From Undecidability.Synthetic Require Import DecidabilityFacts SemiDecidabilityFacts EnumerabilityFacts.
From Undecidability Require Import Shared.embed_nat.
Require Import List.
Import ListNotations.

Definition cumulative {X} (L: nat -> list X) :=
  forall n, exists A, L (S n) = L n ++ A.
Hint Extern 0 (cumulative _) => intros ?; cbn; eauto : core.

Lemma cum_ge {X} {L: nat -> list X} {n m} :
  cumulative L -> m >= n -> exists A, L m = L n ++ A.
Proof.
  induction 2 as [|m _ IH].
  - exists nil. now rewrite app_nil_r.
  - destruct (H m) as (A&->), IH as [B ->].
    exists (B ++ A). now rewrite app_assoc.
Qed.

Lemma cum_ge' {X} {L: nat -> list X} {x n m} :
  cumulative L -> In x (L n) -> m >= n -> In x (L m).
Proof.
  intros ? H [A ->] % (cum_ge (L := L)). apply in_app_iff. eauto. eauto.
Qed.

Definition list_enumerator {X} (L: nat -> list X) (p : X -> Prop) :=
  forall x, p x <-> exists m, In x (L m).
Definition list_enumerable {X} (p : X -> Prop) :=
  exists L, list_enumerator L p.

Definition list_enumerator__T' X f := forall x : X, exists n : nat, In x (f n).
Notation list_enumerator__T f X := (list_enumerator__T' X f).
Definition list_enumerable__T X := exists f : nat -> list X, list_enumerator__T f X.
Definition inf_list_enumerable__T X := { f : nat -> list X | list_enumerator__T f X }.

Section enumerator_list_enumerator.

  Variable X : Type.
  Variable p : X -> Prop.
  Variables (e : nat -> option X).

  Let T (n : nat) : list X := match e n with Some x => [x] | None => [] end.

  Lemma enumerator_to_list_enumerator : forall x, (exists n, e n = Some x) <-> (exists n, In x (T n)).
  Proof.
    split; intros [n H].
    - exists n. unfold T. rewrite H. firstorder.
    - unfold T in *. destruct (e n) eqn:E. inversion H; subst. eauto. inversion H0. inversion H.
  Qed.

End enumerator_list_enumerator.

Lemma enumerable_list_enumerable {X} {p : X -> Prop} :
  enumerable p -> list_enumerable p.
Proof.
  intros [f Hf]. eexists.
  unfold list_enumerator.
  intros x. rewrite <- enumerator_to_list_enumerator.
  eapply Hf.
Qed.

Lemma enumerable__T_list_enumerable {X} :
  enumerable__T X -> list_enumerable__T X.
Proof.
  intros [f Hf]. eexists.
  unfold list_enumerator.
  intros x. rewrite <- enumerator_to_list_enumerator.
  eapply Hf.
Qed.

Section enumerator_list_enumerator.

  Variable X : Type.
  Variables (T : nat -> list X).

  Let e (n : nat) : option X :=
    let (n, m) := unembed n in
    nth_error (T n) m.

  Lemma list_enumerator_to_enumerator : forall x, (exists n, e n = Some x) <-> (exists n, In x (T n)).
  Proof.
    split; intros [k H].
    - unfold e in *.
      destruct (unembed k) as (n, m).
      exists n. eapply (nth_error_In _ _ H).
    - unfold e in *.
      eapply In_nth_error in H as [m].
      exists (embed (k, m)). now rewrite embedP, H.
  Qed.

End enumerator_list_enumerator.

Lemma list_enumerator_enumerator {X} {p : X -> Prop} {T} :
  list_enumerator T p -> enumerator (fun n => let (n, m) := unembed n in
    nth_error (T n) m) p.
Proof.
  unfold list_enumerator.
  intros H x. rewrite list_enumerator_to_enumerator. eauto.
Qed.

Lemma list_enumerable_enumerable {X} {p : X -> Prop} :
  list_enumerable p -> enumerable p.
Proof.
  intros [T HT]. eexists.
  unfold list_enumerator.
  intros x. rewrite list_enumerator_to_enumerator.
  eapply HT.
Qed.

Lemma list_enumerable__T_enumerable {X} :
  list_enumerable__T X -> enumerable__T X.
Proof.
  intros [T HT]. eexists.
  unfold list_enumerator.
  intros x. rewrite list_enumerator_to_enumerator.
  eapply HT.
Qed.

Lemma enum_enumT {X} :
  enumerable__T X <-> list_enumerable__T X.
Proof.
  split.
  eapply enumerable__T_list_enumerable.
  eapply list_enumerable__T_enumerable.
Qed.

Definition to_cumul {X} (L : nat -> list X) := fix f n :=
  match n with 0 => [] | S n => f n ++ L n end.

Lemma to_cumul_cumulative {X} (L : nat -> list X) :
  cumulative (to_cumul L).
Proof.
  eauto.
Qed.

Lemma to_cumul_spec {X} (L : nat -> list X) x :
  (exists n, In x (L n)) <-> exists n, In x (to_cumul L n).
Proof.
  split.
  - intros [n H].
    exists (S n). cbn. eapply in_app_iff. eauto.
  - intros [n H].
    induction n; cbn in *.
    + inversion H.
    + eapply in_app_iff in H as [H | H]; eauto.
Qed.

Lemma cumul_In {X} (L : nat -> list X) x n :
  In x (L n) -> In x (to_cumul L (S n)).
Proof.
  intros H. cbn. eapply in_app_iff. eauto.
Qed.

Lemma In_cumul {X} (L : nat -> list X) x n :
  In x (to_cumul L n) -> exists n, In x (L n).
Proof.
  intros H. eapply to_cumul_spec. eauto.
Qed.

