Timings for Arrow.v
(* -*- mode: coq; mode: visual-line -*- *)
(** * Theorems about Non-dependent function types *)
Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable
Basics.Equivalences Basics.Trunc Basics.Tactics.
Require Import Types.Forall.
Local Open Scope path_scope.
Generalizable Variables A B C D f g n.
Definition arrow@{u u0} (A : Type@{u}) (B : Type@{u0}) := A -> B.
#[export] Instance IsReflexive_arrow : Reflexive arrow :=
fun _ => idmap.
#[export] Instance IsTransitive_arrow : Transitive arrow :=
fun _ _ _ f g => compose g f.
(** ** Paths *)
(** As for dependent functions, paths [p : f = g] in a function type [A -> B] are equivalent to functions taking values in path types, [H : forall x:A, f x = g x], or concisely [H : f == g]. These are all given in the [Overture], but we can give them separate names for clarity in the non-dependent case. *)
Definition path_arrow {A B : Type} (f g : A -> B)
: (f == g) -> (f = g)
:= path_forall f g.
(** There are a number of combinations of dependent and non-dependent for [apD10_path_forall]; we list all of the combinations as helpful lemmas for rewriting. *)
Definition ap10_path_arrow {A B : Type} (f g : A -> B) (h : f == g)
: ap10 (path_arrow f g h) == h
:= apD10_path_forall f g h.
Definition apD10_path_arrow {A B : Type} (f g : A -> B) (h : f == g)
: apD10 (path_arrow f g h) == h
:= apD10_path_forall f g h.
Definition ap10_path_forall {A B : Type} (f g : A -> B) (h : f == g)
: ap10 (path_forall f g h) == h
:= apD10_path_forall f g h.
Definition eta_path_arrow {A B : Type} (f g : A -> B) (p : f = g)
: path_arrow f g (ap10 p) = p
:= eta_path_forall f g p.
Definition path_arrow_1 {A B : Type} (f : A -> B)
: (path_arrow f f (fun x => 1)) = 1
:= eta_path_arrow f f 1.
Definition equiv_ap10 {A B : Type} f g
: (f = g) <~> (f == g)
:= Build_Equiv _ _ (@ap10 A B f g) _.
Global Instance isequiv_path_arrow {A B : Type} (f g : A -> B)
: IsEquiv (path_arrow f g) | 0
:= isequiv_path_forall f g.
Definition equiv_path_arrow {A B : Type} (f g : A -> B)
: (f == g) <~> (f = g)
:= equiv_path_forall f g.
(** Function extensionality for two-variable functions *)
Definition equiv_path_arrow2 {X Y Z: Type} (f g : X -> Y -> Z)
: (forall x y, f x y = g x y) <~> f = g.
refine (equiv_path_arrow _ _ oE _).
apply equiv_functor_forall_id; intro x.
Definition ap100_path_arrow2 {X Y Z : Type} {f g : X -> Y -> Z}
(h : forall x y, f x y = g x y) (x : X) (y : Y)
: ap100 (equiv_path_arrow2 f g h) x y = h x y.
refine (ap (fun p => ap10 p y) _ @ _).
1: apply apD10_path_arrow.
Definition path_arrow_pp {A B : Type} (f g h : A -> B)
(p : f == g) (q : g == h)
: path_arrow f h (fun x => p x @ q x) = path_arrow f g p @ path_arrow g h q
:= path_forall_pp f g h p q.
(** ** Transport *)
(** Transporting in non-dependent function types is somewhat simpler than in dependent ones. *)
Definition transport_arrow {A : Type} {B C : A -> Type}
{x1 x2 : A} (p : x1 = x2) (f : B x1 -> C x1) (y : B x2)
: (transport (fun x => B x -> C x) p f) y = p # (f (p^ # y)).
(** This is an improvement to [transport_arrow]. That result only shows that the functions are homotopic, but even without funext, we can prove that these functions are equal. *)
Definition transport_arrow' {A : Type} {B C : A -> Type}
{x1 x2 : A} (p : x1 = x2) (f : B x1 -> C x1)
: transport (fun x => B x -> C x) p f = transport _ p o f o transport _ p^.
Definition transport_arrow_toconst {A : Type} {B : A -> Type} {C : Type}
{x1 x2 : A} (p : x1 = x2) (f : B x1 -> C) (y : B x2)
: (transport (fun x => B x -> C) p f) y = f (p^ # y).
(** This is an improvement to [transport_arrow_toconst]. That result shows that the functions are homotopic, but even without funext, we can prove that these functions are equal. *)
Definition transport_arrow_toconst' {A : Type} {B : A -> Type} {C : Type}
{x1 x2 : A} (p : x1 = x2) (f : B x1 -> C)
: transport (fun x => B x -> C) p f = f o transport B p^.
Definition transport_arrow_fromconst {A B : Type} {C : A -> Type}
{x1 x2 : A} (p : x1 = x2) (f : B -> C x1) (y : B)
: (transport (fun x => B -> C x) p f) y = p # (f y).
(** And some naturality and coherence for these laws. *)
Definition ap_transport_arrow_toconst {A : Type} {B : A -> Type} {C : Type}
{x1 x2 : A} (p : x1 = x2) (f : B x1 -> C) {y1 y2 : B x2} (q : y1 = y2)
: ap (transport (fun x => B x -> C) p f) q
@ transport_arrow_toconst p f y2
= transport_arrow_toconst p f y1
@ ap (fun y => f (p^ # y)) q.
destruct p, q; reflexivity.
