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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id$ i*)
Require Import Arith_base.
Require Import Compare_dec.
Require Import Sumbool.
Require Import Div2.
Require Import Min.
Require Import Max.
Require Import BinPos.
Require Import BinNat.
Require Import Pnat.
(** Translation from [N] to [nat] and back. *)
Definition nat_of_N (a:N) :=
match a with
| N0 => 0%nat
| Npos p => nat_of_P p
end.
Definition N_of_nat (n:nat) :=
match n with
| O => N0
| S n' => Npos (P_of_succ_nat n')
end.
Lemma N_of_nat_of_N : forall a:N, N_of_nat (nat_of_N a) = a.
Proof.
destruct a as [| p]. reflexivity.
simpl in |- *. elim (ZL4 p). intros n H. rewrite H. simpl in |- *.
rewrite <- nat_of_P_o_P_of_succ_nat_eq_succ in H.
rewrite nat_of_P_inj with (1 := H). reflexivity.
Qed.
Lemma nat_of_N_of_nat : forall n:nat, nat_of_N (N_of_nat n) = n.
Proof.
induction n. trivial.
intros. simpl in |- *. apply nat_of_P_o_P_of_succ_nat_eq_succ.
Qed.
(** Interaction of this translation and usual operations. *)
Lemma nat_of_Ndouble : forall a, nat_of_N (Ndouble a) = 2*(nat_of_N a).
Proof.
destruct a; simpl nat_of_N; auto.
apply nat_of_P_xO.
Qed.
Lemma N_of_double : forall n, N_of_nat (2*n) = Ndouble (N_of_nat n).
Proof.
intros.
pattern n at 1; rewrite <- (nat_of_N_of_nat n).
rewrite <- nat_of_Ndouble.
apply N_of_nat_of_N.
Qed.
Lemma nat_of_Ndouble_plus_one :
forall a, nat_of_N (Ndouble_plus_one a) = S (2*(nat_of_N a)).
Proof.
destruct a; simpl nat_of_N; auto.
apply nat_of_P_xI.
Qed.
Lemma N_of_double_plus_one :
forall n, N_of_nat (S (2*n)) = Ndouble_plus_one (N_of_nat n).
Proof.
intros.
pattern n at 1; rewrite <- (nat_of_N_of_nat n).
rewrite <- nat_of_Ndouble_plus_one.
apply N_of_nat_of_N.
Qed.
Lemma nat_of_Nsucc : forall a, nat_of_N (Nsucc a) = S (nat_of_N a).
Proof.
destruct a; simpl.
apply nat_of_P_xH.
apply nat_of_P_succ_morphism.
Qed.
Lemma N_of_S : forall n, N_of_nat (S n) = Nsucc (N_of_nat n).
Proof.
intros.
pattern n at 1; rewrite <- (nat_of_N_of_nat n).
rewrite <- nat_of_Nsucc.
apply N_of_nat_of_N.
Qed.
Lemma nat_of_Nplus :
forall a a', nat_of_N (Nplus a a') = (nat_of_N a)+(nat_of_N a').
Proof.
destruct a; destruct a'; simpl; auto.
apply nat_of_P_plus_morphism.
Qed.
Lemma N_of_plus :
forall n n', N_of_nat (n+n') = Nplus (N_of_nat n) (N_of_nat n').
Proof.
intros.
pattern n at 1; rewrite <- (nat_of_N_of_nat n).
pattern n' at 1; rewrite <- (nat_of_N_of_nat n').
rewrite <- nat_of_Nplus.
apply N_of_nat_of_N.
Qed.
Lemma nat_of_Nminus :
forall a a', nat_of_N (Nminus a a') = ((nat_of_N a)-(nat_of_N a'))%nat.
Proof.
destruct a; destruct a'; simpl; auto with arith.
case_eq (Pcompare p p0 Eq); simpl; intros.
rewrite (Pcompare_Eq_eq _ _ H); auto with arith.
symmetry; apply not_le_minus_0.
generalize (nat_of_P_lt_Lt_compare_morphism _ _ H); auto with arith.
apply nat_of_P_minus_morphism; auto.
Qed.
Lemma N_of_minus :
forall n n', N_of_nat (n-n') = Nminus (N_of_nat n) (N_of_nat n').
Proof.
intros.
pattern n at 1; rewrite <- (nat_of_N_of_nat n).
pattern n' at 1; rewrite <- (nat_of_N_of_nat n').
rewrite <- nat_of_Nminus.
apply N_of_nat_of_N.
Qed.