Hint Resolve cumul_In In_cumul : core.

Lemma list_enumerator_to_cumul {X} {p : X -> Prop} {L} :
  list_enumerator L p -> list_enumerator (to_cumul L) p.
Proof.
  unfold list_enumerator.
  intros. rewrite H.
  eapply to_cumul_spec.
Qed.

Lemma cumul_spec__T {X} {L} :
  list_enumerator__T L X -> list_enumerator__T (to_cumul L) X.
Proof.
  unfold list_enumerator__T.
  intros. now rewrite <- to_cumul_spec.
Qed.

Lemma cumul_spec {X} {L} {p : X -> Prop} :
  list_enumerator L p -> list_enumerator (to_cumul L) p.
Proof.
  unfold list_enumerator.
  intros. now rewrite <- to_cumul_spec.
Qed.

Require Import Undecidability.Shared.ListAutomation.
Import ListAutomationNotations.

Notation cumul := (to_cumul).

Section L_list_def.
  Context {X : Type}.
  Variable (L : nat -> list X).

Fixpoint L_list (n : nat) : list (list X) :=
  match n
  with
  | 0 => [ [] ]
  | S n => L_list n ++ [ x :: L | (x,L) ∈ (cumul L n × L_list n) ]
  end.
End L_list_def.

Lemma L_list_cumulative {X} L : cumulative (@L_list X L).
Proof.
  intros ?; cbn; eauto.
Qed.

Lemma enumerator__T_list {X} L :
  list_enumerator__T L X -> list_enumerator__T (L_list L) (list X).
Proof.
  intros H l.
  induction l.
  - exists 0. cbn. eauto.
  - destruct IHl as [n IH].
    destruct (cumul_spec__T H a) as [m ?].
    exists (1 + n + m). cbn. intros. in_app 2.
    in_collect (a,l).
    all: eapply cum_ge'; eauto using L_list_cumulative; lia.
Qed.

Lemma enumerable_list {X} : list_enumerable__T X -> list_enumerable__T (list X).
Proof.
  intros [L H].
  eexists. now eapply enumerator__T_list.
Qed.

Hint Extern 4 => match goal with [H : list_enumerator _ ?p |- ?p _ ] => eapply H end : core.

Lemma enumerable_conj X (p q : X -> Prop) :
  discrete X -> enumerable p -> enumerable q -> enumerable (fun x => p x /\ q x).
Proof.
  intros [] % discrete_iff [Lp] % enumerable_list_enumerable [Lq] % enumerable_list_enumerable.
  eapply list_enumerable_enumerable.
  exists (fix f n := match n with 0 => [] | S n => f n ++ [ x | x ∈ cumul Lp n, x el cumul Lq n] end).
  intros. split.
  + intros []. eapply (cumul_spec H) in H1 as [m1]. eapply (cumul_spec H0) in H2 as [m2].
    exists (1 + m1 + m2). cbn. in_app 2. in_collect x.
    eapply cum_ge'; eauto. lia.
    eapply cum_ge'; eauto. lia.
  + intros [m]. induction m.
    * inv H1.
    * inv_collect; eauto.
Qed.

Lemma projection X Y (p : X * Y -> Prop) :
  enumerable p -> enumerable (fun x => exists y, p (x,y)).
Proof.
  intros [f].
  exists (fun n => match f n with Some (x, y) => Some x | None => None end).
  intros; split.
  - intros [y ?]. eapply H in H0 as [n]. exists n. now rewrite H0.
  - intros [n ?]. destruct (f n) as [ [] | ] eqn:E; inv H0.
    exists y. eapply H. eauto.
Qed.

Lemma projection' X Y (p : X * Y -> Prop) :
  enumerable p -> enumerable (fun y => exists x, p (x,y)).
Proof.
  intros [f].
  exists (fun n => match f n with Some (x, y) => Some y | None => None end).
  intros y; split.
  - intros [x ?]. eapply H in H0 as [n]. exists n. now rewrite H0.
  - intros [n ?]. destruct (f n) as [ [] | ] eqn:E; inv H0.
    exists x. eapply H. eauto.
Qed.


Definition L_T {X : Type} {f : nat -> list X} {H : list_enumerator__T f X} : nat -> list X.
Proof.
  exact (cumul f).
Defined.
Arguments L_T _ {_ _} _, {_ _ _}.

Hint Unfold L_T : core.
Hint Resolve cumul_In : core.

Existing Class list_enumerator__T'.

Definition el_T {X} {f} `{list_enumerator__T f X} : list_enumerator__T L_T X.
Proof.
  now eapply cumul_spec__T.
Defined.

Existing Instance enumerator__T_list.

Instance enumerator__T_to_list {X} {f} :
  list_enumerator__T f X -> enumerator__T (fun n => let (n, m) := unembed n in nth_error (f n) m) X | 100.
Proof.
  intros H x. eapply list_enumerator_to_enumerator in H. exact H.
Qed.

Instance enumerator__T_of_list {X} {f} :
  enumerator__T f X -> list_enumerator__T (fun n => match f n with Some x => [x] | None => [] end) X | 100.
Proof.
  intros H x. eapply enumerator_to_list_enumerator. eauto.
Qed.

Existing Class inf_list_enumerable__T.
Instance inf_to_enumerator {X} :
  forall H : inf_list_enumerable__T X, list_enumerator__T (proj1_sig H) X | 100.
Proof.
  intros [? H]. eapply H.
Defined.


Hint Unfold enumerable list_enumerable : core.

Hint Resolve enumerable_list_enumerable
     list_enumerable_enumerable : core.

Lemma enumerable_enum {X} {p : X -> Prop} :
  enumerable p <-> list_enumerable p.
Proof.
  split; eauto.
Qed.