# Library stdpp.infinite

From stdpp Require Export list.

From stdpp Require Import relations pretty.

From stdpp Require Import options.

Set Default Proof Using "Type*".

From stdpp Require Import relations pretty.

From stdpp Require Import options.

Set Default Proof Using "Type*".

# Generic constructions

If A is infinite, and there is an injection from A to B, then B is also infinite. Note that due to constructivity we need a rather strong notion of being injective, we also need a partial function B → option A back.
Program Definition inj_infinite `{Infinite A} {B}

(f : A → B) (g : B → option A) (Hgf : ∀ x, g (f x) = Some x) :

Infinite B := {| infinite_fresh := f ∘ fresh ∘ omap g |}.

Next Obligation.

intros A ? B f g Hfg xs Hxs; simpl in ×.

apply (infinite_is_fresh (omap g xs)), elem_of_list_omap; eauto.

Qed.

Next Obligation. solve_proper. Qed.

(f : A → B) (g : B → option A) (Hgf : ∀ x, g (f x) = Some x) :

Infinite B := {| infinite_fresh := f ∘ fresh ∘ omap g |}.

Next Obligation.

intros A ? B f g Hfg xs Hxs; simpl in ×.

apply (infinite_is_fresh (omap g xs)), elem_of_list_omap; eauto.

Qed.

Next Obligation. solve_proper. Qed.

If there is an injective function f : nat → B, then B is infinite. This
construction works as follows: to obtain an element not in xs, we return the
first element f 0, f 1, f 2, ... that is not in xs.
This construction has a nice computational behavior to e.g. find fresh strings.
Given some prefix s, we use f n := if n is S n' then s +:+ pretty n else s.
The construction then finds the first string starting with s followed by a
number that's not in the input list. For example, given ["H", "H1", "H4"] and
s := "H", it would find "H2".

Section search_infinite.

Context {B} (f : nat → B).

Let R (xs : list B) (n1 n2 : nat) :=

n2 < n1 ∧ (f (n1 - 1)) ∈ xs.

Lemma search_infinite_step xs n : f n ∈ xs → R xs (1 + n) n.

Proof. split; [lia|]. replace (1 + n - 1) with n by lia; eauto. Qed.

Context `{!Inj (=) (=) f, !EqDecision B}.

Lemma search_infinite_R_wf xs : wf (R xs).

Proof.

revert xs. assert (help : ∀ xs n n',

Acc (R (filter (.≠ f n') xs)) n → n' < n → Acc (R xs) n).

{ induction 1 as [n _ IH]. constructor; intros n2 [??]. apply IH; [|lia].

split; [done|]. apply elem_of_list_filter; naive_solver lia. }

intros xs. induction (well_founded_ltof _ length xs) as [xs _ IH].

intros n1; constructor; intros n2 [Hn Hs].

apply help with (n2 - 1); [|lia]. apply IH. eapply filter_length_lt; eauto.

Qed.

Definition search_infinite_go (xs : list B) (n : nat)

(go : ∀ n', R xs n' n → B) : B :=

let x := f n in

match decide (x ∈ xs) with

| left Hx ⇒ go (S n) (search_infinite_step xs n Hx)

| right _ ⇒ x

end.

Program Definition search_infinite : Infinite B := {|

infinite_fresh xs :=

Fix_F _ (search_infinite_go xs) (wf_guard 32 (search_infinite_R_wf xs) 0)

|}.

Next Obligation.

intros xs. unfold fresh.

generalize 0 (wf_guard 32 (search_infinite_R_wf xs) 0). revert xs.

fix FIX 3; intros xs n [?]; simpl; unfold search_infinite_go at 1; simpl.

destruct (decide _); auto.

Qed.

Next Obligation.

intros xs1 xs2 Hxs. unfold fresh.

generalize (wf_guard 32 (search_infinite_R_wf xs1) 0).

generalize (wf_guard 32 (search_infinite_R_wf xs2) 0). generalize 0.

fix FIX 2. intros n [acc1] [acc2]; simpl; unfold search_infinite_go.

destruct (decide ( _ ∈ xs1)) as [H1|H1], (decide (_ ∈ xs2)) as [H2|H2]; auto.

- destruct H2. by rewrite <-Hxs.

- destruct H1. by rewrite Hxs.

Qed.

End search_infinite.

