [??] Hxy; apply Hxy. Qed. Proof. by intros [??] Hxy; apply Hxy. Qed.
Lemma agree_cauchy (x : agree A) n i : {n} x i n x i ={i}= x n. Lemma agree_cauchy (x : agree A) n i : {n} x i n x i {i} x n.
Proof. by intros [? Hx]; apply Hx. Qed. Proof. by intros [? Hx]; apply Hx. Qed.
Program Instance agree_op : Op (agree A) := λ x y, Program Instance agree_op : Op (agree A) := λ x y,
{| agree_car := x; {| agree_car := x;
agree_is_valid n := agree_is_valid x n agree_is_valid y n x ={n}= y |}. agree_is_valid n := agree_is_valid x n agree_is_valid y n x {n} y |}.
Next Obligation. by intros; simpl; split_ands; try apply agree_valid_0. Qed. Next Obligation. by intros; simpl; split_ands; try apply agree_valid_0. Qed.
Next Obligation. naive_solver eauto using agree_valid_S, dist_S. Qed. Next Obligation. naive_solver eauto using agree_valid_S, dist_S. Qed.
Instance agree_unit : Unit (agree A) := id. Instance agree_unit : Unit (agree A) := id.
...@@ -91,7 +91,7 @@ Proof. ...@@ -91,7 +91,7 @@ Proof.
repeat match goal with H : agree_is_valid _ _ |- _ => clear H end; repeat match goal with H : agree_is_valid _ _ |- _ => clear H end;
by cofe_subst; rewrite !agree_idempotent. by cofe_subst; rewrite !agree_idempotent.
Qed. Qed.
Lemma agree_includedN (x y : agree A) n : x {n} y y ={n}= x y. Lemma agree_includedN (x y : agree A) n : x {n} y y {n} x y.
Proof. Proof.
split; [|by intros ?; exists y]. split; [|by intros ?; exists y].
by intros [z Hz]; rewrite Hz (associative _) agree_idempotent. by intros [z Hz]; rewrite Hz (associative _) agree_idempotent.
...@@ -109,9 +109,9 @@ Proof. ...@@ -109,9 +109,9 @@ Proof.
* by intros x y n [(?&?&?) ?]. * by intros x y n [(?&?&?) ?].
* by intros x y n; rewrite agree_includedN. * by intros x y n; rewrite agree_includedN.
Qed. Qed.
Lemma agree_op_inv (x1 x2 : agree A) n : {n} (x1 x2) x1 ={n}= x2. Lemma agree_op_inv (x1 x2 : agree A) n : {n} (x1 x2) x1 {n} x2.
Proof. intros Hxy; apply Hxy. Qed. Proof. intros Hxy; apply Hxy. Qed.
Lemma agree_valid_includedN (x y : agree A) n : {n} y x {n} y x ={n}= y. Lemma agree_valid_includedN (x y : agree A) n : {n} y x {n} y x {n} y.
Proof. Proof.
move=> Hval [z Hy]; move: Hval; rewrite Hy. move=> Hval [z Hy]; move: Hval; rewrite Hy.
by move=> /agree_op_inv->; rewrite agree_idempotent. by move=> /agree_op_inv->; rewrite agree_idempotent.
...@@ -133,7 +133,7 @@ Proof. intros x1 x2 Hx; split; naive_solver eauto using @dist_le. Qed. ...@@ -133,7 +133,7 @@ Proof. intros x1 x2 Hx; split; naive_solver eauto using @dist_le. Qed.
Global Instance to_agree_proper : Proper (() ==> ()) to_agree := ne_proper _. Global Instance to_agree_proper : Proper (() ==> ()) to_agree := ne_proper _.
Global Instance to_agree_inj n : Injective (dist n) (dist n) (to_agree). Global Instance to_agree_inj n : Injective (dist n) (dist n) (to_agree).
Proof. by intros x y [_ Hxy]; apply Hxy. Qed. Proof. by intros x y [_ Hxy]; apply Hxy. Qed.
