Commit 37e95231 by Robbert Krebbers

### Rename solve_elem_of into set_solver.

```It is doing much more than just dealing with ∈, it solves all kinds
of goals involving set operations (including ≡ and ⊆).```
parent 20690605
 ... ... @@ -207,7 +207,7 @@ Ltac decompose_empty := repeat occurrences of [(∪)], [(∩)], [(∖)], [(<\$>)], [∅], [{[_]}], [(≡)], and [(⊆)], by rewriting these into logically equivalent propositions. For example we rewrite [A → x ∈ X ∪ ∅] into [A → x ∈ X ∨ False]. *) Ltac unfold_elem_of := Ltac set_unfold := repeat_on_hyps (fun H => repeat match type of H with | context [ _ ⊆ _ ] => setoid_rewrite elem_of_subseteq in H ... ... @@ -251,21 +251,21 @@ Ltac unfold_elem_of := end. (** Since [firstorder] fails or loops on very small goals generated by [solve_elem_of] already. We use the [naive_solver] tactic as a substitute. [set_solver] already. We use the [naive_solver] tactic as a substitute. This tactic either fails or proves the goal. *) Tactic Notation "solve_elem_of" tactic3(tac) := Tactic Notation "set_solver" tactic3(tac) := setoid_subst; decompose_empty; unfold_elem_of; set_unfold; naive_solver tac. Tactic Notation "solve_elem_of" "-" hyp_list(Hs) "/" tactic3(tac) := clear Hs; solve_elem_of tac. Tactic Notation "solve_elem_of" "+" hyp_list(Hs) "/" tactic3(tac) := revert Hs; clear; solve_elem_of tac. Tactic Notation "solve_elem_of" := solve_elem_of eauto. Tactic Notation "solve_elem_of" "-" hyp_list(Hs) := clear Hs; solve_elem_of. Tactic Notation "solve_elem_of" "+" hyp_list(Hs) := revert Hs; clear; solve_elem_of. Tactic Notation "set_solver" "-" hyp_list(Hs) "/" tactic3(tac) := clear Hs; set_solver tac. Tactic Notation "set_solver" "+" hyp_list(Hs) "/" tactic3(tac) := revert Hs; clear; set_solver tac. Tactic Notation "set_solver" := set_solver eauto. Tactic Notation "set_solver" "-" hyp_list(Hs) := clear Hs; set_solver. Tactic Notation "set_solver" "+" hyp_list(Hs) := revert Hs; clear; set_solver. (** * More theorems *) Section collection. ... ... @@ -273,7 +273,7 @@ Section collection. Implicit Types X Y : C. Global Instance: Lattice C. Proof. split. apply _. firstorder auto. solve_elem_of. Qed. Proof. split. apply _. firstorder auto. set_solver. Qed. Global Instance difference_proper : Proper ((≡) ==> (≡) ==> (≡)) (@difference C _). Proof. ... ... @@ -281,23 +281,23 @@ Section collection. by rewrite !elem_of_difference, HX, HY. Qed. Lemma non_empty_inhabited x X : x ∈ X → X ≢ ∅. Proof. solve_elem_of. Qed. Proof. set_solver. Qed. Lemma intersection_singletons x : ({[x]} : C) ∩ {[x]} ≡ {[x]}. Proof. solve_elem_of. Qed. Proof. set_solver. Qed. Lemma difference_twice X Y : (X ∖ Y) ∖ Y ≡ X ∖ Y. Proof. solve_elem_of. Qed. Proof. set_solver. Qed. Lemma subseteq_empty_difference X Y : X ⊆ Y → X ∖ Y ≡ ∅. Proof. solve_elem_of. Qed. Proof. set_solver. Qed. Lemma difference_diag X : X ∖ X ≡ ∅. Proof. solve_elem_of. Qed. Proof. set_solver. Qed. Lemma difference_union_distr_l X Y Z : (X ∪ Y) ∖ Z ≡ X ∖ Z ∪ Y ∖ Z. Proof. solve_elem_of. Qed. Proof. set_solver. Qed. Lemma difference_union_distr_r X Y Z : Z ∖ (X ∪ Y) ≡ (Z ∖ X) ∩ (Z ∖ Y). Proof. solve_elem_of. Qed. Proof. set_solver. Qed. Lemma difference_intersection_distr_l X Y Z : (X ∩ Y) ∖ Z ≡ X ∖ Z ∩ Y ∖ Z. Proof. solve_elem_of. Qed. Proof. set_solver. Qed. Lemma disjoint_union_difference X Y : X ∩ Y ≡ ∅ → (X ∪ Y) ∖ X ≡ Y. Proof. solve_elem_of. Qed. Proof. set_solver. Qed. Section leibniz. Context `{!LeibnizEquiv C}. ... ... @@ -334,10 +334,10 @@ Section collection. Lemma non_empty_difference X Y : X ⊂ Y → Y ∖ X ≢ ∅. Proof. intros [HXY1 HXY2] Hdiff. destruct HXY2. intros x. destruct (decide (x ∈ X)); solve_elem_of. destruct (decide (x ∈ X)); set_solver. Qed. Lemma empty_difference_subseteq X Y : X ∖ Y ≡ ∅ → X ⊆ Y. Proof. intros ? x ?; apply dec_stable; solve_elem_of. Qed. Proof. intros ? x ?; apply dec_stable; set_solver. Qed. Context `{!LeibnizEquiv C}. Lemma union_difference_L X Y : X ⊆ Y → Y = X ∪ Y ∖ X. Proof. unfold_leibniz. apply union_difference. Qed. ... ... @@ -396,33 +396,33 @@ Section NoDup. Proof. firstorder. Qed. Lemma elem_of_upto_elem_of x X : x ∈ X → elem_of_upto x X. Proof. unfold elem_of_upto. solve_elem_of. Qed. Proof. unfold elem_of_upto. set_solver. Qed. Lemma elem_of_upto_empty x : ¬elem_of_upto x ∅. Proof. unfold elem_of_upto. solve_elem_of. Qed. Proof. unfold elem_of_upto. set_solver. Qed. Lemma elem_of_upto_singleton x y : elem_of_upto x {[ y ]} ↔ R x y. Proof. unfold elem_of_upto. solve_elem_of. Qed. Proof. unfold elem_of_upto. set_solver. Qed. Lemma elem_of_upto_union X Y x : elem_of_upto x (X ∪ Y) ↔ elem_of_upto x X ∨ elem_of_upto x Y. Proof. unfold elem_of_upto. solve_elem_of. Qed. Proof. unfold elem_of_upto. set_solver. Qed. Lemma not_elem_of_upto x X : ¬elem_of_upto x X → ∀ y, y ∈ X → ¬R x y. Proof. unfold elem_of_upto. solve_elem_of. Qed. Proof. unfold elem_of_upto. set_solver. Qed. Lemma set_NoDup_empty: set_NoDup ∅. Proof. unfold set_NoDup. solve_elem_of. Qed. Proof. unfold set_NoDup. set_solver. Qed. Lemma set_NoDup_add x X : ¬elem_of_upto x X → set_NoDup X → set_NoDup ({[ x ]} ∪ X). Proof. unfold set_NoDup, elem_of_upto. solve_elem_of. Qed. Proof. unfold set_NoDup, elem_of_upto. set_solver. Qed. Lemma set_NoDup_inv_add x X : x ∉ X → set_NoDup ({[ x ]} ∪ X) → ¬elem_of_upto x X. Proof. intros Hin Hnodup [y [??]]. rewrite (Hnodup x y) in Hin; solve_elem_of. rewrite (Hnodup x y) in Hin; set_solver. Qed. Lemma set_NoDup_inv_union_l X Y : set_NoDup (X ∪ Y) → set_NoDup X. Proof. unfold set_NoDup. solve_elem_of. Qed. Proof. unfold set_NoDup. set_solver. Qed. Lemma set_NoDup_inv_union_r X Y : set_NoDup (X ∪ Y) → set_NoDup Y. Proof. unfold set_NoDup. solve_elem_of. Qed. Proof. unfold set_NoDup. set_solver. Qed. End NoDup. (** * Quantifiers *) ... ... @@ -433,27 +433,27 @@ Section quantifiers. Definition set_Exists X := ∃ x, x ∈ X ∧ P x. Lemma set_Forall_empty : set_Forall ∅. Proof. unfold set_Forall. solve_elem_of. Qed. Proof. unfold set_Forall. set_solver. Qed. Lemma set_Forall_singleton x : set_Forall {[ x ]} ↔ P x. Proof. unfold set_Forall. solve_elem_of. Qed. Proof. unfold set_Forall. set_solver. Qed. Lemma set_Forall_union X