Context {B} (f : nat → B).

Let R (xs : list B) (n1 n2 : nat) :=

n2 < n1 ∧ (f (n1 - 1)) ∈ xs.

Lemma search_infinite_step xs n : f n ∈ xs → R xs (1 + n) n.

Proof. split; [lia|]. replace (1 + n - 1) with n by lia; eauto. Qed.

Context `{!Inj (=) (=) f, !EqDecision B}.

Lemma search_infinite_R_wf xs : wf (R xs).

Proof.

revert xs. assert (help : ∀ xs n n',

Acc (R (filter (.≠ f n') xs)) n → n' < n → Acc (R xs) n).

{ induction 1 as [n _ IH]. constructor; intros n2 [??]. apply IH; [|lia].

split; [done|]. apply elem_of_list_filter; naive_solver lia. }

intros xs. induction (well_founded_ltof _ length xs) as [xs _ IH].

intros n1; constructor; intros n2 [Hn Hs].

apply help with (n2 - 1); [|lia]. apply IH. eapply filter_length_lt; eauto.

Qed.

Definition search_infinite_go (xs : list B) (n : nat)

(go : ∀ n', R xs n' n → B) : B :=

let x := f n in

match decide (x ∈ xs) with

| left Hx ⇒ go (S n) (search_infinite_step xs n Hx)

| right _ ⇒ x

end.

Program Definition search_infinite : Infinite B := {|

infinite_fresh xs :=

Fix_F _ (search_infinite_go xs) (wf_guard 32 (search_infinite_R_wf xs) 0)

|}.

Next Obligation.

intros xs. unfold fresh.

generalize 0 (wf_guard 32 (search_infinite_R_wf xs) 0). revert xs.

fix FIX 3; intros xs n [?]; simpl; unfold search_infinite_go at 1; simpl.

destruct (decide _); auto.

Qed.

Next Obligation.

intros xs1 xs2 Hxs. unfold fresh.

generalize (wf_guard 32 (search_infinite_R_wf xs1) 0).

generalize (wf_guard 32 (search_infinite_R_wf xs2) 0). generalize 0.

fix FIX 2. intros n [acc1] [acc2]; simpl; unfold search_infinite_go.

destruct (decide ( _ ∈ xs1)) as [H1|H1], (decide (_ ∈ xs2)) as [H2|H2]; auto.

- destruct H2. by rewrite <-Hxs.

- destruct H1. by rewrite Hxs.

Qed.

End search_infinite.

To select a fresh number from a given list x₀ ... xₙ, a possible approach
is to compute (S x₀) `max` ... `max` (S xₙ) `max` 0. For non-empty lists of
non-negative numbers this is equal to taking the maximal element xᵢ and adding
one.
The construction below generalizes this construction to any type A, function
f : A → A → A. and initial value a. The fresh element is computed as
x₀ `f` ... `f` xₙ `f` a. For numbers, the prototypical instance is
f x y := S x `max` y and a:=0, which gives the behavior described before.
Note that this gives a (i.e. 0 for numbers) for the empty list.

Section max_infinite.

Context {A} (f : A → A → A) (a : A) (lt : relation A).

Context (HR : ∀ x, ¬lt x x).

Context (HR1 : ∀ x y, lt x (f x y)).

Context (HR2 : ∀ x x' y, lt x x' → lt x (f y x')).

Context (Hf : ∀ x x' y, f x (f x' y) = f x' (f x y)).

Program Definition max_infinite: Infinite A := {|

infinite_fresh := foldr f a

|}.

Next Obligation.

cut (∀ xs x, x ∈ xs → lt x (foldr f a xs)).

{ intros help xs []%help%HR. }

induction 1; simpl; auto.

Qed.

Next Obligation. by apply (foldr_permutation_proper _ _ _). Qed.

End max_infinite.

Context {A} (f : A → A → A) (a : A) (lt : relation A).

Context (HR : ∀ x, ¬lt x x).

Context (HR1 : ∀ x y, lt x (f x y)).

Context (HR2 : ∀ x x' y, lt x x' → lt x (f y x')).

Context (Hf : ∀ x x' y, f x (f x' y) = f x' (f x y)).

Program Definition max_infinite: Infinite A := {|

infinite_fresh := foldr f a

|}.

