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(* Copyright (c) 2012-2013, Robbert Krebbers. *)
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(* This file is distributed under the terms of the BSD license. *)
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(** This file collects some trivial facts on the Coq types [nat] and [N] for
natural numbers, and the type [Z] for integers. It also declares some useful
notations. *)
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Require Export Eqdep PArith NArith ZArith NPeano.
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Require Import Qcanon.
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Require Export base decidable.
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Open Scope nat_scope.
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Coercion Z.of_nat : nat >-> Z.

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(** * Notations and properties of [nat] *)
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Reserved Notation "x ≤ y ≤ z" (at level 70, y at next level).
Reserved Notation "x ≤ y < z" (at level 70, y at next level).
Reserved Notation "x < y < z" (at level 70, y at next level).
Reserved Notation "x < y ≤ z" (at level 70, y at next level).

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Infix "≤" := le : nat_scope.
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Notation "x ≤ y ≤ z" := (x  y  y  z)%nat : nat_scope.
Notation "x ≤ y < z" := (x  y  y < z)%nat : nat_scope.
Notation "x < y < z" := (x < y  y < z)%nat : nat_scope.
Notation "x < y ≤ z" := (x < y  y  z)%nat : nat_scope.
Notation "(≤)" := le (only parsing) : nat_scope.
Notation "(<)" := lt (only parsing) : nat_scope.

Infix "`div`" := NPeano.div (at level 35) : nat_scope.
Infix "`mod`" := NPeano.modulo (at level 35) : nat_scope.

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Instance nat_eq_dec:  x y : nat, Decision (x = y) := eq_nat_dec.
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Instance nat_le_dec:  x y : nat, Decision (x  y) := le_dec.
Instance nat_lt_dec:  x y : nat, Decision (x < y) := lt_dec.
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Instance nat_inhabited: Inhabited nat := populate 0%nat.
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Instance: Injective (=) (=) S.
Proof. by injection 1. Qed.
Instance: PartialOrder ().
Proof. repeat split; repeat intro; auto with lia. Qed.
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Instance nat_le_pi:  x y : nat, ProofIrrel (x  y).
Proof.
  assert ( x y (p : x  y) y' (q : x  y'),
    y = y'  eq_dep nat (le x) y p y' q) as aux.
  { fix 3. intros x ? [|y p] ? [|y' q].
    * done.
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    * clear nat_le_pi. intros; exfalso; auto with lia.
    * clear nat_le_pi. intros; exfalso; auto with lia.
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    * injection 1. intros Hy. by case (nat_le_pi x y p y' q Hy). }
  intros x y p q.
  by apply (eq_dep_eq_dec (λ x y, decide (x = y))), aux.
Qed.
Instance nat_lt_pi:  x y : nat, ProofIrrel (x < y).
Proof. apply _. Qed.

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Definition sum_list_with {A} (f : A  nat) : list A  nat :=
  fix go l :=
  match l with
  | [] => 0
  | x :: l => f x + go l
  end.
Notation sum_list := (sum_list_with id).

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Lemma Nat_lt_succ_succ n : n < S (S n).
Proof. auto with arith. Qed.
Lemma Nat_mul_split_l n x1 x2 y1 y2 :
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  x2 < n  y2 < n  x1 * n + x2 = y1 * n + y2  x1 = y1  x2 = y2.
Proof.
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  intros Hx2 Hy2 E. cut (x1 = y1); [intros; subst;lia |].
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  revert y1 E. induction x1; simpl; intros [|?]; simpl; auto with lia.
Qed.
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Lemma Nat_mul_split_r n x1 x2 y1 y2 :
  x1 < n  y1 < n  x1 + x2 * n = y1 + y2 * n  x1 = y1  x2 = y2.
Proof. intros. destruct (Nat_mul_split_l n x2 x1 y2 y1); auto with lia. Qed.
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(** * Notations and properties of [positive] *)
Open Scope positive_scope.

