Library MetaCoq.Template.utils.All_Forall

From Coq Require Import List Bool Arith ssreflect Morphisms Lia Utf8.
From MetaCoq.Template Require Import MCPrelude MCReflect MCList MCRelations MCProd MCOption.
From Equations Require Import Equations.
Import ListNotations.

Derive Signature for Forall Forall2.

Inductive All {A} (P : A → Type) : list A → Type :=
    All_nil : All P []
  | All_cons : ∀ (x : A) (l : list A),
                  P x → All P l → All P (x :: l).
Arguments All {A} P%type _.
Arguments All_nil {_ _}.
Arguments All_cons {_ _ _ _}.
Derive Signature NoConfusion for All.
#[global] Hint Constructors All : core.

Inductive Alli {A} (P : nat → A → Type) (n : nat) : list A → Type :=
| Alli_nil : Alli P n []
| Alli_cons hd tl : P n hd → Alli P (S n) tl → Alli P n (hd :: tl).
Arguments Alli_nil {_ _ _}.
Arguments Alli_cons {_ _ _ _ _}.
Derive Signature for Alli.
Derive NoConfusionHom for Alli.

Inductive All2 {A B : Type} (R : A → B → Type) : list A → list B → Type :=
  All2_nil : All2 R [] []
| All2_cons : ∀ (x : A) (y : B) (l : list A) (l' : list B),
    R x y → All2 R l l' → All2 R (x :: l) (y :: l').
Arguments All2_nil {_ _ _}.
Arguments All2_cons {_ _ _ _ _ _ _}.
Derive Signature for All2.
Derive NoConfusionHom for All2.
#[global] Hint Constructors All2 : core.

Inductive All2i {A B : Type} (R : nat → A → B → Type) (n : nat)
  : list A → list B → Type :=
| All2i_nil : All2i R n [] []
| All2i_cons :
    ∀ x y l r,
      R n x y →
      All2i R (S n) l r →
      All2i R n (x :: l) (y :: r).
Arguments All2i_nil {_ _ _ _}.
Arguments All2i_cons {_ _ _ _ _ _ _ _}.

Derive Signature NoConfusionHom for All2i.

Inductive All3 {A B C : Type} (R : A → B → C → Type) : list A → list B → list C → Type :=
  All3_nil : All3 R [] [] []
| All3_cons : ∀ (x : A) (y : B) (z : C) (l : list A) (l' : list B) (l'' : list C),
    R x y z → All3 R l l' l'' → All3 R (x :: l) (y :: l') (z :: l'').
Arguments All3_nil {_ _ _ _}.
Arguments All3_cons {_ _ _ _ _ _ _ _ _ _}.
Derive Signature NoConfusionHom for All3.

Section alli.
  Context {A} (p : nat → A → bool).
  Fixpoint alli (n : nat) (l : list A) : bool :=
  match l with
  | [] ⇒ true
  | hd :: tl ⇒ p n hd && alli (S n) tl
  end.
End alli.

Lemma alli_ext {A} (p q : nat → A → bool) n (l : list A) :
  (∀ i, p i =1 q i) →
  alli p n l = alli q n l.
Proof.
  intros hfg.
  induction l in n |- *; simpl; auto.
  now rewrite IHl.
Qed.

#[global] Instance alli_proper {A} :
   Proper ((pointwise_relation nat (pointwise_relation A eq)) ==> eq ==> eq ==> eq) alli.
Proof.
  intros f g fg.
  intros ? ? → ? ? →.
  now apply alli_ext.
Qed.

Lemma alli_impl {A} (p q : nat → A → bool) n (l : list A) :
  (∀ i x, p i x → q i x) →
  alli p n l → alli q n l.
Proof.
  intros hpq. induction l in n |- *; simpl; auto.
  move/andb_and ⇒ [pna a'].
  rewrite (hpq _ _ pna).
  now apply IHl.
Qed.

Lemma allbiP {A} (P : nat → A → Type) (p : nat → A → bool) n l :
  (∀ i x, reflectT (P i x) (p i x)) →
  reflectT (Alli P n l) (alli p n l).
Proof.
  intros Hp.
  apply equiv_reflectT.
  - induction 1; rewrite /= // IHX // andb_true_r.
    now destruct (Hp n hd).
  - induction l in n |- *; rewrite /= //. constructor.
    move/andb_and ⇒ [pa pl].
    constructor; auto. now destruct (Hp n a).
Qed.

Lemma alli_Alli {A} (p : nat → A → bool) n l :
  alli p n l <~> Alli p n l.
Proof.
  destruct (allbiP p p n l).
  - intros. destruct (p i x); now constructor.
  - split; eauto.
  - split; eauto. by [].
Qed.

Lemma alli_shiftn {A} n k p (l : list A) :
  alli p (n + k) l = alli (fun i ⇒ p (n + i)) k l.
Proof.
  induction l in n, k, p |- *; simpl; auto. f_equal.
  rewrite (IHl (S n) k p) (IHl 1 k _).
  apply alli_ext ⇒ x.
  now rewrite Nat.add_succ_r.
Qed.

Section alli.
  Context {A} (p q : nat → A → bool) (l l' : list A).

  Lemma alli_app n :
    alli p n (l ++ l') =
    alli p n l && alli p (#|l| + n) l'.
  Proof using Type.
    induction l in n |- *; simpl; auto.
    now rewrite IHl0 Nat.add_succ_r andb_assoc.
  Qed.

  Lemma alli_shift n :
    alli p n l = alli (fun i ⇒ p (n + i)) 0 l.
  Proof using Type.
    induction l in n, p |- *; simpl; auto.
    rewrite IHl0 (IHl0 _ 1) Nat.add_0_r.
    f_equal. apply alli_ext ⇒ x.
    now rewrite Nat.add_succ_r.
  Qed.

  Lemma alli_map {B} (f : B → A) n bs : alli p n (map f bs) = alli (fun i ⇒ p i ∘ f) n bs.
  Proof using Type.
    induction bs in n |- *; simpl; auto.
    now rewrite IHbs.
  Qed.
End alli.

Lemma alli_mapi {A B} (f : nat → A → bool) (g : nat → B → A) n l :
  alli f n (mapi_rec g l n) = alli (fun i x ⇒ f i (g i x)) n l.
Proof.
  revert n; induction l ⇒ n; simpl; auto.
  now rewrite IHl.
Qed.

Section Forallb2.
  Context {A B} (f : A → B → bool).

  Fixpoint forallb2 l l' :=
    match l, l' with
    | hd :: tl, hd' :: tl' ⇒ f hd hd' && forallb2 tl tl'
    | [], [] ⇒ true
    | _, _ ⇒ false
    end.

End Forallb2.

Lemma forallb2_refl :
  ∀ A (R : A → A → bool),
    (∀ x, R x x) →
    ∀ l, forallb2 R l l.
Proof.
  intros A R R_refl l.
  induction l.
  - reflexivity.
  - simpl. rewrite R_refl. assumption.
Qed.

Lemma forallb2_map :
  ∀ A B C D (R : A → B → bool) f g (l : list C) (l' : list D),
    forallb2 R (map f l) (map g l') =
    forallb2 (fun x y ⇒ R (f x) (g y)) l l'.
Proof.
  intros A B C D R f g l l'.
  induction l in l' |- ×.
  - destruct l' ⇒ //.
  - destruct l' ⇒ /= //; rewrite IHl //.
Qed.

Lemma forall_map_spec {A B} {l} {f g : A → B} :
  Forall (fun x ⇒ f x = g x) l ↔
  map f l = map g l.
Proof.
  split.
  induction 1; simpl; trivial.
  now rewrite IHForall H.
  induction l ⇒ /= // [=] Ha Hl; constructor; auto.
Qed.

Lemma forall_map_id_spec {A} {l} {f : A → A} :
  Forall (fun x ⇒ f x = x) l ↔ map f l = l.
Proof.
  rewrite -{3}(map_id l). apply forall_map_spec.
Qed.

Lemma forallb_Forall {A} (p : A → bool) l
  : Forall p l ↔ is_true (forallb p l).
Proof.
  split.
  induction 1; rewrite /= // H IHForall //.
  induction l; rewrite /= //. rewrite andb_and.
  intros [pa pl].
  constructor; auto.
Qed.

Lemma forallbP {A} (P : A → Prop) (p : A → bool) l :
  (∀ x, reflect (P x) (p x)) →
  reflect (Forall P l) (forallb p l).
Proof.
  intros Hp.
  apply iff_reflect; change (forallb p l = true) with (forallb p l : Prop); split.
  - induction 1; rewrite /= // IHForall // andb_true_r.
    now destruct (Hp x).
  - induction l; rewrite /= //. rewrite andb_and.
    intros [pa pl].
    constructor; auto. now destruct (Hp a).
Qed.

Lemma forallb_ext {A} (p q : A → bool) : p =1 q → forallb p =1 forallb q.
Proof.
  intros hpq l.
  induction l; simpl; auto.
  now rewrite (hpq a) IHl.
Qed.

#[global] Instance forallb_proper {A} : Proper (`=1` ==> eq ==> eq) (@forallb A).
Proof.
  intros f g Hfg ? ? →. now apply forallb_ext.
Qed.

Lemma forallbP_cond {A} (P Q : A → Prop) (p : A → bool) l :
  Forall Q l →
  (∀ x, Q x → reflect (P x) (p x)) → reflect (Forall P l) (forallb p l).
Proof.
  intros HQ Hp.
  apply iff_reflect; split.
  - induction HQ; intros HP; depelim HP; rewrite /= // IHHQ // andb_true_r.
    now destruct (Hp x H).
  - induction HQ; rewrite /= //. move/andb_and ⇒ [pa pl].
    constructor; auto. now destruct (Hp _ H).
Qed.

Lemma nth_error_forallb {A} {p : A → bool} {l : list A} {n x} :
  nth_error l n = Some x → forallb p l → p x.
Proof.
  intros Hnth HPl.
  induction l in n, Hnth, HPl |- × ⇒ //.
  - rewrite nth_error_nil in Hnth ⇒ //.
  - destruct n ⇒ /=; noconf Hnth.
    × now move: HPl ⇒ /= /andb_and.
    × eapply IHl; tea. now move: HPl ⇒ /andb_and.
Qed.

Lemma forallb_nth_error {A} P l n :
  @forallb A P l → on_Some_or_None P (nth_error l n).
Proof.
  induction l in n |- ×.
  - intros _. destruct n; constructor.
  - intro H. apply forallb_Forall in H.
    inv H. destruct n; cbn; auto.
    now apply forallb_Forall in H1; eauto.
Qed.

Lemma map_eq_inj {A B} (f g : A → B) l: map f l = map g l →
                                         All (fun x ⇒ f x = g x) l.
Proof.
  induction l. simpl. constructor. simpl. intros [=]. constructor; auto.
Qed.

Lemma Forall_mix {A} (P Q : A → Prop) : ∀ l, Forall P l → Forall Q l → Forall (fun x ⇒ P x ∧ Q x) l.
Proof.
  intros l Hl Hq. induction Hl; inv Hq; constructor; auto.
Qed.

Lemma Forall_skipn {A} (P : A → Prop) n l : Forall P l → Forall P (skipn n l).
Proof.
  intros H. revert n; induction H; intros n. rewrite skipn_nil; auto.
  destruct n; simpl.
  - rewrite /skipn. constructor; auto.
  - now auto.
Qed.

Lemma Forall_firstn {A} (P : A → Prop) n l : Forall P l → Forall P (firstn n l).
Proof.
  intros H. revert n; induction H; intros n. rewrite firstn_nil; auto.
  destruct n; simpl.
  - constructor; auto.
  - constructor; auto.
Qed.

Lemma forallb2_All2 {A B : Type} {p : A → B → bool}
      {l : list A} {l' : list B}:
  is_true (forallb2 p l l') → All2 (fun x y ⇒ is_true (p x y)) l l'.
Proof.
  induction l in l' |- *; destruct l'; simpl; intros; try congruence.
  - constructor.
  - constructor. revert H; rewrite andb_and; intros [px pl]. auto.
    apply IHl. revert H; rewrite andb_and; intros [px pl]. auto.
Qed.

Lemma All2_forallb2 {A B : Type} {p : A → B → bool}
      {l : list A} {l' : list B} :
  All2 (fun x y ⇒ is_true (p x y)) l l' → is_true (forallb2 p l l').
Proof.
  induction 1; simpl; intros; try congruence.
  rewrite andb_and. intuition auto.
Qed.

Lemma All2P {A B : Type} {p : A → B → bool} {l l'} :
  reflectT (All2 p l l') (forallb2 p l l').
Proof.
  apply equiv_reflectT. apply All2_forallb2. apply forallb2_All2.
Qed.

Lemma All2_refl {A} {P : A → A → Type} l :
  (∀ x, P x x) →
  All2 P l l.
Proof.
  intros HP. induction l; constructor; auto.
Qed.

Lemma forallb2_app {A B} (p : A → B → bool) l l' q q' :
  is_true (forallb2 p l l' && forallb2 p q q')
  → is_true (forallb2 p (l ++ q) (l' ++ q')).
Proof.
  induction l in l' |- *; destruct l'; simpl; try congruence.
  move⇒ /andb_and[/andb_and[pa pl] pq]. now rewrite pa IHl // pl pq.
Qed.

