statement stringlengths 1 2.04k | proof stringlengths 0 6.2k | type stringclasses 20
values | symbolic_name stringlengths 1 91 | library stringclasses 18
values | filename stringclasses 208
values | imports listlengths 3 22 | deps listlengths 0 43 | docstring stringlengths 0 2.22k | source_url stringclasses 1
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agree (A : Type) : Type | := {
agree_car : list A;
agree_not_nil : bool_decide (agree_car = []) = false
}. | Record | agree | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"Type",
"agree_car"
] | Note that the projection [agree_car] is not non-expansive, so it cannot be
used in the logic. If you need to get a witness out, you should use the
lemma [to_agree_uninjN] instead. In general, [agree_car] should ONLY be used
internally in this file. | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 |
agree_car : agree >-> list. | Coercion | agree_car | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | ||
to_agree {A} (a : A) : agree A | :=
{| agree_car := [a]; agree_not_nil := eq_refl |}. | Definition | to_agree | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree",
"agree_car"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
elem_of_agree {A} (x : agree A) : ∃ a, a ∈ agree_car x. | Proof. destruct x as [[|a ?] ?]; set_solver+. Qed. | Lemma | elem_of_agree | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree",
"agree_car"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_eq {A} (x y : agree A) : agree_car x = agree_car y → x = y. | Proof.
destruct x as [a ?], y as [b ?]; simpl.
intros ->; f_equal. apply (proof_irrel _).
Qed. | Lemma | agree_eq | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree",
"agree_car"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_dist : Dist (agree A) | := λ n x y,
(∀ a, a ∈ agree_car x → ∃ b, b ∈ agree_car y ∧ a ≡{n}≡ b) ∧
(∀ b, b ∈ agree_car y → ∃ a, a ∈ agree_car x ∧ a ≡{n}≡ b). | Instance | agree_dist | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"Dist",
"agree",
"agree_car"
] | OFE | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 |
agree_equiv : Equiv (agree A) | := λ x y, ∀ n, x ≡{n}≡ y. | Instance | agree_equiv | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_ofe_mixin : OfeMixin (agree A). | Proof.
split.
- done.
- intros n; split; rewrite /dist /agree_dist.
+ intros x; split; eauto.
+ intros x y [??]. naive_solver eauto.
+ intros x y z [H1 H1'] [H2 H2']; split.
* intros a ?. destruct (H1 a) as (b&?&?); auto.
destruct (H2 b) as (c&?&?); eauto. by exists c; split; last etrans... | Definition | agree_ofe_mixin | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"OfeMixin",
"agree",
"agree_dist",
"dist_le"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agreeO | := Ofe (agree A) agree_ofe_mixin. | Canonical | agreeO | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree",
"agree_ofe_mixin"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_validN_instance : ValidN (agree A) | := λ n x,
match agree_car x with
| [a] => True
| _ => ∀ a b, a ∈ agree_car x → b ∈ agree_car x → a ≡{n}≡ b
end. | Instance | agree_validN_instance | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"ValidN",
"agree",
"agree_car"
] | agree_validN is carefully written such that, when applied to a singleton, it
is convertible to True. This makes working with agreement much more pleasant. | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 |
agree_valid_instance : Valid (agree A) | := λ x, ∀ n, ✓{n} x. | Instance | agree_valid_instance | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"Valid",
"agree"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_op_instance : Op (agree A) := λ x y,
{| agree_car := agree_car x ++ agree_car y |}. | Next Obligation. by intros [[|??]] y. Qed. | Instance | agree_op_instance | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"Op",
"agree",
"agree_car"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_pcore_instance : PCore (agree A) | := Some. | Instance | agree_pcore_instance | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"PCore",
"agree"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_validN_def n x :
✓{n} x ↔ ∀ a b, a ∈ agree_car x → b ∈ agree_car x → a ≡{n}≡ b. | Proof.
rewrite /validN /agree_validN_instance. destruct (agree_car _) as [|? [|??]]; auto.
setoid_rewrite list_elem_of_singleton; naive_solver.
Qed. | Lemma | agree_validN_def | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree_car",
"agree_validN_instance"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_comm : Comm (≡) (@op (agree A) _). | Proof. intros x y n; split=> a /=; setoid_rewrite elem_of_app; naive_solver. Qed. | Instance | agree_comm | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_assoc : Assoc (≡) (@op (agree A) _). | Proof.
intros x y z n; split=> a /=; repeat setoid_rewrite elem_of_app; naive_solver.
