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Commit e79e91f7 authored by Robbert Krebbers's avatar Robbert Krebbers
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Add documentation, add license, simplify build process, some reorganization, 

improve some definitions, simplify some proofs.
parent fb763df2
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theories/trs.v → theories/ars.v
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Require Export base.
Definition red `(R : relation A) (x : A) := ∃ y, R x y.
Definition nf `(R : relation A) (x : A) := ¬red R x.
(* The reflexive transitive closure *)
Inductive rtc `(R : relation A) : relation A :=
| rtc_refl x : rtc R x x
| rtc_l x y z : R x y → rtc R y z → rtc R x z.
(* A reduction of exactly n steps *)
Inductive nsteps `(R : relation A) : nat → relation A :=
| nsteps_O x : nsteps R 0 x x
| nsteps_l n x y z : R x y → nsteps R n y z → nsteps R (S n) x z.
(* A reduction whose length is bounded by n *)
Inductive bsteps `(R : relation A) : nat → relation A :=
| bsteps_refl n x : bsteps R n x x
| bsteps_l n x y z : R x y → bsteps R n y z → bsteps R (S n) x z.
(* The transitive closure *)
Inductive tc `(R : relation A) : relation A :=
| tc_once x y : R x y → tc R x y
| tc_l x y z : R x y → tc R y z → tc R x z.
Hint Constructors rtc nsteps bsteps tc : trs.
Arguments rtc_l {_ _ _ _ _} _ _.
Arguments nsteps_l {_ _ _ _ _ _} _ _.
Arguments bsteps_refl {_ _} _ _.
Arguments bsteps_l {_ _ _ _ _ _} _ _.
Arguments tc_once {_ _ _} _ _.
Arguments tc_l {_ _ _ _ _} _ _.
Ltac generalize_rtc H :=
match type of H with
| rtc ?R ?x ?y =>
let Hx := fresh in let Hy := fresh in
let Heqx := fresh in let Heqy := fresh in
remember x as (Hx,Heqx); remember y as (Hy,Heqy);
revert Heqx Heqy; repeat
match x with
| context [ ?z ] => revert z
end; repeat
match y with
| context [ ?z ] => revert z
end
end.
(* Copyright (c) 2012, Robbert Krebbers. *)
(* This file is distributed under the terms of the BSD license. *)
(** This file collects definitions and theorems on abstract rewriting systems.
These are particularly useful as we define the operational semantics as a
small step semantics. This file defines a hint database [ars] containing
some theorems on abstract rewriting systems. *)
Require Export tactics base.
(** * Definitions *)
Section definitions.
Context `(R : relation A).
(** An element is reducible if a step is possible. *)
Definition red (x : A) := ∃ y, R x y.
(** An element is in normal form if no further steps are possible. *)
Definition nf (x : A) := ¬red x.
(** The reflexive transitive closure. *)
Inductive rtc : relation A :=
| rtc_refl x : rtc x x
| rtc_l x y z : R x y → rtc y z → rtc x z.
(** Reductions of exactly [n] steps. *)
Inductive nsteps : nat → relation A :=
| nsteps_O x : nsteps 0 x x
| nsteps_l n x y z : R x y → nsteps n y z → nsteps (S n) x z.
(** Reduction of at most [n] steps. *)
Inductive bsteps : nat → relation A :=
| bsteps_refl n x : bsteps n x x
| bsteps_l n x y z : R x y → bsteps n y z → bsteps (S n) x z.
(** The transitive closure. *)
Inductive tc : relation A :=
| tc_once x y : R x y → tc x y
| tc_l x y z : R x y → tc y z → tc x z.
(** An element [x] is looping if all paths starting at [x] are infinite. *)
CoInductive looping : A → Prop :=
| looping_do_step x : red x → (∀ y, R x y → looping y) → looping x.
End definitions.
Hint Constructors rtc nsteps bsteps tc : ars.
(** * General theorems *)
Section rtc.
Context `{R : relation A}.
Global Instance: Reflexive (rtc R).
Proof rtc_refl R.
Global Instance rtc_trans: Transitive (rtc R).
Proof. red; induction 1; eauto with trs. Qed.
Lemma rtc_once {x y} : R x y → rtc R x y.
Proof. eauto with trs. Qed.
Proof. red; induction 1; eauto with ars. Qed.
Lemma rtc_once x y : R x y → rtc R x y.
Proof. eauto with ars. Qed.
Global Instance: subrelation R (rtc R).
Proof. exact @rtc_once. Qed.
