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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* Evgeny Makarov, INRIA, 2007 *)
(************************************************************************)
Require Import
Bool Peano Peano_dec Compare_dec Plus Minus Le EqNat NAxioms NProperties.
(** Functions not already defined *)
Fixpoint pow n m :=
match m with
| O => 1
| S m => n * (pow n m)
end.
Infix "^" := pow : nat_scope.
Lemma pow_0_r : forall a, a^0 = 1.
Proof. reflexivity. Qed.
Lemma pow_succ_r : forall a b, 0<=b -> a^(S b) = a * a^b.
Proof. reflexivity. Qed.
Definition Even n := exists m, n = 2*m.
Definition Odd n := exists m, n = 2*m+1.
Fixpoint even n :=
match n with
| O => true
| 1 => false
| S (S n') => even n'
end.
Definition odd n := negb (even n).
Lemma even_spec : forall n, even n = true <-> Even n.
Proof.
fix 1.
destruct n as [|[|n]]; simpl; try rewrite even_spec; split.
now exists 0.
trivial.
discriminate.
intros (m,H). destruct m. discriminate.
simpl in H. rewrite <- plus_n_Sm in H. discriminate.
intros (m,H). exists (S m). rewrite H. simpl. now rewrite plus_n_Sm.
intros (m,H). destruct m. discriminate. exists m.
simpl in H. rewrite <- plus_n_Sm in H. inversion H. reflexivity.
Qed.
Lemma odd_spec : forall n, odd n = true <-> Odd n.
Proof.
unfold odd.
fix 1.
destruct n as [|[|n]]; simpl; try rewrite odd_spec; split.
discriminate.
intros (m,H). rewrite <- plus_n_Sm in H; discriminate.
now exists 0.
trivial.
intros (m,H). exists (S m). rewrite H. simpl. now rewrite <- (plus_n_Sm m).
intros (m,H). destruct m. discriminate. exists m.
simpl in H. rewrite <- plus_n_Sm in H. inversion H. simpl.
now rewrite <- !plus_n_Sm, <- !plus_n_O.
Qed.
(* A linear, tail-recursive, division for nat.
In [divmod], [y] is the predecessor of the actual divisor,
and [u] is [y] minus the real remainder
*)
Fixpoint divmod x y q u :=
match x with
| 0 => (q,u)
| S x' => match u with
| 0 => divmod x' y (S q) y
| S u' => divmod x' y q u'
end
end.
Definition div x y :=
match y with
| 0 => 0
| S y' => fst (divmod x y' 0 y')
end.
Definition modulo x y :=
match y with
| 0 => 0
| S y' => y' - snd (divmod x y' 0 y')
end.
Infix "/" := div : nat_scope.
Infix "mod" := modulo (at level 40, no associativity) : nat_scope.
Lemma divmod_spec : forall x y q u, u <= y ->
let (q',u') := divmod x y q u in
x + (S y)*q + (y-u) = (S y)*q' + (y-u') /\ u' <= y.
Proof.
induction x. simpl. intuition.
intros y q u H. destruct u; simpl divmod.
generalize (IHx y (S q) y (le_n y)). destruct divmod as (q',u').
intros (EQ,LE); split; trivial.
rewrite <- EQ, <- minus_n_O, minus_diag, <- plus_n_O.
now rewrite !plus_Sn_m, plus_n_Sm, <- plus_assoc, mult_n_Sm.
generalize (IHx y q u (le_Sn_le _ _ H)). destruct divmod as (q',u').
intros (EQ,LE); split; trivial.
rewrite <- EQ.
rewrite !plus_Sn_m, plus_n_Sm. f_equal. now apply minus_Sn_m.
Qed.
Lemma div_mod : forall x y, y<>0 -> x = y*(x/y) + x mod y.
Proof.
intros x y Hy.
destruct y; [ now elim Hy | clear Hy ].
unfold div, modulo.
generalize (divmod_spec x y 0 y (le_n y)).
destruct divmod as (q,u).
intros (U,V).
simpl in *.
now rewrite <- mult_n_O, minus_diag, <- !plus_n_O in U.
Qed.
Lemma mod_upper_bound : forall x y, y<>0 -> x mod y < y.
Proof.
intros x y Hy.
destruct y; [ now elim Hy | clear Hy ].
unfold modulo.
apply le_n_S, le_minus.
Qed.
(** * Implementation of [NAxiomsSig] by [nat] *)
Module Nat
<: NAxiomsSig <: UsualDecidableTypeFull <: OrderedTypeFull <: TotalOrder.
(** Bi-directional induction. *)
Theorem bi_induction :
forall A : nat -> Prop, Proper (eq==>iff) A ->
A 0 -> (forall n : nat, A n <-> A (S n)) -> forall n : nat, A n.
Proof.
intros A A_wd A0 AS. apply nat_ind. assumption. intros; now apply -> AS.
Qed.
(** Basic operations. *)
Definition eq_equiv : Equivalence (@eq nat) := eq_equivalence.
Local Obligation Tactic := simpl_relation.
Program Instance succ_wd : Proper (eq==>eq) S.
Program Instance pred_wd : Proper (eq==>eq) pred.
