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Commit 8c96ad4e authored by Robbert Krebbers's avatar Robbert Krebbers
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Shorter names for common math notions. 

Also do some minor clean up.
parent ba6d9390
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algebra/agree.v
View file @ 8c96ad4e
......@@ -59,9 +59,9 @@ Program Instance agree_op : Op (agree A) := λ x y,
Next Obligation. naive_solver eauto using agree_valid_S, dist_S. Qed.
Instance agree_unit : Unit (agree A) := id.
Instance agree_minus : Minus (agree A) := λ x y, x.
Instance: Commutative (≡) (@op (agree A) _).
Instance: Comm (≡) (@op (agree A) _).
Proof. intros x y; split; [naive_solver|by intros n (?&?&Hxy); apply Hxy]. Qed.
Definition agree_idempotent (x : agree A) : x ⋅ x ≡ x.
Definition agree_idemp (x : agree A) : x ⋅ x ≡ x.
Proof. split; naive_solver. Qed.
Instance: ∀ n : nat, Proper (dist n ==> impl) (@validN (agree A) _ n).
Proof.
......@@ -79,18 +79,18 @@ Proof.
eauto using agree_valid_le.
Qed.
Instance: Proper (dist n ==> dist n ==> dist n) (@op (agree A) _).
Proof. by intros n x1 x2 Hx y1 y2 Hy; rewrite Hy !(commutative _ _ y2) Hx. Qed.
Proof. by intros n x1 x2 Hx y1 y2 Hy; rewrite Hy !(comm _ _ y2) Hx. Qed.
Instance: Proper ((≡) ==> (≡) ==> (≡)) op := ne_proper_2 _.
Instance: Associative (≡) (@op (agree A) _).
Instance: Assoc (≡) (@op (agree A) _).
Proof.
intros x y z; split; simpl; intuition;
repeat match goal with H : agree_is_valid _ _ |- _ => clear H end;
by cofe_subst; rewrite !agree_idempotent.
by cofe_subst; rewrite !agree_idemp.
Qed.
Lemma agree_includedN (x y : agree A) n : x ≼{n} y ↔ y ≡{n}≡ x ⋅ y.
Proof.
split; [|by intros ?; exists y].
by intros [z Hz]; rewrite Hz (associative _) agree_idempotent.
by intros [z Hz]; rewrite Hz assoc agree_idemp.
Qed.
Definition agree_cmra_mixin : CMRAMixin (agree A).
Proof.
......@@ -99,7 +99,7 @@ Proof.
* intros n x [? Hx]; split; [by apply agree_valid_S|intros n' ?].
rewrite (Hx n'); last auto.
symmetry; apply dist_le with n; try apply Hx; auto.
* intros x; apply agree_idempotent.
* intros x; apply agree_idemp.
* by intros x y n [(?&?&?) ?].
* by intros x y n; rewrite agree_includedN.
Qed.
......@@ -108,13 +108,13 @@ Proof. intros Hxy; apply Hxy. Qed.
Lemma agree_valid_includedN (x y : agree A) n : ✓{n} y → x ≼{n} y → x ≡{n}≡ y.
Proof.
move=> Hval [z Hy]; move: Hval; rewrite Hy.
by move=> /agree_op_inv->; rewrite agree_idempotent.
by move=> /agree_op_inv->; rewrite agree_idemp.
Qed.
Definition agree_cmra_extend_mixin : CMRAExtendMixin (agree A).
Proof.
intros n x y1 y2 Hval Hx; exists (x,x); simpl; split.
* by rewrite agree_idempotent.
* by move: Hval; rewrite Hx; move=> /agree_op_inv->; rewrite agree_idempotent.
* by rewrite agree_idemp.
* by move: Hval; rewrite Hx; move=> /agree_op_inv->; rewrite agree_idemp.
Qed.
Canonical Structure agreeRA : cmraT :=
CMRAT agree_cofe_mixin agree_cmra_mixin agree_cmra_extend_mixin.
......@@ -125,7 +125,7 @@ Solve Obligations with done.
Global Instance to_agree_ne n : Proper (dist n ==> dist n) to_agree.
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_inj n : Injective (dist n) (dist n) (to_agree).
Global Instance to_agree_inj n : Inj (dist n) (dist n) (to_agree).
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.
Proof. intros [??]; split; naive_solver eauto using agree_valid_le. Qed.
......
algebra/auth.v
View file @ 8c96ad4e
......@@ -106,10 +106,10 @@ Proof.