Qed. | Instance | agree_assoc | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_idemp x : x ⋅ x ≡ x. | Proof. intros n; split=> a /=; setoid_rewrite elem_of_app; naive_solver. Qed. | Lemma | agree_idemp | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_validN_ne n :
Proper (dist n ==> impl) (@validN SI (agree A) _ n). | Proof.
intros x y [H H']; rewrite /impl !agree_validN_def; intros Hv a b Ha Hb.
destruct (H' a) as (a'&?&<-); auto. destruct (H' b) as (b'&?&<-); auto.
Qed. | Instance | agree_validN_ne | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree",
"agree_validN_def"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_validN_proper n :
Proper (equiv ==> iff) (@validN SI (agree A) _ n). | Proof. move=> x y /equiv_dist H. by split; rewrite (H n). Qed. | Instance | agree_validN_proper | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree",
"equiv_dist"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_op_ne' x : NonExpansive (op x). | Proof.
intros n y1 y2 [H H']; split=> a /=; setoid_rewrite elem_of_app; naive_solver.
Qed. | Instance | agree_op_ne' | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"NonExpansive"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_op_ne : NonExpansive2 (@op (agree A) _). | Proof. by intros n x1 x2 Hx y1 y2 Hy; rewrite Hy !(comm _ _ y2) Hx. Qed. | Instance | agree_op_ne | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"NonExpansive2",
"agree"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_op_proper : Proper ((≡) ==> (≡) ==> (≡)) (op (A := agree A)) | :=
ne_proper_2 _. | Instance | agree_op_proper | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree",
"ne_proper_2"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_included x y : x ≼ y ↔ y ≡ x ⋅ y. | Proof.
split; [|by intros ?; exists y].
by intros [z Hz]; rewrite Hz assoc agree_idemp.
Qed. | Lemma | agree_included | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree_idemp"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_includedN n x y : x ≼{n} y ↔ y ≡{n}≡ x ⋅ y. | Proof.
split; [|by intros ?; exists y].
by intros [z Hz]; rewrite Hz assoc agree_idemp.
Qed. | Lemma | agree_includedN | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree_idemp"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_op_invN n x1 x2 : ✓{n} (x1 ⋅ x2) → x1 ≡{n}≡ x2. | Proof.
rewrite agree_validN_def /=. setoid_rewrite elem_of_app=> Hv; split=> a Ha.
- destruct (elem_of_agree x2); naive_solver.
- destruct (elem_of_agree x1); naive_solver.
Qed. | Lemma | agree_op_invN | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree_validN_def",
"elem_of_agree"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_cmra_mixin : CmraMixin (agree A). | Proof.
apply cmra_total_mixin; try apply _ || by eauto.
- intros n m x; rewrite !agree_validN_def; eauto using dist_le.
- intros x. apply agree_idemp.
- intros n x y; rewrite !agree_validN_def /=.
setoid_rewrite elem_of_app; naive_solver.
- intros n x y1 y2 Hval Hx; exists x, x; simpl; split.
+ by rew... | Definition | agree_cmra_mixin | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"CmraMixin",
"agree",
"agree_idemp",
"agree_op_invN",
"agree_validN_def",
"cmra_total_mixin",
"dist_le"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agreeR : cmra | := Cmra (agree A) agree_cmra_mixin. | Canonical | agreeR | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"Cmra",
"agree",
"agree_cmra_mixin",
"cmra"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_cmra_total : CmraTotal agreeR. | Proof. rewrite /CmraTotal; eauto. Qed. | Instance | agree_cmra_total | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"CmraTotal",
"agreeR"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_core_id x : CoreId x. | Proof. by constructor. Qed. | Instance | agree_core_id | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"CoreId"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_pcore x : pcore x = Some x. | Proof. done. Qed. | Lemma | agree_pcore | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_cmra_discrete : OfeDiscrete A → CmraDiscrete agreeR. | Proof.
intros HD. split.
- intros x y [H H'] n; split=> a; setoid_rewrite <-(discrete_iff_0 _ _ _); auto.
