Library iris.algebra.gset
From stdpp Require Export sets gmap mapset.
From iris.algebra Require Export cmra.
From iris.algebra Require Import updates local_updates big_op.
Section gset.
Context {SI : sidx} `{Countable K}.
Implicit Types X Y : gset K.
Canonical Structure gsetO := discreteO (gset K).
Local Instance gset_valid_instance : Valid (gset K) := λ _, True.
Local Instance gset_unit_instance : Unit (gset K) := (∅ : gset K).
Local Instance gset_op_instance : Op (gset K) := union.
Local Instance gset_pcore_instance : PCore (gset K) := λ X, Some X.
Lemma gset_op X Y : X ⋅ Y = X ∪ Y.
Proof. done. Qed.
Lemma gset_core X : core X = X.
Proof. done. Qed.
Lemma gset_included X Y : X ≼ Y ↔ X ⊆ Y.
Proof.
split.
- intros [Z ->]. rewrite gset_op. set_solver.
- intros (Z&->&?)%subseteq_disjoint_union_L. by ∃ Z.
Qed.
Lemma gset_ra_mixin : RAMixin (gset K).
Proof.
apply ra_total_mixin; apply _ || eauto; [].
intros X. by rewrite gset_core idemp_L.
Qed.
Canonical Structure gsetR := discreteR (gset K) gset_ra_mixin.
Global Instance gset_cmra_discrete : CmraDiscrete gsetR.
Proof. apply discrete_cmra_discrete. Qed.
Lemma gset_ucmra_mixin : UcmraMixin (gset K).
Proof. split; [ done | | done ]. intros X. by rewrite gset_op left_id_L. Qed.
Canonical Structure gsetUR := Ucmra (gset K) gset_ucmra_mixin.
Lemma gset_opM X mY : X ⋅? mY = X ∪ default ∅ mY.
Proof using SI. destruct mY; by rewrite /= ?right_id_L. Qed.
Lemma gset_update X Y : X ~~> Y.
Proof. done. Qed.
Lemma gset_local_update X Y X' : X ⊆ X' → (X,Y) ¬l~> (X',X').
Proof.
intros (Z&->&?)%subseteq_disjoint_union_L.
rewrite local_update_unital_discrete⇒ Z' _ /leibniz_equiv_iff→.
split; [done|]. rewrite gset_op. set_solver.
Qed.
Global Instance gset_core_id X : CoreId X.
Proof. by apply core_id_total; rewrite gset_core. Qed.
Lemma big_opS_singletons X :
([^op set] x ∈ X, {[ x ]}) = X.
Proof.
induction X as [|x X Hx IH] using set_ind_L.
- rewrite big_opS_empty. done.
- unfold_leibniz. rewrite big_opS_insert // IH //.
Qed.
From iris.algebra Require Export cmra.
From iris.algebra Require Import updates local_updates big_op.
Section gset.
Context {SI : sidx} `{Countable K}.
Implicit Types X Y : gset K.
Canonical Structure gsetO := discreteO (gset K).
Local Instance gset_valid_instance : Valid (gset K) := λ _, True.
Local Instance gset_unit_instance : Unit (gset K) := (∅ : gset K).
Local Instance gset_op_instance : Op (gset K) := union.
Local Instance gset_pcore_instance : PCore (gset K) := λ X, Some X.
Lemma gset_op X Y : X ⋅ Y = X ∪ Y.
Proof. done. Qed.
Lemma gset_core X : core X = X.
Proof. done. Qed.
Lemma gset_included X Y : X ≼ Y ↔ X ⊆ Y.
Proof.
split.
- intros [Z ->]. rewrite gset_op. set_solver.
- intros (Z&->&?)%subseteq_disjoint_union_L. by ∃ Z.
Qed.
Lemma gset_ra_mixin : RAMixin (gset K).
Proof.
apply ra_total_mixin; apply _ || eauto; [].
intros X. by rewrite gset_core idemp_L.
Qed.
Canonical Structure gsetR := discreteR (gset K) gset_ra_mixin.
Global Instance gset_cmra_discrete : CmraDiscrete gsetR.
Proof. apply discrete_cmra_discrete. Qed.
Lemma gset_ucmra_mixin : UcmraMixin (gset K).
Proof. split; [ done | | done ]. intros X. by rewrite gset_op left_id_L. Qed.
Canonical Structure gsetUR := Ucmra (gset K) gset_ucmra_mixin.
Lemma gset_opM X mY : X ⋅? mY = X ∪ default ∅ mY.
