Commit d87f8b34 authored by Robbert Krebbers's avatar Robbert Krebbers

Reorganize prelude/base: group stuff that belongs together.

parent ef14edb5
......@@ -10,41 +10,19 @@ Global Set Asymmetric Patterns.
From Coq Require Export Morphisms RelationClasses List Bool Utf8 Program Setoid.
Obligation Tactic := idtac.
(** * General *)
(** Zipping lists. *)
Definition zip_with {A B C} (f : A B C) : list A list B list C :=
fix go l1 l2 :=
match l1, l2 with x1 :: l1, x2 :: l2 => f x1 x2 :: go l1 l2 | _ , _ => [] end.
Notation zip := (zip_with pair).
(** Ensure that [simpl] unfolds [id], [compose], and [flip] when fully
applied. *)
Arguments id _ _ /.
Arguments compose _ _ _ _ _ _ /.
Arguments flip _ _ _ _ _ _ /.
Arguments const _ _ _ _ /.
Typeclasses Transparent id compose flip const.
Instance: Params (@pair) 2.
(** Throughout this development we use [C_scope] for all general purpose
notations that do not belong to a more specific scope. *)
Delimit Scope C_scope with C.
Global Open Scope C_scope.
(** Change [True] and [False] into notations in order to enable overloading.
We will use this in the file [assertions] to give [True] and [False] a
different interpretation in [assert_scope] used for assertions of our axiomatic
semantics. *)
We will use this to give [True] and [False] a different interpretation for
embedded logics. *)
Notation "'True'" := True : type_scope.
Notation "'False'" := False : type_scope.
Notation curry := prod_curry.
Notation uncurry := prod_uncurry.
Definition curry3 {A B C D} (f : A B C D) (p : A * B * C) : D :=
let '(a,b,c) := p in f a b c.
Definition curry4 {A B C D E} (f : A B C D E) (p : A * B * C * D) : E :=
let '(a,b,c,d) := p in f a b c d.
(** Throughout this development we use [C_scope] for all general purpose
notations that do not belong to a more specific scope. *)
Delimit Scope C_scope with C.
Global Open Scope C_scope.
(** * Equality *)
(** Introduce some Haskell style like notations. *)
Notation "(=)" := eq (only parsing) : C_scope.
Notation "( x =)" := (eq x) (only parsing) : C_scope.
......@@ -56,6 +34,295 @@ Notation "(≠ x )" := (λ y, y ≠ x) (only parsing) : C_scope.
Hint Extern 0 (_ = _) => reflexivity.
Hint Extern 100 (_ _) => discriminate.
Instance: @PreOrder A (=).
Proof. split; repeat intro; congruence. Qed.
(** ** Setoid equality *)
(** We define an operational type class for setoid equality. This is based on
(Spitters/van der Weegen, 2011). *)
Class Equiv A := equiv: relation A.
Infix "≡" := equiv (at level 70, no associativity) : C_scope.
Notation "(≡)" := equiv (only parsing) : C_scope.
Notation "( X ≡)" := (equiv X) (only parsing) : C_scope.
Notation "(≡ X )" := (λ Y, Y X) (only parsing) : C_scope.
Notation "(≢)" := (λ X Y, ¬X Y) (only parsing) : C_scope.
Notation "X ≢ Y":= (¬X Y) (at level 70, no associativity) : C_scope.
Notation "( X ≢)" := (λ Y, X Y) (only parsing) : C_scope.
Notation "(≢ X )" := (λ Y, Y X) (only parsing) : C_scope.
(** The type class [LeibnizEquiv] collects setoid equalities that coincide
with Leibniz equality. We provide the tactic [fold_leibniz] to transform such
setoid equalities into Leibniz equalities, and [unfold_leibniz] for the
reverse. *)
Class LeibnizEquiv A `{Equiv A} := leibniz_equiv x y : x y x = y.
Lemma leibniz_equiv_iff `{LeibnizEquiv A, !Reflexive (@equiv A _)} (x y : A) :
x y x = y.
