From Undecidability.Shared.Libs.PSL Require Import Base.
Section Retract.
Variable X Y : Type.
Definition retract (f : X -> Y) (g : Y -> option X) := forall x y, g y = Some x <-> y = f x.
Class Retract :=
{
Retr_f : X -> Y;
Retr_g : Y -> option X;
Retr_retr : retract Retr_f Retr_g;
}.
End Retract.
Arguments Retr_f { _ _ _ }.
Arguments Retr_g { _ _ _ }.
Section Retract_Properties.
Variable X Y : Type.
Hypothesis I : Retract X Y.
Definition retract_g_adjoint : forall x, Retr_g (Retr_f x) = Some x.
Proof. intros. pose proof @Retr_retr _ _ I. hnf in H. now rewrite H. Qed.
Definition retract_g_inv : forall x y, Retr_g y = Some x -> y = Retr_f x.
Proof. intros. now apply Retr_retr. Qed.
Lemma retract_g_surjective : forall x, { y | Retr_g y = Some x }.
Proof. intros x. pose proof retract_g_adjoint x. cbn in H. eauto. Defined.
Lemma retract_f_injective : forall x1 x2, Retr_f x1 = Retr_f x2 -> x1 = x2.
Proof.
intros x1 x2 H.
enough (Some x1 = Some x2) by congruence.
erewrite <- !retract_g_adjoint.
now rewrite H.
Qed.
Lemma retract_g_Some x y :
Retr_g (Retr_f x) = Some y ->
x = y.
Proof. now intros H % retract_g_inv % retract_f_injective. Qed.
Lemma retract_g_None b :
Retr_g b = None ->
forall a, Retr_f a <> b.
Proof.
intros H a <-.
enough (Retr_g (Retr_f a) = Some a) by congruence.
apply retract_g_adjoint.
Qed.
End Retract_Properties.
Ltac retract_adjoint :=
match goal with
| [ H : context [ Retr_g (Retr_f _) ] |- _ ] => rewrite retract_g_adjoint in H
| [ |- context [ Retr_g (Retr_f _) ] ] => rewrite retract_g_adjoint
end.
Section ComposeRetracts.
Variable A B C : Type.
Definition retr_comp_f (f1 : B -> C) (f2 : A -> B) : A -> C := fun a => f1 (f2 a).
Definition retr_comp_g (g1 : C -> option B) (g2 : B -> option A) : C -> option A :=
fun c => match g1 c with
| Some b => g2 b
| None => None
end.
Program Instance ComposeRetract (retr1 : Retract B C) (retr2 : Retract A B) : Retract A C :=
{|
Retr_f := retr_comp_f Retr_f Retr_f;
Retr_g := retr_comp_g Retr_g Retr_g;
|}.
Next Obligation.
abstract now
unfold retr_comp_f, retr_comp_g; intros a c; split;
[intros H; destruct (Retr_g c) as [ | ] eqn:E;
[ apply retract_g_inv in E as ->; now apply retract_g_inv in H as ->
| congruence
]
| intros ->; now do 2 retract_adjoint
].
Defined.
End ComposeRetracts.
Section Usefull_Retracts.
Variable (A B C D : Type).
Global Program Instance Retract_id : Retract A A :=
{|
Retr_f a := a;
Retr_g b := Some b;
|}.
Next Obligation. abstract now hnf; firstorder congruence. Defined.
Global Program Instance Retract_Empty : Retract Empty_set A :=
{|
Retr_f e := @Empty_set_rect (fun _ => A) e;
Retr_g b := None;
|}.
Next Obligation. abstract now intros x; elim x. Defined.
Global Program Instance Retract_Empty_left `{retr: Retract A B} : Retract (A + Empty_set) B :=
{|
Retr_f a := match a with
| inl a => Retr_f a
| inr e => @Empty_set_rect (fun _ => B) e
end;
Retr_g b := match Retr_g b with
| Some a => Some (inl a)
| None => None
end;
|}.