Lemma All2_map_equiv {A B C D} {R : C → D → Type} {f : A → C} {g : B → D} {l l'} :
  All2 (fun x y ⇒ R (f x) (g y)) l l' <~> All2 R (map f l) (map g l').
Proof.
  split.
  - induction 1; simpl; constructor; try congruence.
  - induction l in l' |- *; destruct l'; intros H; depelim H; constructor; auto.
Qed.

Lemma All2_map {A B C D} {R : C → D → Type} {f : A → C} {g : B → D} {l l'} :
  All2 (fun x y ⇒ R (f x) (g y)) l l' → All2 R (map f l) (map g l').
Proof. apply All2_map_equiv. Qed.

Lemma All2_map_inv {A B C D} {R : C → D → Type} {f : A → C} {g : B → D} {l l'} :
  All2 R (map f l) (map g l') → All2 (fun x y ⇒ R (f x) (g y)) l l'.
Proof. apply All2_map_equiv. Qed.



Lemma All2_All_mix_left {A B} {P : A → Type} {Q : A → B → Type}
      {l : list A} {l' : list B} :
  All P l → All2 Q l l' → All2 (fun x y ⇒ (P x × Q x y)%type) l l'.
Proof.
  induction 2; simpl; intros; constructor.
  inv X; intuition auto.
  apply IHX0. inv X; intuition auto.
Qed.

Lemma All2_All_mix_right {A B} {P : B → Type} {Q : A → B → Type}
      {l : list A} {l' : list B} :
  All P l' → All2 Q l l' → All2 (fun x y ⇒ (Q x y × P y)%type) l l'.
Proof.
  induction 2; simpl; intros; constructor.
  inv X; intuition auto.
  apply IHX0. inv X; intuition auto.
Qed.

Lemma All2i_All_mix_left {A B} {P : A → Type} {Q : nat → A → B → Type}
      {n} {l : list A} {l' : list B} :
  All P l → All2i Q n l l' → All2i (fun i x y ⇒ (P x × Q i x y)%type) n l l'.
Proof.
  induction 2; simpl; intros; constructor.
  inv X; intuition auto.
  apply IHX0. inv X; intuition auto.
Qed.

Lemma All2i_All_mix_right {A B} {P : B → Type} {Q : nat → A → B → Type}
      {n} {l : list A} {l' : list B} :
  All P l' → All2i Q n l l' → All2i (fun i x y ⇒ (Q i x y × P y)%type) n l l'.
Proof.
  induction 2; simpl; intros; constructor.
  inv X; intuition auto.
  apply IHX0. inv X; intuition auto.
Qed.

Lemma All2i_All2_mix_left {A B} {P : A → B → Type} {Q : nat → A → B → Type}
      {n} {l : list A} {l' : list B} :
  All2 P l l' → All2i Q n l l' → All2i (fun i x y ⇒ (P x y × Q i x y)%type) n l l'.
Proof.
  induction 2; simpl; intros; constructor.
  inv X; intuition auto.
  apply IHX0. inv X; intuition auto.
Qed.

Lemma Forall_All {A : Type} (P : A → Prop) l :
  Forall P l → All P l.
Proof.
  induction l; intros H; constructor; auto. inv H. auto.
  apply IHl. inv H; auto.
Qed.

Lemma All_Forall {A : Type} (P : A → Prop) l :
  All P l → Forall P l.
Proof. induction 1; constructor; auto. Qed.

Lemma forallb_All {A} (p : A → bool) l : is_true (forallb p l) → All p l.
Proof.
  move/forallb_Forall. apply Forall_All.
Qed.

Lemma All_forallb {A} (p : A → bool) l : All p l → is_true (forallb p l).
Proof.
  move/All_Forall. apply forallb_Forall.
Qed.

Lemma All_firstn {A} {P : A → Type} {l} {n} : All P l → All P (firstn n l).
Proof. intros HPL; induction HPL in n |- × ; simpl; destruct n; try econstructor; eauto. Qed.

Lemma All_skipn {A} {P : A → Type} {l} {n} : All P l → All P (skipn n l).
Proof. intros HPL; induction HPL in n |- × ; simpl; destruct n; try econstructor; eauto. Qed.

Lemma All_app {A} {P : A → Type} {l l'} : All P (l ++ l') → All P l × All P l'.
Proof.
  induction l; simpl; auto. intros H; depelim H; constructor; intuition auto.
Qed.

Lemma All_app_inv {A} (P : A → Type) l l' : All P l → All P l' → All P (l ++ l').
Proof.
  intros Hl Hl'. induction Hl. apply Hl'.
  constructor; intuition auto.
Defined.

Lemma All_True {A} l : All (fun x : A ⇒ True) l.
Proof.
  induction l; intuition auto.
Qed.

Lemma All_tip {A} {P : A → Type} {a : A} : P a <~> All P [a].
Proof. split; intros. repeat constructor; auto. now depelim X. Qed.

Lemma All_mix {A} {P : A → Type} {Q : A → Type} {l} :
  All P l → All Q l → All (fun x ⇒ (P x × Q x)%type) l.
Proof. induction 1; intros Hq; inv Hq; constructor; auto. Qed.

Lemma All2_All_left {A B} {P : A → B → Type} {Q : A → Type} {l l'} :
  All2 P l l' →
  (∀ x y, P x y → Q x) →
  All Q l.
Proof.
  intros HF H. induction HF; constructor; eauto.
Qed.

Lemma All2_All_right {A B} {P : A → B → Type} {Q : B → Type} {l l'} :
  All2 P l l' →
  (∀ x y, P x y → Q y) →
  All Q l'.
Proof.
  intros HF H. induction HF; constructor; eauto.
Qed.

Lemma All2_right {A B} {P : B → Type} {l : list A} {l'} :
  All2 (fun x y ⇒ P y) l l' → All P l'.
Proof.
  induction 1; constructor; auto.
Qed.

Lemma All2_impl {A B} {P Q : A → B → Type} {l l'} :
    All2 P l l' →
    (∀ x y, P x y → Q x y) →
    All2 Q l l'.
Proof.
  induction 1; constructor; auto.
Qed.

Lemma All2_eq_eq {A} (l l' : list A) : l = l' → All2 (fun x y ⇒ x = y) l l'.
Proof.
  intros →. induction l'; constructor; auto.
Qed.

Lemma All2_reflexivity {A} {P : A → A → Type} :
  CRelationClasses.Reflexive P → CRelationClasses.Reflexive (All2 P).
Proof.
  intros hp x. eapply All2_refl. intros; reflexivity.
Qed.

Lemma All2_symmetry {A} (R : A → A → Type) :
  CRelationClasses.Symmetric R →
  CRelationClasses.Symmetric (All2 R).
Proof.
  intros HR x y l.
  induction l; constructor; auto.
Qed.

Lemma All2_transitivity {A} (R : A → A → Type) :
  CRelationClasses.Transitive R →
  CRelationClasses.Transitive (All2 R).
Proof.
  intros HR x y z l; induction l in z |- *; auto.
  intros H; inv H. constructor; eauto.
Qed.

Lemma All2_apply {A B C} {D : A → B → C → Type} {l : list B} {l' : list C} :
  ∀ (a : A),
    All2 (fun x y ⇒ ∀ a : A, D a x y) l l' →
    All2 (fun x y ⇒ D a x y) l l'.
Proof.
  intros a all. eapply (All2_impl all); auto.
Qed.

Lemma All2_apply_arrow {A B C} {D : B → C → Type} {l : list B} {l' : list C} :
  A → All2 (fun x y ⇒ A → D x y) l l' →
  All2 (fun x y ⇒ D x y) l l'.
Proof.
  intros a all. eapply (All2_impl all); auto.
Qed.

Lemma All2_apply_dep_arrow {B C} {A} {D : B → C → Type} {l : list B} {l' : list C} :
  All A l →
  All2 (fun x y ⇒ A x → D x y) l l' →
  All2 D l l'.
Proof.
  intros a all.
  eapply All2_All_mix_left in all; tea.
  eapply (All2_impl all); intuition auto.
Qed.

Lemma All2_apply_dep_All {B C} {A} {D : C → Type} {l : list B} {l' : list C} :
  All A l →
  All2 (fun x y ⇒ A x → D y) l l' →
  All D l'.
Proof.
  intros a all.
  eapply All2_All_mix_left in all; tea.
  eapply All2_impl in all. 2:{ intros x y [ha hd]. exact (hd ha). }
  eapply All2_All_right; tea. auto.
Qed.

Lemma All2i_All_left {A B} {P : nat → A → B → Type} {Q : A → Type} {n l l'} :
  All2i P n l l' →
  (∀ i x y, P i x y → Q x) →
  All Q l.
Proof.
  intros HF H. induction HF; constructor; eauto.
Qed.

Lemma All2i_All_right {A B} {P : nat → A → B → Type} {Q : B → Type} {n l l'} :
  All2i P n l l' →
  (∀ i x y, P i x y → Q y) →
  All Q l'.
Proof.
  intros HF H. induction HF; constructor; eauto.
Qed.

Lemma All_refl {A} (P : A → Type) l : (∀ x, P x) → All P l.
Proof.
  intros Hp; induction l; constructor; auto.
Qed.

Lemma All_rev_map {A B} (P : A → Type) f (l : list B) : All (fun x ⇒ P (f x)) l → All P (rev_map f l).
Proof. induction 1. constructor. rewrite rev_map_cons. apply All_app_inv; auto. Qed.

Lemma All_rev (A : Type) (P : A → Type) (l : list A) : All P l → All P (List.rev l).
Proof.
  induction l using rev_ind. constructor. rewrite rev_app_distr.
  simpl. intros X; apply All_app in X as [? ?]. inversion a0; intuition auto.
Qed.

Lemma All_rev_inv {A} (P : A → Type) (l : list A) : All P (List.rev l) → All P l.
Proof.
  induction l using rev_ind. constructor.
  intros. rewrite rev_app_distr in X. simpl.
  apply All_app in X as [Allx Alll]. inv Allx.
   apply All_app_inv; intuition eauto.
Qed.

Lemma All_impl {A} {P Q} {l : list A} : All P l → (∀ x, P x → Q x) → All Q l.
Proof. induction 1; try constructor; intuition auto. Qed.

Lemma Alli_impl {A} {P Q} (l : list A) {n} : Alli P n l → (∀ n x, P n x → Q n x) → Alli Q n l.
Proof. induction 1; try constructor; intuition auto. Defined.

Lemma All_map {A B} {P : B → Type} {f : A → B} {l : list A} :
  All (fun x ⇒ P (f x)) l → All P (map f l).
Proof. induction 1; constructor; auto. Qed.

Lemma All_map_inv {A B} (P : B → Type) (f : A → B) l : All P (map f l) → All (fun x ⇒ P (f x)) l.
Proof. induction l; intros Hf; inv Hf; try constructor; eauto. Qed.

Lemma In_All {A} {P : A → Type} l :
    (∀ x : A, In x l → P x) → All P l.
Proof.
  induction l; cbn; constructor; auto.
Qed.

Lemma All_nth_error :
  ∀ A P l i x,
    @All A P l →
    nth_error l i = Some x →
    P x.
Proof.
  intros A P l i x h e.
  induction h in i, x, e |- ×.
  - destruct i. all: discriminate.
  - destruct i.
    + simpl in e. inversion e. subst. clear e.
      assumption.
    + simpl in e. eapply IHh in e.
      assumption.
Qed.

Lemma Alli_mix {A} {P : nat → A → Type} {Q : nat → A → Type} {n l} :
  Alli P n l → Alli Q n l → Alli (fun n x ⇒ (P n x × Q n x)%type) n l.
Proof. induction 1; intros Hq; inv Hq; constructor; auto. Qed.

Lemma Alli_All {A} {P : nat → A → Type} {Q : A → Type} {l n} :
  Alli P n l →
  (∀ n x, P n x → Q x) →
  All Q l.
Proof. induction 1; constructor; eauto. Qed.

Lemma Alli_app {A} (P : nat → A → Type) n l l' : Alli P n (l ++ l') → Alli P n l × Alli P (length l + n) l'.
Proof.
  induction l in n, l' |- *; intros H. split; try constructor. apply H.
  inversion_clear H. split; intuition auto. constructor; auto. eapply IHl; eauto.
  simpl. replace (S (#|l| + n)) with (#|l| + S n) by lia.
  eapply IHl; eauto.
Qed.

Lemma Alli_nth_error :
  ∀ A P k l i x,
    @Alli A P k l →
    nth_error l i = Some x →
    P (k + i) x.
Proof.
  intros A P k l i x h e.
  induction h in i, x, e |- ×.
  - destruct i. all: discriminate.
  - destruct i.
    + simpl in e. inversion e. subst. clear e.
      replace (n + 0) with n by lia.
      assumption.
    + simpl in e. eapply IHh in e.
      replace (n + S i) with (S n + i) by lia.
      assumption.
Qed.