Y : set_Forall X → set_Forall Y → set_Forall (X ∪ Y). Proof. unfold set_Forall. solve_elem_of. Qed. Proof. unfold set_Forall. set_solver. Qed. Lemma set_Forall_union_inv_1 X Y : set_Forall (X ∪ Y) → set_Forall X. Proof. unfold set_Forall. solve_elem_of. Qed. Proof. unfold set_Forall. set_solver. Qed. Lemma set_Forall_union_inv_2 X Y : set_Forall (X ∪ Y) → set_Forall Y. Proof. unfold set_Forall. solve_elem_of. Qed. Proof. unfold set_Forall. set_solver. Qed. Lemma set_Exists_empty : ¬set_Exists ∅. Proof. unfold set_Exists. solve_elem_of. Qed. Proof. unfold set_Exists. set_solver. Qed. Lemma set_Exists_singleton x : set_Exists {[ x ]} ↔ P x. Proof. unfold set_Exists. solve_elem_of. Qed. Proof. unfold set_Exists. set_solver. Qed. Lemma set_Exists_union_1 X Y : set_Exists X → set_Exists (X ∪ Y). Proof. unfold set_Exists. solve_elem_of. Qed. Proof. unfold set_Exists. set_solver. Qed. Lemma set_Exists_union_2 X Y : set_Exists Y → set_Exists (X ∪ Y). Proof. unfold set_Exists. solve_elem_of. Qed. Proof. unfold set_Exists. set_solver. Qed. Lemma set_Exists_union_inv X Y : set_Exists (X ∪ Y) → set_Exists X ∨ set_Exists Y. Proof. unfold set_Exists. solve_elem_of. Qed. Proof. unfold set_Exists. set_solver. Qed. End quantifiers. Section more_quantifiers. ... ... @@ -510,7 +510,7 @@ Section fresh. Qed. Lemma Forall_fresh_subseteq X Y xs : Forall_fresh X xs → Y ⊆ X → Forall_fresh Y xs. Proof. rewrite !Forall_fresh_alt; solve_elem_of. Qed. Proof. rewrite !Forall_fresh_alt; set_solver. Qed. Lemma fresh_list_length n X : length (fresh_list n X) = n. Proof. revert X. induction n; simpl; auto. Qed. ... ... @@ -518,12 +518,12 @@ Section fresh. Proof. revert X. induction n as [|n IH]; intros X; simpl;[by rewrite elem_of_nil|]. rewrite elem_of_cons; intros [->| Hin]; [apply is_fresh|]. apply IH in Hin; solve_elem_of. apply IH in Hin; set_solver. Qed. Lemma NoDup_fresh_list n X : NoDup (fresh_list n X). Proof. revert X. induction n; simpl; constructor; auto. intros Hin; apply fresh_list_is_fresh in Hin; solve_elem_of. intros Hin; apply fresh_list_is_fresh in Hin; set_solver. Qed. Lemma Forall_fresh_list X n : Forall_fresh X (fresh_list n X). Proof. ... ... @@ -537,50 +537,50 @@ Section collection_monad. Global Instance collection_fmap_mono {A B} : Proper (pointwise_relation _ (=) ==> (⊆) ==> (⊆)) (@fmap M _ A B). Proof. intros f g ? X Y ?; solve_elem_of. Qed. Proof. intros f g ? X Y ?; set_solver. Qed. Global Instance collection_fmap_proper {A B} : Proper (pointwise_relation _ (=) ==> (≡) ==> (≡)) (@fmap M _ A B). Proof. intros f g ? X Y [??]; split; solve_elem_of. Qed. Proof. intros f g ? X Y [??]; split; set_solver. Qed. Global Instance collection_bind_mono {A B} : Proper (((=) ==> (⊆)) ==> (⊆) ==> (⊆)) (@mbind M _ A B). Proof. unfold respectful; intros f g Hfg X Y ?; solve_elem_of. Qed. Proof. unfold respectful; intros f g Hfg X Y ?; set_solver. Qed. Global Instance collection_bind_proper {A B} : Proper (((=) ==> (≡)) ==> (≡) ==> (≡)) (@mbind M _ A B). Proof. unfold respectful; intros f g Hfg X Y [??]; split; solve_elem_of. Qed. Proof. unfold