* by intros n x1 x2 [Hx Hx'] y1 y2 [Hy Hy'];
split; simpl; rewrite ?Hy ?Hy' ?Hx ?Hx'.
* intros n [[] ?] ?; naive_solver eauto using cmra_includedN_S, cmra_validN_S.
* by split; simpl; rewrite associative.
* by split; simpl; rewrite commutative.
* by split; simpl; rewrite assoc.
* by split; simpl; rewrite comm.
* by split; simpl; rewrite ?cmra_unit_l.
* by split; simpl; rewrite ?cmra_unit_idempotent.
* by split; simpl; rewrite ?cmra_unit_idemp.
* intros n ??; rewrite! auth_includedN; intros [??].
by split; simpl; apply cmra_unit_preservingN.
* assert (∀ n (a b1 b2 : A), b1 ⋅ b2 ≼{n} a → b1 ≼{n} a).
......@@ -153,8 +153,8 @@ Lemma auth_update a a' b b' :
Proof.
move=> Hab [[?| |] bf1] n // =>-[[bf2 Ha] ?]; do 2 red; simpl in *.
destruct (Hab n (bf1 ⋅ bf2)) as [Ha' ?]; auto.
{ by rewrite Ha left_id associative. }
split; [by rewrite Ha' left_id associative; apply cmra_includedN_l|done].
{ by rewrite Ha left_id assoc. }
split; [by rewrite Ha' left_id assoc; apply cmra_includedN_l|done].
Qed.
Lemma auth_local_update L `{!LocalUpdate Lv L} a a' :
......@@ -170,7 +170,7 @@ Lemma auth_update_op_l a a' b :
Proof. by intros; apply (auth_local_update _). Qed.
Lemma auth_update_op_r a a' b :
✓ (a ⋅ b) → ● a ⋅ ◯ a' ~~> ● (a ⋅ b) ⋅ ◯ (a' ⋅ b).
Proof. rewrite -!(commutative _ b); apply auth_update_op_l. Qed.
Proof. rewrite -!(comm _ b); apply auth_update_op_l. Qed.
(* This does not seem to follow from auth_local_update.
The trouble is that given ✓ (L a ⋅ a'), Lv a
......
algebra/cmra.v
View file @ 8c96ad4e
......@@ -43,10 +43,10 @@ Record CMRAMixin A `{Dist A, Equiv A, Unit A, Op A, ValidN A, Minus A} := {
(* valid *)
mixin_cmra_validN_S n x : ✓{S n} x → ✓{n} x;
(* monoid *)
mixin_cmra_associative : Associative (≡) (⋅);
mixin_cmra_commutative : Commutative (≡) (⋅);
mixin_cmra_assoc : Assoc (≡) (⋅);
mixin_cmra_comm : Comm (≡) (⋅);
mixin_cmra_unit_l x : unit x ⋅ x ≡ x;
mixin_cmra_unit_idempotent x : unit (unit x) ≡ unit x;
mixin_cmra_unit_idemp x : unit (unit x) ≡ unit x;
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_op_minus n x y : x ≼{n} y → x ⋅ y ⩪ x ≡{n}≡ y
......@@ -101,14 +101,14 @@ Section cmra_mixin.
Proof. apply (mixin_cmra_minus_ne _ (cmra_mixin A)). Qed.
Lemma cmra_validN_S n x : ✓{S n} x → ✓{n} x.
Proof. apply (mixin_cmra_validN_S _ (cmra_mixin A)). Qed.
Global Instance cmra_associative : Associative (≡) (@op A _).
Proof. apply (mixin_cmra_associative _ (cmra_mixin A)). Qed.
Global Instance cmra_commutative : Commutative (≡) (@op A _).
Proof. apply (mixin_cmra_commutative _ (cmra_mixin A)). Qed.
Global Instance cmra_assoc : Assoc (≡) (@op A _).
Proof. apply (mixin_cmra_assoc _ (cmra_mixin A)). Qed.
Global Instance cmra_comm : Comm (≡) (@op A _).
Proof. apply (mixin_cmra_comm _ (cmra_mixin A)). Qed.
Lemma cmra_unit_l x : unit x ⋅ x ≡ x.
Proof. apply (mixin_cmra_unit_l _ (cmra_mixin A)). Qed.
Lemma cmra_unit_idempotent x : unit (unit x) ≡ unit x.
Proof. apply (mixin_cmra_unit_idempotent _ (cmra_mixin A)). Qed.