Lemma forall_nth_error_All :
  ∀ {A} (P : A → Type) l,
    (∀ i x, nth_error l i = Some x → P x) →
    All P l.
Proof.
  intros A P l h.
  induction l.
  - constructor.
  - constructor.
    + specialize (h 0 a eq_refl). assumption.
    + eapply IHl. intros i x e. eapply (h (S i)). assumption.
Qed.

Lemma forall_nth_error_Alli :
  ∀ {A} (P : nat → A → Type) k l,
    (∀ i x, nth_error l i = Some x → P (k + i) x) →
    Alli P k l.
Proof.
  intros A P k l h.
  induction l in k, h |- ×.
  - constructor.
  - constructor.
    + specialize (h 0 a eq_refl). now rewrite Nat.add_0_r in h.
    + apply IHl. intros. specialize (h (S i) x H).
      simpl. now replace (S (k + i)) with (k + S i) by lia.
Qed.

Lemma Alli_mapi {A B} {P : nat → B → Type} (f : nat → A → B) k l :
  Alli (fun n a ⇒ P n (f n a)) k l <~> Alli P k (mapi_rec f l k).
Proof.
  split.
  { induction 1. simpl. constructor.
    simpl. constructor; eauto. }
  { induction l in k |- ×. simpl. constructor.
    simpl. intros. inversion X. constructor; eauto. }
Qed.

Lemma Alli_shift {A} {P : nat → A → Type} k l :
  Alli (fun x ⇒ P (S x)) k l →
  Alli P (S k) l.
Proof.
  induction 1; simpl; constructor; auto.
Qed.

Lemma Alli_rev {A} {P : nat → A → Type} k l :
  Alli P k l →
  Alli (fun k' ⇒ P (Nat.pred #|l| - k' + k)) 0 (List.rev l).
Proof.
  revert k.
  induction l using rev_ind; simpl; intros; try constructor.
  eapply Alli_app in X. intuition.
  rewrite rev_app_distr. rewrite app_length.
  simpl. constructor.
  replace (Nat.pred (#|l| + 1) - 0) with #|l| by lia.
  inversion b. eauto. specialize (IHl _ a).
  eapply Alli_shift. eapply Alli_impl. eauto.
  simpl; intros.
  now replace (Nat.pred (#|l| + 1) - S n) with (Nat.pred #|l| - n) by lia.
Qed.

Lemma Alli_rev_inv {A: Type} (P : nat → A → Type) (k : nat) (l : list A) :
  Alli P k (List.rev l) →
  Alli (fun k' : nat ⇒ P (Nat.pred #|l| - k' + k)) 0 l.
Proof.
  intros alli.
  eapply Alli_rev in alli. rewrite List.rev_involutive in alli.
  now len in alli.
Qed.

Lemma Alli_app_inv {A} {P} {l l' : list A} {n} : Alli P n l → Alli P (n + #|l|) l' → Alli P n (l ++ l').
Proof.
  induction 1; simpl; auto. now rewrite Nat.add_0_r.
  rewrite Nat.add_succ_r. simpl in IHX.
  intros H; specialize (IHX H).
  constructor; auto.
Qed.

Lemma Alli_rev_nth_error {A} (l : list A) n P x :
  Alli P 0 (List.rev l) →
  nth_error l n = Some x →
  P (#|l| - S n) x.
Proof.
  induction l in x, n |- *; simpl.
  { rewrite nth_error_nil; discriminate. }
  move/Alli_app ⇒ [Alll Alla]. inv Alla. clear X0.
  destruct n as [|n'].
  - move⇒ [=] <-. rewrite List.rev_length Nat.add_0_r in X.
    now rewrite Nat.sub_0_r.
  - simpl. eauto.
Qed.

Lemma Alli_shiftn {A} {P : nat → A → Type} k l n :
  Alli (fun x ⇒ P (n + x)) k l → Alli P (n + k) l.
Proof.
  induction 1; simpl; constructor; auto.
  now rewrite Nat.add_succ_r in IHX.
Qed.

Lemma Alli_shiftn_inv {A} {P : nat → A → Type} k l n :
  Alli P (n + k) l → Alli (fun x ⇒ P (n + x)) k l.
Proof.
  induction l in n, k |- *; simpl; constructor; auto.
  inv X; auto. inv X; auto. apply IHl.
  now rewrite Nat.add_succ_r.
Qed.

Lemma Alli_All_mix {A} {P : nat → A → Type} (Q : A → Type) k l :
  Alli P k l → All Q l → Alli (fun k x ⇒ (P k x) × Q x)%type k l.
Proof.
  induction 1; constructor; try inversion X0; intuition auto.
Qed.

Inductive OnOne2 {A : Type} (P : A → A → Type) : list A → list A → Type :=
| OnOne2_hd hd hd' tl : P hd hd' → OnOne2 P (hd :: tl) (hd' :: tl)
| OnOne2_tl hd tl tl' : OnOne2 P tl tl' → OnOne2 P (hd :: tl) (hd :: tl').
Derive Signature NoConfusion for OnOne2.

Lemma OnOne2_All_mix_left {A} {P : A → A → Type} {Q : A → Type} {l l'} :
  All Q l → OnOne2 P l l' → OnOne2 (fun x y ⇒ (P x y × Q x)%type) l l'.
Proof.
  intros H; induction 1; constructor; try inv H; intuition.
Qed.

Lemma OnOne2_app {A} (P : A → A → Type) l tl tl' : OnOne2 P tl tl' → OnOne2 P (l ++ tl) (l ++ tl').
Proof. induction l; simpl; try constructor; auto. Qed.

Lemma OnOne2_app_r {A} (P : A → A → Type) l l' tl :
  OnOne2 P l l' →
  OnOne2 P (l ++ tl) (l' ++ tl).
Proof. induction 1; constructor; auto. Qed.

Lemma OnOne2_length {A} {P} {l l' : list A} : OnOne2 P l l' → #|l| = #|l'|.
Proof. induction 1; simpl; congruence. Qed.

Lemma OnOne2_mapP {A B} {P} {l l' : list A} (f : A → B) :
  OnOne2 (on_rel P f) l l' → OnOne2 P (map f l) (map f l').
Proof. induction 1; simpl; constructor; try congruence. apply p. Qed.

Lemma OnOne2_map {A B} {P : B → B → Type} {l l' : list A} (f : A → B) :
  OnOne2 (on_Trel P f) l l' → OnOne2 P (map f l) (map f l').
Proof. induction 1; simpl; constructor; try congruence. apply p. Qed.

Lemma OnOne2_sym {A} (P : A → A → Type) l l' : OnOne2 (fun x y ⇒ P y x) l' l → OnOne2 P l l'.
Proof.
  induction 1; constructor; auto.
Qed.

Lemma OnOne2_exist {A} (P : A → A → Type) (Q : A → A → Type) l l' :
  OnOne2 P l l' →
  (∀ x y, P x y → ∑ z, Q x z × Q y z) →
  ∑ r, (OnOne2 Q l r × OnOne2 Q l' r).
Proof.
  intros H HPQ. induction H.
  - destruct (HPQ _ _ p). destruct p0.
    now ∃ (x :: tl); intuition constructor.
               - destruct IHOnOne2 as [r [? ?]].
                 now ∃ (hd :: r); intuition constructor.
Qed.

Lemma OnOne2_ind_l :
  ∀ A (R : list A → A → A → Type)
    (P : ∀ L l l', OnOne2 (R L) l l' → Type),
    (∀ L x y l (r : R L x y), P L (x :: l) (y :: l) (OnOne2_hd _ _ _ l r)) →
    (∀ L x l l' (h : OnOne2 (R L) l l'),
        P L l l' h →
        P L (x :: l) (x :: l') (OnOne2_tl _ x _ _ h)
    ) →
    ∀ l l' h, P l l l' h.
Proof.
  intros A R P hhd htl l l' h. induction h ; eauto.
Qed.

Lemma OnOne2_impl_exist_and_All :
  ∀ A (l1 l2 l3 : list A) R1 R2 R3,
    OnOne2 R1 l1 l2 →
    All2 R2 l3 l2 →
    (∀ x x' y, R1 x y → R2 x' y → ∑ z : A, R3 x z × R2 x' z) →
    ∑ l4, OnOne2 R3 l1 l4 × All2 R2 l3 l4.
Proof.
  intros A l1 l2 l3 R1 R2 R3 h1 h2 h.
  induction h1 in l3, h2 |- ×.
  - destruct l3.
    + inversion h2.
    + inversion h2. subst.
    specialize (h _ _ _ p X) as hh.
    destruct hh as [? [? ?]].
    eexists. constructor.
      × constructor. eassumption.
      × constructor ; eauto.
  - destruct l3.
    + inversion h2.
    + inversion h2. subst.
    specialize (IHh1 _ X0). destruct IHh1 as [? [? ?]].
    eexists. constructor.
      × eapply OnOne2_tl. eassumption.
      × constructor ; eauto.
Qed.

Lemma OnOne2_impl_exist_and_All_r :
  ∀ A (l1 l2 l3 : list A) R1 R2 R3,
    OnOne2 R1 l1 l2 →
    All2 R2 l2 l3 →
    (∀ x x' y, R1 x y → R2 y x' → ∑ z : A, R3 x z × R2 z x') →
    ∑ l4, ( OnOne2 R3 l1 l4 × All2 R2 l4 l3 ).
Proof.
  intros A l1 l2 l3 R1 R2 R3 h1 h2 h.
  induction h1 in l3, h2 |- ×.
  - destruct l3.
    + inversion h2.
    + inversion h2. subst.
      specialize (h _ _ _ p X) as hh.
      destruct hh as [? [? ?]].
      eexists. split.
      × constructor. eassumption.
      × constructor ; eauto.
  - destruct l3.
    + inversion h2.
    + inversion h2. subst.
      specialize (IHh1 _ X0). destruct IHh1 as [? [? ?]].
      eexists. split.
      × eapply OnOne2_tl. eassumption.
      × constructor ; eauto.
Qed.

Lemma OnOne2_split :
  ∀ A (P : A → A → Type) l l',
    OnOne2 P l l' →
    ∑ x y u v,
      P x y ×
      (l = u ++ x :: v ∧
      l' = u ++ y :: v).
Proof.
  intros A P l l' h.
  induction h.
  - ∃ hd, hd', [], tl.
    intuition eauto.
  - destruct IHh as [x [y [u [v ?]]]].
    ∃ x, y, (hd :: u), v.
    intuition eauto. all: subst. all: reflexivity.
Qed.

Lemma OnOne2_impl {A} {P Q} {l l' : list A} :
  OnOne2 P l l' →
  (∀ x y, P x y → Q x y) →
  OnOne2 Q l l'.
Proof.
  induction 1; constructor; intuition eauto.
Qed.

Lemma OnOne2_nth_error {A} (l l' : list A) n t P :
  OnOne2 P l l' →
  nth_error l n = Some t →
  ∑ t', (nth_error l' n = Some t') ×
  ((t = t') + (P t t')).
Proof.
  induction 1 in n |- ×.
  destruct n; simpl.
  - intros [= ->]. ∃ hd'; intuition auto.
  - ∃ t. intuition auto.
  - destruct n; simpl; auto.
    intros [= ->]. ∃ t; intuition auto.
Qed.

Lemma OnOne2_nth_error_r {A} (l l' : list A) n t' P :
  OnOne2 P l l' →
  nth_error l' n = Some t' →
  ∑ t, (nth_error l n = Some t) ×
  ((t = t') + (P t t')).
Proof.
  induction 1 in n |- ×.
  destruct n; simpl.
  - intros [= ->]. ∃ hd; intuition auto.
  - ∃ t'. intuition auto.
  - destruct n; simpl; auto.
    intros [= ->]. ∃ t'; intuition auto.
Qed.

Lemma OnOne2_impl_All_r {A} (P : A → A → Type) (Q : A → Type) l l' :
  (∀ x y, Q x → P x y → Q y) →
  OnOne2 P l l' → All Q l → All Q l'.
Proof.
  intros HPQ.
  induction 1; intros H; depelim H; constructor; auto.
  now eapply HPQ.
Qed.

Lemma OnOne2_All2_All2 {A : Type} {l1 l2 l3 : list A} {R1 R2 R3 : A → A → Type} :
  OnOne2 R1 l1 l2 →
  All2 R2 l1 l3 →
  (∀ x y, R2 x y → R3 x y) →
  (∀ x y z : A, R1 x y → R2 x z → R3 y z) →
  All2 R3 l2 l3.
Proof.
  intros o. induction o in l3 |- ×.
  intros H; depelim H.
  intros Hf Hf'. specialize (Hf' _ _ _ p r). constructor; auto.
  eapply All2_impl; eauto.
  intros H; depelim H.
  intros Hf. specialize (IHo _ H Hf).
  constructor; auto.
Qed.

Lemma OnOne2_All_All {A : Type} {l1 l2 : list A} {R1 : A → A → Type} {R2 R3 : A → Type} :
  OnOne2 R1 l1 l2 →
  All R2 l1 →
  (∀ x, R2 x → R3 x) →
  (∀ x y : A, R1 x y → R2 x → R3 y) →
  All R3 l2.
Proof.
  intros o. induction o.
  intros H; depelim H.
  intros Hf Hf'. specialize (Hf' _ _ p r). constructor; auto.
  eapply All_impl; eauto.
  intros H; depelim H.
  intros Hf. specialize (IHo H Hf).
  constructor; auto.
Qed.