- intros x; rewrite agree_validN_def=> Hv n. apply agree_validN_def=> a b ??.
apply (discrete_iff_0 _ _ _); auto.
Qed. | Instance | agree_cmra_discrete | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"CmraDiscrete",
"OfeDiscrete",
"agreeR",
"agree_validN_def",
"discrete_iff_0"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
to_agree_ne : NonExpansive to_agree. | Proof.
intros n a1 a2 Hx; split=> b /=;
setoid_rewrite list_elem_of_singleton; naive_solver.
Qed. | Instance | to_agree_ne | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"NonExpansive",
"to_agree"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
to_agree_proper : Proper ((≡) ==> (≡)) to_agree | := ne_proper _. | Instance | to_agree_proper | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"ne_proper",
"to_agree"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
to_agree_discrete a : Discrete a → Discrete (to_agree a). | Proof.
intros ? y [H H'] n; split.
- intros a' ->%list_elem_of_singleton. destruct (H a) as [b ?]; first by left.
exists b. by rewrite -discrete_iff_0.
- intros b Hb. destruct (H' b) as (b'&->%list_elem_of_singleton&?); auto.
exists a. by rewrite list_elem_of_singleton -discrete_iff_0.
Qed. | Instance | to_agree_discrete | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"Discrete",
"discrete_iff_0",
"to_agree"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_op_inv x y : ✓ (x ⋅ y) → x ≡ y. | Proof.
intros ?. apply equiv_dist=> n. by apply agree_op_invN, cmra_valid_validN.
Qed. | Lemma | agree_op_inv | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree_op_invN",
"cmra_valid_validN",
"equiv_dist"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
to_agree_injN n : Inj (dist n) (dist n) (to_agree). | Proof.
move=> a b [_] /=. setoid_rewrite list_elem_of_singleton. naive_solver.
Qed. | Instance | to_agree_injN | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"to_agree"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
to_agree_inj : Inj (≡) (≡) (to_agree). | Proof. intros a b ?. apply equiv_dist=>n. by apply (inj to_agree), equiv_dist. Qed. | Instance | to_agree_inj | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"equiv_dist",
"to_agree"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
to_agree_uninjN n x : ✓{n} x → ∃ a, to_agree a ≡{n}≡ x. | Proof.
rewrite agree_validN_def=> Hv.
destruct (elem_of_agree x) as [a ?].
exists a; split=> b /=; setoid_rewrite list_elem_of_singleton; naive_solver.
Qed. | Lemma | to_agree_uninjN | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree_validN_def",
"elem_of_agree",
"to_agree"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
to_agree_uninj x : ✓ x → ∃ a, to_agree a ≡ x. | Proof.
rewrite /valid /agree_valid_instance; setoid_rewrite agree_validN_def.
destruct (elem_of_agree x) as [a ?].
exists a; split=> b /=; setoid_rewrite list_elem_of_singleton; naive_solver.
Qed. | Lemma | to_agree_uninj | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree_validN_def",
"agree_valid_instance",
"elem_of_agree",
"to_agree"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_valid_includedN n x y : ✓{n} y → x ≼{n} y → x ≡{n}≡ y. | Proof.
move=> Hval [z Hy]; move: Hval; rewrite Hy.
by move=> /agree_op_invN->; rewrite agree_idemp.
Qed. | Lemma | agree_valid_includedN | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree_idemp",
"agree_op_invN"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_valid_included x y : ✓ y → x ≼ y → x ≡ y. | Proof.
move=> Hval [z Hy]; move: Hval; rewrite Hy.
by move=> /agree_op_inv->; rewrite agree_idemp.
Qed. | Lemma | agree_valid_included | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree_idemp",
"agree_op_inv"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
to_agree_includedN n a b : to_agree a ≼{n} to_agree b ↔ a ≡{n}≡ b. | Proof.
split; last by intros ->.
intros. by apply (inj to_agree), agree_valid_includedN.
Qed. | Lemma | to_agree_includedN | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree_valid_includedN",
"to_agree"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
to_agree_included a b : to_agree a ≼ to_agree b ↔ a ≡ b. | Proof.
split; last by intros ->.
intros. by apply (inj to_agree), agree_valid_included.