Proof. split. apply leibniz_equiv. intros ->; reflexivity. Qed.
Ltac fold_leibniz := repeat
match goal with
| H : context [ @equiv ?A _ _ _ ] |- _ =>
setoid_rewrite (leibniz_equiv_iff (A:=A)) in H
| |- context [ @equiv ?A _ _ _ ] =>
setoid_rewrite (leibniz_equiv_iff (A:=A))
end.
Ltac unfold_leibniz := repeat
match goal with
| H : context [ @eq ?A _ _ ] |- _ =>
setoid_rewrite <-(leibniz_equiv_iff (A:=A)) in H
| |- context [ @eq ?A _ _ ] =>
setoid_rewrite <-(leibniz_equiv_iff (A:=A))
end.
Definition equivL {A} : Equiv A := (=).
(** A [Params f n] instance forces the setoid rewriting mechanism not to
rewrite in the first [n] arguments of the function [f]. We will declare such
instances for all operational type classes in this development. *)
Instance: Params (@equiv) 2.
(** The following instance forces [setoid_replace] to use setoid equality
(for types that have an [Equiv] instance) rather than the standard Leibniz
equality. *)
Instance equiv_default_relation `{Equiv A} : DefaultRelation () | 3.
Hint Extern 0 (_ _) => reflexivity.
Hint Extern 0 (_ _) => symmetry; assumption.
(** * Type classes *)
(** ** Provable propositions *)
(** This type class collects provable propositions. It is useful to constraint
type classes by arbitrary propositions. *)
Class PropHolds (P : Prop) := prop_holds: P.
Hint Extern 0 (PropHolds _) => assumption : typeclass_instances.
Instance: Proper (iff ==> iff) PropHolds.
Proof. repeat intro; trivial. Qed.
Ltac solve_propholds :=
match goal with
| |- PropHolds (?P) => apply _
| |- ?P => change (PropHolds P); apply _
end.
(** ** Decidable propositions *)
(** This type class by (Spitters/van der Weegen, 2011) collects decidable
propositions. For example to declare a parameter expressing decidable equality
on a type [A] we write [`{∀ x y : A, Decision (x = y)}] and use it by writing
[decide (x = y)]. *)
Class Decision (P : Prop) := decide : {P} + {¬P}.
Arguments decide _ {_}.
(** ** Inhabited types *)
(** This type class collects types that are inhabited. *)
Class Inhabited (A : Type) : Type := populate { inhabitant : A }.
Arguments populate {_} _.
(** ** Proof irrelevant types *)
(** This type class collects types that are proof irrelevant. That means, all
elements of the type are equal. We use this notion only used for propositions,
but by universe polymorphism we can generalize it. *)
Class ProofIrrel (A : Type) : Prop := proof_irrel (x y : A) : x = y.
(** ** Common properties *)
(** These operational type classes allow us to refer to common mathematical
properties in a generic way. For example, for injectivity of [(k ++)] it
allows us to write [inj (k ++)] instead of [app_inv_head k]. *)
Class Inj {A B} (R : relation A) (S : relation B) (f : A B) : Prop :=
inj x y : S (f x) (f y) R x y.
Class Inj2 {A B C} (R1 : relation A) (R2 : relation B)
(S : relation C) (f : A B C) : Prop :=
inj2 x1 x2 y1 y2 : S (f x1 x2) (f y1 y2) R1 x1 y1 R2 x2 y2.
Class Cancel {A B} (S : relation B) (f : A B) (g : B A) : Prop :=
cancel : x, S (f (g x)) x.
Class Surj {A B} (R : relation B) (f : A B) :=
surj y : x, R (f x) y.
Class IdemP {A} (R : relation A) (f : A A A) : Prop :=
idemp x : R (f x x) x.
Class Comm {A B} (R : relation A) (f : B B A) : Prop :=
comm x y : R (f x y) (f y x).