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Infix "≤" := Pos.le : positive_scope.
Notation "x ≤ y ≤ z" := (x  y  y  z)%positive : positive_scope.
Notation "x ≤ y < z" := (x  y  y < z)%positive : positive_scope.
Notation "x < y < z" := (x < y  y < z)%positive : positive_scope.
Notation "x < y ≤ z" := (x < y  y  z)%positive : positive_scope.
Notation "(≤)" := Pos.le (only parsing) : positive_scope.
Notation "(<)" := Pos.lt (only parsing) : positive_scope.
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Notation "(~0)" := xO (only parsing) : positive_scope.
Notation "(~1)" := xI (only parsing) : positive_scope.

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Arguments Pos.of_nat _ : simpl never.
Instance positive_eq_dec:  x y : positive, Decision (x = y) := Pos.eq_dec.
Instance positive_inhabited: Inhabited positive := populate 1.

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Instance: Injective (=) (=) (~0).
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Proof. by injection 1. Qed.
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Instance: Injective (=) (=) (~1).
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Proof. by injection 1. Qed.

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(** Since [positive] represents lists of bits, we define list operations
on it. These operations are in reverse, as positives are treated as snoc
lists instead of cons lists. *)
Fixpoint Papp (p1 p2 : positive) : positive :=
  match p2 with
  | 1 => p1
  | p2~0 => (Papp p1 p2)~0
  | p2~1 => (Papp p1 p2)~1
  end.
Infix "++" := Papp : positive_scope.
Notation "(++)" := Papp (only parsing) : positive_scope.
Notation "( p ++)" := (Papp p) (only parsing) : positive_scope.
Notation "(++ q )" := (λ p, Papp p q) (only parsing) : positive_scope.

Fixpoint Preverse_go (p1 p2 : positive) : positive :=
  match p2 with
  | 1 => p1
  | p2~0 => Preverse_go (p1~0) p2
  | p2~1 => Preverse_go (p1~1) p2
  end.
Definition Preverse : positive  positive := Preverse_go 1.

Global Instance: LeftId (=) 1 (++).
Proof. intros p. induction p; simpl; intros; f_equal; auto. Qed.
Global Instance: RightId (=) 1 (++).
Proof. done. Qed.
Global Instance: Associative (=) (++).
Proof. intros ?? p. induction p; simpl; intros; f_equal; auto. Qed.
Global Instance:  p : positive, Injective (=) (=) (++ p).
Proof. intros p ???. induction p; simplify_equality; auto. Qed.

Lemma Preverse_go_app_cont p1 p2 p3 :
  Preverse_go (p2 ++ p1) p3 = p2 ++ Preverse_go p1 p3.
Proof.
  revert p1. induction p3; simpl; intros.
  * apply (IHp3 (_~1)).
  * apply (IHp3 (_~0)).
  * done.
Qed.
Lemma Preverse_go_app p1 p2 p3 :
  Preverse_go p1 (p2 ++ p3) = Preverse_go p1 p3 ++ Preverse_go 1 p2.
Proof.
  revert p1. induction p3; intros p1; simpl; auto.
  by rewrite <-Preverse_go_app_cont.
Qed.
Lemma Preverse_app p1 p2 :
  Preverse (p1 ++ p2) = Preverse p2 ++ Preverse p1.
Proof. unfold Preverse. by rewrite Preverse_go_app. Qed.

Lemma Preverse_xO p : Preverse (p~0) = (1~0) ++ Preverse p.
Proof Preverse_app p (1~0).
Lemma Preverse_xI p : Preverse (p~1) = (1~1) ++ Preverse p.
Proof Preverse_app p (1~1).

Fixpoint Plength (p : positive) : nat :=
  match p with
  | 1 => 0%nat
  | p~0 | p~1 => S (Plength p)
  end.
Lemma Papp_length p1 p2 :
  Plength (p1 ++ p2) = (Plength p2 + Plength p1)%nat.
Proof. induction p2; simpl; f_equal; auto. Qed.