Lemma to_agree_car n (x : agree A) : {n} x to_agree (x n) ={n}= x. Lemma to_agree_car n (x : agree A) : {n} x to_agree (x n) {n} x.
Proof. intros [??]; split; naive_solver eauto using agree_valid_le. Qed. Proof. intros [??]; split; naive_solver eauto using agree_valid_le. Qed.
End agree. End agree.
......
...@@ -19,7 +19,7 @@ Implicit Types x y : auth A. ...@@ -19,7 +19,7 @@ Implicit Types x y : auth A.
Instance auth_equiv : Equiv (auth A) := λ x y, Instance auth_equiv : Equiv (auth A) := λ x y,
authoritative x authoritative y own x own y. authoritative x authoritative y own x own y.
Instance auth_dist : Dist (auth A) := λ n x y, Instance auth_dist : Dist (auth A) := λ n x y,
authoritative x ={n}= authoritative y own x ={n}= own y. authoritative x {n} authoritative y own x {n} own y.
Global Instance Auth_ne : Proper (dist n ==> dist n ==> dist n) (@Auth A). Global Instance Auth_ne : Proper (dist n ==> dist n ==> dist n) (@Auth A).
Proof. by split. Qed. Proof. by split. Qed.
...@@ -148,7 +148,7 @@ Lemma auth_frag_op a b : ◯ (a ⋅ b) ≡ ◯ a ⋅ ◯ b. ...@@ -148,7 +148,7 @@ Lemma auth_frag_op a b : ◯ (a ⋅ b) ≡ ◯ a ⋅ ◯ b.
Proof. done. Qed. Proof. done. Qed.
Lemma auth_update a a' b b' : Lemma auth_update a a' b b' :
( n af, {S n} a a ={S n}= a' af b ={S n}= b' af {S n} b) ( n af, {S n} a a {S n} a' af b {S n} b' af {S n} b)
a a' ~~> b b'. a a' ~~> b b'.
Proof. Proof.
move=> Hab [[?| |] bf1] n // =>-[[bf2 Ha] ?]; do 2 red; simpl in *. move=> Hab [[?| |] bf1] n // =>-[[bf2 Ha] ?]; do 2 red; simpl in *.
......
...@@ -27,7 +27,7 @@ Instance: Params (@valid) 2. ...@@ -27,7 +27,7 @@ Instance: Params (@valid) 2.
Notation "✓" := valid (at level 1). Notation "✓" := valid (at level 1).
Instance validN_valid `{ValidN A} : Valid A := λ x, n, {n} x. Instance validN_valid `{ValidN A} : Valid A := λ x, n, {n} x.
Definition includedN `{Dist A, Op A} (n : nat) (x y : A) := z, y ={n}= x z. Definition includedN `{Dist A, Op A} (n : nat) (x y : A) := z, y {n} x z.
Notation "x ≼{ n } y" := (includedN n x y) Notation "x ≼{ n } y" := (includedN n x y)
(at level 70, format "x ≼{ n } y") : C_scope. (at level 70, format "x ≼{ n } y") : C_scope.
Instance: Params (@includedN) 4. Instance: Params (@includedN) 4.
...@@ -49,11 +49,11 @@ Record CMRAMixin A `{Dist A, Equiv A, Unit A, Op A, ValidN A, Minus A} := { ...@@ -49,11 +49,11 @@ Record CMRAMixin A `{Dist A, Equiv A, Unit A, Op A, ValidN A, Minus A} := {
mixin_cmra_unit_idempotent x : unit (unit x) unit x; mixin_cmra_unit_idempotent x : unit (unit x) unit x;
mixin_cmra_unit_preservingN n x y : x {n} y unit x {n} unit y; mixin_cmra_unit_preservingN n x y : x {n} y unit x {n} unit y;
mixin_cmra_validN_op_l n x y : {n} (x y) {n} x; mixin_cmra_validN_op_l n x y : {n} (x y) {n} x;
mixin_cmra_op_minus n x y : x {n} y x y x ={n}= y mixin_cmra_op_minus n x y : x {n} y x y x {n} y
}. }.