(** ** Dependent paths *)
(** Usually, a dependent path over [p:x1=x2] in [P:A->Type] between [y1:P x1] and [y2:P x2] is a path [transport P p y1 = y2] in [P x2]. However, when [P] is a function space, these dependent paths have a more convenient description: rather than transporting the argument of [y1] forwards and backwards, we transport only forwards but on both sides of the equation, yielding a "naturality square". *)
Definition dpath_arrow
{A:Type} (B C : A -> Type) {x1 x2:A} (p:x1=x2)
(f : B x1 -> C x1) (g : B x2 -> C x2)
: (forall (y1:B x1), transport C p (f y1) = g (transport B p y1))
<~>
(transport (fun x => B x -> C x) p f = g).
Definition ap10_dpath_arrow
{A:Type} (B C : A -> Type) {x1 x2:A} (p:x1=x2)
(f : B x1 -> C x1) (g : B x2 -> C x2)
(h : forall (y1:B x1), transport C p (f y1) = g (transport B p y1))
(u : B x1)
: ap10 (dpath_arrow B C p f g h) (p # u)
= transport_arrow p f (p # u)
@ ap (fun x => p # (f x)) (transport_Vp B p u)
@ h u.
destruct p; simpl; unfold ap10.
exact (apD10_path_forall f g h u @ (concat_1p _)^).
(** ** Maps on paths *)
(** The action of maps given by application. *)
Definition ap_apply_l {A B : Type} {x y : A -> B} (p : x = y) (z : A) :
ap (fun f => f z) p = ap10 p z
:= 1.
Definition ap_apply_Fl {A B C : Type} {x y : A} (p : x = y) (M : A -> B -> C) (z : B) :
ap (fun a => (M a) z) p = ap10 (ap M p) z
:= match p with 1 => 1 end.
Definition ap_apply_Fr {A B C : Type} {x y : A} (p : x = y) (z : B -> C) (N : A -> B) :
ap (fun a => z (N a)) p = ap01 z (ap N p)
:= (ap_compose N _ _).
Definition ap_apply_FlFr {A B C : Type} {x y : A} (p : x = y) (M : A -> B -> C) (N : A -> B) :
ap (fun a => (M a) (N a)) p = ap11 (ap M p) (ap N p)
:= match p with 1 => 1 end.
(** The action of maps given by lambda. *)
Definition ap_lambda {A B C : Type} {x y : A} (p : x = y) (M : A -> B -> C) :
ap (fun a b => M a b) p =
path_arrow _ _ (fun b => ap (fun a => M a b) p).
destruct p;
symmetry;
simpl; apply path_arrow_1.
(** ** Functorial action *)
Definition functor_arrow `(f : B -> A) `(g : C -> D)
: (A -> C) -> (B -> D)
:= @functor_forall A (fun _ => C) B (fun _ => D) f (fun _ => g).
Definition not_contrapositive `(f : B -> A)
: not A -> not B
:= functor_arrow f idmap.
Definition ap_functor_arrow `(f : B -> A) `(g : C -> D)
(h h' : A -> C) (p : h == h')
: ap (functor_arrow f g) (path_arrow _ _ p)
= path_arrow _ _ (fun b => ap g (p (f b)))
:= @ap_functor_forall _ A (fun _ => C) B (fun _ => D)
f (fun _ => g) h h' p.
(** ** Truncatedness: functions into an n-type is an n-type *)
Global Instance contr_arrow {A B : Type} `{Contr B}
: Contr (A -> B) | 100
:= contr_forall.
Global Instance istrunc_arrow {A B : Type} `{IsTrunc n B}
: IsTrunc n (A -> B) | 100
:= istrunc_forall.
(** ** Functions from a contractible type *)
(** This also follows from [equiv_contr_forall], but this proof has a better inverse map. *)
Definition equiv_arrow_from_contr (A B : Type) `{Contr A}
: (A -> B) <~> B.
srapply (equiv_adjointify (fun f => f (center A)) const).
intro f; funext a; unfold const; simpl.
Global Instance isequiv_functor_arrow `{IsEquiv B A f} `{IsEquiv C D g}
: IsEquiv (functor_arrow f g) | 1000
:= @isequiv_functor_forall _ A (fun _ => C) B (fun _ => D)
_ _ _ _.
Definition equiv_functor_arrow `{IsEquiv B A f} `{IsEquiv C D g}
: (A -> C) <~> (B -> D)
:= @equiv_functor_forall _ A (fun _ => C) B (fun _ => D)
f _ (fun _ => g) _.
Definition equiv_functor_arrow' `(f : B <~> A) `(g : C <~> D)
: (A -> C) <~> (B -> D)
:= @equiv_functor_forall' _ A (fun _ => C) B (fun _ => D)
f (fun _ => g).
(* We could do something like this notation, but it's not clear that it would be that useful, and might be confusing. *)
(* Notation "f -> g" := (equiv_functor_arrow' f g) : equiv_scope. *)
(** What remains is really identical to that in [Forall]. *)
(** ** Decidability *)
(** This doesn't require funext *)
Global Instance decidable_arrow {A B : Type}
`{Decidable A} `{Decidable B}
: Decidable (A -> B).
destruct (dec B) as [x2|y2].
exact (inl (fun _ => x2)).
destruct (dec A) as [x1|y1].