Lemma cmra_unit_idemp x : unit (unit x) ≡ unit x.
Proof. apply (mixin_cmra_unit_idemp _ (cmra_mixin A)). Qed.
Lemma cmra_unit_preservingN n x y : x ≼{n} y → unit x ≼{n} unit y.
Proof. apply (mixin_cmra_unit_preservingN _ (cmra_mixin A)). Qed.
Lemma cmra_validN_op_l n x y : ✓{n} (x ⋅ y) → ✓{n} x.
......@@ -166,7 +166,7 @@ Proof. apply (ne_proper _). Qed.
Global Instance cmra_op_ne' n : Proper (dist n ==> dist n ==> dist n) (@op A _).
Proof.
intros x1 x2 Hx y1 y2 Hy.
by rewrite Hy (commutative _ x1) Hx (commutative _ y2).
by rewrite Hy (comm _ x1) Hx (comm _ y2).
Qed.
Global Instance ra_op_proper' : Proper ((≡) ==> (≡) ==> (≡)) (@op A _).
Proof. apply (ne_proper_2 _). Qed.
......@@ -217,15 +217,15 @@ Proof. induction 2; eauto using cmra_validN_S. Qed.
Lemma cmra_valid_op_l x y : ✓ (x ⋅ y) → ✓ x.
Proof. rewrite !cmra_valid_validN; eauto using cmra_validN_op_l. Qed.
Lemma cmra_validN_op_r x y n : ✓{n} (x ⋅ y) → ✓{n} y.
Proof. rewrite (commutative _ x); apply cmra_validN_op_l. Qed.
Proof. rewrite (comm _ x); apply cmra_validN_op_l. Qed.
Lemma cmra_valid_op_r x y : ✓ (x ⋅ y) → ✓ y.
Proof. rewrite !cmra_valid_validN; eauto using cmra_validN_op_r. Qed.
(** ** Units *)
Lemma cmra_unit_r x : x ⋅ unit x ≡ x.
Proof. by rewrite (commutative _ x) cmra_unit_l. Qed.
Proof. by rewrite (comm _ x) cmra_unit_l. Qed.
Lemma cmra_unit_unit x : unit x ⋅ unit x ≡ unit x.
Proof. by rewrite -{2}(cmra_unit_idempotent x) cmra_unit_r. Qed.
Proof. by rewrite -{2}(cmra_unit_idemp x) cmra_unit_r. Qed.
Lemma cmra_unit_validN x n : ✓{n} x → ✓{n} unit x.
Proof. rewrite -{1}(cmra_unit_l x); apply cmra_validN_op_l. Qed.
Lemma cmra_unit_valid x : ✓ x → ✓ unit x.
......@@ -243,7 +243,7 @@ Proof.
split.
* by intros x; exists (unit x); rewrite cmra_unit_r.
* intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2).
by rewrite (associative _) -Hy -Hz.
by rewrite assoc -Hy -Hz.
Qed.
Global Instance cmra_included_preorder: PreOrder (@included A _ _).
Proof.
......@@ -265,22 +265,22 @@ Proof. by exists y. Qed.
Lemma cmra_included_l x y : x ≼ x ⋅ y.
Proof. by exists y. Qed.
Lemma cmra_includedN_r n x y : y ≼{n} x ⋅ y.
Proof. rewrite (commutative op); apply cmra_includedN_l. Qed.
Proof. rewrite (comm op); apply cmra_includedN_l. Qed.
Lemma cmra_included_r x y : y ≼ x ⋅ y.
Proof. rewrite (commutative op); apply cmra_included_l. Qed.
Proof. rewrite (comm op); apply cmra_included_l. Qed.
Lemma cmra_unit_preserving x y : x ≼ y → unit x ≼ unit y.
Proof. rewrite !cmra_included_includedN; eauto using cmra_unit_preservingN. Qed.
Lemma cmra_included_unit x : unit x ≼ x.
Proof. by exists x; rewrite cmra_unit_l. Qed.
Lemma cmra_preservingN_l n x y z : x ≼{n} y → z ⋅ x ≼{n} z ⋅ y.
Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (associative op). Qed.
Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed.
Lemma cmra_preserving_l x y z : x ≼ y → z ⋅ x ≼ z ⋅ y.
Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (associative op). Qed.
Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed.
Lemma cmra_preservingN_r n x y z : x ≼{n} y → x ⋅ z ≼{n} y ⋅ z.
Proof. by intros; rewrite -!(commutative _ z); apply cmra_preservingN_l. Qed.