Qed. | Lemma | to_agree_included | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree_valid_included",
"to_agree"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
to_agree_included_L `{!LeibnizEquiv A} a b : to_agree a ≼ to_agree b ↔ a = b. | Proof. unfold_leibniz. apply to_agree_included. Qed. | Lemma | to_agree_included_L | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"to_agree",
"to_agree_included"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_cancelable x : Cancelable x. | Proof.
intros n y z Hv Heq.
destruct (to_agree_uninjN n x) as [x' EQx]; first by eapply cmra_validN_op_l.
destruct (to_agree_uninjN n y) as [y' EQy]; first by eapply cmra_validN_op_r.
destruct (to_agree_uninjN n z) as [z' EQz].
{ eapply (cmra_validN_op_r n x z). by rewrite -Heq. }
assert (Hx'y' : x' ≡{n}≡ y... | Instance | agree_cancelable | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"Cancelable",
"agree_op_invN",
"assert",
"cmra_validN_op_l",
"cmra_validN_op_r",
"to_agree",
"to_agree_uninjN"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
to_agree_op_invN a b n : ✓{n} (to_agree a ⋅ to_agree b) → a ≡{n}≡ b. | Proof. by intros ?%agree_op_invN%(inj to_agree). Qed. | Lemma | to_agree_op_invN | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree_op_invN",
"to_agree"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
to_agree_op_inv a b : ✓ (to_agree a ⋅ to_agree b) → a ≡ b. | Proof. by intros ?%agree_op_inv%(inj to_agree). Qed. | Lemma | to_agree_op_inv | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree_op_inv",
"to_agree"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
to_agree_op_inv_L `{!LeibnizEquiv A} a b : ✓ (to_agree a ⋅ to_agree b) → a = b. | Proof. by intros ?%to_agree_op_inv%leibniz_equiv. Qed. | Lemma | to_agree_op_inv_L | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"to_agree",
"to_agree_op_inv"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
to_agree_op_validN a b n : ✓{n} (to_agree a ⋅ to_agree b) ↔ a ≡{n}≡ b. | Proof.
split; first by apply to_agree_op_invN.
intros ->. rewrite agree_idemp //.
Qed. | Lemma | to_agree_op_validN | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree_idemp",
"to_agree",
"to_agree_op_invN"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
to_agree_op_valid a b : ✓ (to_agree a ⋅ to_agree b) ↔ a ≡ b. | Proof.
split; first by apply to_agree_op_inv.
intros ->. rewrite agree_idemp //.
Qed. | Lemma | to_agree_op_valid | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree_idemp",
"to_agree",
"to_agree_op_inv"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
to_agree_op_valid_L `{!LeibnizEquiv A} a b : ✓ (to_agree a ⋅ to_agree b) ↔ a = b. | Proof. rewrite to_agree_op_valid. by fold_leibniz. Qed. | Lemma | to_agree_op_valid_L | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"to_agree",
"to_agree_op_valid"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
: Params (@to_agree) 1 | := {}. | Instance | Params_to_agree | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"to_agree"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_map {A B} (f : A → B) (x : agree A) : agree B :=
{| agree_car := f <$> agree_car x |}. | Next Obligation. by intros A B f [[|??] ?]. Qed. | Definition | agree_map | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree",
"agree_car"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_map_id {A} (x : agree A) : agree_map id x = x. | Proof. apply agree_eq. by rewrite /= list_fmap_id. Qed. | Lemma | agree_map_id | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree",
"agree_eq",
"agree_map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_map_compose {A B C} (f : A → B) (g : B → C) (x : agree A) :
agree_map (g ∘ f) x = agree_map g (agree_map f x). | Proof. apply agree_eq. by rewrite /= list_fmap_compose. Qed. | Lemma | agree_map_compose | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree",
"agree_eq",
"agree_map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_map_to_agree {A B} (f : A → B) (x : A) :
agree_map f (to_agree x) = to_agree (f x). | Proof. by apply agree_eq. Qed. | Lemma | agree_map_to_agree | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree_eq",
"agree_map",
"to_agree"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_map_ne : NonExpansive (agree_map f). | Proof using Type*.
intros n x y [H H']; split=> b /=; setoid_rewrite list_elem_of_fmap.
- intros (a&->&?). destruct (H a) as (a'&?&?); auto. naive_solver.
- intros (a&->&?). destruct (H' a) as (a'&?&?); auto. naive_solver.