Proof using SI. destruct mY; by rewrite /= ?right_id_L. Qed.
Lemma gset_update X Y : X ~~> Y.
Proof. done. Qed.
Lemma gset_local_update X Y X' : X ⊆ X' → (X,Y) ¬l~> (X',X').
Proof.
intros (Z&->&?)%subseteq_disjoint_union_L.
rewrite local_update_unital_discrete⇒ Z' _ /leibniz_equiv_iff→.
split; [done|]. rewrite gset_op. set_solver.
Qed.
Global Instance gset_core_id X : CoreId X.
Proof. by apply core_id_total; rewrite gset_core. Qed.
Lemma big_opS_singletons X :
([^op set] x ∈ X, {[ x ]}) = X.
Proof.
induction X as [|x X Hx IH] using set_ind_L.
- rewrite big_opS_empty. done.
- unfold_leibniz. rewrite big_opS_insert // IH //.
Qed.
Add support X ≼ Y to set_solver. (We get support for ⋅ for free
because it is definitionally equal to ∪).
Global Instance set_unfold_gset_included X Y Q :
SetUnfold (X ⊆ Y) Q → SetUnfold (X ≼ Y) Q.
Proof. intros [?]; constructor. by rewrite gset_included. Qed.
End gset.
Global Arguments gsetO {_} _ {_ _}.
Global Arguments gsetR {_} _ {_ _}.
Global Arguments gsetUR {_} _ {_ _}.
Inductive gset_disj K `{Countable K} :=
| GSet : gset K → gset_disj K
| GSetInvalid : gset_disj K.
Global Arguments GSet {_ _ _} _.
Global Arguments GSetInvalid {_ _ _}.
Section gset_disj.
Context {SI : sidx} `{Countable K}.
Local Arguments op _ _ !_ !_ /.
Local Arguments cmra_op _ !_ !_ /.
Local Arguments ucmra_op _ !_ !_ /.
Global Instance GSet_inj : Inj (=@{gset K}) (=) GSet.
Proof. intros ???. naive_solver. Qed.
Canonical Structure gset_disjO := leibnizO (gset_disj K).
Local Instance gset_disj_valid_instance : Valid (gset_disj K) := λ X,
match X with GSet _ ⇒ True | GSetInvalid ⇒ False end.
Local Instance gset_disj_unit_instance : Unit (gset_disj K) := GSet ∅.
Local Instance gset_disj_op_instance : Op (gset_disj K) := λ X Y,
match X, Y with
| GSet X, GSet Y ⇒ if decide (X ## Y) then GSet (X ∪ Y) else GSetInvalid
| _, _ ⇒ GSetInvalid
end.
Local Instance gset_disj_pcore_instance : PCore (gset_disj K) := λ _, Some ε.
Ltac gset_disj_solve :=
repeat (simpl || case_decide);
first [apply (f_equal GSet)|done|exfalso]; set_solver by eauto.
Lemma gset_disj_included X Y : GSet X ≼ GSet Y ↔ X ⊆ Y.
Proof.
split.
- move⇒ [[Z|]]; simpl; try case_decide; set_solver.
- intros (Z&->&?)%subseteq_disjoint_union_L.
∃ (GSet Z). gset_disj_solve.
Qed.
Lemma gset_disj_valid_inv_l X Y : ✓ (GSet X ⋅ Y) → ∃ Y', Y = GSet Y' ∧ X ## Y'.
Proof. destruct Y; repeat (simpl || case_decide); by eauto. Qed.
Lemma gset_disj_union X Y : X ## Y → GSet X ⋅ GSet Y = GSet (X ∪ Y).
Proof. intros. by rewrite /= decide_True. Qed.
Lemma gset_disj_valid_op X Y : ✓ (GSet X ⋅ GSet Y) ↔ X ## Y.
Proof. simpl. case_decide; by split. Qed.
Lemma gset_disj_ra_mixin : RAMixin (gset_disj K).
Proof.
apply ra_total_mixin; eauto.
- intros [?|]; destruct 1; gset_disj_solve.
- by constructor.
- by destruct 1.
- intros [X1|] [X2|] [X3|]; gset_disj_solve.
- intros [X1|] [X2|]; gset_disj_solve.
- intros [X|]; gset_disj_solve.
- ∃ (GSet ∅); gset_disj_solve.
- intros [X1|] [X2|]; gset_disj_solve.
Qed.
Canonical Structure gset_disjR := discreteR (gset_disj K) gset_disj_ra_mixin.