Proof. by intros; rewrite -!(comm _ z); apply cmra_preservingN_l. Qed.
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 -!(comm _ z); apply cmra_preserving_l. Qed.
Lemma cmra_included_dist_l x1 x2 x1' n :
x1 ≼ x2 → x1' ≡{n}≡ x1 → ∃ x2', x1' ≼ x2' ∧ x2' ≡{n}≡ x2.
......@@ -321,7 +321,7 @@ Section identity.
Lemma cmra_empty_least x : ∅ ≼ x.
Proof. by exists x; rewrite left_id. Qed.
Global Instance cmra_empty_right_id : RightId (≡) ∅ (⋅).
Proof. by intros x; rewrite (commutative op) left_id. Qed.
Proof. by intros x; rewrite (comm op) left_id. Qed.
Lemma cmra_unit_empty : unit ∅ ≡ ∅.
Proof. by rewrite -{2}(cmra_unit_l ∅) right_id. Qed.
End identity.
......@@ -336,7 +336,7 @@ Lemma local_update L `{!LocalUpdate Lv L} x y :
Proof. by rewrite equiv_dist=>?? n; apply (local_updateN L). Qed.
Global Instance local_update_op x : LocalUpdate (λ _, True) (op x).
Proof. split. apply _. by intros n y1 y2 _ _; rewrite associative. Qed.
Proof. split. apply _. by intros n y1 y2 _ _; rewrite assoc. Qed.
(** ** Updates *)
Global Instance cmra_update_preorder : PreOrder (@cmra_update A).
......@@ -366,10 +366,10 @@ Lemma cmra_updateP_op (P1 P2 Q : A → Prop) x1 x2 :
x1 ~~>: P1 → x2 ~~>: P2 → (∀ y1 y2, P1 y1 → P2 y2 → Q (y1 ⋅ y2)) → x1 ⋅ x2 ~~>: Q.
Proof.
intros Hx1 Hx2 Hy z n ?.
destruct (Hx1 (x2 ⋅ z) n) as (y1&?&?); first by rewrite associative.
destruct (Hx1 (x2 ⋅ z) n) as (y1&?&?); first by rewrite assoc.
destruct (Hx2 (y1 ⋅ z) n) as (y2&?&?);
first by rewrite associative (commutative _ x2) -associative.
exists (y1 ⋅ y2); split; last rewrite (commutative _ y1) -associative; auto.
first by rewrite assoc (comm _ x2) -assoc.
exists (y1 ⋅ y2); split; last rewrite (comm _ y1) -assoc; auto.
Qed.
Lemma cmra_updateP_op' (P1 P2 : A → Prop) x1 x2 :
x1 ~~>: P1 → x2 ~~>: P2 → x1 ⋅ x2 ~~>: λ y, ∃ y1 y2, y = y1 ⋅ y2 ∧ P1 y1 ∧ P2 y2.
......@@ -448,10 +448,10 @@ Class RA A `{Equiv A, Unit A, Op A, Valid A, Minus A} := {
ra_validN_ne :> Proper ((≡) ==> impl) valid;
ra_minus_ne :> Proper ((≡) ==> (≡) ==> (≡)) minus;
(* monoid *)
ra_associative :> Associative (≡) (⋅);
ra_commutative :> Commutative (≡) (⋅);
ra_assoc :> Assoc (≡) (⋅);
ra_comm :> Comm (≡) (⋅);
ra_unit_l x : unit x ⋅ x ≡ x;
ra_unit_idempotent x : unit (unit x) ≡ unit x;
ra_unit_idemp x : unit (unit x) ≡ unit x;
ra_unit_preserving x y : x ≼ y → unit x ≼ unit y;
ra_valid_op_l x y : ✓ (x ⋅ y) → ✓ x;
ra_op_minus x y : x ≼ y → x ⋅ y ⩪ x ≡ y
......@@ -524,10 +524,10 @@ Section prod.
* by intros n x1 x2 [Hx1 Hx2] y1 y2 [Hy1 Hy2];
split; rewrite /= ?Hx1 ?Hx2 ?Hy1 ?Hy2.
* by intros n x [??]; split; apply cmra_validN_S.
* split; simpl; apply (associative _).
* split; simpl; apply (commutative _).
* split; simpl; apply cmra_unit_l.
* split; simpl; apply cmra_unit_idempotent.
* by split; rewrite /= assoc.
* by split; rewrite /= comm.
* by split; rewrite /= cmra_unit_l.