Lemma rtc_r {x y z} : rtc R x y → R y z → rtc R x z.
Proof. intros. etransitivity; eauto with trs. Qed.
Lemma rtc_r x y z : rtc R x y → R y z → rtc R x z.
Proof. intros. etransitivity; eauto with ars. Qed.
Lemma rtc_inv {x z} : rtc R x z → x = z ∨ ∃ y, R x y ∧ rtc R y z.
Lemma rtc_inv x z : rtc R x z → x = z ∨ ∃ y, R x y ∧ rtc R y z.
Proof. inversion_clear 1; eauto. Qed.
Lemma rtc_ind_r (P : A → A → Prop)
Lemma rtc_ind_r (P : A → A → Prop)
(Prefl : ∀ x, P x x) (Pstep : ∀ x y z, rtc R x y → R y z → P x y → P x z) :
∀ y z, rtc R y z → P y z.
Proof.
......@@ -70,58 +70,76 @@ Section rtc.
induction 1; eauto using rtc_r.
Qed.
Lemma rtc_inv_r {x z} : rtc R x z → x = z ∨ ∃ y, rtc R x y ∧ R y z.
Lemma rtc_inv_r x z : rtc R x z → x = z ∨ ∃ y, rtc R x y ∧ R y z.
Proof. revert x z. apply rtc_ind_r; eauto. Qed.
Lemma nsteps_once {x y} : R x y → nsteps R 1 x y.
Proof. eauto with trs. Qed.
Lemma nsteps_trans {n m x y z} :
Lemma nsteps_once x y : R x y → nsteps R 1 x y.
Proof. eauto with ars. Qed.
Lemma nsteps_trans n m x y z :
nsteps R n x y → nsteps R m y z → nsteps R (n + m) x z.
Proof. induction 1; simpl; eauto with trs. Qed.
Lemma nsteps_r {n x y z} : nsteps R n x y → R y z → nsteps R (S n) x z.
Proof. induction 1; eauto with trs. Qed.
Lemma nsteps_rtc {n x y} : nsteps R n x y → rtc R x y.
Proof. induction 1; eauto with trs. Qed.
Lemma rtc_nsteps {x y} : rtc R x y → ∃ n, nsteps R n x y.
Proof. induction 1; firstorder eauto with trs. Qed.
Lemma bsteps_once {n x y} : R x y → bsteps R (S n) x y.
Proof. eauto with trs. Qed.
Lemma bsteps_plus_r {n m x y} :
Proof. induction 1; simpl; eauto with ars. Qed.
Lemma nsteps_r n x y z : nsteps R n x y → R y z → nsteps R (S n) x z.
Proof. induction 1; eauto with ars. Qed.
Lemma nsteps_rtc n x y : nsteps R n x y → rtc R x y.
Proof. induction 1; eauto with ars. Qed.
Lemma rtc_nsteps x y : rtc R x y → ∃ n, nsteps R n x y.
Proof. induction 1; firstorder eauto with ars. Qed.
Lemma bsteps_once n x y : R x y → bsteps R (S n) x y.
Proof. eauto with ars. Qed.
Lemma bsteps_plus_r n m x y :
bsteps R n x y → bsteps R (n + m) x y.
Proof. induction 1; simpl; eauto with trs. Qed.
Lemma bsteps_weaken {n m x y} :
Proof. induction 1; simpl; eauto with ars. Qed.
Lemma bsteps_weaken n m x y :
n ≤ m → bsteps R n x y → bsteps R m x y.
Proof.
intros. rewrite (Minus.le_plus_minus n m); auto using bsteps_plus_r.
Qed.
Lemma bsteps_plus_l {n m x y} :
Lemma bsteps_plus_l n m x y :
bsteps R n x y → bsteps R (m + n) x y.
Proof. apply bsteps_weaken. auto with arith. Qed.
Lemma bsteps_S {n x y} : bsteps R n x y → bsteps R (S n) x y.
Lemma bsteps_S n x y : bsteps R n x y → bsteps R (S n) x y.
Proof. apply bsteps_weaken. auto with arith. Qed.
Lemma bsteps_trans {n m x y z} :
Lemma bsteps_trans n m x y z :
bsteps R n x y → bsteps R m y z → bsteps R (n + m) x z.
Proof. induction 1; simpl; eauto using bsteps_plus_l with trs. Qed.
Lemma bsteps_r {n x y z} : bsteps R n x y → R y z → bsteps R (S n) x z.
Proof. induction 1; eauto with trs. Qed.