Global Instance gset_disj_cmra_discrete : CmraDiscrete gset_disjR.
Proof. apply discrete_cmra_discrete. Qed.
Lemma gset_disj_ucmra_mixin : UcmraMixin (gset_disj K).
Proof. split; try apply _ || done. intros [X|]; gset_disj_solve. Qed.
Canonical Structure gset_disjUR := Ucmra (gset_disj K) gset_disj_ucmra_mixin.
Local Arguments op _ _ _ _ : simpl never.
Lemma gset_disj_alloc_updateP_strong P (Q : gset_disj K → Prop) X :
(∀ Y, X ⊆ Y → ∃ j, j ∉ Y ∧ P j) →
(∀ i, i ∉ X → P i → Q (GSet ({[i]} ∪ X))) →
GSet X ~~>: Q.
Proof.
intros Hfresh HQ.
apply cmra_discrete_total_updateP⇒ ? /gset_disj_valid_inv_l [Y [->?]].
destruct (Hfresh (X ∪ Y)) as (i&?&?); first set_solver.
∃ (GSet ({[ i ]} ∪ X)); split.
- apply HQ; set_solver by eauto.
- apply gset_disj_valid_op. set_solver by eauto.
Qed.
Lemma gset_disj_alloc_updateP_strong' P X :
(∀ Y, X ⊆ Y → ∃ j, j ∉ Y ∧ P j) →
GSet X ~~>: λ Y, ∃ i, Y = GSet ({[ i ]} ∪ X) ∧ i ∉ X ∧ P i.
Proof. eauto using gset_disj_alloc_updateP_strong. Qed.
Lemma gset_disj_alloc_empty_updateP_strong P (Q : gset_disj K → Prop) :
(∀ Y : gset K, ∃ j, j ∉ Y ∧ P j) →
(∀ i, P i → Q (GSet {[i]})) → GSet ∅ ~~>: Q.
Proof.
intros. apply (gset_disj_alloc_updateP_strong P); eauto.
intros i; rewrite right_id_L; auto.
Qed.
Lemma gset_disj_alloc_empty_updateP_strong' P :
(∀ Y : gset K, ∃ j, j ∉ Y ∧ P j) →
GSet ∅ ~~>: λ Y, ∃ i, Y = GSet {[ i ]} ∧ P i.
Proof. eauto using gset_disj_alloc_empty_updateP_strong. Qed.
Section fresh_updates.
Local Set Default Proof Using "Type*".
Context `{!Infinite K}.
Lemma gset_disj_alloc_updateP (Q : gset_disj K → Prop) X :
(∀ i, i ∉ X → Q (GSet ({[i]} ∪ X))) → GSet X ~~>: Q.
Proof.
intro; eapply gset_disj_alloc_updateP_strong with (λ _, True); eauto.
intros Y ?; ∃ (fresh Y). split; [|done]. apply is_fresh.
Qed.
Lemma gset_disj_alloc_updateP' X :
GSet X ~~>: λ Y, ∃ i, Y = GSet ({[ i ]} ∪ X) ∧ i ∉ X.
Proof. eauto using gset_disj_alloc_updateP. Qed.
Lemma gset_disj_alloc_empty_updateP (Q : gset_disj K → Prop) :
(∀ i, Q (GSet {[i]})) → GSet ∅ ~~>: Q.
Proof.
intro. apply gset_disj_alloc_updateP. intros i; rewrite right_id_L; auto.
Qed.
Lemma gset_disj_alloc_empty_updateP' : GSet ∅ ~~>: λ Y, ∃ i, Y = GSet {[ i ]}.
Proof. eauto using gset_disj_alloc_empty_updateP. Qed.
End fresh_updates.
Lemma gset_disj_dealloc_local_update X Y :
(GSet X, GSet Y) ¬l~> (GSet (X ∖ Y), GSet ∅).
Proof.
apply local_update_total_valid⇒ _ _ /gset_disj_included HYX.
rewrite local_update_unital_discrete⇒ -[Xf|] _ /leibniz_equiv_iff //=.
rewrite {1}/op /=. destruct (decide _) as [HXf|]; [intros[= ->]|done].
by rewrite difference_union_distr_l_L
difference_diag_L !left_id_L difference_disjoint_L.
Qed.
Lemma gset_disj_dealloc_empty_local_update X Z :
(GSet Z ⋅ GSet X, GSet Z) ¬l~> (GSet X, GSet ∅).