Close Scope positive_scope.

(** * Notations and properties of [N] *)
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Infix "≤" := N.le : N_scope.
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Notation "x ≤ y ≤ z" := (x  y  y  z)%N : N_scope.
Notation "x ≤ y < z" := (x  y  y < z)%N : N_scope.
Notation "x < y < z" := (x < y  y < z)%N : N_scope.
Notation "x < y ≤ z" := (x < y  y  z)%N : N_scope.
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Notation "(≤)" := N.le (only parsing) : N_scope.
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Notation "(<)" := N.lt (only parsing) : N_scope.
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Infix "`div`" := N.div (at level 35) : N_scope.
Infix "`mod`" := N.modulo (at level 35) : N_scope.

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Instance: Injective (=) (=) Npos.
Proof. by injection 1. Qed.

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Instance N_eq_dec:  x y : N, Decision (x = y) := N.eq_dec.
Program Instance N_le_dec (x y : N) : Decision (x  y)%N :=
  match Ncompare x y with
  | Gt => right _
  | _ => left _
  end.
Next Obligation. congruence. Qed.
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Program Instance N_lt_dec (x y : N) : Decision (x < y)%N :=
  match Ncompare x y with
  | Lt => left _
  | _ => right _
  end.
Next Obligation. congruence. Qed.
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Instance N_inhabited: Inhabited N := populate 1%N.
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Instance: PartialOrder ()%N.
Proof.
  repeat split; red. apply N.le_refl. apply N.le_trans. apply N.le_antisymm.
Qed.
Hint Extern 0 (_  _)%N => reflexivity.
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(** * Notations and properties of [Z] *)
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Open Scope Z_scope.

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Infix "≤" := Z.le : Z_scope.
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Notation "x ≤ y ≤ z" := (x  y  y  z) : Z_scope.
Notation "x ≤ y < z" := (x  y  y < z) : Z_scope.
Notation "x < y < z" := (x < y  y < z) : Z_scope.
Notation "x < y ≤ z" := (x < y  y  z) : Z_scope.
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Notation "(≤)" := Z.le (only parsing) : Z_scope.
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Notation "(<)" := Z.lt (only parsing) : Z_scope.
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Infix "`div`" := Z.div (at level 35) : Z_scope.
Infix "`mod`" := Z.modulo (at level 35) : Z_scope.
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Infix "`quot`" := Z.quot (at level 35) : Z_scope.
Infix "`rem`" := Z.rem (at level 35) : Z_scope.
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Infix "≪" := Z.shiftl (at level 35) : Z_scope.
Infix "≫" := Z.shiftr (at level 35) : Z_scope.
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Instance Z_eq_dec:  x y : Z, Decision (x = y) := Z.eq_dec.
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Instance Z_le_dec:  x y : Z, Decision (x  y) := Z_le_dec.
Instance Z_lt_dec:  x y : Z, Decision (x < y) := Z_lt_dec.
Instance Z_inhabited: Inhabited Z := populate 1.
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Instance: PartialOrder ().
Proof.
  repeat split; red. apply Z.le_refl. apply Z.le_trans. apply Z.le_antisymm.
Qed.
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Lemma Z_pow_pred_r n m : 0 < m  n * n ^ (Z.pred m) = n ^ m.
Proof.
  intros. rewrite <-Z.pow_succ_r, Z.succ_pred. done. by apply Z.lt_le_pred.
Qed.
Lemma Z_quot_range_nonneg k x y : 0  x < k  0 < y  0  x `quot` y < k.
Proof.
  intros [??] ?.
  destruct (decide (y = 1)); subst; [rewrite Z.quot_1_r; auto |].
  destruct (decide (x = 0)); subst; [rewrite Z.quot_0_l; auto with lia |].
  split. apply Z.quot_pos; lia. transitivity x; auto. apply Z.quot_lt; lia.
Qed.
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(* Note that we cannot disable simpl for [Z.of_nat] as that would break
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tactics as [lia]. *)
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Arguments Z.to_nat _ : simpl never.
Arguments Z.mul _ _ : simpl never.
Arguments Z.add _ _ : simpl never.
Arguments Z.opp _ : simpl never.
Arguments Z.pow _ _ : simpl never.
Arguments Z.div _ _ : simpl never.
Arguments Z.modulo _ _ : simpl never.
Arguments Z.quot _ _ : simpl never.
Arguments Z.rem _ _ : simpl never.

