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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 *)
(************************************************************************)
(** The type [nat] of Peano natural numbers (built from [O] and [S])
is defined in [Datatypes.v] *)
(** This module defines the following operations on natural numbers :
- predecessor [pred]
- addition [plus]
- multiplication [mult]
- less or equal order [le]
- less [lt]
- greater or equal [ge]
- greater [gt]
It states various lemmas and theorems about natural numbers,
including Peano's axioms of arithmetic (in Coq, these are provable).
Case analysis on [nat] and induction on [nat * nat] are provided too
*)
Require Import Notations.
Require Import Datatypes.
Require Import Logic.
Open Scope nat_scope.
Definition eq_S := f_equal S.
Hint Resolve (f_equal S): v62.
Hint Resolve (f_equal (A:=nat)): core.
(** The predecessor function *)
Definition pred (n:nat) : nat := match n with
| O => n
| S u => u
end.
Hint Resolve (f_equal pred): v62.
Theorem pred_Sn : forall n:nat, n = pred (S n).
Proof.
simpl; reflexivity.
Qed.
(** Injectivity of successor *)
Definition eq_add_S n m (H: S n = S m): n = m := f_equal pred H.
Hint Immediate eq_add_S: core.
Theorem not_eq_S : forall n m:nat, n <> m -> S n <> S m.
Proof.
red; auto.
Qed.
Hint Resolve not_eq_S: core.
Definition IsSucc (n:nat) : Prop :=
match n with
| O => False
| S p => True
end.
(** Zero is not the successor of a number *)
Theorem O_S : forall n:nat, 0 <> S n.
Proof.
discriminate.
Qed.
Hint Resolve O_S: core.
Theorem n_Sn : forall n:nat, n <> S n.
Proof.
induction n; auto.
Qed.
Hint Resolve n_Sn: core.
(** Addition *)
Fixpoint plus (n m:nat) : nat :=
match n with
| O => m
| S p => S (p + m)
end
where "n + m" := (plus n m) : nat_scope.
Hint Resolve (f_equal2 plus): v62.
Hint Resolve (f_equal2 (A1:=nat) (A2:=nat)): core.
Lemma plus_n_O : forall n:nat, n = n + 0.
Proof.
induction n; simpl; auto.
Qed.
Hint Resolve plus_n_O: core.
Lemma plus_O_n : forall n:nat, 0 + n = n.
Proof.
auto.
Qed.
Lemma plus_n_Sm : forall n m:nat, S (n + m) = n + S m.
Proof.
intros n m; induction n; simpl; auto.
Qed.
Hint Resolve plus_n_Sm: core.
Lemma plus_Sn_m : forall n m:nat, S n + m = S (n + m).
Proof.
auto.
Qed.
(** Standard associated names *)
Notation plus_0_r_reverse := plus_n_O (compat "8.2").
Notation plus_succ_r_reverse := plus_n_Sm (compat "8.2").
(** Multiplication *)
Fixpoint mult (n m:nat) : nat :=
match n with
| O => 0
| S p => m + p * m
end
where "n * m" := (mult n m) : nat_scope.
Hint Resolve (f_equal2 mult): core.
Lemma mult_n_O : forall n:nat, 0 = n * 0.
Proof.
induction n; simpl; auto.
Qed.
Hint Resolve mult_n_O: core.
Lemma mult_n_Sm : forall n m:nat, n * m + n = n * S m.
Proof.
intros; induction n as [| p H]; simpl; auto.
destruct H; rewrite <- plus_n_Sm; apply eq_S.
pattern m at 1 3; elim m; simpl; auto.
Qed.
Hint Resolve mult_n_Sm: core.
(** Standard associated names *)
Notation mult_0_r_reverse := mult_n_O (compat "8.2").
Notation mult_succ_r_reverse := mult_n_Sm (compat "8.2").
(** Truncated subtraction: [m-n] is [0] if [n>=m] *)
Fixpoint minus (n m:nat) : nat :=
match n, m with
| O, _ => n
| S k, O => n
| S k, S l => k - l
end
where "n - m" := (minus n m) : nat_scope.
(** Definition of the usual orders, the basic properties of [le] and [lt]
can be found in files Le and Lt *)
Inductive le (n:nat) : nat -> Prop :=
| le_n : n <= n
| le_S : forall m:nat, n <= m -> n <= S m
where "n <= m" := (le n m) : nat_scope.