Class LeftId {A} (R : relation A) (i : A) (f : A A A) : Prop :=
left_id x : R (f i x) x.
Class RightId {A} (R : relation A) (i : A) (f : A A A) : Prop :=
right_id x : R (f x i) x.
Class Assoc {A} (R : relation A) (f : A A A) : Prop :=
assoc x y z : R (f x (f y z)) (f (f x y) z).
Class LeftAbsorb {A} (R : relation A) (i : A) (f : A A A) : Prop :=
left_absorb x : R (f i x) i.
Class RightAbsorb {A} (R : relation A) (i : A) (f : A A A) : Prop :=
right_absorb x : R (f x i) i.
Class AntiSymm {A} (R S : relation A) : Prop :=
anti_symm x y : S x y S y x R x y.
Class Total {A} (R : relation A) := total x y : R x y R y x.
Class Trichotomy {A} (R : relation A) :=
trichotomy x y : R x y x = y R y x.
Class TrichotomyT {A} (R : relation A) :=
trichotomyT x y : {R x y} + {x = y} + {R y x}.
Arguments irreflexivity {_} _ {_} _ _.
Arguments inj {_ _ _ _} _ {_} _ _ _.
Arguments inj2 {_ _ _ _ _ _} _ {_} _ _ _ _ _.
Arguments cancel {_ _ _} _ _ {_} _.
Arguments surj {_ _ _} _ {_} _.
Arguments idemp {_ _} _ {_} _.
Arguments comm {_ _ _} _ {_} _ _.
Arguments left_id {_ _} _ _ {_} _.
Arguments right_id {_ _} _ _ {_} _.
Arguments assoc {_ _} _ {_} _ _ _.
Arguments left_absorb {_ _} _ _ {_} _.
Arguments right_absorb {_ _} _ _ {_} _.
Arguments anti_symm {_ _} _ {_} _ _ _ _.
Arguments total {_} _ {_} _ _.
Arguments trichotomy {_} _ {_} _ _.
Arguments trichotomyT {_} _ {_} _ _.
Instance left_id_propholds {A} (R : relation A) i f :
LeftId R i f x, PropHolds (R (f i x) x).
Proof. red. trivial. Qed.
Instance right_id_propholds {A} (R : relation A) i f :
RightId R i f x, PropHolds (R (f x i) x).
Proof. red. trivial. Qed.
Instance left_absorb_propholds {A} (R : relation A) i f :
LeftAbsorb R i f x, PropHolds (R (f i x) i).
Proof. red. trivial. Qed.
Instance right_absorb_propholds {A} (R : relation A) i f :
RightAbsorb R i f x, PropHolds (R (f x i) i).
Proof. red. trivial. Qed.
Instance idem_propholds {A} (R : relation A) f :
IdemP R f x, PropHolds (R (f x x) x).
Proof. red. trivial. Qed.
Lemma not_symmetry `{R : relation A, !Symmetric R} x y : ¬R x y ¬R y x.
Proof. intuition. Qed.
Lemma symmetry_iff `(R : relation A) `{!Symmetric R} x y : R x y R y x.
Proof. intuition. Qed.
Lemma not_inj `{Inj A B R R' f} x y : ¬R x y ¬R' (f x) (f y).
Proof. intuition. Qed.
Lemma not_inj2_1 `{Inj2 A B C R R' R'' f} x1 x2 y1 y2 :
¬R x1 x2 ¬R'' (f x1 y1) (f x2 y2).
Proof. intros HR HR''. destruct (inj2 f x1 y1 x2 y2); auto. Qed.
Lemma not_inj2_2 `{Inj2 A B C R R' R'' f} x1 x2 y1 y2 :
¬R' y1 y2 ¬R'' (f x1 y1) (f x2 y2).
Proof. intros HR' HR''. destruct (inj2 f x1 y1 x2 y2); auto. Qed.
Lemma inj_iff {A B} {R : relation A} {S : relation B} (f : A B)
`{!Inj R S f} `{!Proper (R ==> S) f} x y : S (f x) (f y) R x y.
