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Commit 5eabf109 authored by Robbert Krebbers's avatar Robbert Krebbers
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Merge branch 'robbert/disj_union' into 'master' 

Rename multiset "union" into "disjoint union"

Closes #13

See merge request iris/stdpp!57
parents d1363b24 a7ee334e
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theories/base.v
+ 8
− 0
View file @ 5eabf109
...@@ -782,6 +782,14 @@ Definition union_list `{Empty A} `{Union A} : list A → A := fold_right (∪) ...@@ -782,6 +782,14 @@ Definition union_list `{Empty A} `{Union A} : list A → A := fold_right (∪)
Arguments union_list _ _ _ !_ / : assert. Arguments union_list _ _ _ !_ / : assert.
Notation "⋃ l" := (union_list l) (at level 20, format "⋃ l") : stdpp_scope. Notation "⋃ l" := (union_list l) (at level 20, format "⋃ l") : stdpp_scope.
Class DisjUnion A := disj_union: A → A → A.
Hint Mode DisjUnion ! : typeclass_instances.
Instance: Params (@disj_union) 2 := {}.
Infix "⊎" := disj_union (at level 50, left associativity) : stdpp_scope.
Notation "(⊎)" := disj_union (only parsing) : stdpp_scope.
Notation "( x ⊎)" := (disj_union x) (only parsing) : stdpp_scope.
Notation "(⊎ x )" := (λ y, disj_union y x) (only parsing) : stdpp_scope.
Class Intersection A := intersection: A → A → A. Class Intersection A := intersection: A → A → A.
Hint Mode Intersection ! : typeclass_instances. Hint Mode Intersection ! : typeclass_instances.
Instance: Params (@intersection) 2 := {}. Instance: Params (@intersection) 2 := {}.
......
theories/gmultiset.v
+ 148
− 29
View file @ 5eabf109
...@@ -36,6 +36,13 @@ Section definitions. ...@@ -36,6 +36,13 @@ Section definitions.
Global Instance gmultiset_singleton : Singleton A (gmultiset A) := λ x, Global Instance gmultiset_singleton : Singleton A (gmultiset A) := λ x,
GMultiSet {[ x := 0 ]}. GMultiSet {[ x := 0 ]}.
Global Instance gmultiset_union : Union (gmultiset A) := λ X Y, Global Instance gmultiset_union : Union (gmultiset A) := λ X Y,
let (X) := X in let (Y) := Y in
GMultiSet $ union_with (λ x y, Some (x `max` y)) X Y.
Global Instance gmultiset_intersection : Intersection (gmultiset A) := λ X Y,
let (X) := X in let (Y) := Y in
GMultiSet $ intersection_with (λ x y, Some (x `min` y)) X Y.
(** Often called the "sum" *)
Global Instance gmultiset_disj_union : DisjUnion (gmultiset A) := λ X Y,
let (X) := X in let (Y) := Y in let (X) := X in let (Y) := Y in
GMultiSet $ union_with (λ x y, Some (S (x + y))) X Y. GMultiSet $ union_with (λ x y, Some (S (x + y))) X Y.
Global Instance gmultiset_difference : Difference (gmultiset A) := λ X Y, Global Instance gmultiset_difference : Difference (gmultiset A) := λ X Y,
...@@ -77,7 +84,19 @@ Proof. unfold multiplicity; simpl. by rewrite lookup_singleton. Qed. ...@@ -77,7 +84,19 @@ Proof. unfold multiplicity; simpl. by rewrite lookup_singleton. Qed.
Lemma multiplicity_singleton_ne x y : x ≠ y → multiplicity x {[ y ]} = 0. Lemma multiplicity_singleton_ne x y : x ≠ y → multiplicity x {[ y ]} = 0.
Proof. intros. unfold multiplicity; simpl. by rewrite lookup_singleton_ne. Qed. Proof. intros. unfold multiplicity; simpl. by rewrite lookup_singleton_ne. Qed.
Lemma multiplicity_union X Y x : Lemma multiplicity_union X Y x :
multiplicity x (X ∪ Y) = multiplicity x X + multiplicity x Y. multiplicity x (X ∪ Y) = multiplicity x X `max` multiplicity x Y.
Proof.
destruct X as [X], Y as [Y]; unfold multiplicity; simpl.
rewrite lookup_union_with. destruct (X !! _), (Y !! _); simpl; lia.
Qed.
Lemma multiplicity_intersection X Y x :
multiplicity x (X ∩ Y) = multiplicity x X `min` multiplicity x Y.