Next Obligation.

cut (∀ xs x, x ∈ xs → lt x (foldr f a xs)).

{ intros help xs []%help%HR. }

induction 1; simpl; auto.

Qed.

Next Obligation. by apply (foldr_permutation_proper _ _ _). Qed.

End max_infinite.

Instances for number types

Program Instance nat_infinite : Infinite nat :=

max_infinite (Nat.max ∘ S) 0 (<) _ _ _ _.

Solve Obligations with (intros; simpl; lia).

Program Instance N_infinite : Infinite N :=

max_infinite (N.max ∘ N.succ) 0%N N.lt _ _ _ _.

Solve Obligations with (intros; simpl; lia).

Program Instance positive_infinite : Infinite positive :=

max_infinite (Pos.max ∘ Pos.succ) 1%positive Pos.lt _ _ _ _.

Solve Obligations with (intros; simpl; lia).

Program Instance Z_infinite: Infinite Z :=

max_infinite (Z.max ∘ Z.succ) 0%Z Z.lt _ _ _ _.

Solve Obligations with (intros; simpl; lia).

max_infinite (Nat.max ∘ S) 0 (<) _ _ _ _.

Solve Obligations with (intros; simpl; lia).

Program Instance N_infinite : Infinite N :=

max_infinite (N.max ∘ N.succ) 0%N N.lt _ _ _ _.

Solve Obligations with (intros; simpl; lia).

Program Instance positive_infinite : Infinite positive :=

max_infinite (Pos.max ∘ Pos.succ) 1%positive Pos.lt _ _ _ _.

Solve Obligations with (intros; simpl; lia).

Program Instance Z_infinite: Infinite Z :=

max_infinite (Z.max ∘ Z.succ) 0%Z Z.lt _ _ _ _.

Solve Obligations with (intros; simpl; lia).

Instances for option, sum, products

Instance option_infinite `{Infinite A} : Infinite (option A) :=

inj_infinite Some id (λ _, eq_refl).

Instance sum_infinite_l `{Infinite A} {B} : Infinite (A + B) :=

inj_infinite inl (maybe inl) (λ _, eq_refl).

Instance sum_infinite_r {A} `{Infinite B} : Infinite (A + B) :=

inj_infinite inr (maybe inr) (λ _, eq_refl).

Instance prod_infinite_l `{Infinite A, Inhabited B} : Infinite (A × B) :=

inj_infinite (., inhabitant) (Some ∘ fst) (λ _, eq_refl).

Instance prod_infinite_r `{Inhabited A, Infinite B} : Infinite (A × B) :=

inj_infinite (inhabitant ,.) (Some ∘ snd) (λ _, eq_refl).

inj_infinite Some id (λ _, eq_refl).

Instance sum_infinite_l `{Infinite A} {B} : Infinite (A + B) :=

inj_infinite inl (maybe inl) (λ _, eq_refl).

Instance sum_infinite_r {A} `{Infinite B} : Infinite (A + B) :=

inj_infinite inr (maybe inr) (λ _, eq_refl).

Instance prod_infinite_l `{Infinite A, Inhabited B} : Infinite (A × B) :=

inj_infinite (., inhabitant) (Some ∘ fst) (λ _, eq_refl).

Instance prod_infinite_r `{Inhabited A, Infinite B} : Infinite (A × B) :=

inj_infinite (inhabitant ,.) (Some ∘ snd) (λ _, eq_refl).

Instance for lists

Program Instance list_infinite `{Inhabited A} : Infinite (list A) := {|

infinite_fresh xxs := replicate (fresh (length <$> xxs)) inhabitant

|}.

Next Obligation.

intros A ? xs ?. destruct (infinite_is_fresh (length <$> xs)).

apply elem_of_list_fmap. eexists; split; [|done].

unfold fresh. by rewrite replicate_length.

Qed.

Next Obligation. unfold fresh. by intros A ? xs1 xs2 →. Qed.

infinite_fresh xxs := replicate (fresh (length <$> xxs)) inhabitant

|}.

Next Obligation.

intros A ? xs ?. destruct (infinite_is_fresh (length <$> xs)).

apply elem_of_list_fmap. eexists; split; [|done].

unfold fresh. by rewrite replicate_length.

Qed.

Next Obligation. unfold fresh. by intros A ? xs1 xs2 →. Qed.

Instance for strings