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.
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.