Definition CMRAExtendMixin A `{Equiv A, Dist A, Op A, ValidN A} := n x y1 y2, Definition CMRAExtendMixin A `{Equiv A, Dist A, Op A, ValidN A} := n x y1 y2,
{n} x x ={n}= y1 y2 {n} x x {n} y1 y2
{ z | x z.1 z.2 z.1 ={n}= y1 z.2 ={n}= y2 }. { z | x z.1 z.2 z.1 {n} y1 z.2 {n} y2 }.
(** Bundeled version *) (** Bundeled version *)
Structure cmraT := CMRAT { Structure cmraT := CMRAT {
...@@ -115,11 +115,11 @@ Section cmra_mixin. ...@@ -115,11 +115,11 @@ Section cmra_mixin.
Proof. apply (mixin_cmra_unit_preservingN _ (cmra_mixin A)). Qed. Proof. apply (mixin_cmra_unit_preservingN _ (cmra_mixin A)). Qed.
Lemma cmra_validN_op_l n x y : {n} (x y) {n} x. Lemma cmra_validN_op_l n x y : {n} (x y) {n} x.
Proof. apply (mixin_cmra_validN_op_l _ (cmra_mixin A)). Qed. Proof. apply (mixin_cmra_validN_op_l _ (cmra_mixin A)). Qed.
Lemma cmra_op_minus n x y : x {n} y x y x ={n}= y. Lemma cmra_op_minus n x y : x {n} y x y x {n} y.
Proof. apply (mixin_cmra_op_minus _ (cmra_mixin A)). Qed. Proof. apply (mixin_cmra_op_minus _ (cmra_mixin A)). Qed.
Lemma cmra_extend_op n x y1 y2 : Lemma cmra_extend_op n x y1 y2 :
{n} x x ={n}= y1 y2 {n} x x {n} y1 y2
{ z | x z.1 z.2 z.1 ={n}= y1 z.2 ={n}= y2 }. { z | x z.1 z.2 z.1 {n} y1 z.2 {n} y2 }.
Proof. apply (cmra_extend_mixin A). Qed. Proof. apply (cmra_extend_mixin A). Qed.
End cmra_mixin. End cmra_mixin.
...@@ -277,7 +277,7 @@ Lemma cmra_preserving_r x y z : x ≼ y → x ⋅ z ≼ y ⋅ z. ...@@ -277,7 +277,7 @@ Lemma cmra_preserving_r x y z : x ≼ y → x ⋅ z ≼ y ⋅ z.
Proof. by intros; rewrite -!(commutative _ z); apply cmra_preserving_l. Qed. Proof. by intros; rewrite -!(commutative _ z); apply cmra_preserving_l. Qed.
Lemma cmra_included_dist_l x1 x2 x1' n : Lemma cmra_included_dist_l x1 x2 x1' n :
x1 x2 x1' ={n}= x1 x2', x1' x2' x2' ={n}= x2. x1 x2 x1' {n} x1 x2', x1' x2' x2' {n} x2.
Proof. Proof.
intros [z Hx2] Hx1; exists (x1' z); split; auto using cmra_included_l. intros [z Hx2] Hx1; exists (x1' z); split; auto using cmra_included_l.
by rewrite Hx1 Hx2. by rewrite Hx1 Hx2.
......
...@@ -3,10 +3,10 @@ Require Export algebra.base. ...@@ -3,10 +3,10 @@ Require Export algebra.base.
(** Unbundeled version *) (** Unbundeled version *)
Class Dist A := dist : nat relation A. Class Dist A := dist : nat relation A.
Instance: Params (@dist) 3. Instance: Params (@dist) 3.
Notation "x ={ n }= y" := (dist n x y) Notation "x { n } y" := (dist n x y)
(at level 70, n at next level, format "x ={ n }= y"). (at level 70, n at next level, format "x { n } y").