Lemma bsteps_rtc {n x y} : bsteps R n x y → rtc R x y.
Proof. induction 1; eauto with trs. Qed.
Lemma rtc_bsteps {x y} : rtc R x y → ∃ n, bsteps R n x y.
Proof. induction 1. exists 0. auto with trs. firstorder eauto with trs. Qed.
Proof. induction 1; simpl; eauto using bsteps_plus_l with ars. Qed.
Lemma bsteps_r n x y z : bsteps R n x y → R y z → bsteps R (S n) x z.
Proof. induction 1; eauto with ars. Qed.
Lemma bsteps_rtc n x y : bsteps R n x y → rtc R x y.
Proof. induction 1; eauto with ars. Qed.
Lemma rtc_bsteps x y : rtc R x y → ∃ n, bsteps R n x y.
Proof. induction 1. exists 0. auto with ars. firstorder eauto with ars. Qed.
Global Instance tc_trans: Transitive (tc R).
Proof. red; induction 1; eauto with trs. Qed.
Lemma tc_r {x y z} : tc R x y → R y z → tc R x z.
Proof. intros. etransitivity; eauto with trs. Qed.
Lemma tc_rtc {x y} : tc R x y → rtc R x y.
Proof. induction 1; eauto with trs. Qed.
Proof. red; induction 1; eauto with ars. Qed.
Lemma tc_r x y z : tc R x y → R y z → tc R x z.
Proof. intros. etransitivity; eauto with ars. Qed.
Lemma tc_rtc x y : tc R x y → rtc R x y.
Proof. induction 1; eauto with ars. Qed.
Global Instance: subrelation (tc R) (rtc R).
Proof. exact @tc_rtc. Qed.
Lemma looping_red x : looping R x → red R x.
Proof. destruct 1; auto. Qed.
Lemma looping_step x y : looping R x → R x y → looping R y.
Proof. destruct 1; auto. Qed.
Lemma looping_rtc x y : looping R x → rtc R x y → looping R y.
Proof. induction 2; eauto using looping_step. Qed.
Lemma looping_alt x :
looping R x ↔ ∀ y, rtc R x y → red R y.
Proof.
split.
* eauto using looping_red, looping_rtc.
* intros H. cut (∀ z, rtc R x z → looping R z).
{ eauto with ars. }
cofix FIX. constructor; eauto using rtc_r with ars.
Qed.
End rtc.
Hint Resolve rtc_once rtc_r tc_r : trs.
Hint Resolve rtc_once rtc_r tc_r : ars.
(** * Theorems on sub relations *)
Section subrel.
Context {A} (R1 R2 : relation A) (Hsub : subrelation R1 R2).
......
theories/base.v
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theories/collections.v
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Require Export base orders.
(* Copyright (c) 2012, Robbert Krebbers. *)
(* This file is distributed under the terms of the BSD license. *)
(** This file collects definitions and theorems on collections. Most
importantly, it implements some tactics to automatically solve goals involving
collections. *)
Require Export base tactics orders.
(** * Theorems *)
Section collection.
Context `{Collection A B}.
Lemma elem_of_empty_iff x : x ∈ ∅ ↔ False.
Proof. split. apply elem_of_empty. easy. Qed.
Lemma elem_of_empty x : x ∈ ∅ ↔ False.
Proof. split. apply not_elem_of_empty. easy. Qed.
Lemma elem_of_union_l x X Y : x ∈ X → x ∈ X ∪ Y.
Proof. intros. apply elem_of_union. auto. Qed.
Lemma elem_of_union_r x X Y : x ∈ Y → x ∈ X ∪ Y.
Proof. intros. apply elem_of_union. auto. Qed.
Lemma not_elem_of_singleton x y : x ∉ {[ y ]} ↔ x ≠ y.
Proof. now rewrite elem_of_singleton. Qed.
Lemma not_elem_of_union x X Y : x ∉ X ∪ Y ↔ x ∉ X ∧ x ∉ Y.
Proof. rewrite elem_of_union. tauto. Qed.
Global Instance collection_subseteq: SubsetEq B := λ X Y, ∀ x, x ∈ X → x ∈ Y.
Global Instance collection_subseteq: SubsetEq B := λ X Y,
∀ x, x ∈ X → x ∈ Y.
Global Instance: BoundedJoinSemiLattice B.
Proof. firstorder. Qed.
Global Instance: MeetSemiLattice B.
......@@ -25,17 +35,20 @@ Section collection.