Proof.
destruct X as [X], Y as [Y]; unfold multiplicity; simpl.
rewrite lookup_intersection_with. destruct (X !! _), (Y !! _); simpl; lia.
Qed.
Lemma multiplicity_disj_union X Y x :
multiplicity x (X ⊎ Y) = multiplicity x X + multiplicity x Y.
Proof. Proof.
destruct X as [X], Y as [Y]; unfold multiplicity; simpl. destruct X as [X], Y as [Y]; unfold multiplicity; simpl.
rewrite lookup_union_with. destruct (X !! _), (Y !! _); simpl; lia. rewrite lookup_union_with. destruct (X !! _), (Y !! _); simpl; lia.
...@@ -107,31 +126,116 @@ Qed. ...@@ -107,31 +126,116 @@ Qed.
Global Instance gmultiset_elem_of_dec : RelDecision (∈@{gmultiset A}). Global Instance gmultiset_elem_of_dec : RelDecision (∈@{gmultiset A}).
Proof. refine (λ x X, cast_if (decide (0 < multiplicity x X))); done. Defined. Proof. refine (λ x X, cast_if (decide (0 < multiplicity x X))); done. Defined.
Global Instance set_unfold_gmultiset_disj_union x X Y P Q :
SetUnfold (x ∈ X) P → SetUnfold (x ∈ Y) Q → SetUnfold (x ∈ X ⊎ Y) (P ∨ Q).
Proof.
intros ??; constructor.
rewrite <-(set_unfold (x ∈ X) P), <-(set_unfold (x ∈ Y) Q).
rewrite !elem_of_multiplicity, multiplicity_disj_union. lia.
Qed.
(* Algebraic laws *) (* Algebraic laws *)
Global Instance gmultiset_comm : Comm (=@{gmultiset A}) (∪). (** For union *)
Global Instance gmultiset_union_comm : Comm (=@{gmultiset A}) (∪).
Proof. Proof.
intros X Y. apply gmultiset_eq; intros x. rewrite !multiplicity_union; lia. intros X Y. apply gmultiset_eq; intros x. rewrite !multiplicity_union; lia.
Qed. Qed.
Global Instance gmultiset_assoc : Assoc (=@{gmultiset A}) (∪). Global Instance gmultiset_union_assoc : Assoc (=@{gmultiset A}) (∪).
Proof. Proof.
intros X Y Z. apply gmultiset_eq; intros x. rewrite !multiplicity_union; lia. intros X Y Z. apply gmultiset_eq; intros x. rewrite !multiplicity_union; lia.
Qed. Qed.
Global Instance gmultiset_left_id : LeftId (=@{gmultiset A}) ∅ (∪). Global Instance gmultiset_union_left_id : LeftId (=@{gmultiset A}) ∅ (∪).
Proof. Proof.
intros X. apply gmultiset_eq; intros x. intros X. apply gmultiset_eq; intros x.
by rewrite multiplicity_union, multiplicity_empty. by rewrite multiplicity_union, multiplicity_empty.
Qed. Qed.
Global Instance gmultiset_right_id : RightId (=@{gmultiset A}) ∅ (∪). Global Instance gmultiset_union_right_id : RightId (=@{gmultiset A}) ∅ (∪).
Proof. intros X. by rewrite (comm_L (∪)), (left_id_L _ _). Qed. Proof. intros X. by rewrite (comm_L (∪)), (left_id_L _ _). Qed.
Global Instance gmultiset_union_idemp : IdemP (=@{gmultiset A}) (∪).
Proof.
intros X. apply gmultiset_eq; intros x. rewrite !multiplicity_union; lia.
Qed.
(** For intersection *)
Global Instance gmultiset_intersection_comm : Comm (=@{gmultiset A}) (∩).
Proof.
intros X Y. apply gmultiset_eq; intros x. rewrite !multiplicity_intersection; lia.
Qed.
Global Instance gmultiset_intersection_assoc : Assoc (=@{gmultiset A}) (∩).
Proof.
intros X Y Z. apply gmultiset_eq; intros x. rewrite !multiplicity_intersection; lia.
Qed.
Global Instance gmultiset_intersection_left_absorb : LeftAbsorb (=@{gmultiset A}) ∅ (∩).
Proof.
intros X. apply gmultiset_eq; intros x.
by rewrite multiplicity_intersection, multiplicity_empty.
Qed.
Global Instance gmultiset_intersection_right_absorb : RightAbsorb (=@{gmultiset A}) ∅ (∩).