Hint Extern 0 (?x ={_}= ?y) => reflexivity. Hint Extern 0 (?x {_} ?y) => reflexivity.
Hint Extern 0 (_ ={_}= _) => symmetry; assumption. Hint Extern 0 (_ {_} _) => symmetry; assumption.
Tactic Notation "cofe_subst" ident(x) := Tactic Notation "cofe_subst" ident(x) :=
repeat match goal with repeat match goal with
...@@ -23,18 +23,18 @@ Tactic Notation "cofe_subst" := ...@@ -23,18 +23,18 @@ Tactic Notation "cofe_subst" :=
Record chain (A : Type) `{Dist A} := { Record chain (A : Type) `{Dist A} := {
chain_car :> nat A; chain_car :> nat A;
chain_cauchy n i : n i chain_car n ={n}= chain_car i chain_cauchy n i : n i chain_car n {n} chain_car i
}. }.
Arguments chain_car {_ _} _ _. Arguments chain_car {_ _} _ _.
Arguments chain_cauchy {_ _} _ _ _ _. Arguments chain_cauchy {_ _} _ _ _ _.
Class Compl A `{Dist A} := compl : chain A A. Class Compl A `{Dist A} := compl : chain A A.
Record CofeMixin A `{Equiv A, Compl A} := { Record CofeMixin A `{Equiv A, Compl A} := {
mixin_equiv_dist x y : x y n, x ={n}= y; mixin_equiv_dist x y : x y n, x {n} y;
mixin_dist_equivalence n : Equivalence (dist n); mixin_dist_equivalence n : Equivalence (dist n);
mixin_dist_S n x y : x ={S n}= y x ={n}= y; mixin_dist_S n x y : x {S n} y x {n} y;
mixin_dist_0 x y : x ={0}= y; mixin_dist_0 x y : x {0} y;
mixin_conv_compl (c : chain A) n : compl c ={n}= c n mixin_conv_compl (c : chain A) n : compl c {n} c n
}. }.
Class Contractive `{Dist A, Dist B} (f : A -> B) := Class Contractive `{Dist A, Dist B} (f : A -> B) :=
contractive n : Proper (dist n ==> dist (S n)) f. contractive n : Proper (dist n ==> dist (S n)) f.
...@@ -60,19 +60,19 @@ Arguments cofe_mixin : simpl never. ...@@ -60,19 +60,19 @@ Arguments cofe_mixin : simpl never.
Section cofe_mixin. Section cofe_mixin.
Context {A : cofeT}. Context {A : cofeT}.
Implicit Types x y : A. Implicit Types x y : A.
Lemma equiv_dist x y : x y n, x ={n}= y. Lemma equiv_dist x y : x y n, x {n} y.
Proof. apply (mixin_equiv_dist _ (cofe_mixin A)). Qed. Proof. apply (mixin_equiv_dist _ (cofe_mixin A)). Qed.
Global Instance dist_equivalence n : Equivalence (@dist A _ n). Global Instance dist_equivalence n : Equivalence (@dist A _ n).
Proof. apply (mixin_dist_equivalence _ (cofe_mixin A)). Qed. Proof. apply (mixin_dist_equivalence _ (cofe_mixin A)). Qed.
Lemma dist_S n x y : x ={S n}= y x ={n}= y. Lemma dist_S n x y : x {S n} y x {n} y.
Proof. apply (mixin_dist_S _ (cofe_mixin A)). Qed. Proof. apply (mixin_dist_S _ (cofe_mixin A)). Qed.
Lemma dist_0 x y : x ={0}= y. Lemma dist_0 x y : x {0} y.
Proof. apply (mixin_dist_0 _ (cofe_mixin A)). Qed. Proof. apply (mixin_dist_0 _ (cofe_mixin A)). Qed.
Lemma conv_compl (c : chain A) n : compl c ={n}= c n. Lemma conv_compl (c : chain A) n : compl c {n} c n.
Proof. apply (mixin_conv_compl _ (cofe_mixin A)). Qed. Proof. apply (mixin_conv_compl _ (cofe_mixin A)). Qed.