Proof. firstorder. Qed.
Instance inj2_inj_1 `{Inj2 A B C R1 R2 R3 f} y : Inj R1 R3 (λ x, f x y).
Proof. repeat intro; edestruct (inj2 f); eauto. Qed.
Instance inj2_inj_2 `{Inj2 A B C R1 R2 R3 f} x : Inj R2 R3 (f x).
Proof. repeat intro; edestruct (inj2 f); eauto. Qed.
Lemma cancel_inj `{Cancel A B R1 f g, !Equivalence R1, !Proper (R2 ==> R1) f} :
Inj R1 R2 g.
Proof.
intros x y E. rewrite <-(cancel f g x), <-(cancel f g y), E. reflexivity.
Qed.
Lemma cancel_surj `{Cancel A B R1 f g} : Surj R1 f.
Proof. intros y. exists (g y). auto. Qed.
(** The following lemmas are specific versions of the projections of the above
type classes for Leibniz equality. These lemmas allow us to enforce Coq not to
use the setoid rewriting mechanism. *)
Lemma idemp_L {A} f `{!@IdemP A (=) f} x : f x x = x.
Proof. auto. Qed.
Lemma comm_L {A B} f `{!@Comm A B (=) f} x y : f x y = f y x.
Proof. auto. Qed.
Lemma left_id_L {A} i f `{!@LeftId A (=) i f} x : f i x = x.
Proof. auto. Qed.
Lemma right_id_L {A} i f `{!@RightId A (=) i f} x : f x i = x.
Proof. auto. Qed.
Lemma assoc_L {A} f `{!@Assoc A (=) f} x y z : f x (f y z) = f (f x y) z.
Proof. auto. Qed.
Lemma left_absorb_L {A} i f `{!@LeftAbsorb A (=) i f} x : f i x = i.
Proof. auto. Qed.
Lemma right_absorb_L {A} i f `{!@RightAbsorb A (=) i f} x : f x i = i.
Proof. auto. Qed.
(** ** Generic orders *)
(** The classes [PreOrder], [PartialOrder], and [TotalOrder] use an arbitrary
relation [R] instead of [⊆] to support multiple orders on the same type. *)
Definition strict {A} (R : relation A) : relation A := λ X Y, R X Y ¬R Y X.
Instance: Params (@strict) 2.
Class PartialOrder {A} (R : relation A) : Prop := {
partial_order_pre :> PreOrder R;
partial_order_anti_symm :> AntiSymm (=) R
}.
Class TotalOrder {A} (R : relation A) : Prop := {
total_order_partial :> PartialOrder R;
total_order_trichotomy :> Trichotomy (strict R)
}.
(** * Logic *)
Notation "(∧)" := and (only parsing) : C_scope.
Notation "( A ∧)" := (and A) (only parsing) : C_scope.
Notation "(∧ B )" := (λ A, A B) (only parsing) : C_scope.
Notation "(∨)" := or (only parsing) : C_scope.
Notation "( A ∨)" := (or A) (only parsing) : C_scope.
Notation "(∨ B )" := (λ A, A B) (only parsing) : C_scope.
Notation "(↔)" := iff (only parsing) : C_scope.
Notation "( A ↔)" := (iff A) (only parsing) : C_scope.
Notation "(↔ B )" := (λ A, A B) (only parsing) : C_scope.
Hint Extern 0 (_ _) => reflexivity.
Hint Extern 0 (_ _) => symmetry; assumption.
Lemma or_l P Q : ¬Q P Q P.
Proof. tauto. Qed.
Lemma or_r P Q : ¬P P Q Q.
Proof. tauto. Qed.
Lemma and_wlog_l (P Q : Prop) : (Q P) Q (P Q).
Proof. tauto. Qed.
Lemma and_wlog_r (P Q : Prop) : P (P Q) (P Q).
Proof. tauto. Qed.