* by split; rewrite /= cmra_unit_idemp.
* intros n x y; rewrite !prod_includedN.
by intros [??]; split; apply cmra_unit_preservingN.
* intros n x y [??]; split; simpl in *; eauto using cmra_validN_op_l.
......
algebra/cmra_big_op.v
View file @ 8c96ad4e
......@@ -22,21 +22,21 @@ Global Instance big_op_permutation : Proper ((≡ₚ) ==> (≡)) big_op.
Proof.
induction 1 as [|x xs1 xs2 ? IH|x y xs|xs1 xs2 xs3]; simpl; auto.
* by rewrite IH.
* by rewrite !(associative _) (commutative _ x).
* by rewrite !assoc (comm _ x).
* by transitivity (big_op xs2).
Qed.
Global Instance big_op_proper : Proper ((≡) ==> (≡)) big_op.
Proof. by induction 1; simpl; repeat apply (_ : Proper (_ ==> _ ==> _) op). Qed.
Lemma big_op_app xs ys : big_op (xs ++ ys) ≡ big_op xs ⋅ big_op ys.
Proof.
induction xs as [|x xs IH]; simpl; first by rewrite ?(left_id _ _).
by rewrite IH (associative _).
induction xs as [|x xs IH]; simpl; first by rewrite ?left_id.
by rewrite IH assoc.
Qed.
Lemma big_op_contains xs ys : xs `contains` ys → big_op xs ≼ big_op ys.
Proof.
induction 1 as [|x xs ys|x y xs|x xs ys|xs ys zs]; rewrite //=.
* by apply cmra_preserving_l.
* by rewrite !associative (commutative _ y).
* by rewrite !assoc (comm _ y).
* by transitivity (big_op ys); last apply cmra_included_r.
* by transitivity (big_op ys).
Qed.
......@@ -58,7 +58,7 @@ Qed.
Lemma big_opM_singleton i x : big_opM ({[i ↦ x]} : M A) ≡ x.
Proof.
rewrite -insert_empty big_opM_insert /=; last auto using lookup_empty.
by rewrite big_opM_empty (right_id _ _).
by rewrite big_opM_empty right_id.
Qed.
Global Instance big_opM_proper : Proper ((≡) ==> (≡)) (big_opM : M A → _).
Proof.
......
algebra/cmra_tactics.v
View file @ 8c96ad4e
......@@ -25,7 +25,7 @@ Module ra_reflection. Section ra_reflection.
eval Σ e ≡ big_op ((λ n, from_option ∅ (Σ !! n)) <$> flatten e).
Proof.
by induction e as [| |e1 IH1 e2 IH2];
rewrite /= ?(right_id _ _) ?fmap_app ?big_op_app ?IH1 ?IH2.
rewrite /= ?right_id ?fmap_app ?big_op_app ?IH1 ?IH2.
Qed.
Lemma flatten_correct Σ e1 e2 :
flatten e1 `contains` flatten e2 → eval Σ e1 ≼ eval Σ e2.
......
algebra/cofe.v
View file @ 8c96ad4e
......@@ -337,7 +337,7 @@ Section later.
Canonical Structure laterC : cofeT := CofeT later_cofe_mixin.
Global Instance Next_contractive : Contractive (@Next A).
Proof. intros [|n] x y Hxy; [done|]; apply Hxy; lia. Qed.
Global Instance Later_inj n : Injective (dist n) (dist (S n)) (@Next A).
Global Instance Later_inj n : Inj (dist n) (dist (S n)) (@Next A).
Proof. by intros x y. Qed.
End later.
......
algebra/dra.v
View file @ 8c96ad4e
......@@ -46,14 +46,14 @@ Class DRA A `{Equiv A, Valid A, Unit A, Disjoint A, Op A, Minus A} := {
dra_unit_valid x : ✓ x → ✓ unit x;
dra_minus_valid x y : ✓ x → ✓ y → x ≼ y → ✓ (y ⩪ x);
(* monoid *)
dra_associative :> Associative (≡) (⋅);
dra_assoc :> Assoc (≡) (⋅);
dra_disjoint_ll x y z : ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → x ⊥ z;
dra_disjoint_move_l x y z : ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → x ⊥ y ⋅ z;
dra_symmetric :> Symmetric (@disjoint A _);
dra_commutative x y : ✓ x → ✓ y → x ⊥ y → x ⋅ y ≡ y ⋅ x;
dra_comm x y : ✓ x → ✓ y → x ⊥ y → x ⋅ y ≡ y ⋅ x;
dra_unit_disjoint_l x : ✓ x → unit x ⊥ x;
dra_unit_l x : ✓ x → unit x ⋅ x ≡ x;
dra_unit_idempotent x : ✓ x → unit (unit x) ≡ unit x;
dra_unit_idemp x : ✓ x → unit (unit x) ≡ unit x;
dra_unit_preserving x y : ✓ x → ✓ y → x ≼ y → unit x ≼ unit y;
dra_disjoint_minus x y : ✓ x → ✓ y → x ≼ y → x ⊥ y ⩪ x;
dra_op_minus x y : ✓ x → ✓ y → x ≼ y → x ⋅ y ⩪ x ≡ y
......@@ -73,12 +73,12 @@ Qed.