End cofe_mixin. End cofe_mixin.
Hint Extern 0 (_ ={0}= _) => apply dist_0. Hint Extern 0 (_ {0} _) => apply dist_0.
(** General properties *) (** General properties *)
Section cofe. Section cofe.
...@@ -97,7 +97,7 @@ Section cofe. ...@@ -97,7 +97,7 @@ Section cofe.
Qed. Qed.
Global Instance dist_proper_2 n x : Proper (() ==> iff) (dist n x). Global Instance dist_proper_2 n x : Proper (() ==> iff) (dist n x).
Proof. by apply dist_proper. Qed. Proof. by apply dist_proper. Qed.
Lemma dist_le (x y : A) n n' : x ={n}= y n' n x ={n'}= y. Lemma dist_le (x y : A) n n' : x {n} y n' n x {n'} y.
Proof. induction 2; eauto using dist_S. Qed. Proof. induction 2; eauto using dist_S. Qed.
Instance ne_proper {B : cofeT} (f : A B) Instance ne_proper {B : cofeT} (f : A B)
`{! n, Proper (dist n ==> dist n) f} : Proper (() ==> ()) f | 100. `{! n, Proper (dist n ==> dist n) f} : Proper (() ==> ()) f | 100.
...@@ -109,7 +109,7 @@ Section cofe. ...@@ -109,7 +109,7 @@ Section cofe.
unfold Proper, respectful; setoid_rewrite equiv_dist. unfold Proper, respectful; setoid_rewrite equiv_dist.
by intros x1 x2 Hx y1 y2 Hy n; rewrite (Hx n) (Hy n). by intros x1 x2 Hx y1 y2 Hy n; rewrite (Hx n) (Hy n).
Qed. Qed.
Lemma compl_ne (c1 c2: chain A) n : c1 n ={n}= c2 n compl c1 ={n}= compl c2. Lemma compl_ne (c1 c2: chain A) n : c1 n {n} c2 n compl c1 {n} compl c2.
Proof. intros. by rewrite (conv_compl c1 n) (conv_compl c2 n). Qed. Proof. intros. by rewrite (conv_compl c1 n) (conv_compl c2 n). Qed.
Lemma compl_ext (c1 c2 : chain A) : ( i, c1 i c2 i) compl c1 compl c2. Lemma compl_ext (c1 c2 : chain A) : ( i, c1 i c2 i) compl c1 compl c2.
Proof. setoid_rewrite equiv_dist; naive_solver eauto using compl_ne. Qed. Proof. setoid_rewrite equiv_dist; naive_solver eauto using compl_ne. Qed.
...@@ -127,9 +127,9 @@ Program Definition chain_map `{Dist A, Dist B} (f : A → B) ...@@ -127,9 +127,9 @@ Program Definition chain_map `{Dist A, Dist B} (f : A → B)
Next Obligation. by intros ? A ? B f Hf c n i ?; apply Hf, chain_cauchy. Qed. Next Obligation. by intros ? A ? B f Hf c n i ?; apply Hf, chain_cauchy. Qed.
(** Timeless elements *) (** Timeless elements *)
Class Timeless {A : cofeT} (x : A) := timeless y : x ={1}= y x y. Class Timeless {A : cofeT} (x : A) := timeless y : x {1} y x y.
Arguments timeless {_} _ {_} _ _. Arguments timeless {_} _ {_} _ _.
Lemma timeless_S {A : cofeT} (x y : A) n : Timeless x x y x ={S n}= y. Lemma timeless_S {A : cofeT} (x y : A) n : Timeless x x y x {S n} y.
Proof. Proof.
split; intros; [by apply equiv_dist|]. split; intros; [by apply equiv_dist|].
apply (timeless _), dist_le with (S n); auto with lia. apply (timeless _), dist_le with (S n); auto with lia.