Inductive OnOne2i {A : Type} (P : nat → A → A → Type) : nat → list A → list A → Type :=
| OnOne2i_hd i hd hd' tl : P i hd hd' → OnOne2i P i (hd :: tl) (hd' :: tl)
| OnOne2i_tl i hd tl tl' : OnOne2i P (S i) tl tl' → OnOne2i P i (hd :: tl) (hd :: tl').
Derive Signature NoConfusion for OnOne2i.

Lemma OnOne2i_All_mix_left {A} {P : nat → A → A → Type} {Q : A → Type} {i l l'} :
  All Q l → OnOne2i P i l l' → OnOne2i (fun i x y ⇒ (P i x y × Q x)%type) i l l'.
Proof.
  intros H; induction 1; constructor; try inv H; intuition.
Qed.

Lemma OnOne2i_app {A} (P : nat → A → A → Type) {i l tl tl'} :
  OnOne2i P (#|l| + i) tl tl' →
  OnOne2i P i (l ++ tl) (l ++ tl').
Proof. induction l in i |- *; simpl; try constructor; eauto.
  eapply IHl. now rewrite Nat.add_succ_r.
Qed.

Lemma OnOne2i_app_r {A} (P : nat → A → A → Type) i l l' tl :
  OnOne2i P i l l' →
  OnOne2i P i (l ++ tl) (l' ++ tl).
Proof. induction 1; constructor; auto. Qed.

Lemma OnOne2i_length {A} {P} {i} {l l' : list A} : OnOne2i P i l l' → #|l| = #|l'|.
Proof. induction 1; simpl; congruence. Qed.

Lemma OnOne2i_mapP {A B} {P} {i} {l l' : list A} (f : A → B) :
  OnOne2i (fun i ⇒ on_rel (P i) f) i l l' → OnOne2i P i (map f l) (map f l').
Proof. induction 1; simpl; constructor; try congruence. apply p. Qed.

Lemma OnOne2i_map {A B} {P : nat → B → B → Type} {i} {l l' : list A} (f : A → B) :
  OnOne2i (fun i ⇒ on_Trel (P i) f) i l l' → OnOne2i P i (map f l) (map f l').
Proof. induction 1; simpl; constructor; try congruence. apply p. Qed.

Lemma OnOne2i_sym {A} (P : nat → A → A → Type) i l l' : OnOne2i (fun i x y ⇒ P i y x) i l' l → OnOne2i P i l l'.
Proof.
  induction 1; constructor; auto.
Qed.

Lemma OnOne2i_exist {A} (P : nat → A → A → Type) (Q : nat → A → A → Type) i l l' :
  OnOne2i P i l l' →
  (∀ i x y, P i x y → ∑ z, Q i x z × Q i y z) →
  ∑ r, (OnOne2i Q i l r × OnOne2i Q i l' r).
Proof.
  intros H HPQ. induction H.
  - destruct (HPQ _ _ _ p). destruct p0.
    now ∃ (x :: tl); intuition constructor.
               - destruct IHOnOne2i as [r [? ?]].
                 now ∃ (hd :: r); intuition constructor.
Qed.

Lemma OnOne2i_ind_l :
  ∀ A (R : list A → nat → A → A → Type)
    (P : ∀ L i l l', OnOne2i (R L) i l l' → Type),
    (∀ L i x y l (r : R L i x y), P L i (x :: l) (y :: l) (OnOne2i_hd _ _ _ _ l r)) →
    (∀ L i x l l' (h : OnOne2i (R L) (S i) l l'),
        P L (S i) l l' h →
        P L i (x :: l) (x :: l') (OnOne2i_tl _ i x _ _ h)
    ) →
    ∀ i l l' h, P l i l l' h.
Proof.
  intros A R P hhd htl i l l' h. induction h ; eauto.
Qed.

Lemma OnOne2i_impl_exist_and_All :
  ∀ A i (l1 l2 l3 : list A) R1 R2 R3,
    OnOne2i R1 i l1 l2 →
    All2 R2 l3 l2 →
    (∀ i x x' y, R1 i x y → R2 x' y → ∑ z : A, R3 i x z × R2 x' z) →
    ∑ l4, OnOne2i R3 i l1 l4 × All2 R2 l3 l4.
Proof.
  intros A i l1 l2 l3 R1 R2 R3 h1 h2 h.
  induction h1 in l3, h2 |- ×.
  - destruct l3.
    + inversion h2.
    + inversion h2. subst.
    specialize (h _ _ _ _ p X) as hh.
    destruct hh as [? [? ?]].
    eexists. constructor.
      × constructor. eassumption.
      × constructor ; eauto.
  - destruct l3.
    + inversion h2.
    + inversion h2. subst.
    specialize (IHh1 _ X0). destruct IHh1 as [? [? ?]].
    eexists. constructor.
      × eapply OnOne2i_tl. eassumption.
      × constructor ; eauto.
Qed.

Lemma OnOne2i_impl_exist_and_All_r :
  ∀ A i (l1 l2 l3 : list A) R1 R2 R3,
    OnOne2i R1 i l1 l2 →
    All2 R2 l2 l3 →
    (∀ i x x' y, R1 i x y → R2 y x' → ∑ z : A, R3 i x z × R2 z x') →
    ∑ l4, ( OnOne2i R3 i l1 l4 × All2 R2 l4 l3 ).
Proof.
  intros A i l1 l2 l3 R1 R2 R3 h1 h2 h.
  induction h1 in l3, h2 |- ×.
  - destruct l3.
    + inversion h2.
    + inversion h2. subst.
      specialize (h _ _ _ _ p X) as hh.
      destruct hh as [? [? ?]].
      eexists. split.
      × constructor. eassumption.
      × constructor ; eauto.
  - destruct l3.
    + inversion h2.
    + inversion h2. subst.
      specialize (IHh1 _ X0). destruct IHh1 as [? [? ?]].
      eexists. split.
      × eapply OnOne2i_tl. eassumption.
      × constructor ; eauto.
Qed.

Lemma OnOne2i_split :
  ∀ A (P : nat → A → A → Type) i l l',
    OnOne2i P i l l' →
    ∑ i x y u v,
      P i x y ×
      (l = u ++ x :: v ∧
      l' = u ++ y :: v).
Proof.
  intros A P i l l' h.
  induction h.
  - ∃ i, hd, hd', [], tl.
    intuition eauto.
  - destruct IHh as [i' [x [y [u [v ?]]]]].
    ∃ i', x, y, (hd :: u), v.
    intuition eauto. all: subst. all: reflexivity.
Qed.

Lemma OnOne2i_impl {A} {P Q} {i} {l l' : list A} :
  OnOne2i P i l l' →
  (∀ i x y, P i x y → Q i x y) →
  OnOne2i Q i l l'.
Proof.
  induction 1; constructor; intuition eauto.
Qed.

Lemma OnOne2i_nth_error {A} (l l' : list A) i n t P :
  OnOne2i P i l l' →
  nth_error l n = Some t →
  ∑ t', (nth_error l' n = Some t') ×
  ((t = t') + (P (i + n)%nat t t')).
Proof.
  induction 1 in n |- ×.
  destruct n; simpl.
  - intros [= ->]. ∃ hd'; rewrite Nat.add_0_r; intuition auto.
  - ∃ t. intuition auto.
  - destruct n; simpl; rewrite ?Nat.add_succ_r /=; auto.
    intros [= ->]. ∃ t; intuition auto.
    apply IHX.
Qed.

Lemma OnOne2i_nth_error_r {A} i (l l' : list A) n t' P :
  OnOne2i P i l l' →
  nth_error l' n = Some t' →
  ∑ t, (nth_error l n = Some t) ×
  ((t = t') + (P (i + n)%nat t t')).
Proof.
  induction 1 in n |- ×.
  destruct n; simpl.
  - intros [= ->]. rewrite Nat.add_0_r; ∃ hd; intuition auto.
  - ∃ t'. intuition auto.
  - destruct n; simpl; auto.
    intros [= ->]. ∃ t'; intuition auto.
    rewrite Nat.add_succ_r; apply IHX.
Qed.

Inductive OnOne2All {A B : Type} (P : B → A → A → Type) : list B → list A → list A → Type :=
| OnOne2All_hd b bs hd hd' tl : P b hd hd' → #|bs| = #|tl| → OnOne2All P (b :: bs) (hd :: tl) (hd' :: tl)
| OnOne2All_tl b bs hd tl tl' : OnOne2All P bs tl tl' → OnOne2All P (b :: bs) (hd :: tl) (hd :: tl').
Derive Signature NoConfusion for OnOne2All.

Lemma OnOne2All_All_mix_left {A B} {P : B → A → A → Type} {Q : A → Type} {i l l'} :
  All Q l → OnOne2All P i l l' → OnOne2All (fun i x y ⇒ (P i x y × Q x)%type) i l l'.
Proof.
  intros H; induction 1; constructor; try inv H; intuition.
Qed.

Lemma OnOne2All_All2_mix_left {A B} {P : B → A → A → Type} {Q : B → A → Type} {i l l'} :
  All2 Q i l → OnOne2All P i l l' → OnOne2All (fun i x y ⇒ (P i x y × Q i x)%type) i l l'.
Proof.
  intros a; induction 1; constructor; try inv a; intuition.
Qed.

Lemma OnOne2All_app {A B} (P : B → A → A → Type) {i i' l tl tl'} :
  OnOne2All P i tl tl' →
  #|i'| = #|l| →
  OnOne2All P (i' ++ i) (l ++ tl) (l ++ tl').
Proof. induction l in i, i' |- *; simpl; try constructor; eauto.
  destruct i' ⇒ //.
  intros. destruct i' ⇒ //. simpl. constructor.
  eapply IHl; auto.
Qed.
Lemma OnOne2All_length {A B} {P} {i : list B} {l l' : list A} : OnOne2All P i l l' → #|l| = #|l'|.
Proof. induction 1; simpl; congruence. Qed.

Lemma OnOne2All_length2 {A B} {P} {i : list B} {l l' : list A} : OnOne2All P i l l' → #|i| = #|l|.
Proof. induction 1; simpl; congruence. Qed.

Lemma OnOne2All_mapP {A B I} {P} {i : list I} {l l' : list A} (f : A → B) :
  OnOne2All (fun i ⇒ on_rel (P i) f) i l l' → OnOne2All P i (map f l) (map f l').
Proof. induction 1; simpl; constructor; try congruence. apply p. now rewrite map_length. Qed.

Lemma OnOne2All_map {A I B} {P : I → B → B → Type} {i : list I} {l l' : list A} (f : A → B) :
  OnOne2All (fun i ⇒ on_Trel (P i) f) i l l' → OnOne2All P i (map f l) (map f l').
Proof. induction 1; simpl; constructor; try congruence. apply p. now rewrite map_length. Qed.

Lemma OnOne2All_map_all {A B I I'} {P} {i : list I} {l l' : list A} (g : I → I') (f : A → B) :
  OnOne2All (fun i ⇒ on_Trel (P (g i)) f) i l l' → OnOne2All P (map g i) (map f l) (map f l').
Proof. induction 1; simpl; constructor; try congruence. apply p. now rewrite !map_length. Qed.

Lemma OnOne2All_sym {A B} (P : B → A → A → Type) i l l' : OnOne2All (fun i x y ⇒ P i y x) i l' l → OnOne2All P i l l'.
Proof.
  induction 1; constructor; auto.
Qed.

Lemma OnOne2All_exist {A B} (P : B → A → A → Type) (Q : B → A → A → Type) i l l' :
  OnOne2All P i l l' →
  (∀ i x y, P i x y → ∑ z, Q i x z × Q i y z) →
  ∑ r, (OnOne2All Q i l r × OnOne2All Q i l' r).
Proof.
  intros H HPQ. induction H.
  - destruct (HPQ _ _ _ p). destruct p0.
    now ∃ (x :: tl); intuition constructor.
               - destruct IHOnOne2All as [r [? ?]].
                 now ∃ (hd :: r); intuition constructor.
Qed.

Lemma OnOne2All_ind_l :
  ∀ A B (R : list A → B → A → A → Type)
    (P : ∀ L i l l', OnOne2All (R L) i l l' → Type),
    (∀ L b bs x y l (r : R L b x y) (len : #|bs| = #|l|),
      P L (b :: bs) (x :: l) (y :: l) (OnOne2All_hd _ _ _ _ _ l r len)) →
    (∀ L b bs x l l' (h : OnOne2All (R L) bs l l'),
        P L bs l l' h →
        P L (b :: bs) (x :: l) (x :: l') (OnOne2All_tl _ _ _ x _ _ h)
    ) →
    ∀ i l l' h, P l i l l' h.
Proof.
  intros A B R P hhd htl i l l' h. induction h ; eauto.
Qed.