Proof. intros X. by rewrite (comm_L (∩)), (left_absorb_L _ _). Qed.
Global Instance gmultiset_intersection_idemp : IdemP (=@{gmultiset A}) (∩).
Proof.
intros X. apply gmultiset_eq; intros x. rewrite !multiplicity_intersection; lia.
Qed.
Global Instance gmultiset_union_inj_1 X : Inj (=) (=) (X ∪). Lemma gmultiset_union_intersection_l X Y Z : X ∪ (Y ∩ Z) = (X ∪ Y) ∩ (X ∪ Z).
Proof.
apply gmultiset_eq; intros y.
rewrite multiplicity_union, !multiplicity_intersection, !multiplicity_union. lia.
Qed.
Lemma gmultiset_union_intersection_r X Y Z : (X ∩ Y) ∪ Z = (X ∪ Z) ∩ (Y ∪ Z).
Proof. by rewrite <-!(comm_L _ Z), gmultiset_union_intersection_l. Qed.
Lemma gmultiset_intersection_union_l X Y Z : X ∩ (Y ∪ Z) = (X ∩ Y) ∪ (X ∩ Z).
Proof.
apply gmultiset_eq; intros y.
rewrite multiplicity_union, !multiplicity_intersection, !multiplicity_union. lia.
Qed.
Lemma gmultiset_intersection_union_r X Y Z : (X ∪ Y) ∩ Z = (X ∩ Z) ∪ (Y ∩ Z).
Proof. by rewrite <-!(comm_L _ Z), gmultiset_intersection_union_l. Qed.
(** For disjoint union (aka sum) *)
Global Instance gmultiset_disj_union_comm : Comm (=@{gmultiset A}) (⊎).
Proof.
intros X Y. apply gmultiset_eq; intros x. rewrite !multiplicity_disj_union; lia.
Qed.
Global Instance gmultiset_disj_union_assoc : Assoc (=@{gmultiset A}) (⊎).
Proof.
intros X Y Z. apply gmultiset_eq; intros x. rewrite !multiplicity_disj_union; lia.
Qed.
Global Instance gmultiset_disj_union_left_id : LeftId (=@{gmultiset A}) ∅ (⊎).
Proof.
intros X. apply gmultiset_eq; intros x.
by rewrite multiplicity_disj_union, multiplicity_empty.
Qed.
Global Instance gmultiset_disj_union_right_id : RightId (=@{gmultiset A}) ∅ (⊎).
Proof. intros X. by rewrite (comm_L (∪)), (left_id_L _ _). Qed.
Global Instance gmultiset_disj_union_inj_1 X : Inj (=) (=) (X ⊎).
Proof. Proof.
intros Y1 Y2. rewrite !gmultiset_eq. intros HX x; generalize (HX x). intros Y1 Y2. rewrite !gmultiset_eq. intros HX x; generalize (HX x).
rewrite !multiplicity_union. lia. rewrite !multiplicity_disj_union. lia.
Qed. Qed.
Global Instance gmultiset_union_inj_2 X : Inj (=) (=) (∪ X). Global Instance gmultiset_disj_union_inj_2 X : Inj (=) (=) (⊎ X).
Proof. intros Y1 Y2. rewrite <-!(comm_L _ X). apply (inj _). Qed. Proof. intros Y1 Y2. rewrite <-!(comm_L _ X). apply (inj _). Qed.
Lemma gmultiset_disj_union_intersection_l X Y Z : X ⊎ (Y ∩ Z) = (X ⊎ Y) ∩ (X ⊎ Z).
Proof.
apply gmultiset_eq; intros y.
rewrite multiplicity_disj_union, !multiplicity_intersection,
!multiplicity_disj_union. lia.
Qed.
Lemma gmultiset_disj_union_intersection_r X Y Z : (X ∩ Y) ⊎ Z = (X ⊎ Z) ∩ (Y ⊎ Z).
Proof. by rewrite <-!(comm_L _ Z), gmultiset_disj_union_intersection_l. Qed.
Lemma gmultiset_disj_union_union_l X Y Z : X ⊎ (Y ∪ Z) = (X ⊎ Y) ∪ (X ⊎ Z).
Proof.
apply gmultiset_eq; intros y.
rewrite multiplicity_disj_union, !multiplicity_union,
!multiplicity_disj_union. lia.
Qed.
Lemma gmultiset_disj_union_union_r X Y Z : (X ∪ Y) ⊎ Z = (X ⊎ Z) ∪ (Y ⊎ Z).