...@@ -154,7 +154,7 @@ Section fixpoint. ...@@ -154,7 +154,7 @@ Section fixpoint.
by rewrite {1}(chain_cauchy (fixpoint_chain f) n (S n)); last lia. by rewrite {1}(chain_cauchy (fixpoint_chain f) n (S n)); last lia.
Qed. Qed.
Lemma fixpoint_ne (g : A A) `{!Contractive g} n : Lemma fixpoint_ne (g : A A) `{!Contractive g} n :
( z, f z ={n}= g z) fixpoint f ={n}= fixpoint g. ( z, f z {n} g z) fixpoint f {n} fixpoint g.
Proof. Proof.
intros Hfg; unfold fixpoint. intros Hfg; unfold fixpoint.
rewrite (conv_compl (fixpoint_chain f) n) (conv_compl (fixpoint_chain g) n). rewrite (conv_compl (fixpoint_chain f) n) (conv_compl (fixpoint_chain g) n).
...@@ -181,7 +181,7 @@ Section cofe_mor. ...@@ -181,7 +181,7 @@ Section cofe_mor.
Global Instance cofe_mor_proper (f : cofeMor A B) : Proper (() ==> ()) f. Global Instance cofe_mor_proper (f : cofeMor A B) : Proper (() ==> ()) f.
Proof. apply ne_proper, cofe_mor_ne. Qed. Proof. apply ne_proper, cofe_mor_ne. Qed.
Instance cofe_mor_equiv : Equiv (cofeMor A B) := λ f g, x, f x g x. Instance cofe_mor_equiv : Equiv (cofeMor A B) := λ f g, x, f x g x.
Instance cofe_mor_dist : Dist (cofeMor A B) := λ n f g, x, f x ={n}= g x. Instance cofe_mor_dist : Dist (cofeMor A B) := λ n f g, x, f x {n} g x.
Program Definition fun_chain `(c : chain (cofeMor A B)) (x : A) : chain B := Program Definition fun_chain `(c : chain (cofeMor A B)) (x : A) : chain B :=
{| chain_car n := c n x |}. {| chain_car n := c n x |}.
Next Obligation. intros c x n i ?. by apply (chain_cauchy c). Qed. Next Obligation. intros c x n i ?. by apply (chain_cauchy c). Qed.
...@@ -230,7 +230,7 @@ Definition ccompose {A B C} ...@@ -230,7 +230,7 @@ Definition ccompose {A B C}
Instance: Params (@ccompose) 3. Instance: Params (@ccompose) 3.
Infix "◎" := ccompose (at level 40, left associativity). Infix "◎" := ccompose (at level 40, left associativity).
Lemma ccompose_ne {A B C} (f1 f2 : B -n> C) (g1 g2 : A -n> B) n : Lemma ccompose_ne {A B C} (f1 f2 : B -n> C) (g1 g2 : A -n> B) n :
f1 ={n}= f2 g1 ={n}= g2 f1 g1 ={n}= f2 g2. f1 {n} f2 g1 {n} g2 f1 g1 {n} f2 g2.
Proof. by intros Hf Hg x; rewrite /= (Hg x) (Hf (g2 x)). Qed. Proof. by intros Hf Hg x; rewrite /= (Hg x) (Hf (g2 x)). Qed.
(** unit *) (** unit *)
...@@ -325,7 +325,7 @@ Section later. ...@@ -325,7 +325,7 @@ Section later.
Context {A : cofeT}. Context {A : cofeT}.
Instance later_equiv : Equiv (later A) := λ x y, later_car x later_car y. Instance later_equiv : Equiv (later A) := λ x y, later_car x later_car y.
Instance later_dist : Dist (later A) := λ n x y, Instance later_dist : Dist (later A) := λ n x y,
match n with 0 => True | S n => later_car x ={n}= later_car y end. match n with 0 => True | S n => later_car x {n} later_car y end.
Program Definition later_chain (c : chain (later A)) : chain A := Program Definition later_chain (c : chain (later A)) : chain A :=
{| chain_car n := later_car (c (S n)) |}. {| chain_car n := later_car (c (S n)) |}.