Lemma OnOne2All_impl_exist_and_All :
  ∀ A B (i : list B) (l1 l2 l3 : list A) R1 R2 R3,
    OnOne2All R1 i l1 l2 →
    All2 R2 l3 l2 →
    (∀ i x x' y, R1 i x y → R2 x' y → ∑ z : A, R3 i x z × R2 x' z) →
    ∑ l4, OnOne2All R3 i l1 l4 × All2 R2 l3 l4.
Proof.
  intros A B i l1 l2 l3 R1 R2 R3 h1 h2 h.
  induction h1 in l3, h2 |- ×.
  - destruct l3.
    + inversion h2.
    + inversion h2. subst.
    specialize (h _ _ _ _ p X) as hh.
    destruct hh as [? [? ?]].
    eexists. constructor.
      × constructor; eassumption.
      × constructor ; eauto.
  - destruct l3.
    + inversion h2.
    + inversion h2. subst.
    specialize (IHh1 _ X0). destruct IHh1 as [? [? ?]].
    eexists. constructor.
      × eapply OnOne2All_tl. eassumption.
      × constructor ; eauto.
Qed.

Lemma OnOne2All_impl_exist_and_All_r :
  ∀ A B (i : list B) (l1 l2 l3 : list A) R1 R2 R3,
    OnOne2All R1 i l1 l2 →
    All2 R2 l2 l3 →
    (∀ i x x' y, R1 i x y → R2 y x' → ∑ z : A, R3 i x z × R2 z x') →
    ∑ l4, ( OnOne2All R3 i l1 l4 × All2 R2 l4 l3 ).
Proof.
  intros A B i l1 l2 l3 R1 R2 R3 h1 h2 h.
  induction h1 in l3, h2 |- ×.
  - destruct l3.
    + inversion h2.
    + inversion h2. subst.
      specialize (h _ _ _ _ p X) as hh.
      destruct hh as [? [? ?]].
      eexists. split.
      × constructor; eassumption.
      × constructor ; eauto.
  - destruct l3.
    + inversion h2.
    + inversion h2. subst.
      specialize (IHh1 _ X0). destruct IHh1 as [? [? ?]].
      eexists. split.
      × eapply OnOne2All_tl. eassumption.
      × constructor ; eauto.
Qed.

Lemma OnOne2All_split :
  ∀ A B (P : B → A → A → Type) i l l',
    OnOne2All P i l l' →
    ∑ i x y u v,
      P i x y ×
      (l = u ++ x :: v ∧
      l' = u ++ y :: v).
Proof.
  intros A B P i l l' h.
  induction h.
  - ∃ b, hd, hd', [], tl.
    intuition eauto.
  - destruct IHh as [i' [x [y [u [v ?]]]]].
    ∃ i', x, y, (hd :: u), v.
    intuition eauto. all: subst. all: reflexivity.
Qed.

Lemma OnOne2All_impl {A B} {P Q} {i : list B} {l l' : list A} :
  OnOne2All P i l l' →
  (∀ i x y, P i x y → Q i x y) →
  OnOne2All Q i l l'.
Proof.
  induction 1; constructor; intuition eauto.
Qed.

Lemma OnOne2All_nth_error {A B} {i : list B} (l l' : list A) n t P :
  OnOne2All P i l l' →
  nth_error l n = Some t →
  ∑ t', (nth_error l' n = Some t') ×
  ((t = t') + (∑ i', (nth_error i n = Some i') × P i' t t')).
Proof.
  induction 1 in n |- ×.
  destruct n; simpl.
  - intros [= ->]. ∃ hd'. intuition auto. now right; ∃ b.
  - intros hnth. ∃ t; intuition auto.
  - destruct n; simpl; rewrite ?Nat.add_succ_r /=; auto.
    intros [= ->]. ∃ t; intuition auto.
Qed.

Lemma OnOne2All_nth_error_r {A B} (i : list B) (l l' : list A) n t' P :
  OnOne2All P i l l' →
  nth_error l' n = Some t' →
  ∑ t, (nth_error l n = Some t) ×
  ((t = t') + (∑ i', (nth_error i n = Some i') × P i' t t')).
Proof.
  induction 1 in n |- ×.
  destruct n; simpl.
  - intros [= ->]. ∃ hd; intuition auto.
    now right; ∃ b.
  - ∃ t'. intuition auto.
  - destruct n; simpl; auto.
    intros [= ->]. ∃ t'; intuition auto.
Qed.

Lemma OnOne2All_impl_All_r {A B} (P : B → A → A → Type) (Q : A → Type) i l l' :
  (∀ i x y, Q x → P i x y → Q y) →
  OnOne2All P i l l' → All Q l → All Q l'.
Proof.
  intros HPQ.
  induction 1; intros H; depelim H; constructor; auto.
  now eapply HPQ.
Qed.

Lemma OnOne2All_nth_error_impl {A B} (P : A → B → B → Type) il l l' :
  OnOne2All P il l l' →
  OnOne2All (fun i x y ⇒ (∑ ni, nth_error il ni = Some i) × P i x y) il l l'.
Proof.
  induction 1.
  - econstructor ⇒ //.
    split ⇒ //.
    ∃ 0; reflexivity.
  - constructor. eapply (OnOne2All_impl IHX).
    intros i x y [[ni hni] ?].
    split; auto. ∃ (S ni). apply hni.
Qed.

Lemma All2_Forall2 {A B} {P : A → B → Prop} {l l'} :
  All2 P l l' → Forall2 P l l'.
Proof.
  induction 1; eauto.
Qed.

Lemma Forall2_All2 {A B} {P : A → B → Prop} l l' : Forall2 P l l' → All2 P l l'.
Proof.
  intros f; induction l in l', f |- *; destruct l'; try constructor; auto.
  elimtype False. inv f.
  elimtype False. inv f.
  inv f; auto.
  apply IHl. inv f; auto.
Qed.

Lemma All2_map_left_equiv {A B C} {P : A → C → Type} {l l'} {f : B → A} :
  All2 (fun x y ⇒ P (f x) y) l l' <~> All2 P (map f l) l'.
Proof. intros. rewrite -{2}(map_id l'). eapply All2_map_equiv; eauto. Qed.

Lemma All2_map_right_equiv {A B C} {P : A → C → Type} {l l'} {f : B → C} :
  All2 (fun x y ⇒ P x (f y)) l l' <~> All2 P l (map f l').
Proof. intros. rewrite -{2}(map_id l). eapply All2_map_equiv; eauto. Qed.

Lemma All2_map_left {A B C} {P : A → C → Type} {l l'} {f : B → A} :
  All2 (fun x y ⇒ P (f x) y) l l' → All2 P (map f l) l'.
Proof. apply All2_map_left_equiv. Qed.

Lemma All2_map_right {A B C} {P : A → C → Type} {l l'} {f : B → C} :
  All2 (fun x y ⇒ P x (f y)) l l' → All2 P l (map f l').
Proof. apply All2_map_right_equiv. Qed.

Definition All2_map_left_inv {A B C} {P : A → C → Type} {l l'} {f : B → A} :
  All2 P (map f l) l' → All2 (fun x y ⇒ P (f x) y) l l' := (snd All2_map_left_equiv).
Definition All2_map_right_inv {A B C} {P : A → C → Type} {l l'} {f : B → C} :
  All2 P l (map f l') → All2 (fun x y ⇒ P x (f y)) l l' := (snd All2_map_right_equiv).

Ltac toAll :=
  match goal with
  | H : is_true (forallb _ _) |- _ ⇒ apply forallb_All in H

  | |- is_true (forallb _ _) ⇒ apply All_forallb

  | H : Forall _ ?x |- _ ⇒ apply Forall_All in H

  | |- Forall _ _ ⇒ apply All_Forall

  | H : Forall2 _ _ _ |- _ ⇒ apply Forall2_All2 in H

  | |- Forall2 _ _ _ ⇒ apply All2_Forall2

  | H : is_true (forallb2 _ _ _) |- _ ⇒ apply forallb2_All2 in H

  | |- is_true (forallb2 _ _ _) ⇒ apply All2_forallb2

  | H : All _ ?x, H' : All _ ?x |- _ ⇒
    apply (All_mix H) in H'; clear H

  | H : Alli _ _ ?x, H' : Alli _ _ ?x |- _ ⇒
    apply (Alli_mix H) in H'; clear H

  | H : All _ ?x, H' : All2 _ ?x _ |- _ ⇒
    apply (All2_All_mix_left H) in H'; clear H

  | H : All _ ?x, H' : All2 _ _ ?x |- _ ⇒
    apply (All2_All_mix_right H) in H'; clear H

  | H : All _ ?x, H' : All2i _ _ ?x _ |- _ ⇒
    apply (All2i_All_mix_left H) in H'; clear H

  | H : All _ ?x, H' : All2i _ _ _ ?x |- _ ⇒
    apply (All2i_All_mix_right H) in H'; clear H

  | |- All _ (map _ _) ⇒ apply All_map

  | H : All _ (map _ _) |- _ ⇒ apply All_map_inv in H

  | |- All _ (rev_map _ _) ⇒ apply All_rev_map

  | |- All _ (List.rev _) ⇒ apply All_rev

  | |- All2 _ (map _ _) (map _ _) ⇒ apply All2_map

  | |- All2 _ _ (map _ _) ⇒ apply All2_map_right

  | |- All2 _ (map _ _) _ ⇒ apply All2_map_left
  end.

Lemma All_map_eq {A B} {l} {f g : A → B} :
  All (fun x ⇒ f x = g x) l →
  map f l = map g l.
Proof.
  induction 1; simpl; trivial.
  now rewrite IHX p.
Qed.

Lemma All_map_id {A} {l} {f : A → A} :
  All (fun x ⇒ f x = x) l →
  map f l = l.
Proof.
  induction 1; simpl; trivial.
  now rewrite IHX p.
Qed.

Ltac All_map :=
  match goal with
    |- map _ _ = map _ _ ⇒ apply All_map_eq
  | |- map _ _ = _ ⇒ apply All_map_id
  end.

Lemma forall_forallb_map_spec {A B : Type} {P : A → Prop} {p : A → bool}
      {l : list A} {f g : A → B} :
    Forall P l → is_true (forallb p l) →
    (∀ x : A, P x → is_true (p x) → f x = g x) → map f l = map g l.
Proof.
  induction 1; simpl; trivial.
  rewrite andb_and. intros [px pl] Hx.
  f_equal. now apply Hx. now apply IHForall.
Qed.

Lemma forall_forallb_forallb_spec {A : Type} {P : A → Prop} {p : A → bool}
      {l : list A} {f : A → bool} :
    Forall P l → is_true (forallb p l) →
    (∀ x : A, P x → is_true (p x) → is_true (f x)) → is_true (forallb f l).
Proof.
  induction 1; simpl; trivial.
  rewrite !andb_and. intros [px pl] Hx. eauto.
Qed.

Lemma Forall_map {A B} (P : B → Prop) (f : A → B) l : Forall (fun x ⇒ P (f x)) l → Forall P (map f l).
Proof.
  induction 1; constructor; auto.
Qed.

Lemma Forall_map_inv {A B} (P : B → Prop) (f : A → B) l : Forall P (map f l) → Forall (fun x ⇒ P (f x)) l.
Proof. induction l; intros Hf; inv Hf; try constructor; eauto. Qed.

Lemma Forall_impl {A} {P Q : A → Prop} {l} :
  Forall P l → (∀ x, P x → Q x) → Forall Q l.
Proof.
  induction 1; constructor; auto.
Qed.

Lemma Forall_app {A} (P : A → Prop) l l' : Forall P (l ++ l') → Forall P l ∧ Forall P l'.
Proof.
  induction l; intros H. split; try constructor. apply H.
  inversion_clear H. split; intuition auto.
Qed.

Lemma Forall_last {A} (P : A → Prop) a l : l ≠ [] → Forall P l → P (last l a).
Proof.
  intros. induction H0.
  - congruence.
  - destruct l.
    + cbn. eauto.
    + cbn. eapply IHForall. congruence.
Qed.

Lemma All_safe_nth {A} {P : A → Type} {Γ n} (isdecl : n < length Γ) : All P Γ →
   P (safe_nth Γ (exist _ n isdecl)).
Proof.
  induction 1 in n, isdecl |- ×.
  exfalso. inversion isdecl.
  destruct n. simpl. auto.
  simpl in ×. eapply IHX.
Qed.

Definition size := nat.


Section All_size.
  Context {A} (P : A → Type) (fn : ∀ x1, P x1 → size).
  Fixpoint all_size {l1 : list A} (f : All P l1) : size :=
  match f with
  | All_nil ⇒ 0
  | All_cons px pl ⇒ fn _ px + all_size pl
  end.
End All_size.

Section Alli_size.
  Context {A} (P : nat → A → Type) (fn : ∀ n x1, P n x1 → size).
  Fixpoint alli_size {l1 : list A} {n} (f : Alli P n l1) : size :=
  match f with
  | Alli_nil ⇒ 0
  | Alli_cons px pl ⇒ fn _ _ px + alli_size pl
  end.
End Alli_size.

Section All2_size.
  Context {A B} (P : A → B → Type) (fn : ∀ x1 x2, P x1 x2 → size).
  Fixpoint all2_size {l1 l2} (f : All2 P l1 l2) : size :=
  match f with
  | All2_nil ⇒ 0
  | All2_cons rxy rll' ⇒ fn _ _ rxy + all2_size rll'
  end.
End All2_size.