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Lemma Z_mod_pos a b : 0 < b  0  a `mod` b.
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Proof. apply Z.mod_pos_bound. Qed.

Hint Resolve Z.lt_le_incl : zpos.
Hint Resolve Z.add_nonneg_pos Z.add_pos_nonneg Z.add_nonneg_nonneg : zpos.
Hint Resolve Z.mul_nonneg_nonneg Z.mul_pos_pos : zpos.
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Hint Resolve Z.pow_pos_nonneg Z.pow_nonneg: zpos.
Hint Resolve Z_mod_pos Z.div_pos : zpos.
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Hint Extern 1000 => lia : zpos.

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Lemma Z2Nat_inj_pow (x y : nat) : Z.of_nat (x ^ y) = x ^ y.
Proof.
  induction y as [|y IH].
  * by rewrite Z.pow_0_r, Nat.pow_0_r.
  * by rewrite Nat.pow_succ_r, Nat2Z.inj_succ, Z.pow_succ_r,
      Nat2Z.inj_mul, IH by auto with zpos.
Qed.
Lemma Z2Nat_inj_div x y : Z.of_nat (x `div` y) = x `div` y.
Proof.
  destruct (decide (y = 0%nat)); [by subst; destruct x |].
  apply Z.div_unique with (x `mod` y)%nat.
  { left. rewrite <-(Nat2Z.inj_le 0), <-Nat2Z.inj_lt.
    apply Nat.mod_bound_pos; lia. }
  by rewrite <-Nat2Z.inj_mul, <-Nat2Z.inj_add, <-Nat.div_mod.
Qed.
Lemma Z2Nat_inj_mod x y : Z.of_nat (x `mod` y) = x `mod` y.
Proof.
  destruct (decide (y = 0%nat)); [by subst; destruct x |].
  apply Z.mod_unique with (x `div` y)%nat.
  { left. rewrite <-(Nat2Z.inj_le 0), <-Nat2Z.inj_lt.
    apply Nat.mod_bound_pos; lia. }
  by rewrite <-Nat2Z.inj_mul, <-Nat2Z.inj_add, <-Nat.div_mod.
Qed.
Close Scope Z_scope.

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(** * Notations and properties of [Qc] *)
Notation "2" := (1+1)%Qc : Qc_scope.
Infix "≤" := Qcle : Qc_scope.
Notation "x ≤ y ≤ z" := (x  y  y  z)%Qc : Qc_scope.
Notation "x ≤ y < z" := (x  y  y < z)%Qc : Qc_scope.
Notation "x < y < z" := (x < y  y < z)%Qc : Qc_scope.
Notation "x < y ≤ z" := (x < y  y  z)%Qc : Qc_scope.
Notation "(≤)" := Qcle (only parsing) : Qc_scope.
Notation "(<)" := Qclt (only parsing) : Qc_scope.

Instance Qc_eq_dec:  x y : Qc, Decision (x = y) := Qc_eq_dec.
Program Instance Qc_le_dec (x y : Qc) : Decision (x  y)%Qc :=
  if Qclt_le_dec y x then right _ else left _.
Next Obligation. by apply Qclt_not_le. Qed.
Program Instance Qc_lt_dec (x y : Qc) : Decision (x < y)%Qc :=
  if Qclt_le_dec x y then left _ else right _.
Next Obligation. by apply Qcle_not_lt. Qed.