Next Obligation. intros c n i ?; apply (chain_cauchy c (S n)); lia. Qed. Next Obligation. intros c n i ?; apply (chain_cauchy c (S n)); lia. Qed.
......
...@@ -42,7 +42,7 @@ Proof. ...@@ -42,7 +42,7 @@ Proof.
induction k as [|k IH]; simpl in *; [by destruct x|]. induction k as [|k IH]; simpl in *; [by destruct x|].
rewrite -map_comp -{2}(map_id _ _ x); by apply map_ext. rewrite -map_comp -{2}(map_id _ _ x); by apply map_ext.
Qed. Qed.
Lemma fg {n k} (x : A (S k)) : n k f (g x) ={n}= x. Lemma fg {n k} (x : A (S k)) : n k f (g x) {n} x.
Proof. Proof.
intros Hnk; apply dist_le with k; auto; clear Hnk. intros Hnk; apply dist_le with k; auto; clear Hnk.
induction k as [|k IH]; simpl; [apply dist_0|]. induction k as [|k IH]; simpl; [apply dist_0|].
...@@ -57,7 +57,7 @@ Record tower := { ...@@ -57,7 +57,7 @@ Record tower := {
g_tower k : g (tower_car (S k)) tower_car k g_tower k : g (tower_car (S k)) tower_car k
}. }.
Instance tower_equiv : Equiv tower := λ X Y, k, X k Y k. Instance tower_equiv : Equiv tower := λ X Y, k, X k Y k.
Instance tower_dist : Dist tower := λ n X Y, k, X k ={n}= Y k. Instance tower_dist : Dist tower := λ n X Y, k, X k {n} Y k.
Program Definition tower_chain (c : chain tower) (k : nat) : chain (A k) := Program Definition tower_chain (c : chain tower) (k : nat) : chain (A k) :=
{| chain_car i := c i k |}. {| chain_car i := c i k |}.
Next Obligation. intros c k n i ?; apply (chain_cauchy c n); lia. Qed. Next Obligation. intros c k n i ?; apply (chain_cauchy c n); lia. Qed.
...@@ -91,9 +91,9 @@ Fixpoint gg {k} (i : nat) : A (i + k) -n> A k := ...@@ -91,9 +91,9 @@ Fixpoint gg {k} (i : nat) : A (i + k) -n> A k :=
match i with 0 => cid | S i => gg i g end. match i with 0 => cid | S i => gg i g end.
Lemma ggff {k i} (x : A k) : gg i (ff i x) x. Lemma ggff {k i} (x : A k) : gg i (ff i x) x.
Proof. induction i as [|i IH]; simpl; [done|by rewrite (gf (ff i x)) IH]. Qed. Proof. induction i as [|i IH]; simpl; [done|by rewrite (gf (ff i x)) IH]. Qed.
Lemma f_tower {n k} (X : tower) : n k f (X k) ={n}= X (S k). Lemma f_tower {n k} (X : tower) : n k f (X k) {n} X (S k).
Proof. intros. by rewrite -(fg (X (S k))) // -(g_tower X). Qed. Proof. intros. by rewrite -(fg (X (S k))) // -(g_tower X). Qed.
Lemma ff_tower {n} k i (X : tower) : n k ff i (X k) ={n}= X (i + k). Lemma ff_tower {n} k i (X : tower) : n k ff i (X k) {n} X (i + k).
Proof. Proof.
intros; induction i as [|i IH]; simpl; [done|]. intros; induction i as [|i IH]; simpl; [done|].
by rewrite IH (f_tower X); last lia. by rewrite IH (f_tower X); last lia.
...@@ -170,7 +170,7 @@ Proof. ...@@ -170,7 +170,7 @@ Proof.
* assert (H : (i - S k) + (1 + k) = i) by lia; rewrite (ff_ff _ H) /=. * assert (H : (i - S k) + (1 + k) = i) by lia; rewrite (ff_ff _ H) /=.
by erewrite coerce_proper by done. by erewrite coerce_proper by done.