Section All2i_size.
  Context {A B} (P : nat → A → B → Type) (fn : ∀ i x1 x2, P i x1 x2 → size).
  Fixpoint all2i_size {n l1 l2} (f : All2i P n l1 l2) : size :=
  match f with
  | All2i_nil ⇒ 0
  | All2i_cons rxy rll' ⇒ fn _ _ _ rxy + all2i_size rll'
  end.
End All2i_size.

Lemma All2i_impl {A B R R' n l l'} :
    @All2i A B R n l l' →
    (∀ i x y, R i x y → R' i x y) →
    All2i R' n l l'.
Proof.
  intros ha h.
  induction ha. 1: constructor.
  constructor. 2: assumption.
  eapply h. assumption.
Qed.

Ltac close_Forall :=
  match goal with
  | H : Forall _ _ |- Forall _ _ ⇒ apply (Forall_impl H); clear H; simpl
  | H : All _ _ |- All _ _ ⇒ apply (All_impl H); clear H; simpl
  | H : OnOne2 _ _ _ |- OnOne2 _ _ _ ⇒ apply (OnOne2_impl H); clear H; simpl
  | H : OnOne2i _ _ _ _ |- OnOne2i _ _ _ _ ⇒ apply (OnOne2_impl H); clear H; simpl
  | H : OnOne2All _ _ _ _ |- OnOne2All _ _ _ _ ⇒ apply (OnOne2All_impl H); clear H; simpl
  | H : All2 _ _ _ |- All2 _ _ _ ⇒ apply (All2_impl H); clear H; simpl
  | H : All2i _ _ _ _ |- All2i _ _ _ _ ⇒ apply (All2i_impl H); clear H; simpl
  | H : All2 _ _ _ |- All _ _ ⇒
    (apply (All2_All_left H) || apply (All2_All_right H)); clear H; simpl
  | H : All2i _ _ _ _ |- All _ _ ⇒
    (apply (All2i_All_left H) || apply (All2i_All_right H)); clear H; simpl
  end.

Lemma All2_non_nil {A B} (P : A → B → Type) (l : list A) (l' : list B) :
  All2 P l l' → l ≠ nil → l' ≠ nil.
Proof.
  induction 1; congruence.
Qed.

Lemma nth_error_forall {A} {P : A → Prop} {l : list A} {n x} :
  nth_error l n = Some x → Forall P l → P x.
Proof.
  intros Hnth HPl. induction HPl in n, Hnth |- ×. destruct n; discriminate.
  revert Hnth. destruct n. now intros [= ->].
  intros H'; eauto.
Qed.

Lemma nth_error_all {A} {P : A → Type} {l : list A} {n x} :
  nth_error l n = Some x → All P l → P x.
Proof.
  intros Hnth HPl.
  induction l in n, Hnth, HPl |- ×. destruct n; discriminate.
  destruct n; cbn in Hnth.
  - inversion Hnth. subst. inversion HPl. eauto.
  - eapply IHl. eassumption. inversion HPl. eassumption.
Defined.

Lemma nth_error_alli {A} {P : nat → A → Type} {l : list A} {n x} {k} :
  nth_error l n = Some x → Alli P k l → P (k + n) x.
Proof.
  intros Hnth HPl. induction HPl in n, Hnth |- ×.
  destruct n; discriminate.
  revert Hnth. destruct n. intros [= ->]. now rewrite Nat.add_0_r.
  intros H'; eauto. rewrite <- Nat.add_succ_comm. eauto.
Qed.

Lemma All_map_id' {A} {P : A → Type} {l} {f} :
  All P l →
  (∀ x, P x → f x = x) →
  map f l = l.
Proof.
  induction 1; simpl; f_equal; intuition auto.
  f_equal; auto.
Qed.

Lemma Alli_mapi_spec {A B} {P : nat → A → Type} {l} {f g : nat → A → B} {n} :
  Alli P n l → (∀ n x, P n x → f n x = g n x) →
  mapi_rec f l n = mapi_rec g l n.
Proof.
  induction 1; simpl; trivial.
  intros Heq. rewrite Heq; f_equal; auto.
Qed.

Lemma Alli_mapi_id {A} {P : nat → A → Type} {l} {f} {n} :
  Alli P n l →
  (∀ n x, P n x → f n x = x) →
  mapi_rec f l n = l.
Proof.
  induction 1; simpl; f_equal; intuition auto.
  f_equal; eauto.
Qed.

Lemma Alli_map_id {A} {P : nat → A → Type} {l} {f} {n} :
  Alli P n l →
  (∀ n x, P n x → f x = x) →
  map f l = l.
Proof.
  induction 1; simpl; f_equal; intuition auto.
  f_equal; eauto.
Qed.

Lemma forallb_map {A B} (f : A → B) (l : list A) p :
  forallb p (map f l) = forallb (fun x ⇒ p (f x)) l.
Proof.
  induction l in p, f |- *; simpl; rewrite ?andb_and;
    intuition (f_equal; auto).
Qed.

Lemma All_forallb' {A} P (l : list A) (p : A → bool) :
  All P l →
  (∀ x, P x → is_true (p x)) →
  is_true (forallb p l).
Proof.
  induction 1 in p |- *; simpl; rewrite ?andb_and;
    intuition auto.
Qed.

Lemma forallb_Forall' {A} (P : A → Prop) (l : list A) p :
  is_true (forallb p l) →
  (∀ x, is_true (p x) → P x) →
  Forall P l.
Proof.
  induction l in p |- *; simpl. constructor.
  intros. constructor; eauto; rewrite → andb_and in H; intuition eauto.
Qed.

Lemma forallb_skipn {A} (p : A → bool) n l :
  is_true (forallb p l) →
  is_true (forallb p (skipn n l)).
Proof.
  induction l in n |- *; destruct n; simpl; try congruence.
  intros. apply IHl. rewrite → andb_and in H; intuition.
Qed.

Lemma Forall_forallb_eq_forallb {A} (P : A → Prop) (p q : A → bool) l :
  Forall P l →
  (∀ x, P x → p x = q x) →
  forallb p l = forallb q l.
Proof.
  induction 1; simpl; intuition (f_equal; auto).
Qed.

Lemma forallb2_length {A B} (p : A → B → bool) l l' : is_true (forallb2 p l l') → length l = length l'.
Proof.
  induction l in l' |- *; destruct l'; simpl; try congruence.
  rewrite !andb_and. intros [Hp Hl]. erewrite IHl; eauto.
Qed.

Lemma map_option_Some X (L : list (option X)) t : map_option_out L = Some t → All2 (fun x y ⇒ x = Some y) L t.
Proof.
  intros. induction L in t, H |- *; cbn in ×.
  - inv H. econstructor.
  - destruct a. destruct (map_option_out L). all:inv H. eauto.
Qed.

Lemma Alli_map_option_out_mapi_Some_spec {A B} (f g : nat → A → option B) l t P :
  Alli P 0 l →
  (∀ i x t, P i x → f i x = Some t → g i x = Some t) →
  map_option_out (mapi f l) = Some t →
  map_option_out (mapi g l) = Some t.
Proof.
  unfold mapi. generalize 0.
  move ⇒ n Ha Hfg. move: t.
  induction Ha; try constructor; auto.
  move⇒ t /=. case E: (f n hd) ⇒ [b|]; try congruence.
  rewrite (Hfg n _ _ p E).
  case E' : map_option_out ⇒ [b'|]; try congruence.
  move⇒ [= <-]. now rewrite (IHHa _ E').
Qed.

Lemma All_mapi {A B} P f l k :
  Alli (fun i x ⇒ P (f i x)) k l → All P (@mapi_rec A B f l k).
Proof.
  induction 1; simpl; constructor; auto.
Qed.

Lemma All_Alli {A} {P : A → Type} {Q : nat → A → Type} {l n} :
  All P l →
  (∀ n x, P x → Q n x) →
  Alli Q n l.
Proof.
  intro H. revert n. induction H; constructor; eauto.
Qed.

Lemma All2_All_left_pack {A B} {P : A → B → Type} {l l'} :
  All2 P l l' → Alli (fun i x ⇒ ∑ y, (nth_error l i = Some x ∧ nth_error l' i = Some y) × P x y) 0 l.
Proof.
  intros HF. induction HF; constructor; intuition eauto.
  ∃ y; intuition eauto. clear -IHHF.
  revert IHHF. generalize l at 1 3. intros. apply Alli_shift.
  now simpl.
Qed.

Lemma map_option_out_All {A} P (l : list (option A)) l' :
  (All (on_Some_or_None P) l) →
  map_option_out l = Some l' →
  All P l'.
Proof.
  induction 1 in l' |- *; cbn; inversion 1; subst; try constructor.
  destruct x; [|discriminate].
  case_eq (map_option_out l); [|intro e; rewrite e in H1; discriminate].
  intros l0 e; rewrite e in H1; inversion H1; subst.
  constructor; auto.
Qed.

Lemma Forall_forallb {A} P (l : list A) (p : A → bool) :
  Forall P l →
  (∀ x, P x → is_true (p x)) →
  is_true (forallb p l).
Proof.
  induction 1 in p |- *; simpl; rewrite ?andb_and;
    intuition auto.
Qed.

Lemma Forall2_right {A B} (P : B → Prop) (l : list A) (l' : list B) :
  Forall2 (fun x y ⇒ P y) l l' → List.Forall (fun x ⇒ P x) l'.
Proof.
  induction 1; constructor; auto.
Qed.

Lemma Forall2_non_nil {A B} (P : A → B → Prop) (l : list A) (l' : list B) :
  Forall2 P l l' → l ≠ nil → l' ≠ nil.
Proof.
  induction 1; congruence.
Qed.

Lemma Forall2_length {A B} {P : A → B → Prop} {l l'} : Forall2 P l l' → #|l| = #|l'|.
Proof. induction 1; simpl; auto. Qed.

Lemma Forall2_app_r {A} (P : A → A → Prop) l l' r r' : Forall2 P (l ++ [r]) (l' ++ [r']) →
                                                         (Forall2 P l l') ∧ (P r r').
Proof.
  induction l in l', r |- *; simpl; intros; destruct l'; simpl in *; inversion H; subst.
  - intuition.
  - destruct l'; cbn in ×.
    + inversion H5.
    + inversion H5.
  - destruct l; cbn in ×.
    + inversion H5.
    + inversion H5.
  - specialize (IHl _ _ H5). intuition auto.
Qed.

Lemma Forall2_map_inv :
  ∀ {A B A' B'} (R : A' → B' → Prop) (f : A → A')
    (g : B → B') (l : list A) (l' : list B),
    Forall2 R (map f l) (map g l') →
    Forall2 (fun x y ⇒ R (f x) (g y)) l l'.
Proof.
  intros A B A' B' R f g l l' h.
  induction l in l', h |- × ; destruct l' ; try solve [ inversion h ].
  - constructor.
  - constructor.
    + inversion h. subst. assumption.
    + eapply IHl. inversion h. assumption.
Qed.

Lemma All2_length {A B} {P : A → B → Type} {l l'} : All2 P l l' → #|l| = #|l'|.
Proof. induction 1; simpl; auto. Qed.

Lemma All2_nth_error_None {A B} {P : A → B → Type} {l l'} n :
  All2 P l l' →
  nth_error l n = None →
  nth_error l' n = None.
Proof.
  intros Hall. revert n.
  induction Hall; destruct n; simpl; try congruence. auto.
Qed.

Lemma All2_All2_mix {A B} {P Q : A → B → Type} l l' :
  All2 P l l' →
  All2 Q l l' →
  All2 (fun x y ⇒ P x y × Q x y) l l'.
Proof.
  induction 1; intros H; depelim H; constructor; auto.
Qed.

Lemma All2_app_inv_l : ∀ (A B : Type) (R : A → B → Type),
    ∀ l1 l2 r,
      All2 R (l1 ++ l2) r →
      ∑ r1 r2,
        (r = r1 ++ r2)%list ×
        All2 R l1 r1 ×
        All2 R l2 r2.
Proof.
  intros. revert l2 r X. induction l1; intros; cbn in ×.
  - ∃ [], r. eauto.
  - depelim X.
    apply IHl1 in X as (?&?&?&?&?).
    subst.
    eexists _, _.
    split; [|split; eauto]; auto.
Qed.

Lemma All2_app_inv_r :
  ∀ A B R l r1 r2,
    @All2 A B R l (r1 ++ r2) →
    ∑ l1 l2,
      (l = l1 ++ l2)%list ×
      All2 R l1 r1 ×
      All2 R l2 r2.
Proof.
  intros A B R l r1 r2 h.
  induction r1 in l, r1, r2, h |- ×.
  - ∃ [], l; eauto.
  - depelim h.
    apply IHr1 in h as (?&?&?&?&?).
    subst.
    eexists _, _.
    split; [|split; eauto]; auto.
Qed.