Lemma impl_transitive (P Q R : Prop) : (P Q) (Q R) (P R).
Proof. tauto. Qed.
Instance: Comm () (@eq A).
Proof. red; intuition. Qed.
Instance: Comm () (λ x y, @eq A y x).
Proof. red; intuition. Qed.
Instance: Comm () ().
Proof. red; intuition. Qed.
Instance: Comm () ().
Proof. red; intuition. Qed.
Instance: Assoc () ().
Proof. red; intuition. Qed.
Instance: IdemP () ().
Proof. red; intuition. Qed.
Instance: Comm () ().
Proof. red; intuition. Qed.
Instance: Assoc () ().
Proof. red; intuition. Qed.
Instance: IdemP () ().
Proof. red; intuition. Qed.
Instance: LeftId () True ().
Proof. red; intuition. Qed.
Instance: RightId () True ().
Proof. red; intuition. Qed.
Instance: LeftAbsorb () False ().
Proof. red; intuition. Qed.
Instance: RightAbsorb () False ().
Proof. red; intuition. Qed.
Instance: LeftId () False ().
Proof. red; intuition. Qed.
Instance: RightId () False ().
Proof. red; intuition. Qed.
Instance: LeftAbsorb () True ().
Proof. red; intuition. Qed.
Instance: RightAbsorb () True ().
Proof. red; intuition. Qed.
Instance: LeftId () True impl.
Proof. unfold impl. red; intuition. Qed.
Instance: RightAbsorb () True impl.
Proof. unfold impl. red; intuition. Qed.
(** * Common data types *)
(** ** Functions *)
Notation "(→)" := (λ A B, A B) (only parsing) : C_scope.
Notation "( A →)" := (λ B, A B) (only parsing) : C_scope.
Notation "(→ B )" := (λ A, A B) (only parsing) : C_scope.
......@@ -70,141 +337,196 @@ Notation "(∘)" := compose (only parsing) : C_scope.
Notation "( f ∘)" := (compose f) (only parsing) : C_scope.
Notation "(∘ f )" := (λ g, compose g f) (only parsing) : C_scope.
Notation "(∧)" := and (only parsing) : C_scope.
Notation "( A ∧)" := (and A) (only parsing) : C_scope.
Notation "(∧ B )" := (λ A, A B) (only parsing) : C_scope.
(** Ensure that [simpl] unfolds [id], [compose], and [flip] when fully
applied. *)
Arguments id _ _ /.
Arguments compose _ _ _ _ _ _ /.
Arguments flip _ _ _ _ _ _ /.
Arguments const _ _ _ _ /.
Typeclasses Transparent id compose flip const.
Instance: Params (@pair) 2.
Definition fun_map {A A' B B'} (f: A' A) (g: B B') (h : A B) : A' B' :=
g h f.
Instance const_proper `{R1 : relation A, R2 : relation B} (x : B) :
Reflexive R2 Proper (R1 ==> R2) (λ _, x).
Proof. intros ? y1 y2; reflexivity. Qed.
Instance id_inj {A} : Inj (=) (=) (@id A).
Proof. intros ??; auto. Qed.
Instance compose_inj {A B C} R1 R2 R3 (f : A B) (g : B C) :
Inj R1 R2 f Inj R2 R3 g Inj R1 R3 (g f).
Proof. red; intuition. Qed.
Instance id_surj {A} : Surj (=) (@id A).
Proof. intros y; exists y; reflexivity. Qed.
Instance compose_surj {A B C} R (f : A B) (g : B C) :
Surj (=) f Surj R g Surj R (g f).
Proof.
intros ?? x. unfold compose. destruct (surj g x) as [y ?].
destruct (surj f y) as [z ?]. exists z. congruence.
Qed.
Instance id_comm {A B} (x : B) : Comm (=) (λ _ _ : A, x).
Proof. intros ?; reflexivity. Qed.
Instance id_assoc {A} (x : A) : Assoc (=) (λ _ _ : A, x).