Qed. Qed.
Lemma embed_tower j n (X : T) : n j embed j (X j) ={n}= X. Lemma embed_tower j n (X : T) : n j embed j (X j) {n} X.
Proof. Proof.
move=> Hn i; rewrite /= /embed'; destruct (le_lt_dec i j) as [H|H]; simpl. move=> Hn i; rewrite /= /embed'; destruct (le_lt_dec i j) as [H|H]; simpl.
* rewrite -(gg_tower i (j - i) X). * rewrite -(gg_tower i (j - i) X).
......
...@@ -23,10 +23,10 @@ Inductive excl_equiv : Equiv (excl A) := ...@@ -23,10 +23,10 @@ Inductive excl_equiv : Equiv (excl A) :=
| ExclBot_equiv : ExclBot ExclBot. | ExclBot_equiv : ExclBot ExclBot.
Existing Instance excl_equiv. Existing Instance excl_equiv.
Inductive excl_dist `{Dist A} : Dist (excl A) := Inductive excl_dist `{Dist A} : Dist (excl A) :=
| excl_dist_0 (x y : excl A) : x ={0}= y | excl_dist_0 (x y : excl A) : x {0} y
| Excl_dist (x y : A) n : x ={n}= y Excl x ={n}= Excl y | Excl_dist (x y : A) n : x {n} y Excl x {n} Excl y
| ExclUnit_dist n : ExclUnit ={n}= ExclUnit | ExclUnit_dist n : ExclUnit {n} ExclUnit
| ExclBot_dist n : ExclBot ={n}= ExclBot. | ExclBot_dist n : ExclBot {n} ExclBot.
Existing Instance excl_dist. Existing Instance excl_dist.
Global Instance Excl_ne : Proper (dist n ==> dist n) (@Excl A). Global Instance Excl_ne : Proper (dist n ==> dist n) (@Excl A).
Proof. by constructor. Qed. Proof. by constructor. Qed.
...@@ -138,7 +138,7 @@ Lemma excl_validN_inv_l n x y : ✓{S n} (Excl x ⋅ y) → y = ∅. ...@@ -138,7 +138,7 @@ Lemma excl_validN_inv_l n x y : ✓{S n} (Excl x ⋅ y) → y = ∅.
Proof. by destruct y. Qed. Proof. by destruct y. Qed.
Lemma excl_validN_inv_r n x y : {S n} (x Excl y) x = . Lemma excl_validN_inv_r n x y : {S n} (x Excl y) x = .
Proof. by destruct x. Qed. Proof. by destruct x. Qed.
Lemma Excl_includedN n x y : {n} y Excl x {n} y y ={n}= Excl x. Lemma Excl_includedN n x y : {n} y Excl x {n} y y {n} Excl x.
Proof. Proof.
intros Hvalid; split; [destruct n as [|n]; [done|]|by intros ->]. intros Hvalid; split; [destruct n as [|n]; [done|]|by intros ->].
by intros [z ?]; cofe_subst; rewrite (excl_validN_inv_l n x z). by intros [z ?]; cofe_subst; rewrite (excl_validN_inv_l n x z).
......
...@@ -6,7 +6,7 @@ Context `{Countable K} {A : cofeT}. ...@@ -6,7 +6,7 @@ Context `{Countable K} {A : cofeT}.
Implicit Types m : gmap K A. Implicit Types m : gmap K A.
Instance map_dist : Dist (gmap K A) := λ n m1 m2, Instance map_dist : Dist (gmap K A) := λ n m1 m2,
i, m1 !! i ={n}= m2 !! i. i, m1 !! i {n} m2 !! i.
Program Definition map_chain (c : chain (gmap K A)) Program Definition map_chain (c : chain (gmap K A))
(k : K) : chain (option A) := {| chain_car n := c n !! k |}. (k : K) : chain (option A) := {| chain_car n := c n !! k |}.
Next Obligation. by intros c k n i ?; apply (chain_cauchy c). Qed.