Qed. | Instance | agree_map_ne | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"NonExpansive",
"Type",
"agree_map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_map_proper : Proper ((≡) ==> (≡)) (agree_map f) | := ne_proper _. | Instance | agree_map_proper | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree_map",
"ne_proper"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_map_ext (g : A → B) x :
(∀ a, f a ≡ g a) → agree_map f x ≡ agree_map g x. | Proof using Hf.
intros Hfg n; split=> b /=; setoid_rewrite list_elem_of_fmap.
- intros (a&->&?). exists (g a). rewrite Hfg; eauto.
- intros (a&->&?). exists (f a). rewrite -Hfg; eauto.
Qed. | Lemma | agree_map_ext | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agree_map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agree_map_morphism : CmraMorphism (agree_map f). | Proof using Hf.
split; first apply _.
- intros n x. rewrite !agree_validN_def=> Hv b b' /=.
intros (a&->&?)%list_elem_of_fmap (a'&->&?)%list_elem_of_fmap.
apply Hf; eauto.
- done.
- intros x y n; split=> b /=;
rewrite !fmap_app; setoid_rewrite elem_of_app; eauto.
Qed. | Instance | agree_map_morphism | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"CmraMorphism",
"agree_map",
"agree_validN_def"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agreeO_map {SI : sidx} {A B : ofe}
(f : A -n> B) : agreeO A -n> agreeO B | :=
OfeMor (agree_map f : agreeO A → agreeO B). | Definition | agreeO_map | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agreeO",
"agree_map",
"ofe",
"sidx"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agreeO_map_ne {SI : sidx} A B : NonExpansive (@agreeO_map SI A B). | Proof.
intros n f g Hfg x; split=> b /=;
setoid_rewrite list_elem_of_fmap; naive_solver.
Qed. | Instance | agreeO_map_ne | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"NonExpansive",
"agreeO_map",
"sidx"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agreeRF {SI : sidx} (F : oFunctor) : rFunctor := {|
rFunctor_car A _ B _ := agreeR (oFunctor_car F A B);
rFunctor_map A1 _ A2 _ B1 _ B2 _ fg := agreeO_map (oFunctor_map F fg)
|}. | Next Obligation.
intros ? ? A1 ? A2 ? B1 ? B2 ? n ???; simpl.
by apply agreeO_map_ne, oFunctor_map_ne.
Qed. | Definition | agreeRF | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agreeO_map",
"agreeO_map_ne",
"agreeR",
"fg",
"oFunctor",
"rFunctor",
"sidx"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
agreeRF_contractive {SI : sidx} F :
oFunctorContractive F → rFunctorContractive (agreeRF F). | Proof.
intros ? A1 ? A2 ? B1 ? B2 ? n ???; simpl.
by apply agreeO_map_ne, oFunctor_map_contractive.
Qed. | Instance | agreeRF_contractive | iris.algebra | iris/algebra/agree.v | [
"iris.algebra",
"cmra",
"iris.prelude",
"options"
] | [
"agreeO_map_ne",
"agreeRF",
"oFunctorContractive",
"rFunctorContractive",
"sidx"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_view_rel_raw {SI : sidx} {A : ucmra} (n : SI) (a b : A) : Prop | :=
b ≼{n} a ∧ ✓{n} a. | Definition | auth_view_rel_raw | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"sidx",
"ucmra"
] | The authoritative camera is obtained by instantiating the view camera. | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 |
auth_view_rel_raw_mono {SI : sidx} (A : ucmra) n1 n2 (a1 a2 b1 b2 : A) :
auth_view_rel_raw n1 a1 b1 →
a1 ≡{n2}≡ a2 →
b2 ≼{n2} b1 →
(n2 ≤ n1)%sidx →
auth_view_rel_raw n2 a2 b2. | Proof.
intros [??] Ha12 ??. split.
- trans b1; [done|]. rewrite -Ha12. by apply cmra_includedN_le with n1.
- rewrite -Ha12. by apply cmra_validN_le with n1.