Proof. intros ???; reflexivity. Qed.
Instance const1_assoc {A} : Assoc (=) (λ x _ : A, x).
Proof. intros ???; reflexivity. Qed.
Instance const2_assoc {A} : Assoc (=) (λ _ x : A, x).
Proof. intros ???; reflexivity. Qed.
Instance const1_idemp {A} : IdemP (=) (λ x _ : A, x).
Proof. intros ?; reflexivity. Qed.
Instance const2_idemp {A} : IdemP (=) (λ _ x : A, x).
Proof. intros ?; reflexivity. Qed.
(** ** Lists *)
Instance list_inhabited {A} : Inhabited (list A) := populate [].
Definition zip_with {A B C} (f : A B C) : list A list B list C :=
fix go l1 l2 :=
match l1, l2 with x1 :: l1, x2 :: l2 => f x1 x2 :: go l1 l2 | _ , _ => [] end.
Notation zip := (zip_with pair).
(** ** Booleans *)
(** The following coercion allows us to use Booleans as propositions. *)
Coercion Is_true : bool >-> Sortclass.
Hint Unfold Is_true.
Hint Immediate Is_true_eq_left.
Hint Resolve orb_prop_intro andb_prop_intro.
Notation "(&&)" := andb (only parsing).
Notation "(||)" := orb (only parsing).
Infix "&&*" := (zip_with (&&)) (at level 40).
Infix "||*" := (zip_with (||)) (at level 50).
Instance bool_inhabated : Inhabited bool := populate true.
Notation "(∨)" := or (only parsing) : C_scope.
Notation "( A ∨)" := (or A) (only parsing) : C_scope.
Notation "(∨ B )" := (λ A, A B) (only parsing) : C_scope.
Definition bool_le (β1 β2 : bool) : Prop := negb β1 || β2.
Infix "=.>" := bool_le (at level 70).
Infix "=.>*" := (Forall2 bool_le) (at level 70).
Instance: PartialOrder bool_le.
Proof. repeat split; repeat intros [|]; compute; tauto. Qed.
Notation "(↔)" := iff (only parsing) : C_scope.
Notation "( A ↔)" := (iff A) (only parsing) : C_scope.
Notation "(↔ B )" := (λ A, A B) (only parsing) : C_scope.
Lemma andb_True b1 b2 : b1 && b2 b1 b2.
Proof. destruct b1, b2; simpl; tauto. Qed.
Lemma orb_True b1 b2 : b1 || b2 b1 b2.
Proof. destruct b1, b2; simpl; tauto. Qed.
Lemma negb_True b : negb b ¬b.
Proof. destruct b; simpl; tauto. Qed.
Lemma Is_true_false (b : bool) : b = false ¬b.
Proof. now intros -> ?. Qed.
Hint Extern 0 (_ _) => reflexivity.
Hint Extern 0 (_ _) => symmetry; assumption.
(** ** Unit *)
Instance unit_equiv : Equiv unit := λ _ _, True.
Instance unit_equivalence : Equivalence (@equiv unit _).
Proof. repeat split. Qed.
Instance unit_inhabited: Inhabited unit := populate ().
(** ** Products *)
Notation "( x ,)" := (pair x) (only parsing) : C_scope.
Notation "(, y )" := (λ x, (x,y)) (only parsing) : C_scope.
Notation "p .1" := (fst p) (at level 10, format "p .1").
Notation "p .2" := (snd p) (at level 10, format "p .2").
Definition fun_map {A A' B B'} (f : A' A) (g : B B')
(h : A B) : A' B' := g h f.
Definition prod_map {A A' B B'} (f : A A') (g : B B')
(p : A * B) : A' * B' := (f (p.1), g (p.2)).
Notation curry := prod_curry.
Notation uncurry := prod_uncurry.
Definition curry3 {A B C D} (f : A B C D) (p : A * B * C) : D :=
let '(a,b,c) := p in f a b c.