Program Instance add_wd : Proper (eq==>eq==>eq) plus.
Program Instance sub_wd : Proper (eq==>eq==>eq) minus.
Program Instance mul_wd : Proper (eq==>eq==>eq) mult.
Theorem pred_succ : forall n : nat, pred (S n) = n.
Proof.
reflexivity.
Qed.
Theorem add_0_l : forall n : nat, 0 + n = n.
Proof.
reflexivity.
Qed.
Theorem add_succ_l : forall n m : nat, (S n) + m = S (n + m).
Proof.
reflexivity.
Qed.
Theorem sub_0_r : forall n : nat, n - 0 = n.
Proof.
intro n; now destruct n.
Qed.
Theorem sub_succ_r : forall n m : nat, n - (S m) = pred (n - m).
Proof.
induction n; destruct m; simpl; auto. apply sub_0_r.
Qed.
Theorem mul_0_l : forall n : nat, 0 * n = 0.
Proof.
reflexivity.
Qed.
Theorem mul_succ_l : forall n m : nat, S n * m = n * m + m.
Proof.
assert (add_S_r : forall n m, n+S m = S(n+m)) by (induction n; auto).
assert (add_comm : forall n m, n+m = m+n).
induction n; simpl; auto. intros; rewrite add_S_r; auto.
intros n m; now rewrite add_comm.
Qed.
(** Order on natural numbers *)
Program Instance lt_wd : Proper (eq==>eq==>iff) lt.
Theorem lt_succ_r : forall n m : nat, n < S m <-> n <= m.
Proof.
unfold lt; split. apply le_S_n. induction 1; auto.
Qed.
Theorem lt_eq_cases : forall n m : nat, n <= m <-> n < m \/ n = m.
Proof.
split.
inversion 1; auto. rewrite lt_succ_r; auto.
destruct 1; [|subst; auto]. rewrite <- lt_succ_r; auto.
Qed.
Theorem lt_irrefl : forall n : nat, ~ (n < n).
Proof.
induction n. intro H; inversion H. rewrite lt_succ_r; auto.
Qed.
(** Facts specific to natural numbers, not integers. *)
Theorem pred_0 : pred 0 = 0.
Proof.
reflexivity.
Qed.
(** Recursion fonction *)
Definition recursion {A} : A -> (nat -> A -> A) -> nat -> A :=
nat_rect (fun _ => A).
Instance recursion_wd {A} (Aeq : relation A) :
Proper (Aeq ==> (eq==>Aeq==>Aeq) ==> eq ==> Aeq) recursion.
Proof.
intros a a' Ha f f' Hf n n' Hn. subst n'.
induction n; simpl; auto. apply Hf; auto.
Qed.
Theorem recursion_0 :
forall {A} (a : A) (f : nat -> A -> A), recursion a f 0 = a.
Proof.
reflexivity.
Qed.
Theorem recursion_succ :
forall {A} (Aeq : relation A) (a : A) (f : nat -> A -> A),
Aeq a a -> Proper (eq==>Aeq==>Aeq) f ->
forall n : nat, Aeq (recursion a f (S n)) (f n (recursion a f n)).
Proof.
unfold Proper, respectful in *; induction n; simpl; auto.
Qed.
(** The instantiation of operations.
Placing them at the very end avoids having indirections in above lemmas. *)
Definition t := nat.
Definition eq := @eq nat.
Definition eqb := beq_nat.
Definition compare := nat_compare.
Definition zero := 0.
Definition succ := S.
Definition pred := pred.
Definition add := plus.
Definition sub := minus.
Definition mul := mult.
Definition lt := lt.
Definition le := le.
Definition min := min.
Definition max := max.
Definition max_l := max_l.
Definition max_r := max_r.
Definition min_l := min_l.
Definition min_r := min_r.
Definition eqb_eq := beq_nat_true_iff.
Definition compare_spec := nat_compare_spec.
Definition eq_dec := eq_nat_dec.
Definition Even := Even.
Definition Odd := Odd.
Definition even := even.
Definition odd := odd.
Definition even_spec := even_spec.
Definition odd_spec := odd_spec.
Definition pow := pow.
Program Instance pow_wd : Proper (eq==>eq==>eq) pow.
Definition pow_0_r := pow_0_r.
Definition pow_succ_r := pow_succ_r.
Definition div := div.
Definition modulo := modulo.
Program Instance div_wd : Proper (eq==>eq==>eq) div.
Program Instance mod_wd : Proper (eq==>eq==>eq) modulo.
Definition div_mod := div_mod.
Definition mod_upper_bound := mod_upper_bound.
(** Generic Properties *)
Include NProp
<+ UsualMinMaxLogicalProperties <+ UsualMinMaxDecProperties.
End Nat.
(** [Nat] contains an [order] tactic for natural numbers *)
(** Note that [Nat.order] is domain-agnostic: it will not prove
[1<=2] or [x<=x+x], but rather things like [x<=y -> y<=x -> x=y]. *)
Section TestOrder.
Let test : forall x y, x<=y -> y<=x -> x=y.
Proof.
Nat.order.
Qed.
End TestOrder.
|