Proof. by rewrite <-!(comm_L _ Z), gmultiset_disj_union_union_l. Qed.
(** Misc *)
Lemma gmultiset_non_empty_singleton x : {[ x ]} ≠@{gmultiset A} ∅. Lemma gmultiset_non_empty_singleton x : {[ x ]} ≠@{gmultiset A} ∅.
Proof. Proof.
rewrite gmultiset_eq. intros Hx; generalize (Hx x). rewrite gmultiset_eq. intros Hx; generalize (Hx x).
...@@ -159,8 +263,8 @@ Lemma gmultiset_elements_singleton x : elements ({[ x ]} : gmultiset A) = [ x ]. ...@@ -159,8 +263,8 @@ Lemma gmultiset_elements_singleton x : elements ({[ x ]} : gmultiset A) = [ x ].
Proof. Proof.
unfold elements, gmultiset_elements; simpl. by rewrite map_to_list_singleton. unfold elements, gmultiset_elements; simpl. by rewrite map_to_list_singleton.
Qed. Qed.
Lemma gmultiset_elements_union X Y : Lemma gmultiset_elements_disj_union X Y :
elements (X ∪ Y) ≡ₚ elements X ++ elements Y. elements (X ⊎ Y) ≡ₚ elements X ++ elements Y.
Proof. Proof.
destruct X as [X], Y as [Y]; unfold elements, gmultiset_elements. destruct X as [X], Y as [Y]; unfold elements, gmultiset_elements.
set (f xn := let '(x, n) := xn in replicate (S n) x); simpl. set (f xn := let '(x, n) := xn in replicate (S n) x); simpl.
...@@ -230,10 +334,10 @@ Lemma gmultiset_size_singleton x : size ({[ x ]} : gmultiset A) = 1. ...@@ -230,10 +334,10 @@ Lemma gmultiset_size_singleton x : size ({[ x ]} : gmultiset A) = 1.
Proof. Proof.
unfold size, gmultiset_size; simpl. by rewrite gmultiset_elements_singleton. unfold size, gmultiset_size; simpl. by rewrite gmultiset_elements_singleton.
Qed. Qed.
Lemma gmultiset_size_union X Y : size (X ∪ Y) = size X + size Y. Lemma gmultiset_size_disj_union X Y : size (X ⊎ Y) = size X + size Y.
Proof. Proof.
unfold size, gmultiset_size; simpl. unfold size, gmultiset_size; simpl.
by rewrite gmultiset_elements_union, app_length. by rewrite gmultiset_elements_disj_union, app_length.
Qed. Qed.
(* Order stuff *) (* Order stuff *)
...@@ -271,22 +375,36 @@ Proof. intros x. rewrite multiplicity_union. lia. Qed. ...@@ -271,22 +375,36 @@ Proof. intros x. rewrite multiplicity_union. lia. Qed.
Lemma gmultiset_union_subseteq_r X Y : Y ⊆ X ∪ Y. Lemma gmultiset_union_subseteq_r X Y : Y ⊆ X ∪ Y.
Proof. intros x. rewrite multiplicity_union. lia. Qed. Proof. intros x. rewrite multiplicity_union. lia. Qed.
Lemma gmultiset_union_mono X1 X2 Y1 Y2 : X1 ⊆ X2 → Y1 ⊆ Y2 → X1 ∪ Y1 ⊆ X2 ∪ Y2. Lemma gmultiset_union_mono X1 X2 Y1 Y2 : X1 ⊆ X2 → Y1 ⊆ Y2 → X1 ∪ Y1 ⊆ X2 ∪ Y2.
Proof. intros ?? x. rewrite !multiplicity_union. by apply Nat.add_le_mono. Qed. Proof.
intros HX HY x. rewrite !multiplicity_union.
specialize (HX x); specialize (HY x); lia.
Qed.
Lemma gmultiset_union_mono_l X Y1 Y2 : Y1 ⊆ Y2 → X ∪ Y1 ⊆ X ∪ Y2. Lemma gmultiset_union_mono_l X Y1 Y2 : Y1 ⊆ Y2 → X ∪ Y1 ⊆ X ∪ Y2.
Proof. intros. by apply gmultiset_union_mono. Qed. Proof. intros. by apply gmultiset_union_mono. Qed.
Lemma gmultiset_union_mono_r X1 X2 Y : X1 ⊆ X2 → X1 ∪ Y ⊆ X2 ∪ Y. Lemma gmultiset_union_mono_r X1 X2 Y : X1 ⊆ X2 → X1 ∪ Y ⊆ X2 ∪ Y.