Proof.
apply local_update_total_valid⇒ /gset_disj_valid_op HZX _ _.
assert (X = (Z ∪ X) ∖ Z) as HX by set_solver.
rewrite gset_disj_union // {2}HX. apply gset_disj_dealloc_local_update.
Qed.
Lemma gset_disj_dealloc_op_local_update X Y Z :
(GSet Z ⋅ GSet X, GSet Z ⋅ GSet Y) ¬l~> (GSet X,GSet Y).
Proof.
rewrite -{2}(left_id ε _ (GSet Y)).
apply op_local_update_frame, gset_disj_dealloc_empty_local_update.
Qed.
Lemma gset_disj_alloc_op_local_update X Y Z :
Z ## X → (GSet X,GSet Y) ¬l~> (GSet Z ⋅ GSet X, GSet Z ⋅ GSet Y).
Proof.
intros. apply op_local_update_discrete. by rewrite gset_disj_valid_op.
Qed.
Lemma gset_disj_alloc_local_update X Y Z :
Z ## X → (GSet X,GSet Y) ¬l~> (GSet (Z ∪ X), GSet (Z ∪ Y)).
Proof.
intros. apply local_update_total_valid⇒ _ _ /gset_disj_included ?.
rewrite -!gset_disj_union //; last set_solver.
auto using gset_disj_alloc_op_local_update.
Qed.
Lemma gset_disj_alloc_empty_local_update X Z :
Z ## X → (GSet X, GSet ∅) ¬l~> (GSet (Z ∪ X), GSet Z).
Proof.
intros. rewrite -{2}(right_id_L _ union Z).
apply gset_disj_alloc_local_update; set_solver.
Qed.
SetUnfold (X ⊆ Y) Q → SetUnfold (X ≼ Y) Q.
Proof. intros [?]; constructor. by rewrite gset_included. Qed.
End gset.
Global Arguments gsetO {_} _ {_ _}.
Global Arguments gsetR {_} _ {_ _}.
Global Arguments gsetUR {_} _ {_ _}.
Inductive gset_disj K `{Countable K} :=
| GSet : gset K → gset_disj K
| GSetInvalid : gset_disj K.
Global Arguments GSet {_ _ _} _.
Global Arguments GSetInvalid {_ _ _}.
Section gset_disj.
Context {SI : sidx} `{Countable K}.
Local Arguments op _ _ !_ !_ /.
Local Arguments cmra_op _ !_ !_ /.
Local Arguments ucmra_op _ !_ !_ /.
Global Instance GSet_inj : Inj (=@{gset K}) (=) GSet.
Proof. intros ???. naive_solver. Qed.
Canonical Structure gset_disjO := leibnizO (gset_disj K).
Local Instance gset_disj_valid_instance : Valid (gset_disj K) := λ X,
match X with GSet _ ⇒ True | GSetInvalid ⇒ False end.
Local Instance gset_disj_unit_instance : Unit (gset_disj K) := GSet ∅.
Local Instance gset_disj_op_instance : Op (gset_disj K) := λ X Y,
match X, Y with
| GSet X, GSet Y ⇒ if decide (X ## Y) then GSet (X ∪ Y) else GSetInvalid
| _, _ ⇒ GSetInvalid
end.
Local Instance gset_disj_pcore_instance : PCore (gset_disj K) := λ _, Some ε.
Ltac gset_disj_solve :=
repeat (simpl || case_decide);
first [apply (f_equal GSet)|done|exfalso]; set_solver by eauto.
Lemma gset_disj_included X Y : GSet X ≼ GSet Y ↔ X ⊆ Y.
Proof.
split.
- move⇒ [[Z|]]; simpl; try case_decide; set_solver.
- intros (Z&->&?)%subseteq_disjoint_union_L.
∃ (GSet Z). gset_disj_solve.
Qed.
Lemma gset_disj_valid_inv_l X Y : ✓ (GSet X ⋅ Y) → ∃ Y', Y = GSet Y' ∧ X ## Y'.
Proof. destruct Y; repeat (simpl || case_decide); by eauto. Qed.
Lemma gset_disj_union X Y : X ## Y → GSet X ⋅ GSet Y = GSet (X ∪ Y).
Proof. intros. by rewrite /= decide_True. Qed.
Lemma gset_disj_valid_op X Y : ✓ (GSet X ⋅ GSet Y) ↔ X ## Y.
Proof. simpl. case_decide; by split. Qed.
Lemma gset_disj_ra_mixin : RAMixin (gset_disj K).