respectful; intros f g Hfg X Y [??]; split; set_solver. Qed. Global Instance collection_join_mono {A} : Proper ((⊆) ==> (⊆)) (@mjoin M _ A). Proof. intros X Y ?; solve_elem_of. Qed. Proof. intros X Y ?; set_solver. Qed. Global Instance collection_join_proper {A} : Proper ((≡) ==> (≡)) (@mjoin M _ A). Proof. intros X Y [??]; split; solve_elem_of. Qed. Proof. intros X Y [??]; split; set_solver. Qed. Lemma collection_bind_singleton {A B} (f : A → M B) x : {[ x ]} ≫= f ≡ f x. Proof. solve_elem_of. Qed. Proof. set_solver. Qed. Lemma collection_guard_True {A} `{Decision P} (X : M A) : P → guard P; X ≡ X. Proof. solve_elem_of. Qed. Proof. set_solver. Qed. Lemma collection_fmap_compose {A B C} (f : A → B) (g : B → C) (X : M A) : g ∘ f <\$> X ≡ g <\$> (f <\$> X). Proof. solve_elem_of. Qed. Proof. set_solver. Qed. Lemma elem_of_fmap_1 {A B} (f : A → B) (X : M A) (y : B) : y ∈ f <\$> X → ∃ x, y = f x ∧ x ∈ X. Proof. solve_elem_of. Qed. Proof. set_solver. Qed. Lemma elem_of_fmap_2 {A B} (f : A → B) (X : M A) (x : A) : x ∈ X → f x ∈ f <\$> X. Proof. solve_elem_of. Qed. Proof. set_solver. Qed. Lemma elem_of_fmap_2_alt {A B} (f : A → B) (X : M A) (x : A) (y : B) : x ∈ X → y = f x → y ∈ f <\$> X. Proof. solve_elem_of. Qed. Proof. set_solver. Qed. Lemma elem_of_mapM {A B} (f : A → M B) l k : l ∈ mapM f k ↔ Forall2 (λ x y, x ∈ f y) l k. Proof. split. - revert l. induction k; solve_elem_of. - induction 1; solve_elem_of. - revert l. induction k; set_solver. - induction 1; set_solver. Qed. Lemma collection_mapM_length {A B} (f : A → M B) l k : l ∈ mapM f k → length l = length k. Proof. revert l; induction k; solve_elem_of. Qed. Proof. revert l; induction k; set_solver. Qed. Lemma elem_of_mapM_fmap {A B} (f : A → B) (g : B → M A) l k : Forall (λ x, ∀ y, y ∈ g x → f y = x) l → k ∈ mapM g l → fmap f k = l. Proof. ... ... @@ -606,7 +606,7 @@ Section finite. Context `{SimpleCollection A B}. Global Instance set_finite_subseteq : Proper (flip (⊆) ==> impl) (@set_finite A B _). Proof. intros X Y HX [l Hl]; exists l; solve_elem_of. Qed. Proof. intros X Y HX [l Hl]; exists l; set_solver. Qed. Global Instance set_finite_proper : Proper ((≡) ==> iff) (@set_finite A B _). Proof. by intros X Y [??]; split; apply set_finite_subseteq. Qed. Lemma empty_finite : set_finite ∅. ... ... @@ -619,9 +619,9 @@ Section finite. rewrite elem_of_union, elem_of_app; naive_solver. Qed. Lemma union_finite_inv_l X Y : set_finite (X ∪ Y) → set_finite X. Proof. intros [l ?]; exists l; solve_elem_of. Qed. Proof. intros [l ?]; exists l; set_solver. Qed. Lemma union_finite_inv_r X Y : set_finite (X ∪ Y) → set_finite Y. Proof. intros [l ?]; exists l; solve_elem_of. Qed. Proof. intros [l ?]; exists l; set_solver. Qed. End finite. Section more_finite. ... ... @@ -636,6 +636,6 @@ Section more_finite. set_finite Y → set_finite (X ∖ Y) → set_finite X. Proof. intros [l ?] [k ?]; exists (l ++ k). intros x ?; destruct (decide (x ∈ Y)); rewrite elem_of_app; solve_elem_of. intros x ?; destruct (decide (x ∈ Y)); rewrite elem_of_app; set_solver. Qed. End more_finite.