X ≡ Y ↔ (∀ x, x ∈ X → x ∈ Y) ∧ (∀ x, x ∈ Y → x ∈ X).
Proof. firstorder. Qed.
Global Instance: Proper ((=) ==> (≡) ==> iff) (∈).
Global Instance singleton_proper : Proper ((=) ==> (≡)) singleton.
Proof. repeat intro. now subst. Qed.
Global Instance elem_of_proper: Proper ((=) ==> (≡) ==> iff) (∈).
Proof. intros ???. subst. firstorder. Qed.
Lemma empty_ne_singleton x : ∅ ≢ {[ x ]}.
Lemma empty_ne_singleton x : ∅ ≢ {[ x ]}.
Proof.
intros [_ E]. destruct (elem_of_empty x).
intros [_ E]. apply (elem_of_empty x).
apply E. now apply elem_of_singleton.
Qed.
Qed.
End collection.
Section cmap.
(** * Theorems about map *)
Section map.
Context `{Collection A C}.
Lemma elem_of_map_1 (f : A → A) (X : C) (x : A) :
......@@ -47,19 +60,18 @@ Section cmap.
Lemma elem_of_map_2 (f : A → A) (X : C) (x : A) :
x ∈ map f X → ∃ y, x = f y ∧ y ∈ X.
Proof. intros. now apply (elem_of_map _). Qed.
End cmap.
Definition fresh_sig `{FreshSpec A C} (X : C) : { x : A | x ∉ X } :=
exist (∉ X) (fresh X) (is_fresh X).
Lemma elem_of_fresh_iff `{FreshSpec A C} (X : C) : fresh X ∈ X ↔ False.
Proof. split. apply is_fresh. easy. Qed.
Ltac split_elem_ofs := repeat
End map.
(** * Tactics *)
(** The first pass consists of eliminating all occurrences of [(∪)], [(∩)],
[(∖)], [map], [∅], [{[_]}], [(≡)], and [(⊆)], by rewriting these into
logically equivalent propositions. For example we rewrite [A → x ∈ X ∪ ∅] into
[A → x ∈ X ∨ False]. *)
Ltac unfold_elem_of := repeat
match goal with
| H : context [ _ ⊆ _ ] |- _ => setoid_rewrite elem_of_subseteq in H
| H : context [ _ ≡ _ ] |- _ => setoid_rewrite elem_of_equiv_alt in H
| H : context [ _ ∈ ∅ ] |- _ => setoid_rewrite elem_of_empty_iff in H
| H : context [ _ ∈ ∅ ] |- _ => setoid_rewrite elem_of_empty in H
| H : context [ _ ∈ {[ _ ]} ] |- _ => setoid_rewrite elem_of_singleton in H
| H : context [ _ ∈ _ ∪ _ ] |- _ => setoid_rewrite elem_of_union in H
| H : context [ _ ∈ _ ∩ _ ] |- _ => setoid_rewrite elem_of_intersection in H
......@@ -67,7 +79,7 @@ Ltac split_elem_ofs := repeat
| H : context [ _ ∈ map _ _ ] |- _ => setoid_rewrite elem_of_map in H
| |- context [ _ ⊆ _ ] => setoid_rewrite elem_of_subseteq
| |- context [ _ ≡ _ ] => setoid_rewrite elem_of_equiv_alt
| |- context [ _ ∈ ∅ ] => setoid_rewrite elem_of_empty_iff
| |- context [ _ ∈ ∅ ] => setoid_rewrite elem_of_empty
| |- context [ _ ∈ {[ _ ]} ] => setoid_rewrite elem_of_singleton
| |- context [ _ ∈ _ ∪ _ ] => setoid_rewrite elem_of_union
| |- context [ _ ∈ _ ∩ _ ] => setoid_rewrite elem_of_intersection
......@@ -75,56 +87,49 @@ Ltac split_elem_ofs := repeat
| |- context [ _ ∈ map _ _ ] => setoid_rewrite elem_of_map
end.
Ltac destruct_elem_ofs := repeat
match goal with
| H : context [ @elem_of (_ * _) _ _ ?x _ ] |- _ => is_var x; destruct x
| H : context [ @elem_of (_ + _) _ _ ?x _] |- _ => is_var x; destruct x
end.
Tactic Notation "simplify_elem_of" tactic(t) :=
intros; (* due to bug #2790 *)
(** The tactic [solve_elem_of tac] composes the above tactic with [intuition].