Proof. intros. by apply gmultiset_union_mono. Qed. Proof. intros. by apply gmultiset_union_mono. Qed.
Lemma gmultiset_disj_union_subseteq_l X Y : X ⊆ X ⊎ Y.
Proof. intros x. rewrite multiplicity_disj_union. lia. Qed.
Lemma gmultiset_disj_union_subseteq_r X Y : Y ⊆ X ⊎ Y.
Proof. intros x. rewrite multiplicity_disj_union. lia. Qed.
Lemma gmultiset_disj_union_mono X1 X2 Y1 Y2 : X1 ⊆ X2 → Y1 ⊆ Y2 → X1 ⊎ Y1 ⊆ X2 ⊎ Y2.
Proof. intros ?? x. rewrite !multiplicity_disj_union. by apply Nat.add_le_mono. Qed.
Lemma gmultiset_disj_union_mono_l X Y1 Y2 : Y1 ⊆ Y2 → X ⊎ Y1 ⊆ X ⊎ Y2.
Proof. intros. by apply gmultiset_disj_union_mono. Qed.
Lemma gmultiset_disj_union_mono_r X1 X2 Y : X1 ⊆ X2 → X1 ⊎ Y ⊆ X2 ⊎ Y.
Proof. intros. by apply gmultiset_disj_union_mono. Qed.
Lemma gmultiset_subset X Y : X ⊆ Y → size X < size Y → X ⊂ Y. Lemma gmultiset_subset X Y : X ⊆ Y → size X < size Y → X ⊂ Y.
Proof. intros. apply strict_spec_alt; split; naive_solver auto with lia. Qed. Proof. intros. apply strict_spec_alt; split; naive_solver auto with lia. Qed.
Lemma gmultiset_union_subset_l X Y : Y ≠ ∅ → X ⊂ X ∪ Y. Lemma gmultiset_disj_union_subset_l X Y : Y ≠ ∅ → X ⊂ X ⊎ Y.
Proof. Proof.
intros HY%gmultiset_size_non_empty_iff. intros HY%gmultiset_size_non_empty_iff.
apply gmultiset_subset; auto using gmultiset_union_subseteq_l. apply gmultiset_subset; auto using gmultiset_disj_union_subseteq_l.
rewrite gmultiset_size_union; lia. rewrite gmultiset_size_disj_union; lia.
Qed. Qed.
Lemma gmultiset_union_subset_r X Y : X ≠ ∅ → Y ⊂ X ∪ Y. Lemma gmultiset_union_subset_r X Y : X ≠ ∅ → Y ⊂ X ⊎ Y.
Proof. rewrite (comm_L (∪)). apply gmultiset_union_subset_l. Qed. Proof. rewrite (comm_L (⊎)). apply gmultiset_disj_union_subset_l. Qed.
Lemma gmultiset_elem_of_singleton_subseteq x X : x ∈ X ↔ {[ x ]} ⊆ X. Lemma gmultiset_elem_of_singleton_subseteq x X : x ∈ X ↔ {[ x ]} ⊆ X.
Proof. Proof.
...@@ -300,21 +418,21 @@ Qed. ...@@ -300,21 +418,21 @@ Qed.
Lemma gmultiset_elem_of_subseteq X1 X2 x : x ∈ X1 → X1 ⊆ X2 → x ∈ X2. Lemma gmultiset_elem_of_subseteq X1 X2 x : x ∈ X1 → X1 ⊆ X2 → x ∈ X2.
Proof. rewrite !gmultiset_elem_of_singleton_subseteq. by intros ->. Qed. Proof. rewrite !gmultiset_elem_of_singleton_subseteq. by intros ->. Qed.
Lemma gmultiset_union_difference X Y : X ⊆ Y → Y = X ∪ Y ∖ X. Lemma gmultiset_disj_union_difference X Y : X ⊆ Y → Y = X ⊎ Y ∖ X.
Proof. Proof.
intros HXY. apply gmultiset_eq; intros x; specialize (HXY x). intros HXY. apply gmultiset_eq; intros x; specialize (HXY x).
rewrite multiplicity_union, multiplicity_difference; lia. rewrite multiplicity_disj_union, multiplicity_difference; lia.
Qed. Qed.
Lemma gmultiset_union_difference' x Y : x ∈ Y → Y = {[ x ]} ∪ Y ∖ {[ x ]}. Lemma gmultiset_disj_union_difference' x Y : x ∈ Y → Y = {[ x ]} ⊎ Y ∖ {[ x ]}.