Next Obligation.
abstract now intros [ a | [] ] b; split;
[ intros H; destruct (Retr_g b) eqn:E; inv H; now apply retract_g_inv in E
| intros ->; now retract_adjoint
].
Defined.
Global Program Instance Retract_Empty_right `{retr: Retract A B} : Retract (Empty_set + A) B :=
{|
Retr_f a := match a with
| inl e => @Empty_set_rect (fun _ => B) e
| inr a => Retr_f a
end;
Retr_g b := match Retr_g b with
| Some a => Some (inr a)
| None => None
end;
|}.
Next Obligation.
abstract now intros [ [] | a ] b; split;
[ intros H; destruct (Retr_g b) eqn:E; inv H; now apply retract_g_inv in E
| intros ->; now retract_adjoint
].
Defined.
Global Program Instance Retract_option `{retr: Retract A B} : Retract A (option B) :=
{|
Retr_f a := Some (Retr_f a);
Retr_g ob := match ob with
| Some b => Retr_g b
| None => None
end;
|}.
Next Obligation.
abstract now
split;
[ intros H; destruct y as [b|];
[ now apply retract_g_inv in H as ->
| inv H
]
| intros ->; now retract_adjoint
].
Defined.
Definition retract_inl_f (f : A -> B) : A -> (B + C) := fun a => inl (f a).
Definition retract_inl_g (g : B -> option A) : B+C -> option A :=
fun x => match x with
| inl b => g b
| inr c => None
end.
Global Program Instance Retract_inl (retrAB : Retract A B) : Retract A (B + C) :=
{|
Retr_f := retract_inl_f Retr_f;
Retr_g := retract_inl_g Retr_g;
|}.
Next Obligation.
abstract now
unfold retract_inl_f, retract_inl_g; hnf; intros x y; split;
[ destruct y as [a|b]; [ now intros -> % retract_g_inv | congruence ]
| intros ->; now retract_adjoint
].
Defined.
Definition retract_inr_f (f : A -> B) : A -> (C + B) := fun a => inr (f a).
Definition retract_inr_g (g : B -> option A) : C+B -> option A :=
fun x => match x with
| inr b => g b
| inl c => None
end.
Global Program Instance Retract_inr (retrAB : Retract A B) : Retract A (C + B) :=
{|
Retr_f := retract_inr_f Retr_f;
Retr_g := retract_inr_g Retr_g;
|}.
Next Obligation.
abstract now
unfold retract_inr_f, retract_inr_g; hnf; intros x y; split;
[ destruct y as [a|b]; [ congruence | now intros -> % retract_g_inv ]
| intros ->; now retract_adjoint
].
Defined.
Section Retract_sum.
Definition retract_sum_f (f1: A -> C) (f2: B -> D) : A+B -> C+D :=
fun x => match x with
| inl a => inl (f1 a)
| inr b => inr (f2 b)
end.
Definition retract_sum_g (g1: C -> option A) (g2: D -> option B) : C+D -> option (A+B) :=
fun y => match y with
| inl c => match g1 c with
| Some a => Some (inl a)
| None => None
end
| inr d => match g2 d with
| Some b => Some (inr b)
| None => None
end
end.
Local Program Instance Retract_sum (retr1 : Retract A C) (retr2 : Retract B D) : Retract (A+B) (C+D) :=
{|
Retr_f := retract_sum_f Retr_f Retr_f;
Retr_g := retract_sum_g Retr_g Retr_g;
|}.
Next Obligation.
abstract now
unfold retract_sum_f, retract_sum_g; intros x y; split;
[ intros H; destruct y as [c|d];
[ destruct (Retr_g c) eqn:E1; inv H; f_equal; now apply retract_g_inv
| destruct (Retr_g d) eqn:E1; inv H; f_equal; now apply retract_g_inv
]
| intros ->; destruct x as [a|b]; now retract_adjoint
].
Defined.