For goals that do not involve [≡], [⊆], [map], or quantifiers this tactic is
generally powerful enough. This tactic either fails or proves the goal. *)
Tactic Notation "solve_elem_of" tactic(tac) :=
simpl in *;
split_elem_ofs;
destruct_elem_ofs;
intuition (simplify_eqs; t).
Tactic Notation "simplify_elem_of" := simplify_elem_of auto.
Ltac naive_firstorder t :=
match goal with
(* intros *)
| |- ∀ _, _ => intro; naive_firstorder t
(* destructs without information loss *)
| H : False |- _ => destruct H
| H : ?X, Hneg : ¬?X|- _ => now destruct Hneg
| H : _ ∧ _ |- _ => destruct H; naive_firstorder t
| H : ∃ _, _ |- _ => destruct H; naive_firstorder t
(* simplification *)
| |- _ => progress (simplify_eqs; simpl in *); naive_firstorder t
(* constructs *)
| |- _ ∧ _ => split; naive_firstorder t
(* solve *)
| |- _ => solve [t]
(* dirty destructs *)
| H : context [ ∃ _, _ ] |- _ =>
edestruct H; clear H;naive_firstorder t || clear H; naive_firstorder t
| H : context [ _ ∧ _ ] |- _ =>
destruct H; clear H; naive_firstorder t || clear H; naive_firstorder t
| H : context [ _ ∨ _ ] |- _ =>
edestruct H; clear H; naive_firstorder t || clear H; naive_firstorder t
(* dirty constructs *)
| |- ∃ x, _ => eexists; naive_firstorder t
| |- _ ∨ _ => left; naive_firstorder t || right; naive_firstorder t
| H : _ → False |- _ => destruct H; naive_firstorder t
end.
Tactic Notation "naive_firstorder" tactic(t) :=
unfold iff, not in *;
naive_firstorder t.
Tactic Notation "esimplify_elem_of" tactic(t) :=
(simplify_elem_of t);
try naive_firstorder t.
Tactic Notation "esimplify_elem_of" := esimplify_elem_of (eauto 5).
unfold_elem_of;
solve [intuition (simplify_equality; tac)].
Tactic Notation "solve_elem_of" := solve_elem_of auto.
(** For goals with quantifiers we could use the above tactic but with
[firstorder] instead of [intuition] as finishing tactic. However, [firstorder]
fails or loops on very small goals generated by [solve_elem_of] already. We
use the [naive_solver] tactic as a substitute. This tactic either fails or
proves the goal. *)
Tactic Notation "esolve_elem_of" tactic(tac) :=
simpl in *;
unfold_elem_of;
naive_solver tac.
Tactic Notation "esolve_elem_of" := esolve_elem_of eauto.
(** Given an assumption [H : _ ∈ _], the tactic [destruct_elem_of H] will
recursively split [H] for [(∪)], [(∩)], [(∖)], [map], [∅], [{[_]}]. *)
Tactic Notation "destruct_elem_of" hyp(H) :=
let rec go H :=
lazymatch type of H with
| _ ∈ ∅ => apply elem_of_empty in H; destruct H
| _ ∈ {[ ?l' ]} => apply elem_of_singleton in H; subst l'
| _ ∈ _ ∪ _ =>
let H1 := fresh in let H2 := fresh in apply elem_of_union in H;
destruct H as [H1|H2]; [go H1 | go H2]
| _ ∈ _ ∩ _ =>
let H1 := fresh in let H2 := fresh in apply elem_of_intersection in H;
destruct H as [H1 H2]; go H1; go H2
| _ ∈ _ ∖ _ =>
let H1 := fresh in let H2 := fresh in apply elem_of_difference in H;
destruct H as [H1 H2]; go H1; go H2
| _ ∈ map _ _ =>
let H1 := fresh in apply elem_of_map in H;
destruct H as [?[? H1]]; go H1
| _ => idtac
end in go H.
(** * Sets without duplicates up to an equivalence *)
Section no_dup.
Context `{Collection A B} (R : relation A) `{!Equivalence R}.
......@@ -143,33 +148,34 @@ Section no_dup.
Proof. firstorder. Qed.
Lemma elem_of_upto_elem_of x X : x ∈ X → elem_of_upto x X.
Proof. unfold elem_of_upto. esimplify_elem_of. Qed.
Proof. unfold elem_of_upto. esolve_elem_of. Qed.
Lemma elem_of_upto_empty x : ¬elem_of_upto x ∅.
Proof. unfold elem_of_upto. esimplify_elem_of. Qed.