Instance: Reflexive Qcle.
Proof. red. apply Qcle_refl. Qed.
Instance: Transitive Qcle.
Proof. red. apply Qcle_trans. Qed.

Lemma Qcle_ngt (x y : Qc) : (x  y  ¬y < x)%Qc.
Proof. split; auto using Qcle_not_lt, Qcnot_lt_le. Qed.
Lemma Qclt_nge (x y : Qc) : (x < y  ¬y  x)%Qc.
Proof. split; auto using Qclt_not_le, Qcnot_le_lt. Qed.

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Lemma Qcplus_le_mono_l (x y z : Qc) : (x  y  z + x  z + y)%Qc.
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Proof.
  split; intros.
  * by apply Qcplus_le_compat.
  * replace x with ((0 - z) + (z + x))%Qc by ring.
    replace y with ((0 - z) + (z + y))%Qc by ring.
    by apply Qcplus_le_compat.
Qed.
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Lemma Qcplus_le_mono_r (x y z : Qc) : (x  y  x + z  y + z)%Qc.
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Proof. rewrite !(Qcplus_comm _ z). apply Qcplus_le_mono_l. Qed.
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Lemma Qcplus_lt_mono_l (x y z : Qc) : (x < y  z + x < z + y)%Qc.
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Proof. by rewrite !Qclt_nge, <-Qcplus_le_mono_l. Qed.
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Lemma Qcplus_lt_mono_r (x y z : Qc) : (x < y  x + z < y + z)%Qc.
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Proof. by rewrite !Qclt_nge, <-Qcplus_le_mono_r. Qed.

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(** * Conversions *)
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Lemma Z_to_nat_nonpos x : (x  0)%Z  Z.to_nat x = 0.
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Proof. destruct x; simpl; auto using Z2Nat.inj_neg. by intros []. Qed.
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(** The function [Z_to_option_N] converts an integer [x] into a natural number
by giving [None] in case [x] is negative. *)
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Definition Z_to_option_N (x : Z) : option N :=
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  match x with
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  | Z0 => Some N0 | Zpos p => Some (Npos p) | Zneg _ => None
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  end.
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Definition Z_to_option_nat (x : Z) : option nat :=
  match x with
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  | Z0 => Some 0 | Zpos p => Some (Pos.to_nat p) | Zneg _ => None
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  end.
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Lemma Z_to_option_N_Some x y :
  Z_to_option_N x = Some y  (0  x)%Z  y = Z.to_N x.
Proof.
  split.
  * intros. by destruct x; simpl in *; simplify_equality;
      auto using Zle_0_pos.
  * intros [??]. subst. destruct x; simpl; auto; lia.
Qed.
Lemma Z_to_option_N_Some_alt x y :
  Z_to_option_N x = Some y  (0  x)%Z  x = Z.of_N y.
Proof.
  rewrite Z_to_option_N_Some.
  split; intros [??]; subst; auto using N2Z.id, Z2N.id, eq_sym.
Qed.

Lemma Z_to_option_nat_Some x y :
  Z_to_option_nat x = Some y  (0  x)%Z  y = Z.to_nat x.
Proof.
  split.
  * intros. by destruct x; simpl in *; simplify_equality;
      auto using Zle_0_pos.
  * intros [??]. subst. destruct x; simpl; auto; lia.
Qed.
Lemma Z_to_option_nat_Some_alt x y :
  Z_to_option_nat x = Some y  (0  x)%Z  x = Z.of_nat y.
Proof.
  rewrite Z_to_option_nat_Some.
  split; intros [??]; subst; auto using Nat2Z.id, Z2Nat.id, eq_sym.
Qed.
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Lemma Z_to_option_of_nat x : Z_to_option_nat (Z.of_nat x) = Some x.
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Proof. apply Z_to_option_nat_Some_alt. auto using Nat2Z.is_nonneg. Qed.