Lemma nat_of_Nmult :
forall a a', nat_of_N (Nmult a a') = (nat_of_N a)*(nat_of_N a').
Proof.
destruct a; destruct a'; simpl; auto.
apply nat_of_P_mult_morphism.
Qed.
Lemma N_of_mult :
forall n n', N_of_nat (n*n') = Nmult (N_of_nat n) (N_of_nat n').
Proof.
intros.
pattern n at 1; rewrite <- (nat_of_N_of_nat n).
pattern n' at 1; rewrite <- (nat_of_N_of_nat n').
rewrite <- nat_of_Nmult.
apply N_of_nat_of_N.
Qed.
Lemma nat_of_Ndiv2 :
forall a, nat_of_N (Ndiv2 a) = div2 (nat_of_N a).
Proof.
destruct a; simpl in *; auto.
destruct p; auto.
rewrite nat_of_P_xI.
rewrite div2_double_plus_one; auto.
rewrite nat_of_P_xO.
rewrite div2_double; auto.
Qed.
Lemma N_of_div2 :
forall n, N_of_nat (div2 n) = Ndiv2 (N_of_nat n).
Proof.
intros.
pattern n at 1; rewrite <- (nat_of_N_of_nat n).
rewrite <- nat_of_Ndiv2.
apply N_of_nat_of_N.
Qed.
Lemma nat_of_Ncompare :
forall a a', Ncompare a a' = nat_compare (nat_of_N a) (nat_of_N a').
Proof.
destruct a; destruct a'; simpl.
compute; auto.
generalize (lt_O_nat_of_P p).
unfold nat_compare.
destruct (lt_eq_lt_dec 0 (nat_of_P p)) as [[H|H]|H]; auto.
rewrite <- H; inversion 1.
intros; generalize (lt_trans _ _ _ H0 H); inversion 1.
generalize (lt_O_nat_of_P p).
unfold nat_compare.
destruct (lt_eq_lt_dec (nat_of_P p) 0) as [[H|H]|H]; auto.
intros; generalize (lt_trans _ _ _ H0 H); inversion 1.
rewrite H; inversion 1.
unfold nat_compare.
destruct (lt_eq_lt_dec (nat_of_P p) (nat_of_P p0)) as [[H|H]|H]; auto.
apply nat_of_P_lt_Lt_compare_complement_morphism; auto.
rewrite (nat_of_P_inj _ _ H); apply Pcompare_refl.
apply nat_of_P_gt_Gt_compare_complement_morphism; auto.
Qed.
Lemma N_of_nat_compare :
forall n n', nat_compare n n' = Ncompare (N_of_nat n) (N_of_nat n').
Proof.
intros.
pattern n at 1; rewrite <- (nat_of_N_of_nat n).
pattern n' at 1; rewrite <- (nat_of_N_of_nat n').
symmetry; apply nat_of_Ncompare.
Qed.
Lemma nat_of_Nmin :
forall a a', nat_of_N (Nmin a a') = min (nat_of_N a) (nat_of_N a').
Proof.
intros; unfold Nmin; rewrite nat_of_Ncompare.
unfold nat_compare.
destruct (lt_eq_lt_dec (nat_of_N a) (nat_of_N a')) as [[|]|];
simpl; intros; symmetry; auto with arith.
apply min_l; rewrite e; auto with arith.
Qed.
Lemma N_of_min :
forall n n', N_of_nat (min n n') = Nmin (N_of_nat n) (N_of_nat n').
Proof.
intros.
pattern n at 1; rewrite <- (nat_of_N_of_nat n).
pattern n' at 1; rewrite <- (nat_of_N_of_nat n').
rewrite <- nat_of_Nmin.
apply N_of_nat_of_N.
Qed.
Lemma nat_of_Nmax :
forall a a', nat_of_N (Nmax a a') = max (nat_of_N a) (nat_of_N a').
Proof.
intros; unfold Nmax; rewrite nat_of_Ncompare.
unfold nat_compare.
destruct (lt_eq_lt_dec (nat_of_N a) (nat_of_N a')) as [[|]|];
simpl; intros; symmetry; auto with arith.
apply max_r; rewrite e; auto with arith.
Qed.
Lemma N_of_max :
forall n n', N_of_nat (max n n') = Nmax (N_of_nat n) (N_of_nat n').
Proof.
intros.
pattern n at 1; rewrite <- (nat_of_N_of_nat n).
pattern n' at 1; rewrite <- (nat_of_N_of_nat n').
rewrite <- nat_of_Nmax.
apply N_of_nat_of_N.
Qed.
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