Proof.
apply ra_total_mixin; eauto.
- intros [?|]; destruct 1; gset_disj_solve.
- by constructor.
- by destruct 1.
- intros [X1|] [X2|] [X3|]; gset_disj_solve.
- intros [X1|] [X2|]; gset_disj_solve.
- intros [X|]; gset_disj_solve.
- ∃ (GSet ∅); gset_disj_solve.
- intros [X1|] [X2|]; gset_disj_solve.
Qed.
Canonical Structure gset_disjR := discreteR (gset_disj K) gset_disj_ra_mixin.
Global Instance gset_disj_cmra_discrete : CmraDiscrete gset_disjR.
Proof. apply discrete_cmra_discrete. Qed.
Lemma gset_disj_ucmra_mixin : UcmraMixin (gset_disj K).
Proof. split; try apply _ || done. intros [X|]; gset_disj_solve. Qed.
Canonical Structure gset_disjUR := Ucmra (gset_disj K) gset_disj_ucmra_mixin.
Local Arguments op _ _ _ _ : simpl never.
Lemma gset_disj_alloc_updateP_strong P (Q : gset_disj K → Prop) X :
(∀ Y, X ⊆ Y → ∃ j, j ∉ Y ∧ P j) →
(∀ i, i ∉ X → P i → Q (GSet ({[i]} ∪ X))) →
GSet X ~~>: Q.
Proof.
intros Hfresh HQ.
apply cmra_discrete_total_updateP⇒ ? /gset_disj_valid_inv_l [Y [->?]].
destruct (Hfresh (X ∪ Y)) as (i&?&?); first set_solver.
∃ (GSet ({[ i ]} ∪ X)); split.
- apply HQ; set_solver by eauto.
- apply gset_disj_valid_op. set_solver by eauto.
Qed.
Lemma gset_disj_alloc_updateP_strong' P X :
(∀ Y, X ⊆ Y → ∃ j, j ∉ Y ∧ P j) →
GSet X ~~>: λ Y, ∃ i, Y = GSet ({[ i ]} ∪ X) ∧ i ∉ X ∧ P i.
Proof. eauto using gset_disj_alloc_updateP_strong. Qed.
Lemma gset_disj_alloc_empty_updateP_strong P (Q : gset_disj K → Prop) :
(∀ Y : gset K, ∃ j, j ∉ Y ∧ P j) →
(∀ i, P i → Q (GSet {[i]})) → GSet ∅ ~~>: Q.
Proof.
intros. apply (gset_disj_alloc_updateP_strong P); eauto.
intros i; rewrite right_id_L; auto.
Qed.
Lemma gset_disj_alloc_empty_updateP_strong' P :
(∀ Y : gset K, ∃ j, j ∉ Y ∧ P j) →
GSet ∅ ~~>: λ Y, ∃ i, Y = GSet {[ i ]} ∧ P i.
Proof. eauto using gset_disj_alloc_empty_updateP_strong. Qed.
Section fresh_updates.
Local Set Default Proof Using "Type*".
Context `{!Infinite K}.
Lemma gset_disj_alloc_updateP (Q : gset_disj K → Prop) X :
(∀ i, i ∉ X → Q (GSet ({[i]} ∪ X))) → GSet X ~~>: Q.
Proof.
intro; eapply gset_disj_alloc_updateP_strong with (λ _, True); eauto.
intros Y ?; ∃ (fresh Y). split; [|done]. apply is_fresh.
Qed.
Lemma gset_disj_alloc_updateP' X :
GSet X ~~>: λ Y, ∃ i, Y = GSet ({[ i ]} ∪ X) ∧ i ∉ X.
Proof. eauto using gset_disj_alloc_updateP. Qed.
Lemma gset_disj_alloc_empty_updateP (Q : gset_disj K → Prop) :
(∀ i, Q (GSet {[i]})) → GSet ∅ ~~>: Q.
Proof.
intro. apply gset_disj_alloc_updateP. intros i; rewrite right_id_L; auto.
Qed.
Lemma gset_disj_alloc_empty_updateP' : GSet ∅ ~~>: λ Y, ∃ i, Y = GSet {[ i ]}.
Proof. eauto using gset_disj_alloc_empty_updateP. Qed.
End fresh_updates.
Lemma gset_disj_dealloc_local_update X Y :
(GSet X, GSet Y) ¬l~> (GSet (X ∖ Y), GSet ∅).