Proof. unfold elem_of_upto. esolve_elem_of. Qed.
Lemma elem_of_upto_singleton x y : elem_of_upto x {[ y ]} ↔ R x y.
Proof. unfold elem_of_upto. esimplify_elem_of. Qed.
Proof. unfold elem_of_upto. esolve_elem_of. 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. esimplify_elem_of. Qed.
Proof. unfold elem_of_upto. esolve_elem_of. Qed.
Lemma not_elem_of_upto x X : ¬elem_of_upto x X → ∀ y, y ∈ X → ¬R x y.
Proof. unfold elem_of_upto. esimplify_elem_of. Qed.
Proof. unfold elem_of_upto. esolve_elem_of. Qed.
Lemma no_dup_empty: no_dup ∅.
Proof. unfold no_dup. simplify_elem_of. Qed.
Proof. unfold no_dup. solve_elem_of. Qed.
Lemma no_dup_add x X : ¬elem_of_upto x X → no_dup X → no_dup ({[ x ]} ∪ X).
Proof. unfold no_dup, elem_of_upto. esimplify_elem_of. Qed.
Proof. unfold no_dup, elem_of_upto. esolve_elem_of. Qed.
Lemma no_dup_inv_add x X : x ∉ X → no_dup ({[ x ]} ∪ X) → ¬elem_of_upto x X.
Proof.
intros Hin Hnodup [y [??]].
rewrite (Hnodup x y) in Hin; simplify_elem_of.
rewrite (Hnodup x y) in Hin; solve_elem_of.
Qed.
Lemma no_dup_inv_union_l X Y : no_dup (X ∪ Y) → no_dup X.
Proof. unfold no_dup. simplify_elem_of. Qed.
Proof. unfold no_dup. solve_elem_of. Qed.
Lemma no_dup_inv_union_r X Y : no_dup (X ∪ Y) → no_dup Y.
Proof. unfold no_dup. simplify_elem_of. Qed.
Proof. unfold no_dup. solve_elem_of. Qed.
End no_dup.
(** * Quantifiers *)
Section quantifiers.
Context `{Collection A B} (P : A → Prop).
......@@ -177,48 +183,65 @@ Section quantifiers.
Definition cexists X := ∃ x, x ∈ X ∧ P x.
Lemma cforall_empty : cforall ∅.
Proof. unfold cforall. simplify_elem_of. Qed.
Proof. unfold cforall. solve_elem_of. Qed.
Lemma cforall_singleton x : cforall {[ x ]} ↔ P x.
Proof. unfold cforall. simplify_elem_of. Qed.
Proof. unfold cforall. solve_elem_of. Qed.
Lemma cforall_union X Y : cforall X → cforall Y → cforall (X ∪ Y).
Proof. unfold cforall. simplify_elem_of. Qed.
Proof. unfold cforall. solve_elem_of. Qed.
Lemma cforall_union_inv_1 X Y : cforall (X ∪ Y) → cforall X.
Proof. unfold cforall. simplify_elem_of. Qed.
Proof. unfold cforall. solve_elem_of. Qed.
Lemma cforall_union_inv_2 X Y : cforall (X ∪ Y) → cforall Y.
Proof. unfold cforall. simplify_elem_of. Qed.
Proof. unfold cforall. solve_elem_of. Qed.
Lemma cexists_empty : ¬cexists ∅.
Proof. unfold cexists. esimplify_elem_of. Qed.
Proof. unfold cexists. esolve_elem_of. Qed.
Lemma cexists_singleton x : cexists {[ x ]} ↔ P x.
Proof. unfold cexists. esimplify_elem_of. Qed.
Proof. unfold cexists. esolve_elem_of. Qed.
Lemma cexists_union_1 X Y : cexists X → cexists (X ∪ Y).
Proof. unfold cexists. esimplify_elem_of. Qed.
Proof. unfold cexists. esolve_elem_of. Qed.
Lemma cexists_union_2 X Y : cexists Y → cexists (X ∪ Y).
Proof. unfold cexists. esimplify_elem_of. Qed.
Proof. unfold cexists. esolve_elem_of. Qed.
Lemma cexists_union_inv X Y : cexists (X ∪ Y) → cexists X ∨ cexists Y.
Proof. unfold cexists. esimplify_elem_of. Qed.
Proof. unfold cexists. esolve_elem_of. Qed.
End quantifiers.
Section more_quantifiers.
Context `{Collection A B}.
Lemma cforall_weak (P Q : A → Prop) (Hweak : ∀ x,