Proof. Proof.
intros. by apply gmultiset_union_difference, intros. by apply gmultiset_disj_union_difference,
gmultiset_elem_of_singleton_subseteq. gmultiset_elem_of_singleton_subseteq.
Qed. Qed.
Lemma gmultiset_size_difference X Y : Y ⊆ X → size (X ∖ Y) = size X - size Y. Lemma gmultiset_size_difference X Y : Y ⊆ X → size (X ∖ Y) = size X - size Y.
Proof. Proof.
intros HX%gmultiset_union_difference. intros HX%gmultiset_disj_union_difference.
rewrite HX at 2; rewrite gmultiset_size_union. lia. rewrite HX at 2; rewrite gmultiset_size_disj_union. lia.
Qed. Qed.
Lemma gmultiset_non_empty_difference X Y : X ⊂ Y → Y ∖ X ≠ ∅. Lemma gmultiset_non_empty_difference X Y : X ⊂ Y → Y ∖ X ≠ ∅.
...@@ -327,13 +445,13 @@ Qed. ...@@ -327,13 +445,13 @@ Qed.
Lemma gmultiset_difference_subset X Y : X ≠ ∅ → X ⊆ Y → Y ∖ X ⊂ Y. Lemma gmultiset_difference_subset X Y : X ≠ ∅ → X ⊆ Y → Y ∖ X ⊂ Y.
Proof. Proof.
intros. eapply strict_transitive_l; [by apply gmultiset_union_subset_r|]. intros. eapply strict_transitive_l; [by apply gmultiset_union_subset_r|].
by rewrite <-(gmultiset_union_difference X Y). by rewrite <-(gmultiset_disj_union_difference X Y).
Qed. Qed.
(* Mononicity *) (* Mononicity *)
Lemma gmultiset_elements_submseteq X Y : X ⊆ Y → elements X ⊆+ elements Y. Lemma gmultiset_elements_submseteq X Y : X ⊆ Y → elements X ⊆+ elements Y.
Proof. Proof.
intros ->%gmultiset_union_difference. rewrite gmultiset_elements_union. intros ->%gmultiset_disj_union_difference. rewrite gmultiset_elements_disj_union.
by apply submseteq_inserts_r. by apply submseteq_inserts_r.
Qed. Qed.
...@@ -344,7 +462,8 @@ Lemma gmultiset_subset_size X Y : X ⊂ Y → size X < size Y. ...@@ -344,7 +462,8 @@ Lemma gmultiset_subset_size X Y : X ⊂ Y → size X < size Y.
Proof. Proof.
intros HXY. assert (size (Y ∖ X) ≠ 0). intros HXY. assert (size (Y ∖ X) ≠ 0).
{ by apply gmultiset_size_non_empty_iff, gmultiset_non_empty_difference. } { by apply gmultiset_size_non_empty_iff, gmultiset_non_empty_difference. }
rewrite (gmultiset_union_difference X Y), gmultiset_size_union by auto. lia. rewrite (gmultiset_disj_union_difference X Y),
gmultiset_size_disj_union by auto. lia.
Qed. Qed.
(* Well-foundedness *) (* Well-foundedness *)
...@@ -354,11 +473,11 @@ Proof. ...@@ -354,11 +473,11 @@ Proof.
Qed. Qed.
Lemma gmultiset_ind (P : gmultiset A → Prop) : Lemma gmultiset_ind (P : gmultiset A → Prop) :
P ∅ → (∀ x X, P X → P ({[ x ]} ∪ X)) → ∀ X, P X. P ∅ → (∀ x X, P X → P ({[ x ]} ⊎ X)) → ∀ X, P X.
Proof. Proof.
intros Hemp Hinsert X. induction (gmultiset_wf X) as [X _ IH]. intros Hemp Hinsert X. induction (gmultiset_wf X) as [X _ IH].
destruct (gmultiset_choose_or_empty X) as [[x Hx]| ->]; auto. destruct (gmultiset_choose_or_empty X) as [[x Hx]| ->]; auto.
rewrite (gmultiset_union_difference' x X) by done. rewrite (gmultiset_disj_union_difference' x X) by done.
apply Hinsert, IH, gmultiset_difference_subset, apply Hinsert, IH, gmultiset_difference_subset,
gmultiset_elem_of_singleton_subseteq; auto using gmultiset_non_empty_singleton. gmultiset_elem_of_singleton_subseteq; auto using gmultiset_non_empty_singleton.
Qed. Qed.
......
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