 ... ... @@ -41,7 +41,7 @@ Qed. Lemma elements_singleton x : elements {[ x ]} = [x]. Proof. apply Permutation_singleton. by rewrite <-(right_id ∅ (∪) {[x]}), elements_union_singleton, elements_empty by solve_elem_of. elements_union_singleton, elements_empty by set_solver. Qed. Lemma elements_contains X Y : X ⊆ Y → elements X `contains` elements Y. Proof. ... ... @@ -90,7 +90,7 @@ Proof. intros E. destruct (size_pos_elem_of X); auto with lia. exists x. apply elem_of_equiv. split. - rewrite elem_of_singleton. eauto using size_singleton_inv. - solve_elem_of. - set_solver. Qed. Lemma size_union X Y : X ∩ Y ≡ ∅ → size (X ∪ Y) = size X + size Y. Proof. ... ... @@ -98,7 +98,7 @@ Proof. apply Permutation_length, NoDup_Permutation. - apply NoDup_elements. - apply NoDup_app; repeat split; try apply NoDup_elements. intros x; rewrite !elem_of_elements; solve_elem_of. intros x; rewrite !elem_of_elements; set_solver. - intros. by rewrite elem_of_app, !elem_of_elements, elem_of_union. Qed. Instance elem_of_dec_slow (x : A) (X : C) : Decision (x ∈ X) | 100. ... ... @@ -121,15 +121,15 @@ Next Obligation. Qed. Lemma size_union_alt X Y : size (X ∪ Y) = size X + size (Y ∖ X). Proof. rewrite <-size_union by solve_elem_of. setoid_replace (Y ∖ X) with ((Y ∪ X) ∖ X) by solve_elem_of. rewrite <-union_difference, (comm (∪)); solve_elem_of. rewrite <-size_union by set_solver. setoid_replace (Y ∖ X) with ((Y ∪ X) ∖ X) by set_solver. rewrite <-union_difference, (comm (∪)); set_solver. Qed. Lemma subseteq_size X Y : X ⊆ Y → size X ≤ size Y. Proof. intros. rewrite (union_difference X Y), size_union_alt by done. lia. Qed. Lemma subset_size X Y : X ⊂ Y → size X < size Y. Proof. intros. rewrite (union_difference X Y) by solve_elem_of. intros. rewrite (union_difference X Y) by set_solver. rewrite size_union_alt, difference_twice. cut (size (Y ∖ X) ≠ 0); [lia |]. by apply size_non_empty_iff, non_empty_difference. ... ... @@ -143,8 +143,8 @@ Proof. intros ? Hemp Hadd. apply well_founded_induction with (⊂). { apply collection_wf. } intros X IH. destruct (collection_choose_or_empty X) as [[x ?]|HX]. - rewrite (union_difference {[ x ]} X) by solve_elem_of. apply Hadd. solve_elem_of. apply IH; solve_elem_of. - rewrite (union_difference {[ x ]} X) by set_solver. apply Hadd. set_solver. apply IH; set_solver. - by rewrite HX. Qed. Lemma collection_fold_ind {B} (P : B → C → Prop) (f : A → B → B) (b : B) : ... ... @@ -158,10 +158,10 @@ Proof. symmetry. apply elem_of_elements. } induction 1 as [|x l ?? IH]; simpl. - intros X HX. setoid_rewrite elem_of_nil in HX. rewrite equiv_empty. done. solve_elem_of. rewrite equiv_empty. done. set_solver. - intros X HX. setoid_rewrite elem_of_cons in HX. rewrite (union_difference {[ x ]} X) by solve_elem_of. apply Hadd. solve_elem_of. apply IH. solve_elem_of. rewrite (union_difference {[ x ]} X) by set_solver. apply Hadd. set_solver. apply IH. set_solver. Qed. Lemma collection_fold_proper {B} (R : relation B) `{!Equivalence R} (f : A → B → B) (b : B) `{!Proper ((=) ==> R ==> R) f} ... ...