Lemma All2_app_inv :
  ∀ A B (P : A → B → Type) l1 l2 r1 r2,
    #|l1| = #|r1| →
    All2 P (l1 ++ l2) (r1 ++ r2) →
    All2 P l1 r1 × All2 P l2 r2.
Proof.
  intros A B P l1 l2 r1 r2 e h.
  apply All2_app_inv_l in h as (w1&w2&e1&h1&h2).
  apply app_inj_length_l in e1 as (->&->); auto.
  apply All2_length in h1.
  apply All2_length in h2.
  congruence.
Qed.

Lemma All2_rect_rev : ∀ (A B : Type) (R : A → B → Type) (P : ∀ (l : list A) (l0 : list B), Type),
    P [] [] →
    (∀ (x : A) (y : B) (l : list A) (l' : list B) (r : R x y) (a : All2 R l l'),
        P l l' → P (l ++ [x])%list (l' ++ [y]))%list →
    ∀ (l : list A) (l0 : list B) (a : All2 R l l0), P l l0.
Proof.
  intros. revert l0 a. induction l using rev_ind; cbn; intros.
  - inv a. eauto.
  - eapply All2_app_inv_l in a as (?&?&?&?&?). subst.
    inv a0. inv X2. eauto.
Qed.

Lemma All2_ind_rev : ∀ (A B : Type) (R : A → B → Type) (P : ∀ (l : list A) (l0 : list B), Prop),
    P [] [] →
    (∀ (x : A) (y : B) (l : list A) (l' : list B) (r : R x y) (a : All2 R l l'),
        P l l' → P (l ++ [x])%list (l' ++ [y]))%list →
    ∀ (l : list A) (l0 : list B) (a : All2 R l l0), P l l0.
Proof.
  intros.
  eapply All2_rect_rev; eauto.
Qed.

Lemma All2_app :
  ∀ (A B : Type) (R : A → B → Type),
  ∀ l1 l2 l1' l2',
    All2 R l1 l1' →
    All2 R l2 l2' →
    All2 R (l1 ++ l2) (l1' ++ l2').
Proof.
  induction 1 ; cbn ; eauto.
Qed.

Lemma All2_rev (A B : Type) (P : A → B → Type) l l' :
  All2 P l l' →
  All2 P (List.rev l) (List.rev l').
Proof.
  induction 1. constructor.
  simpl. eapply All2_app; auto.
Qed.

Lemma All_All2_flex {A B} (P : A → Type) (Q : A → B → Type) l l' :
  All P l →
  (∀ x y, In y l' → P x → Q x y) →
  length l' = length l →
  All2 Q l l'.
Proof.
  intros H1 H2 Hl.
  induction H1 in l', H2, Hl |- *; destruct l'; depelim Hl.
  - econstructor.
  - econstructor; firstorder. eapply IHAll; firstorder.
Qed.

Lemma All_All_All2 {A} (P Q : A → Prop) l l' :
  All P l → All Q l' → #|l| = #|l'| → All2 (fun x y ⇒ P x ∧ Q y) l l'.
Proof.
  induction l in l' |- *; destruct l'; simpl; auto ⇒ //.
  intros ha hb. specialize (IHl l'); depelim ha; depelim hb.
  constructor; auto.
Qed.

Lemma All2_impl_In {A B} {P Q : A → B → Type} {l l'} :
  All2 P l l' →
  (∀ x y, In x l → In y l' → P x y → Q x y) →
  All2 Q l l'.
Proof.
  revert l'. induction l; intros; inversion X.
  - econstructor.
  - subst. econstructor. eapply X0. firstorder. firstorder. eauto.
    eapply IHl. eauto. intros.
    eapply X0. now right. now right. eauto.
Qed.

Lemma Forall2_skipn A B (P : A → B → Prop) l l' n:
  Forall2 P l l' → Forall2 P (skipn n l) (skipn n l').
Proof.
  revert l l'; induction n; intros.
  - unfold skipn. eauto.
  - cbv [skipn]. fold (@skipn A n). fold (@skipn B n).
    inversion H; subst. econstructor.
    eauto.
Qed.

Lemma Forall2_nth_error_Some {A B} {P : A → B → Prop} {l l'} n t :
  Forall2 P l l' →
  nth_error l n = Some t →
  ∃ t' : B, (nth_error l' n = Some t') ∧ P t t'.
Proof.
  intros Hall. revert n.
  induction Hall; destruct n; simpl; try congruence. intros [= ->]. ∃ y. intuition auto.
  eauto.
Qed.

Lemma Forall2_impl {A B} {P Q : A → B → Prop} {l l'} :
    Forall2 P l l' →
    (∀ x y, P x y → Q x y) →
    Forall2 Q l l'.
Proof.
  induction 1; constructor; auto.
Qed.

Lemma app_Forall :
  ∀ A P (l1 l2 : list A),
    Forall P l1 →
    Forall P l2 →
    Forall P (l1 ++ l2).
Proof.
  intros A P l1 l2 h1 h2.
  revert l2 h2.
  induction h1 ; intros l2 h2.
  - assumption.
  - cbn. constructor ; try assumption.
    eapply IHh1. assumption.
Qed.

Lemma rev_Forall :
  ∀ A P l,
    Forall P l →
    Forall P (@List.rev A l).
Proof.
  intros A P l h.
  induction l.
  - cbn. constructor.
  - cbn. inversion h; subst.
    specialize (IHl ltac:(assumption)).
    eapply app_Forall ; try assumption.
    repeat constructor. assumption.
Qed.

Lemma Forall2_impl' {A B} {P Q : A → B → Prop} {l l'} :
    Forall2 P l l' →
    Forall (fun x ⇒ ∀ y, P x y → Q x y) l →
    Forall2 Q l l'.
Proof.
  induction 1; constructor;
    inversion H1; intuition.
Qed.

Lemma Forall2_Forall {A R l} : @Forall2 A A R l l → Forall (fun x ⇒ R x x) l.
Proof.
  induction l. constructor.
  inversion 1; constructor; intuition.
Qed.

Lemma All2_All {A R l} : @All2 A A R l l → All (fun x ⇒ R x x) l.
Proof.
  induction l. constructor.
  inversion 1; constructor; intuition.
Qed.

Lemma Forall_Forall2 {A R l} : Forall (fun x ⇒ R x x) l → @Forall2 A A R l l.
Proof.
  induction l. constructor.
  inversion 1; constructor; intuition.
Qed.

Lemma Forall_True {A} {P : A → Prop} l : (∀ x, P x) → Forall P l.
Proof.
  intro H. induction l; now constructor.
Qed.

Lemma Forall2_True {A B} {R : A → B → Prop} l l'
  : (∀ x y, R x y) → #|l| = #|l'| → Forall2 R l l'.
Proof.
  intro H. revert l'; induction l; simpl;
    intros [] e; try discriminate e; constructor.
  easy.
  apply IHl. now apply eq_add_S.
Qed.

Lemma Forall2_map {A B A' B'} (R : A' → B' → Prop) (f : A → A') (g : B → B') l l'
  : Forall2 (fun x y ⇒ R (f x) (g y)) l l' → Forall2 R (map f l) (map g l').
Proof.
  induction 1; constructor; auto.
Qed.

Lemma Forall2_map_right {A B C} (P : A → B → Prop) (f : C → B) (l : list A) (l' : list C) :
  Forall2 P l (map f l') ↔ Forall2 (fun x y ⇒ P x (f y)) l l'.
Proof.
  split; intros.
  + eapply Forall2_map_inv. now rewrite map_id.
  + rewrite -(map_id l). now eapply Forall2_map.
Qed.

Lemma Forall2_and {A B} (R R' : A → B → Prop) l l'
  : Forall2 R l l' → Forall2 R' l l' → Forall2 (fun x y ⇒ R x y ∧ R' x y) l l'.
Proof.
  induction 1.
  intro; constructor.
  inversion_clear 1.
  constructor; intuition.
Defined.

Lemma Forall_Forall2_and {A B} {R : A → B → Prop} {P l l'}
  : Forall2 R l l' → Forall P l → Forall2 (fun x y ⇒ P x ∧ R x y) l l'.
Proof.
  induction 1.
  intro; constructor.
  inversion_clear 1.
  constructor; intuition.
Defined.

Lemma Forall_Forall2_and' {A B} {R : A → B → Prop} {P l l'}
  : Forall2 R l l' → Forall P l' → Forall2 (fun x y ⇒ R x y ∧ P y) l l'.
Proof.
  induction 1.
  intro; constructor.
  inversion_clear 1.
  constructor; intuition.
Defined.

Lemma Forall2_nth :
  ∀ A B P l l' n (d : A) (d' : B),
    Forall2 P l l' →
    P d d' →
    P (nth n l d) (nth n l' d').
Proof.
  intros A B P l l' n d d' h hd.
  induction n in l, l', h |- ×.
  - destruct h.
    + assumption.
    + assumption.
  - destruct h.
    + assumption.
    + simpl. apply IHn. assumption.
Qed.

Lemma Forall2_nth_error_Some_l :
  ∀ A B (P : A → B → Prop) l l' n t,
    nth_error l n = Some t →
    Forall2 P l l' →
    ∃ t',
      nth_error l' n = Some t' ∧
      P t t'.
Proof.
  intros A B P l l' n t e h.
  induction n in l, l', t, e, h |- ×.
  - destruct h.
    + cbn in e. discriminate.
    + cbn in e. inversion e. subst.
      ∃ y. split ; auto.
  - destruct h.
    + cbn in e. discriminate.
    + cbn in e. apply IHn with (l' := l') in e ; assumption.
Qed.

Lemma Forall2_nth_error_Some_r :
  ∀ A B (P : A → B → Prop) l l' n t,
    nth_error l' n = Some t →
    Forall2 P l l' →
    ∃ t',
      nth_error l n = Some t' ∧
      P t' t.
Proof.
  intros A B P l l' n t e h.
  induction n in l, l', t, e, h |- ×.
  - destruct h.
    + cbn in e. discriminate.
    + cbn in e. inversion e. subst.
      ∃ x. split ; auto.
  - destruct h.
    + cbn in e. discriminate.
    + cbn in e. apply IHn with (l := l) in e ; assumption.
Qed.

Lemma Forall2_nth_error_None_l :
  ∀ A B (P : A → B → Prop) l l' n,
    nth_error l n = None →
    Forall2 P l l' →
    nth_error l' n = None.
Proof.
  intros A B P l l' n e h.
  induction n in l, l', e, h |- ×.
  - destruct h.
    + reflexivity.
    + cbn in e. discriminate e.
  - destruct h.
    + reflexivity.
    + cbn in e. cbn. eapply IHn ; eauto.
Qed.

Lemma Forall2_trans :
  ∀ A (P : A → A → Prop),
    RelationClasses.Transitive P →
    RelationClasses.Transitive (Forall2 P).
Proof.
  intros A P hP l1 l2 l3 h1 h2.
  induction h1 in l3, h2 |- ×.
  - inversion h2. constructor.
  - inversion h2. constructor.
    + eapply hP ; eauto.
    + eapply IHh1 ; eauto.
Qed.

Lemma Forall2_rev :
  ∀ A B R l l',
    @Forall2 A B R l l' →
    Forall2 R (List.rev l) (List.rev l').
Proof.
  intros A B R l l' h.
  induction h.
  - constructor.
  - cbn. eapply Forall2_app ; eauto.
Qed.

Lemma Forall2_mapi :
  ∀ A B A' B' (R : A' → B' → Prop) (f : nat → A → A') (g : nat → B → B') l l',
    Forall2 (fun x y ⇒ ∀ i, R (f i x) (g i y)) l l' →
    Forall2 R (mapi f l) (mapi g l').
Proof.
  intros A B A' B' R f g l l' h.
  unfold mapi. generalize 0. intro i.
  induction h in i |- ×.
  - constructor.
  - cbn. constructor ; eauto.
Qed.

Lemma All2_trans {A} (P : A → A → Type) :
  CRelationClasses.Transitive P →
  CRelationClasses.Transitive (All2 P).
Proof.
  intros HP x y z H. induction H in z |- ×.
  intros Hyz. inv Hyz. constructor.
  intros Hyz. inv Hyz. constructor; auto.
  now transitivity y.
Qed.

Lemma All2_trans' {A B C}
      (P : A → B → Type) (Q : B → C → Type) (R : A → C → Type)
      (H : ∀ x y z, P x y × Q y z → R x z) {l1 l2 l3}
  : All2 P l1 l2 → All2 Q l2 l3 → All2 R l1 l3.
Proof.
  induction 1 in l3 |- ×.
  - inversion 1; constructor.
  - inversion 1; subst. constructor; eauto.
Qed.

Lemma All2_nth :
  ∀ A B P l l' n (d : A) (d' : B),
    All2 P l l' →
    P d d' →
    P (nth n l d) (nth n l' d').
Proof.
  intros A B P l l' n d d' h hd.
  induction n in l, l', h |- ×.
  - destruct h.
    + assumption.
    + assumption.
  - destruct h.
    + assumption.
    + simpl. apply IHn. assumption.
Qed.