Lemma dra_disjoint_rl x y z : ✓ x → ✓ y → ✓ z → y ⊥ z → x ⊥ y ⋅ z → x ⊥ y.
Proof. intros ???. rewrite !(symmetry_iff _ x). by apply dra_disjoint_ll. Qed.
Lemma dra_disjoint_lr x y z : ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → y ⊥ z.
Proof. intros ????. rewrite dra_commutative //. by apply dra_disjoint_ll. Qed.
Proof. intros ????. rewrite dra_comm //. by apply dra_disjoint_ll. Qed.
Lemma dra_disjoint_move_r x y z :
✓ x → ✓ y → ✓ z → y ⊥ z → x ⊥ y ⋅ z → x ⋅ y ⊥ z.
Proof.
intros; symmetry; rewrite dra_commutative; eauto using dra_disjoint_rl.
apply dra_disjoint_move_l; auto; by rewrite dra_commutative.
intros; symmetry; rewrite dra_comm; eauto using dra_disjoint_rl.
apply dra_disjoint_move_l; auto; by rewrite dra_comm.
Qed.
Hint Immediate dra_disjoint_move_l dra_disjoint_move_r.
Hint Unfold dra_included.
......@@ -114,11 +114,11 @@ Proof.
+ exists z. by rewrite Hx ?Hy; tauto.
* intros [x px ?] [y py ?] [z pz ?]; split; simpl;
[intuition eauto 2 using dra_disjoint_lr, dra_disjoint_rl
|intros; apply (associative _)].
* intros [x px ?] [y py ?]; split; naive_solver eauto using dra_commutative.
|by intros; rewrite assoc].
* intros [x px ?] [y py ?]; split; naive_solver eauto using dra_comm.
* intros [x px ?]; split;
naive_solver eauto using dra_unit_l, dra_unit_disjoint_l.
* intros [x px ?]; split; naive_solver eauto using dra_unit_idempotent.
* intros [x px ?]; split; naive_solver eauto using dra_unit_idemp.
* intros x y Hxy; exists (unit y ⩪ unit x).
destruct x as [x px ?], y as [y py ?], Hxy as [[z pz ?] [??]]; simpl in *.
assert (py → unit x ≼ unit y)
......
algebra/excl.v
View file @ 8c96ad4e
......@@ -31,9 +31,9 @@ Global Instance Excl_ne : Proper (dist n ==> dist n) (@Excl A).
Proof. by constructor. Qed.
Global Instance Excl_proper : Proper ((≡) ==> (≡)) (@Excl A).
Proof. by constructor. Qed.
Global Instance Excl_inj : Injective (≡) (≡) (@Excl A).
Global Instance Excl_inj : Inj (≡) (≡) (@Excl A).
Proof. by inversion_clear 1. Qed.
Global Instance Excl_dist_inj n : Injective (dist n) (dist n) (@Excl A).
Global Instance Excl_dist_inj n : Inj (dist n) (dist n) (@Excl A).
Proof. by inversion_clear 1. Qed.
Program Definition excl_chain
(c : chain (excl A)) (x : A) (H : maybe Excl (c 1) = Some x) : chain A :=
......
algebra/fin_maps.v
View file @ 8c96ad4e
......@@ -121,10 +121,10 @@ Proof.
* by intros n m1 m2 Hm ? i; rewrite -(Hm i).
* by intros n m1 m1' Hm1 m2 m2' Hm2 i; rewrite !lookup_minus (Hm1 i) (Hm2 i).
* intros n m Hm i; apply cmra_validN_S, Hm.
* by intros m1 m2 m3 i; rewrite !lookup_op associative.
* by intros m1 m2 i; rewrite !lookup_op commutative.
