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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
End of preview. Expand in Data Studio

Coq-Iris

Structured dataset from Iris, a higher-order concurrent separation logic framework for Coq.

Source

Schema

Column Type Description
statement string Declaration signature/claim with the leading keyword removed (verbatim slice); the full declaration minus its proof
proof string Verbatim proof/body, empty if the declaration has none
type string Declaration keyword
symbolic_name string Declaration identifier
library string Sub-library
filename string Repository-relative source path
imports list[string] File-level Require/Import modules
deps list[string] Intra-corpus identifiers referenced
docstring string Preceding documentation comment, empty if absent
source_url string Upstream repository
commit string Upstream commit extracted

Statistics

  • Entries: 9,834
  • With proof: 9,773 (99.4%)
  • With docstring: 1,169 (11.9%)
  • Libraries: 18

By type

Type Count
Lemma 4,780
Instance 2,684
Definition 1,087
Notation 521
Class 204
Ltac 139
Canonical 121
Inductive 74
Record 59
Fixpoint 53
Hypothesis 30
Coercion 24
Ltac2 15
Structure 13
Theorem 10
Let 7
Corollary 6
Parameter 5
Hypotheses 1
Example 1

Example

agree (A : Type) : Type
:= {
  agree_car : list A;
  agree_not_nil : bool_decide (agree_car = []) = false
}.
  • type: Record | symbolic_name: agree | iris/algebra/agree.v

Use

Each declaration is split into a statement (signature/claim) and a proof (body) that are disjoint and together form the complete declaration, for proof modeling, autoformalization, retrieval, and dependency analysis via deps.

Citation

@misc{coq_iris_dataset,
  title  = {Coq-Iris},
  author = {Norton, Charles},
  year   = {2026},
  note   = {Extracted from https://gitlab.mpi-sws.org/iris/iris, commit 49cc21a1d9be},
  url    = {https://huggingface.co/datasets/phanerozoic/Coq-Iris}
}
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