End Retract_sum.
End Usefull_Retracts.
Section Join.
Variable A B Z : Type.
Variable retr1 : Retract A Z.
Variable retr2 : Retract B Z.
Local Arguments Retr_f {_ _} (Retract).
Local Arguments Retr_g {_ _} (Retract).
Definition retract_join_f s :=
match s with
| inl a => Retr_f retr1 a
| inr b => Retr_f retr2 b
end.
Definition retract_join_g z :=
match Retr_g retr1 z with
| Some a => Some (inl a)
| None =>
match Retr_g retr2 z with
| Some b => Some (inr b)
| None => None
end
end.
Hypothesis disjoint : forall (a : A) (b : B), Retr_f _ a <> Retr_f _ b.
Lemma retract_join : retract retract_join_f retract_join_g.
Proof.
unfold retract_join_f, retract_join_g. hnf; intros s z; split.
- destruct s as [a|b]; intros H.
+ destruct (Retr_g retr1 z) eqn:E.
* inv H. now apply retract_g_inv in E.
* destruct (Retr_g retr2 z) eqn:E2; inv H.
+ destruct (Retr_g retr1 z) eqn:E.
* inv H.
* destruct (Retr_g retr2 z) eqn:E2.
-- inv H. now apply retract_g_inv in E2.
-- inv H.
- intros ->. destruct s as [a|b]; retract_adjoint. reflexivity.
destruct (Retr_g retr1 (Retr_f retr2 b)) eqn:E.
+ exfalso. apply retract_g_inv in E. symmetry in E. now apply disjoint in E.
+ reflexivity.
Qed.
Local Instance Retract_join : Retract (A+B) Z := Build_Retract retract_join.
End Join.
Section MoreSums.
Local Instance Retract_sum3 (A A' B B' C C' : Type) (retr1 : Retract A A') (retr2 : Retract B B') (retr3 : Retract C C') :
Retract (A+B+C) (A'+B'+C') := Retract_sum (Retract_sum retr1 retr2) retr3.
Local Instance Retract_sum4 (A A' B B' C C' D D' : Type) (retr1 : Retract A A') (retr2 : Retract B B') (retr3 : Retract C C') (retr4 : Retract D D') :
Retract (A+B+C+D) (A'+B'+C'+D') := Retract_sum (Retract_sum (Retract_sum retr1 retr2) retr3) retr4.
Local Instance Retract_sum5 (A A' B B' C C' D D' E E' : Type) (retr1 : Retract A A') (retr2 : Retract B B') (retr3 : Retract C C') (retr4 : Retract D D') (retr5 : Retract E E') :
Retract (A+B+C+D+E) (A'+B'+C'+D'+E') := Retract_sum (Retract_sum (Retract_sum (Retract_sum retr1 retr2) retr3) retr4) retr5.
Local Instance Retract_sum6 (A A' B B' C C' D D' E E' F F' : Type) (retr1 : Retract A A') (retr2 : Retract B B') (retr3 : Retract C C') (retr4 : Retract D D') (retr5 : Retract E E') (retr6 : Retract F F') :
Retract (A+B+C+D+E+F) (A'+B'+C'+D'+E'+F') := Retract_sum (Retract_sum (Retract_sum (Retract_sum (Retract_sum retr1 retr2) retr3) retr4) retr5) retr6.
Local Instance Retract_sum7 (A A' B B' C C' D D' E E' F F' G G' : Type) (retr1 : Retract A A') (retr2 : Retract B B') (retr3 : Retract C C') (retr4 : Retract D D') (retr5 : Retract E E') (retr6 : Retract F F') (retr7 : Retract G G') :
Retract (A+B+C+D+E+F+G) (A'+B'+C'+D'+E'+F'+G') := Retract_sum (Retract_sum (Retract_sum (Retract_sum (Retract_sum (Retract_sum retr1 retr2) retr3) retr4) retr5) retr6) retr7.
End MoreSums.