* by intros m1 m2 m3 i; rewrite !lookup_op assoc.
* by intros m1 m2 i; rewrite !lookup_op comm.
* by intros m i; rewrite lookup_op !lookup_unit cmra_unit_l.
* by intros m i; rewrite !lookup_unit cmra_unit_idempotent.
* by intros m i; rewrite !lookup_unit cmra_unit_idemp.
* intros n x y; rewrite !map_includedN_spec; intros Hm i.
by rewrite !lookup_unit; apply cmra_unit_preservingN.
* intros n m1 m2 Hm i; apply cmra_validN_op_l with (m2 !! i).
......
algebra/iprod.v
View file @ 8c96ad4e
......@@ -141,10 +141,10 @@ Section iprod_cmra.
* by intros n f1 f2 Hf ? x; rewrite -(Hf x).
* by intros n f f' Hf g g' Hg i; rewrite iprod_lookup_minus (Hf i) (Hg i).
* intros n f Hf x; apply cmra_validN_S, Hf.
* by intros f1 f2 f3 x; rewrite iprod_lookup_op associative.
* by intros f1 f2 x; rewrite iprod_lookup_op commutative.
* by intros f1 f2 f3 x; rewrite iprod_lookup_op assoc.
* by intros f1 f2 x; rewrite iprod_lookup_op comm.
* by intros f x; rewrite iprod_lookup_op iprod_lookup_unit cmra_unit_l.
* by intros f x; rewrite iprod_lookup_unit cmra_unit_idempotent.
* by intros f x; rewrite iprod_lookup_unit cmra_unit_idemp.
* intros n f1 f2; rewrite !iprod_includedN_spec=> Hf x.
by rewrite iprod_lookup_unit; apply cmra_unit_preservingN, Hf.
* intros n f1 f2 Hf x; apply cmra_validN_op_l with (f2 x), Hf.
......
algebra/option.v
View file @ 8c96ad4e
......@@ -45,7 +45,7 @@ Global Instance Some_ne : Proper (dist n ==> dist n) (@Some A).
Proof. by constructor. Qed.
Global Instance is_Some_ne n : Proper (dist n ==> iff) (@is_Some A).
Proof. inversion_clear 1; split; eauto. Qed.
Global Instance Some_dist_inj : Injective (dist n) (dist n) (@Some A).
Global Instance Some_dist_inj : Inj (dist n) (dist n) (@Some A).
Proof. by inversion_clear 1. Qed.
Global Instance None_timeless : Timeless (@None A).
Proof. inversion_clear 1; constructor. Qed.
......@@ -92,10 +92,10 @@ Proof.
* by destruct 1; rewrite /validN /option_validN //=; cofe_subst.
* by destruct 1; inversion_clear 1; constructor; cofe_subst.
* intros n [x|]; unfold validN, option_validN; eauto using cmra_validN_S.
* intros [x|] [y|] [z|]; constructor; rewrite ?associative; auto.
* intros [x|] [y|]; constructor; rewrite 1?commutative; auto.
* intros [x|] [y|] [z|]; constructor; rewrite ?assoc; auto.
* intros [x|] [y|]; constructor; rewrite 1?comm; auto.
* by intros [x|]; constructor; rewrite cmra_unit_l.
* by intros [x|]; constructor; rewrite cmra_unit_idempotent.
* by intros [x|]; constructor; rewrite cmra_unit_idemp.
* intros n mx my; rewrite !option_includedN;intros [->|(x&y&->&->&?)]; auto.
right; exists (unit x), (unit y); eauto using cmra_unit_preservingN.
* intros n [x|] [y|]; rewrite /validN /option_validN /=;
......
algebra/sts.v
View file @ 8c96ad4e
......@@ -151,7 +151,7 @@ Proof.
* intros ???? (z&Hy&?&Hxz); destruct Hxz; inversion Hy;clear Hy; setoid_subst;
rewrite ?disjoint_union_difference; auto using closed_up with sts.
eapply closed_up_set; eauto 2 using closed_disjoint with sts.
* intros [] [] []; constructor; rewrite ?(associative _); auto with sts.
* intros [] [] []; constructor; rewrite ?assoc; auto with sts.
* destruct 4; inversion_clear 1; constructor; auto with sts.
* destruct 4; inversion_clear 1; constructor; auto with sts.
* destruct 1; constructor; auto with sts.