 ... ... @@ -36,13 +36,13 @@ Proof. Qed. Lemma dom_empty {A} : dom D (@empty (M A) _) ≡ ∅. Proof. split; intro; [|solve_elem_of]. split; intro; [|set_solver]. rewrite elem_of_dom, lookup_empty. by inversion 1. Qed. Lemma dom_empty_inv {A} (m : M A) : dom D m ≡ ∅ → m = ∅. Proof. intros E. apply map_empty. intros. apply not_elem_of_dom. rewrite E. solve_elem_of. rewrite E. set_solver. Qed. Lemma dom_alter {A} f (m : M A) i : dom D (alter f i m) ≡ dom D m. Proof. ... ... @@ -54,19 +54,19 @@ Lemma dom_insert {A} (m : M A) i x : dom D (<[i:=x]>m) ≡ {[ i ]} ∪ dom D m. Proof. apply elem_of_equiv. intros j. rewrite elem_of_union, !elem_of_dom. unfold is_Some. setoid_rewrite lookup_insert_Some. destruct (decide (i = j)); solve_elem_of. destruct (decide (i = j)); set_solver. Qed. Lemma dom_insert_subseteq {A} (m : M A) i x : dom D m ⊆ dom D (<[i:=x]>m). Proof. rewrite (dom_insert _). solve_elem_of. Qed. Proof. rewrite (dom_insert _). set_solver. Qed. Lemma dom_insert_subseteq_compat_l {A} (m : M A) i x X : X ⊆ dom D m → X ⊆ dom D (<[i:=x]>m). Proof. intros. transitivity (dom D m); eauto using dom_insert_subseteq. Qed. Lemma dom_singleton {A} (i : K) (x : A) : dom D {[i := x]} ≡ {[ i ]}. Proof. rewrite <-insert_empty, dom_insert, dom_empty; solve_elem_of. Qed. Proof. rewrite <-insert_empty, dom_insert, dom_empty; set_solver. Qed. Lemma dom_delete {A} (m : M A) i : dom D (delete i m) ≡ dom D m ∖ {[ i ]}. Proof. apply elem_of_equiv. intros j. rewrite elem_of_difference, !elem_of_dom. unfold is_Some. setoid_rewrite lookup_delete_Some. solve_elem_of. unfold is_Some. setoid_rewrite lookup_delete_Some. set_solver. Qed. Lemma delete_partial_alter_dom {A} (m : M A) i f : i ∉ dom D m → delete i (partial_alter f i m) = m. ... ...
 ... ... @@ -155,7 +155,7 @@ Proof. - revert x. induction l as [|y l IH]; intros x; simpl. { by rewrite elem_of_empty. } rewrite elem_of_union, elem_of_singleton. intros [->|]; [left|right]; eauto. - induction 1; solve_elem_of. - induction 1; set_solver. Qed. Lemma NoDup_remove_dups_fast l : NoDup (remove_dups_fast l). Proof. ... ...
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