Qed. | Lemma | auth_view_rel_raw_mono | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"auth_view_rel_raw",
"cmra_includedN_le",
"cmra_validN_le",
"sidx",
"ucmra"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_view_rel_raw_valid {SI : sidx} (A : ucmra) n (a b : A) :
auth_view_rel_raw n a b → ✓{n} b. | Proof. intros [??]; eauto using cmra_validN_includedN. Qed. | Lemma | auth_view_rel_raw_valid | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"auth_view_rel_raw",
"cmra_validN_includedN",
"sidx",
"ucmra"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_view_rel_raw_unit {SI : sidx} (A : ucmra) n :
∃ a : A, auth_view_rel_raw n a ε. | Proof. exists ε. split; [done|]. apply ucmra_unit_validN. Qed. | Lemma | auth_view_rel_raw_unit | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"auth_view_rel_raw",
"sidx",
"ucmra",
"ucmra_unit_validN"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_view_rel {SI : sidx} {A : ucmra} : view_rel A A | :=
ViewRel auth_view_rel_raw (auth_view_rel_raw_mono A)
(auth_view_rel_raw_valid A) (auth_view_rel_raw_unit A). | Canonical | auth_view_rel | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"auth_view_rel_raw",
"auth_view_rel_raw_mono",
"auth_view_rel_raw_unit",
"auth_view_rel_raw_valid",
"sidx",
"ucmra",
"view_rel"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_view_rel_unit {SI : sidx} {A : ucmra} n (a : A) :
auth_view_rel n a ε ↔ ✓{n} a. | Proof. split; [by intros [??]|]. split; auto using ucmra_unit_leastN. Qed. | Lemma | auth_view_rel_unit | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"auth_view_rel",
"sidx",
"ucmra",
"ucmra_unit_leastN"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_view_rel_exists {SI : sidx} {A : ucmra} n (b : A) :
(∃ a, auth_view_rel n a b) ↔ ✓{n} b. | Proof.
split; [|intros; exists b; by split].
intros [a Hrel]. eapply auth_view_rel_raw_valid, Hrel.
Qed. | Lemma | auth_view_rel_exists | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"auth_view_rel",
"auth_view_rel_raw_valid",
"sidx",
"ucmra"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_view_rel_discrete {SI : sidx} {A : ucmra} :
CmraDiscrete A → ViewRelDiscrete (auth_view_rel (A:=A)). | Proof.
intros ? n a b [??]; split.
- by apply cmra_discrete_included_iff_0.
- by apply cmra_discrete_valid_iff_0.
Qed. | Instance | auth_view_rel_discrete | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"CmraDiscrete",
"ViewRelDiscrete",
"auth_view_rel",
"cmra_discrete_included_iff_0",
"cmra_discrete_valid_iff_0",
"sidx",
"ucmra"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth A | := (view (A:=A) (B:=A) auth_view_rel_raw). | Notation | auth | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"auth_view_rel_raw",
"view"
] | The type [auth] is not defined as a [Definition], but as a [Notation].
This way, one can use [auth A] with [A : Type] instead of [A : ucmra], and let
canonical structure search determine the corresponding camera instance. | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 |
authO {SI : sidx} (A : ucmra) : ofe | :=
viewO (A:=A) (B:=A) auth_view_rel. | Definition | authO | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"auth_view_rel",
"ofe",
"sidx",
"ucmra",
"viewO"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
authR {SI : sidx} (A : ucmra) : cmra | :=
viewR (A:=A) (B:=A) auth_view_rel. | Definition | authR | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"auth_view_rel",
"cmra",
"sidx",
"ucmra",
"viewR"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
authUR {SI : sidx} (A : ucmra) : ucmra | :=
viewUR (A:=A) (B:=A) auth_view_rel. | Definition | authUR | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"auth_view_rel",
"sidx",
"ucmra",
"viewUR"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_auth {SI : sidx} {A: ucmra} : dfrac → A → auth A | := view_auth. | Definition | auth_auth | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"auth",
"dfrac",
"sidx",
"ucmra",
"view_auth"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_frag {SI : sidx} {A: ucmra} : A → auth A | := view_frag. | Definition | auth_frag | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"auth",
"sidx",
"ucmra",
"view_frag"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
: Params (@auth_auth) 3 | := {}. | Instance | Params_auth_auth | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"auth_auth"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
: Params (@auth_frag) 2 | := {}. | Instance | Params_auth_frag | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"auth_frag"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
"● dq a" | := (auth_auth dq a)
(at level 20, dq custom dfrac at level 1, format "● dq a"). | Notation | ● dq a | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"auth_auth",
"dfrac"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