Lemma All2_mapi :
  ∀ A B A' B' (R : A' → B' → Type) (f : nat → A → A') (g : nat → B → B') l l',
    All2 (fun x y ⇒ ∀ i, R (f i x) (g i y)) l l' →
    All2 R (mapi f l) (mapi g l').
Proof.
  intros A B A' B' R f g l l' h.
  unfold mapi. generalize 0. intro i.
  induction h in i |- ×.
  - constructor.
  - cbn. constructor ; eauto.
Qed.

Lemma All2_skipn :
  ∀ A B R l l' n,
    @All2 A B R l l' →
    @All2 A B R (skipn n l) (skipn n l').
Proof.
  intros A B R l l' n h.
  induction h in n |- ×.
  all: destruct n ; try econstructor ; eauto.
Qed.

Lemma All2_right_triv {A B} {l : list A} {l' : list B} P :
  All P l' → #|l| = #|l'| → All2 (fun _ b ⇒ P b) l l'.
Proof.
  induction 1 in l |- ×.
  all: cbn.
  all: intros.
  all: destruct l.
  all: cbn in ×.
  all: try (exfalso ; lia).
  all: try solve [ econstructor; eauto ].
Qed.

Lemma All_repeat {A} {n P} x :
  P x → @All A P (repeat x n).
Proof.
  induction n; cbn; econstructor; eauto.
Qed.

Lemma Forall2_Forall_right {A B} {P : A → B → Prop} {Q : B → Prop} {l l'} :
  Forall2 P l l' →
  (∀ x y, P x y → Q y) →
  Forall Q l'.
Proof.
  intros HF H. induction HF; constructor; eauto.
Qed.

Lemma All2_from_nth_error A B L1 L2 (P : A → B → Type) :
  #|L1| = #|L2| →
                (∀ n x1 x2, n < #|L1| → nth_error L1 n = Some x1
                                      → nth_error L2 n = Some x2
                                      → P x1 x2) →
                All2 P L1 L2.
Proof.
  revert L2; induction L1; cbn; intros.
  - destruct L2; inv H. econstructor.
  - destruct L2; inversion H. econstructor.
    eapply (X 0); cbn; eauto. lia.
    eapply IHL1. eauto.
    intros. eapply (X (S n)); cbn; eauto. lia.
Qed.

Lemma All2_nth_error {A B} {P : A → B → Type} {l l'} n t t' :
  All2 P l l' →
  nth_error l n = Some t →
  nth_error l' n = Some t' →
  P t t'.
Proof.
  intros Hall. revert n.
  induction Hall; destruct n; simpl; try congruence.
  eauto.
Qed.

Require Import MCSquash.

Lemma All2_swap :
  ∀ A B R l l',
    @All2 A B R l l' →
    All2 (fun x y ⇒ R y x) l' l.
Proof.
  intros A B R l l' h.
  induction h ; eauto.
Qed.

Lemma All2_firstn :
  ∀ A B R l l' n,
    @All2 A B R l l' →
    @All2 A B R (firstn n l) (firstn n l').
Proof.
  intros A B R l l' n h.
  induction h in n |- ×.
  all: destruct n ; try econstructor ; eauto.
Qed.

Lemma All2_impl' {A B} {P Q : A → B → Type} {l : list A} {l' : list B}
  : All2 P l l' → All (fun x ⇒ ∀ y, P x y → Q x y) l → All2 Q l l'.
Proof.
  induction 1. constructor.
  intro XX; inv XX.
  constructor; auto.
Defined.

Lemma All_All2 {A} {P : A → A → Type} {Q} {l : list A} :
  All Q l →
  (∀ x, Q x → P x x) →
  All2 P l l.
Proof.
  induction 1; constructor; auto.
Qed.

Lemma All2_nth_error_Some {A B} {P : A → B → Type} {l l'} n t :
  All2 P l l' →
  nth_error l n = Some t →
  { t' : B & (nth_error l' n = Some t') × P t t'}%type.
Proof.
  intros Hall. revert n.
  induction Hall; destruct n; simpl; try congruence.
  intros [= ->]. ∃ y. intuition auto.
  eauto.
Qed.

Lemma All2_nth_error_Some_r {A B} {P : A → B → Type} {l l'} n t' :
  All2 P l l' →
  nth_error l' n = Some t' →
  ∑ t, nth_error l n = Some t × P t t'.
Proof.
  intros Hall. revert n.
  induction Hall; destruct n; simpl; try congruence.
  intros [= ->]. ∃ x. intuition auto.
  eauto.
Qed.

Lemma All2_same {A} {P : A → A → Type} l : (∀ x, P x x) → All2 P l l.
Proof. induction l; constructor; auto. Qed.

Lemma All2_prod {A} P Q (l l' : list A) : All2 P l l' → All2 Q l l' → All2 (Trel_conj P Q) l l'.
Proof.
  induction 1; inversion 1; subst; auto.
Defined.

Lemma All2_prod_inv :
  ∀ A (P : A → A → Type) Q l l',
    All2 (Trel_conj P Q) l l' →
    All2 P l l' × All2 Q l l'.
Proof.
  intros A P Q l l' h.
  induction h.
  - auto.
  - destruct IHh. destruct r.
    split ; constructor ; auto.
Qed.

Lemma All2_sym {A} (P : A → A → Type) l l' :
  All2 P l l' → All2 (fun x y ⇒ P y x) l' l.
Proof.
  induction 1; constructor; auto.
Qed.

Lemma All2_symP {A} (P : A → A → Type) :
  CRelationClasses.Symmetric P →
  CRelationClasses.Symmetric (All2 P).
Proof.
  intros hP x y h. induction h.
  - constructor.
  - constructor ; eauto.
Qed.

Lemma All_All2_All2_mix {A B} (P : B → B → Type) (Q R : A → B → Type) l l' l'' :
  All (fun x ⇒ ∀ y z, Q x y → R x z → ∑ v, P y v × P z v) l → All2 Q l l' → All2 R l l'' →
  ∑ l''', All2 P l' l''' × All2 P l'' l'''.
Proof.
  intros H; induction H in l', l'' |- *;
  intros H' H''; inversion H'; inversion H''; subst.
  ∃ []; repeat constructor.
  destruct (IHAll _ _ X0 X2) as [? [? ?]]. destruct (p _ _ X X1) as [? [? ?]].
  ∃ (x1 :: x0). split; constructor; intuition auto.
Qed.

Lemma All2_nth_error_Some_right {A} {P : A → A → Type} {l l'} n t :
  All2 P l l' →
  nth_error l' n = Some t →
  { t' : A & (nth_error l n = Some t') × P t' t}%type.
Proof.
  intros Hall. revert n.
  induction Hall; destruct n; simpl; try congruence. intros [= ->]. ∃ x. intuition auto.
  eauto.
Qed.

Lemma All2_mix {A} {P Q : A → A → Type} {l l'} :
  All2 P l l' → All2 Q l l' → All2 (fun x y ⇒ (P x y × Q x y))%type l l'.
Proof.
  induction 1; intros HQ; inv HQ; constructor; eauto.
Qed.

Lemma All2_mix_inv {A} {P Q : A → A → Type} {l l'} :
  All2 (fun x y ⇒ (P x y × Q x y))%type l l' →
  (All2 P l l' × All2 Q l l').
Proof.
  induction 1; split; constructor; intuition eauto.
Qed.

Lemma All_forallb_map_spec {A B : Type} {P : A → Type} {p : A → bool}
      {l : list A} {f g : A → B} :
    All P l → forallb p l →
    (∀ x : A, P x → p x → f x = g x) → map f l = map g l.
Proof.
  induction 1; simpl; trivial.
  rewrite andb_and. intros [px pl] Hx.
  f_equal. now apply Hx. now apply IHX.
Qed.

Lemma All_forallb_forallb_spec {A : Type} {P : A → Type} {p : A → bool}
      {l : list A} {f : A → bool} :
    All P l → forallb p l →
    (∀ x : A, P x → p x → f x) → forallb f l.
Proof.
  induction 1; simpl; trivial.
  rewrite !andb_and. intros [px pl] Hx. eauto.
Qed.

Lemma All_forallb_eq_forallb {A} (P : A → Type) (p q : A → bool) l :
  All P l →
  (∀ x, P x → p x = q x) →
  forallb p l = forallb q l.
Proof.
  induction 1; simpl; intuition (f_equal; auto).
Qed.

Lemma forallb_nth {A} (l : list A) (n : nat) P d :
  forallb P l → n < #|l| → ∃ x, (nth n l d = x) ∧ P x.
Proof.
  induction l in n |- *; destruct n; simpl; auto; try easy.
  move/andb_and ⇒ [pa pl] pn. ∃ a; easy.
  move/andb_and ⇒ [pa pl] pn. specialize (IHl n pl).
  apply IHl; lia.
Qed.

Lemma forallb_nth' {A} {l : list A} {P} n d :
  forallb P l → {∃ x, (nth n l d = x) ∧ P x} + {nth n l d = d}.
Proof.
  intro H. destruct (le_lt_dec #|l| n) as [HH|HH].
  - rewrite nth_overflow; try assumption; now right.
  - left. eapply forallb_nth; assumption.
Qed.

Lemma forallb_impl {A} (p q : A → bool) l :
  (∀ x, In x l → p x → q x) → forallb p l → forallb q l.
Proof.
  intros H0 H1. eapply forallb_forall. intros x H2.
  eapply forallb_forall in H1; try eassumption.
  now eapply H0.
Qed.

Lemma forallb_iff {A} (p q : A → bool) l :
  (∀ x, In x l → p x ↔ q x) → forallb p l = forallb q l.
Proof.
  intros H0. eapply Forall_forallb_eq_forallb with (P:=fun x ⇒ In x l).
  - now apply Forall_forall.
  - intros; eapply eq_true_iff_eq; eauto.
Qed.

Lemma All2_eq :
  ∀ A (l l' : list A),
    All2 eq l l' →
    l = l'.
Proof.
  intros A l l' h.
  induction h ; eauto. subst. reflexivity.
Qed.

Lemma All_prod_inv :
  ∀ A P Q l,
    All (fun x : A ⇒ P x × Q x) l →
    All P l × All Q l.
Proof.
  intros A P Q l h.
  induction h.
  - split ; auto.
  - destruct IHh, p.
    split ; constructor ; auto.
Qed.

Lemma All_pair {A} (P Q : A → Type) l :
  All (fun x ⇒ P x × Q x) l <~> (All P l × All Q l).
Proof.
  split. induction 1; intuition auto.
  move⇒ [] Hl Hl'. induction Hl; depelim Hl'; intuition auto.
Qed.

Lemma All_prod :
  ∀ A P Q l,
    All P l →
    All Q l →
    All (fun x : A ⇒ P x × Q x) l.
Proof.
  intros A P Q l h1 h2.
  induction h1 in h2 |- ×.
  - constructor.
  - inversion h2.
    specialize (IHh1 X0). auto.
Qed.

Lemma All2i_mapi :
  ∀ A B C D R f g l l',
    @All2i A B (fun i x y ⇒ R i (f i x) (g i y)) 0 l l' →
    @All2i C D R 0 (mapi f l) (mapi g l').
Proof.
  intros A B C D R f g l l' h.
  unfold mapi.
  revert h.
  generalize 0. intros n h.
  induction h. 1: constructor.
  simpl. constructor. all: assumption.
Qed.

Lemma All2i_app :
  ∀ A B R n l1 l2 r1 r2,
    @All2i A B R n l1 r1 →
    All2i R (n + #|l1|) l2 r2 →
    All2i R n (l1 ++ l2) (r1 ++ r2).
Proof.
  intros A B R n l1 l2 r1 r2 h1 h2.
  induction h1 in r2, l2, h2 |- ×.
  - simpl in ×. replace (n + 0)%nat with n in h2 by lia. assumption.
  - simpl in ×. constructor. 1: assumption.
    eapply IHh1. replace (S (n + #|l|))%nat with (n + S #|l|)%nat by lia.
    assumption.
Qed.


Lemma All2i_mapi_rec {A B C D} (R : nat → A → B → Type)
      (l : list C) (l' : list D) (f : nat → C → A) (g : nat → D → B) n :
  All2i (fun n x y ⇒ R n (f n x) (g n y)) n l l' →
  All2i R n (mapi_rec f l n) (mapi_rec g l' n).
Proof.
  induction 1; constructor; auto.
Qed.

Lemma All2i_trivial {A B} (R : nat → A → B → Type) (H : ∀ n x y, R n x y) n l l' :
  #|l| = #|l'| → All2i R n l l'.
Proof.
  induction l in n, l' |- *; destruct l'; simpl; try discriminate; intros; constructor; auto.
Qed.

Lemma All2i_rev :
  ∀ A B R l l' n,
    @All2i A B R n l l' →
    All2i (fun i x y ⇒ R (n + #|l| - (S i)) x y) 0 (List.rev l) (List.rev l').
Proof.
  intros A B R l l' n h.
  induction h. 1: constructor.
  simpl. apply All2i_app.
  - eapply All2i_impl. 1: eassumption.
    simpl. intros ? ? ? ?.
    replace (n + S #|l| - S i) with (n + #|l| - i) by lia.
    assumption.
  - simpl. constructor. 2: constructor.
    rewrite List.rev_length.
    replace (n + S #|l| - S #|l|) with n by lia.
    assumption.
Qed.

Section All2i_len.