Proof.
apply local_update_total_valid⇒ _ _ /gset_disj_included HYX.
rewrite local_update_unital_discrete⇒ -[Xf|] _ /leibniz_equiv_iff //=.
rewrite {1}/op /=. destruct (decide _) as [HXf|]; [intros[= ->]|done].
by rewrite difference_union_distr_l_L
difference_diag_L !left_id_L difference_disjoint_L.
Qed.
Lemma gset_disj_dealloc_empty_local_update X Z :
(GSet Z ⋅ GSet X, GSet Z) ¬l~> (GSet X, GSet ∅).
Proof.
apply local_update_total_valid⇒ /gset_disj_valid_op HZX _ _.
assert (X = (Z ∪ X) ∖ Z) as HX by set_solver.
rewrite gset_disj_union // {2}HX. apply gset_disj_dealloc_local_update.
Qed.
Lemma gset_disj_dealloc_op_local_update X Y Z :
(GSet Z ⋅ GSet X, GSet Z ⋅ GSet Y) ¬l~> (GSet X,GSet Y).
Proof.
rewrite -{2}(left_id ε _ (GSet Y)).
apply op_local_update_frame, gset_disj_dealloc_empty_local_update.
Qed.
Lemma gset_disj_alloc_op_local_update X Y Z :
Z ## X → (GSet X,GSet Y) ¬l~> (GSet Z ⋅ GSet X, GSet Z ⋅ GSet Y).
Proof.
intros. apply op_local_update_discrete. by rewrite gset_disj_valid_op.
Qed.
Lemma gset_disj_alloc_local_update X Y Z :
Z ## X → (GSet X,GSet Y) ¬l~> (GSet (Z ∪ X), GSet (Z ∪ Y)).
Proof.
intros. apply local_update_total_valid⇒ _ _ /gset_disj_included ?.
rewrite -!gset_disj_union //; last set_solver.
auto using gset_disj_alloc_op_local_update.
Qed.
Lemma gset_disj_alloc_empty_local_update X Z :
Z ## X → (GSet X, GSet ∅) ¬l~> (GSet (Z ∪ X), GSet Z).
Proof.
intros. rewrite -{2}(right_id_L _ union Z).
apply gset_disj_alloc_local_update; set_solver.
Qed.
Add some basic support for GSet X = GSet Y, GSet X ≼ GSet Y, and
✓ (GSet X ⋅ GSet Y) to set_solver. There are probably more cases we could
cover (e.g., involving GSetInvalid, or nesting of ⋅), but it is not clear
these are useful in practice, nor how to handle them effectively.
Global Instance set_unfold_gset_eq (X Y : gset K) Q :
SetUnfold (X = Y) Q → SetUnfold (GSet X = GSet Y) Q.
Proof. intros [?]; constructor. by rewrite (inj_iff _). Qed.
Global Instance set_unfold_gset_disj_included (X Y : gset K) Q :
SetUnfold (X ⊆ Y) Q → SetUnfold (GSet X ≼ GSet Y) Q.
Proof. intros [?]; constructor. by rewrite gset_disj_included. Qed.
Global Instance set_unfold_gset_disj_valid_op (X Y : gset K) Q :
SetUnfold (X ## Y) Q → SetUnfold (✓ (GSet X ⋅ GSet Y)) Q.
Proof. intros [?]; constructor. by rewrite gset_disj_valid_op. Qed.
End gset_disj.
Global Arguments gset_disjO {_} _ {_ _}.
Global Arguments gset_disjR {_} _ {_ _}.
Global Arguments gset_disjUR {_} _ {_ _}.
SetUnfold (X = Y) Q → SetUnfold (GSet X = GSet Y) Q.
Proof. intros [?]; constructor. by rewrite (inj_iff _). Qed.
Global Instance set_unfold_gset_disj_included (X Y : gset K) Q :
SetUnfold (X ⊆ Y) Q → SetUnfold (GSet X ≼ GSet Y) Q.
Proof. intros [?]; constructor. by rewrite gset_disj_included. Qed.
Global Instance set_unfold_gset_disj_valid_op (X Y : gset K) Q :
SetUnfold (X ## Y) Q → SetUnfold (✓ (GSet X ⋅ GSet Y)) Q.
Proof. intros [?]; constructor. by rewrite gset_disj_valid_op. Qed.
End gset_disj.
Global Arguments gset_disjO {_} _ {_ _}.
Global Arguments gset_disjR {_} _ {_ _}.
Global Arguments gset_disjUR {_} _ {_ _}.