Hint Constructors le: core.
(*i equivalent to : "Hints Resolve le_n le_S : core." i*)
Definition lt (n m:nat) := S n <= m.
Hint Unfold lt: core.
Infix "<" := lt : nat_scope.
Definition ge (n m:nat) := m <= n.
Hint Unfold ge: core.
Infix ">=" := ge : nat_scope.
Definition gt (n m:nat) := m < n.
Hint Unfold gt: core.
Infix ">" := gt : nat_scope.
Notation "x <= y <= z" := (x <= y /\ y <= z) : nat_scope.
Notation "x <= y < z" := (x <= y /\ y < z) : nat_scope.
Notation "x < y < z" := (x < y /\ y < z) : nat_scope.
Notation "x < y <= z" := (x < y /\ y <= z) : nat_scope.
Theorem le_pred : forall n m, n <= m -> pred n <= pred m.
Proof.
induction 1; auto. destruct m; simpl; auto.
Qed.
Theorem le_S_n : forall n m, S n <= S m -> n <= m.
Proof.
intros n m. exact (le_pred (S n) (S m)).
Qed.
(** Case analysis *)
Theorem nat_case :
forall (n:nat) (P:nat -> Prop), P 0 -> (forall m:nat, P (S m)) -> P n.
Proof.
induction n; auto.
Qed.
(** Principle of double induction *)
Theorem nat_double_ind :
forall R:nat -> nat -> Prop,
(forall n:nat, R 0 n) ->
(forall n:nat, R (S n) 0) ->
(forall n m:nat, R n m -> R (S n) (S m)) -> forall n m:nat, R n m.
Proof.
induction n; auto.
destruct m; auto.
Qed.
(** Maximum and minimum : definitions and specifications *)
Fixpoint max n m : nat :=
match n, m with
| O, _ => m
| S n', O => n
| S n', S m' => S (max n' m')
end.
Fixpoint min n m : nat :=
match n, m with
| O, _ => 0
| S n', O => 0
| S n', S m' => S (min n' m')
end.
Theorem max_l : forall n m : nat, m <= n -> max n m = n.
Proof.
induction n; destruct m; simpl; auto. inversion 1.
intros. apply f_equal. apply IHn. apply le_S_n. trivial.
Qed.
Theorem max_r : forall n m : nat, n <= m -> max n m = m.
Proof.
induction n; destruct m; simpl; auto. inversion 1.
intros. apply f_equal. apply IHn. apply le_S_n. trivial.
Qed.
Theorem min_l : forall n m : nat, n <= m -> min n m = n.
Proof.
induction n; destruct m; simpl; auto. inversion 1.
intros. apply f_equal. apply IHn. apply le_S_n. trivial.
Qed.
Theorem min_r : forall n m : nat, m <= n -> min n m = m.
Proof.
induction n; destruct m; simpl; auto. inversion 1.
intros. apply f_equal. apply IHn. apply le_S_n. trivial.
Qed.
(** [n]th iteration of the function [f] *)
Fixpoint nat_iter (n:nat) {A} (f:A->A) (x:A) : A :=
match n with
| O => x
| S n' => f (nat_iter n' f x)
end.
Lemma nat_iter_succ_r n {A} (f:A->A) (x:A) :
nat_iter (S n) f x = nat_iter n f (f x).
Proof.
induction n; intros; simpl; rewrite <- ?IHn; trivial.
Qed.
Theorem nat_iter_plus :
forall (n m:nat) {A} (f:A -> A) (x:A),
nat_iter (n + m) f x = nat_iter n f (nat_iter m f x).
Proof.
induction n; intros; simpl; rewrite ?IHn; trivial.
Qed.
(** Preservation of invariants : if [f : A->A] preserves the invariant [Inv],
then the iterates of [f] also preserve it. *)
Theorem nat_iter_invariant :
forall (n:nat) {A} (f:A -> A) (P : A -> Prop),
(forall x, P x -> P (f x)) ->
forall x, P x -> P (nat_iter n f x).
Proof.
induction n; simpl; trivial.
intros A f P Hf x Hx. apply Hf, IHn; trivial.
Qed.
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