Definition curry4 {A B C D E} (f : A B C D E) (p : A * B * C * D) : E :=
let '(a,b,c,d) := p in f a b c d.
Definition prod_map {A A' B B'} (f: A A') (g: B B') (p : A * B) : A' * B' :=
(f (p.1), g (p.2)).
Arguments prod_map {_ _ _ _} _ _ !_ /.
Definition prod_zip {A A' A'' B B' B''} (f : A A' A'') (g : B B' B'')
(p : A * B) (q : A' * B') : A'' * B'' := (f (p.1) (q.1), g (p.2) (q.2)).
Arguments prod_zip {_ _ _ _ _ _} _ _ !_ !_ /.
(** Set convenient implicit arguments for [existT] and introduce notations. *)
Arguments existT {_ _} _ _.
Arguments proj1_sig {_ _} _.
Notation "x ↾ p" := (exist _ x p) (at level 20) : C_scope.
Notation "` x" := (proj1_sig x) (at level 10, format "` x") : C_scope.
(** * Type classes *)
(** ** Provable propositions *)
(** This type class collects provable propositions. It is useful to constraint
type classes by arbitrary propositions. *)
Class PropHolds (P : Prop) := prop_holds: P.
Hint Extern 0 (PropHolds _) => assumption : typeclass_instances.
Instance: Proper (iff ==> iff) PropHolds.
Proof. repeat intro; trivial. Qed.
Instance prod_inhabited {A B} (iA : Inhabited A)
(iB : Inhabited B) : Inhabited (A * B) :=
match iA, iB with populate x, populate y => populate (x,y) end.
Ltac solve_propholds :=
match goal with
| |- PropHolds (?P) => apply _
| |- ?P => change (PropHolds P); apply _
end.
Instance pair_inj : Inj2 (=) (=) (=) (@pair A B).
Proof. injection 1; auto. Qed.
Instance prod_map_inj {A A' B B'} (f : A A') (g : B B') :
Inj (=) (=) f Inj (=) (=) g Inj (=) (=) (prod_map f g).
Proof.
intros ?? [??] [??] ?; simpl in *; f_equal;
[apply (inj f)|apply (inj g)]; congruence.
Qed.
(** ** Decidable propositions *)
(** This type class by (Spitters/van der Weegen, 2011) collects decidable
propositions. For example to declare a parameter expressing decidable equality
on a type [A] we write [`{∀ x y : A, Decision (x = y)}] and use it by writing
[decide (x = y)]. *)
Class Decision (P : Prop) := decide : {P} + {¬P}.
Arguments decide _ {_}.
Definition prod_relation {A B} (R1 : relation A) (R2 : relation B) :
relation (A * B) := λ x y, R1 (x.1) (y.1) R2 (x.2) (y.2).
Section prod_relation.
Context `{R1 : relation A, R2 : relation B}.
Global Instance prod_relation_refl :
Reflexive R1 Reflexive R2 Reflexive (prod_relation R1 R2).
Proof. firstorder eauto. Qed.
Global Instance prod_relation_sym :
Symmetric R1 Symmetric R2 Symmetric (prod_relation R1 R2).
Proof. firstorder eauto. Qed.
Global Instance prod_relation_trans :
Transitive R1 Transitive R2 Transitive (prod_relation R1 R2).
Proof. firstorder eauto. Qed.
Global Instance prod_relation_equiv :
Equivalence R1 Equivalence R2 Equivalence (prod_relation R1 R2).
Proof. split; apply _. Qed.
Global Instance pair_proper' : Proper (R1 ==> R2 ==> prod_relation R1 R2) pair.
Proof. firstorder eauto. Qed.
Global Instance fst_proper' : Proper (prod_relation R1 R2 ==> R1) fst.
Proof. firstorder eauto. Qed.
Global Instance snd_proper' : Proper (prod_relation R1 R2 ==> R2) snd.
Proof. firstorder eauto. Qed.
End prod_relation.
