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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 *)
(************************************************************************)
(*i $Id: Peano.v 13323 2010-07-24 15:57:30Z herbelin $ i*)
(** 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.
Unset Boxed Definitions.
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 *)
Theorem eq_add_S : forall n m:nat, S n = S m -> n = m.
Proof.
intros n m Sn_eq_Sm.
replace (n=m) with (pred (S n) = pred (S m)) by auto using pred_Sn.
rewrite Sn_eq_Sm; trivial.
Qed.
Hint Immediate eq_add_S: core.
Theorem not_eq_S : forall n m:nat, n <> m -> S n <> S m.
Proof.
red in |- *; 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 in |- *; 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 in |- *; 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 (only parsing).
Notation plus_succ_r_reverse := plus_n_Sm (only parsing).
(** 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 in |- *; 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 in |- *; auto.
destruct H; rewrite <- plus_n_Sm; apply (f_equal S).
pattern m at 1 3 in |- *; elim m; simpl in |- *; auto.
Qed.
Hint Resolve mult_n_Sm: core.
(** Standard associated names *)
Notation mult_0_r_reverse := mult_n_O (only parsing).
Notation mult_succ_r_reverse := mult_n_Sm (only parsing).
(** 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.
(** 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 as [| n0]; auto.
Qed.
|