(** The function [Z_of_sumbool] converts a sumbool [P] into an integer
by yielding one if [P] and zero if [Q]. *)
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Definition Z_of_sumbool {P Q : Prop} (p : {P} + {Q} ) : Z :=
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  (if p then 1 else 0)%Z.
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(** Some correspondence lemmas between [nat] and [N] that are not part of the
standard library. We declare a hint database [natify] to rewrite a goal
involving [N] into a corresponding variant involving [nat]. *)
Lemma N_to_nat_lt x y : N.to_nat x < N.to_nat y  (x < y)%N.
Proof. by rewrite <-N.compare_lt_iff, nat_compare_lt, N2Nat.inj_compare. Qed.
Lemma N_to_nat_le x y : N.to_nat x  N.to_nat y  (x  y)%N.
Proof. by rewrite <-N.compare_le_iff, nat_compare_le, N2Nat.inj_compare. Qed.
Lemma N_to_nat_0 : N.to_nat 0 = 0.
Proof. done. Qed.
Lemma N_to_nat_1 : N.to_nat 1 = 1.
Proof. done. Qed.
Lemma N_to_nat_div x y : N.to_nat (x `div` y) = N.to_nat x `div` N.to_nat y.
Proof.
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  destruct (decide (y = 0%N)); [by subst; destruct x |].
  apply Nat.div_unique with (N.to_nat (x `mod` y)).
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  { by apply N_to_nat_lt, N.mod_lt. }
  rewrite (N.div_unique_exact (x * y) y x), N.div_mul by lia.
  by rewrite <-N2Nat.inj_mul, <-N2Nat.inj_add, <-N.div_mod.
Qed.
(* We have [x `mod` 0 = 0] on [nat], and [x `mod` 0 = x] on [N]. *)
Lemma N_to_nat_mod x y :
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  y  0%N  N.to_nat (x `mod` y) = N.to_nat x `mod` N.to_nat y.
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Proof.
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  intros. apply Nat.mod_unique with (N.to_nat (x `div` y)).
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  { by apply N_to_nat_lt, N.mod_lt. }
  rewrite (N.div_unique_exact (x * y) y x), N.div_mul by lia.
  by rewrite <-N2Nat.inj_mul, <-N2Nat.inj_add, <-N.div_mod.
Qed.

Hint Rewrite <-N2Nat.inj_iff : natify.
Hint Rewrite <-N_to_nat_lt : natify.
Hint Rewrite <-N_to_nat_le : natify.
Hint Rewrite Nat2N.id : natify.
Hint Rewrite N2Nat.inj_add : natify.
Hint Rewrite N2Nat.inj_mul : natify.
Hint Rewrite N2Nat.inj_sub : natify.
Hint Rewrite N2Nat.inj_succ : natify.
Hint Rewrite N2Nat.inj_pred : natify.
Hint Rewrite N_to_nat_div : natify.
Hint Rewrite N_to_nat_0 : natify.
Hint Rewrite N_to_nat_1 : natify.
Ltac natify := repeat autorewrite with natify in *.

Hint Extern 100 (Nlt _ _) => natify : natify.
Hint Extern 100 (Nle _ _) => natify : natify.
Hint Extern 100 (@eq N _ _) => natify : natify.
Hint Extern 100 (lt _ _) => natify : natify.
Hint Extern 100 (le _ _) => natify : natify.
Hint Extern 100 (@eq nat _ _) => natify : natify.

Instance:  x, PropHolds (0 < x)%N  PropHolds (0 < N.to_nat x).
Proof. unfold PropHolds. intros. by natify. Qed.
Instance:  x, PropHolds (0  x)%N  PropHolds (0  N.to_nat x).
Proof. unfold PropHolds. intros. by natify. Qed.