......
algebra/upred.v
View file @ 8c96ad4e
......@@ -140,7 +140,7 @@ Next Obligation.
intros M P Q x y n1 n2 (x1&x2&Hx&?&?) Hxy ??.
assert (∃ x2', y ≡{n2}≡ x1 ⋅ x2' ∧ x2 ≼ x2') as (x2'&Hy&?).
{ destruct Hxy as [z Hy]; exists (x2 ⋅ z); split; eauto using cmra_included_l.
apply dist_le with n1; auto. by rewrite (associative op) -Hx Hy. }
apply dist_le with n1; auto. by rewrite (assoc op) -Hx Hy. }
clear Hxy; cofe_subst y; exists x1, x2'; split_ands; [done| |].
* apply uPred_weaken with x1 n1; eauto using cmra_validN_op_l.
* apply uPred_weaken with x2 n1; eauto using cmra_validN_op_r.
......@@ -179,7 +179,7 @@ Program Definition uPred_ownM {M : cmraT} (a : M) : uPred M :=
Next Obligation. by intros M a x1 x2 n [a' ?] Hx; exists a'; rewrite -Hx. Qed.
Next Obligation.
intros M a x1 x n1 n2 [a' Hx1] [x2 Hx] ??.
exists (a' ⋅ x2). by rewrite (associative op) -(dist_le _ _ _ _ Hx1) // Hx.
exists (a' ⋅ x2). by rewrite (assoc op) -(dist_le _ _ _ _ Hx1) // Hx.
Qed.
Program Definition uPred_valid {M A : cmraT} (a : A) : uPred M :=
{| uPred_holds n x := ✓{n} a |}.
......@@ -245,11 +245,11 @@ Arguments uPred_holds {_} !_ _ _ /.
Global Instance: PreOrder (@uPred_entails M).
Proof. split. by intros P x i. by intros P Q Q' HP HQ x i ??; apply HQ, HP. Qed.
Global Instance: AntiSymmetric (≡) (@uPred_entails M).
Global Instance: AntiSymm (≡) (@uPred_entails M).
Proof. intros P Q HPQ HQP; split; auto. Qed.
Lemma equiv_spec P Q : P ≡ Q ↔ P ⊑ Q ∧ Q ⊑ P.
Proof.
split; [|by intros [??]; apply (anti_symmetric (⊑))].
split; [|by intros [??]; apply (anti_symm (⊑))].
intros HPQ; split; intros x i; apply HPQ.
Qed.
Global Instance entails_proper :
......@@ -434,7 +434,7 @@ Proof. intros; apply const_elim with φ; eauto. Qed.
Lemma const_elim_r φ Q R : (φ → Q ⊑ R) → (Q ∧ ■ φ) ⊑ R.
Proof. intros; apply const_elim with φ; eauto. Qed.
Lemma const_equiv (φ : Prop) : φ → (■ φ : uPred M)%I ≡ True%I.
Proof. intros; apply (anti_symmetric _); auto using const_intro. Qed.
Proof. intros; apply (anti_symm _); auto using const_intro. Qed.
Lemma equiv_eq {A : cofeT} P (a b : A) : a ≡ b → P ⊑ (a ≡ b).
Proof. intros ->; apply eq_refl. Qed.
Lemma eq_sym {A : cofeT} (a b : A) : (a ≡ b) ⊑ (b ≡ a).
......@@ -484,57 +484,57 @@ Global Instance exist_mono' A :
Proper (pointwise_relation _ (⊑) ==> (⊑)) (@uPred_exist M A).
Proof. intros P1 P2; apply exist_mono. Qed.
Global Instance and_idem : Idempotent (≡) (@uPred_and M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance or_idem : Idempotent (≡) (@uPred_or M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance and_comm : Commutative (≡) (@uPred_and M).
Proof. intros P Q; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance and_idem : IdemP (≡) (@uPred_and M).
Proof. intros P; apply (anti_symm (⊑)); auto. Qed.
Global Instance or_idem : IdemP (≡) (@uPred_or M).
Proof. intros P; apply (anti_symm (⊑)); auto. Qed.
Global Instance and_comm : Comm (≡) (@uPred_and M).
Proof. intros P Q; apply (anti_symm (⊑)); auto. Qed.
Global Instance True_and : LeftId (≡) True%I (@uPred_and M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Proof. intros P; apply (anti_symm (⊑)); auto. Qed.
Global Instance and_True : RightId (≡) True%I (@uPred_and M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Proof. intros P; apply (anti_symm (⊑)); auto. Qed.
Global Instance False_and : LeftAbsorb (≡) False%I (@uPred_and M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.