"◯ a" | := (auth_frag a) (at level 20). | Notation | ◯ a | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"auth_frag"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_auth_ne dq : NonExpansive (@auth_auth SI A dq). | Proof. rewrite /auth_auth. apply _. Qed. | Instance | auth_auth_ne | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"NonExpansive",
"auth_auth"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_auth_proper dq : Proper ((≡) ==> (≡)) (@auth_auth SI A dq). | Proof. rewrite /auth_auth. apply _. Qed. | Instance | auth_auth_proper | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"auth_auth"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_frag_ne : NonExpansive (@auth_frag SI A). | Proof. rewrite /auth_frag. apply _. Qed. | Instance | auth_frag_ne | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"NonExpansive",
"auth_frag"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_frag_proper : Proper ((≡) ==> (≡)) (@auth_frag SI A). | Proof. rewrite /auth_frag. apply _. Qed. | Instance | auth_frag_proper | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"auth_frag"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_auth_dist_inj n :
Inj2 (=) (dist n) (dist n) (@auth_auth SI A). | Proof. rewrite /auth_auth. apply _. Qed. | Instance | auth_auth_dist_inj | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"auth_auth"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_auth_inj : Inj2 (=) (≡) (≡) (@auth_auth SI A). | Proof. rewrite /auth_auth. apply _. Qed. | Instance | auth_auth_inj | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"auth_auth"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_frag_dist_inj n : Inj (dist n) (dist n) (@auth_frag SI A). | Proof. rewrite /auth_frag. apply _. Qed. | Instance | auth_frag_dist_inj | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"auth_frag"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_frag_inj : Inj (≡) (≡) (@auth_frag SI A). | Proof. rewrite /auth_frag. apply _. Qed. | Instance | auth_frag_inj | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"auth_frag"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_ofe_discrete : OfeDiscrete A → OfeDiscrete (authO A). | Proof. apply _. Qed. | Instance | auth_ofe_discrete | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"OfeDiscrete",
"authO"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_auth_discrete dq a :
Discrete a → Discrete (ε : A) → Discrete (●{dq} a). | Proof. rewrite /auth_auth. apply _. Qed. | Instance | auth_auth_discrete | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"Discrete",
"auth_auth"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_frag_discrete a : Discrete a → Discrete (◯ a). | Proof. rewrite /auth_frag. apply _. Qed. | Instance | auth_frag_discrete | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"Discrete",
"auth_frag"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_cmra_discrete : CmraDiscrete A → CmraDiscrete (authR A). | Proof. apply _. Qed. | Instance | auth_cmra_discrete | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"CmraDiscrete",
"authR"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_auth_dfrac_op dq1 dq2 a : ●{dq1 ⋅ dq2} a ≡ ●{dq1} a ⋅ ●{dq2} a. | Proof. apply view_auth_dfrac_op. Qed. | Lemma | auth_auth_dfrac_op | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"view_auth_dfrac_op"
] | Operation | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 |
auth_auth_dfrac_is_op dq dq1 dq2 a :
IsOp dq dq1 dq2 → IsOp' (●{dq} a) (●{dq1} a) (●{dq2} a). | Proof. rewrite /auth_auth. apply _. Qed. | Instance | auth_auth_dfrac_is_op | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"IsOp",
"IsOp'",
"auth_auth"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_frag_op a b : ◯ (a ⋅ b) = ◯ a ⋅ ◯ b. | Proof. apply view_frag_op. Qed. | Lemma | auth_frag_op | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"view_frag_op"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_frag_mono a b : a ≼ b → ◯ a ≼ ◯ b. | Proof. apply view_frag_mono. Qed. | Lemma | auth_frag_mono | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"view_frag_mono"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_frag_core a : core (◯ a) = ◯ (core a). | Proof. apply view_frag_core. Qed. | Lemma | auth_frag_core | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"core",
"view_frag_core"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
auth_both_core_discarded a b :
core (●□ a ⋅ ◯ b) ≡ ●□ a ⋅ ◯ (core b). | Proof. apply view_both_core_discarded. Qed. | Lemma | auth_both_core_discarded | iris.algebra | iris/algebra/auth.v | [
"iris.algebra",
"view",
"frac",
"proofmode_classes",
"big_op",
"iris.prelude",
"options"
] | [
"core",
"view_both_core_discarded"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 |
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