(** ** Inhabited types *)
(** This type class collects types that are inhabited. *)
Class Inhabited (A : Type) : Type := populate { inhabitant : A }.
Arguments populate {_} _.
Instance prod_equiv `{Equiv A,Equiv B} : Equiv (A * B) := prod_relation () ().
Instance pair_proper `{Equiv A, Equiv B} :
Proper (() ==> () ==> ()) (@pair A B) | 0 := _.
Instance fst_proper `{Equiv A, Equiv B} : Proper (() ==> ()) (@fst A B) := _.
Instance snd_proper `{Equiv A, Equiv B} : Proper (() ==> ()) (@snd A B) := _.
Typeclasses Opaque prod_equiv.
Instance unit_inhabited: Inhabited unit := populate ().
Instance bool_inhabated : Inhabited bool := populate true.
Instance list_inhabited {A} : Inhabited (list A) := populate [].
Instance prod_inhabited {A B} (iA : Inhabited A)
(iB : Inhabited B) : Inhabited (A * B) :=
match iA, iB with populate x, populate y => populate (x,y) end.
(** ** Sums *)
Instance sum_inhabited_l {A B} (iA : Inhabited A) : Inhabited (A + B) :=
match iA with populate x => populate (inl x) end.
Instance sum_inhabited_r {A B} (iB : Inhabited A) : Inhabited (A + B) :=
match iB with populate y => populate (inl y) end.
Instance option_inhabited {A} : Inhabited (option A) := populate None.
(** ** Proof irrelevant types *)
(** This type class collects types that are proof irrelevant. That means, all
elements of the type are equal. We use this notion only used for propositions,
but by universe polymorphism we can generalize it. *)
Class ProofIrrel (A : Type) : Prop := proof_irrel (x y : A) : x = y.
Instance inl_inj : Inj (=) (=) (@inl A B).
Proof. injection 1; auto. Qed.
Instance inr_inj : Inj (=) (=) (@inr A B).
Proof. injection 1; auto. Qed.
(** ** Setoid equality *)
(** We define an operational type class for setoid equality. This is based on
(Spitters/van der Weegen, 2011). *)
Class Equiv A := equiv: relation A.
Infix "≡" := equiv (at level 70, no associativity) : C_scope.
Notation "(≡)" := equiv (only parsing) : C_scope.
Notation "( X ≡)" := (equiv X) (only parsing) : C_scope.
Notation "(≡ X )" := (λ Y, Y X) (only parsing) : C_scope.
Notation "(≢)" := (λ X Y, ¬X Y) (only parsing) : C_scope.
Notation "X ≢ Y":= (¬X Y) (at level 70, no associativity) : C_scope.
Notation "( X ≢)" := (λ Y, X Y) (only parsing) : C_scope.
Notation "(≢ X )" := (λ Y, Y X) (only parsing) : C_scope.
(** ** Option *)
Instance option_inhabited {A} : Inhabited (option A) := populate None.
(** The type class [LeibnizEquiv] collects setoid equalities that coincide
with Leibniz equality. We provide the tactic [fold_leibniz] to transform such
setoid equalities into Leibniz equalities, and [unfold_leibniz] for the
reverse. *)
Class LeibnizEquiv A `{Equiv A} := leibniz_equiv x y : x y x = y.
Lemma leibniz_equiv_iff `{LeibnizEquiv A, !Reflexive (@equiv A _)} (x y : A) :
x y x = y.
Proof. split. apply leibniz_equiv. intros ->; reflexivity. Qed.
Ltac fold_leibniz := repeat
match goal with
| H : context [ @equiv ?A _ _ _ ] |- _ =>
setoid_rewrite (leibniz_equiv_iff (A:=A)) in H
| |- context [ @equiv ?A _ _ _ ] =>
setoid_rewrite (leibniz_equiv_iff (A:=A))
end.
Ltac unfold_leibniz := repeat
match goal with
| H : context [ @eq ?A _ _ ] |- _ =>
setoid_rewrite <-(leibniz_equiv_iff