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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: Peano.v,v 1.1.2.1 2004/07/16 19:31:26 herbelin Exp $ i*)
(** Natural numbers [nat] built from [O] and [S] are 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]
This module states various lemmas and theorems about natural numbers,
including Peano's axioms of arithmetic (in Coq, these are in fact provable)
Case analysis on [nat] and induction on [nat * nat] are provided too *)
Require Notations.
Require Datatypes.
Require Logic.
Open Scope nat_scope.
Definition eq_S := (f_equal nat nat S).
Hint eq_S : v62 := Resolve (f_equal nat nat S).
Hint eq_nat_unary : core := Resolve (f_equal nat).
(** The predecessor function *)
Definition pred : nat->nat := [n:nat](Cases n of O => O | (S u) => u end).
Hint eq_pred : v62 := Resolve (f_equal nat nat pred).
Theorem pred_Sn : (m:nat) m=(pred (S m)).
Proof.
Auto.
Qed.
Theorem eq_add_S : (n,m:nat) (S n)=(S m) -> n=m.
Proof.
Intros n m H ; Change (pred (S n))=(pred (S m)); Auto.
Qed.
Hints Immediate eq_add_S : core v62.
(** A consequence of the previous axioms *)
Theorem not_eq_S : (n,m:nat) ~(n=m) -> ~((S n)=(S m)).
Proof.
Red; Auto.
Qed.
Hints Resolve not_eq_S : core v62.
Definition IsSucc : nat->Prop
:= [n:nat]Cases n of O => False | (S p) => True end.
Theorem O_S : (n:nat)~(O=(S n)).
Proof.
Red;Intros n H.
Change (IsSucc O).
Rewrite <- (sym_eq nat O (S n));[Exact I | Assumption].
Qed.
Hints Resolve O_S : core v62.
Theorem n_Sn : (n:nat) ~(n=(S n)).
Proof.
NewInduction n ; Auto.
Qed.
Hints Resolve n_Sn : core v62.
(** Addition *)
Fixpoint plus [n:nat] : nat -> nat :=
[m:nat]Cases n of
O => m
| (S p) => (S (plus p m)) end.
Hint eq_plus : v62 := Resolve (f_equal2 nat nat nat plus).
Hint eq_nat_binary : core := Resolve (f_equal2 nat nat).
V8Infix "+" plus : nat_scope.
Lemma plus_n_O : (n:nat) n=(plus n O).
Proof.
NewInduction n ; Simpl ; Auto.
Qed.
Hints Resolve plus_n_O : core v62.
Lemma plus_O_n : (n:nat) (plus O n)=n.
Proof.
Auto.
Qed.
Lemma plus_n_Sm : (n,m:nat) (S (plus n m))=(plus n (S m)).
Proof.
Intros n m; NewInduction n; Simpl; Auto.
Qed.
Hints Resolve plus_n_Sm : core v62.
Lemma plus_Sn_m : (n,m:nat)(plus (S n) m)=(S (plus n m)).
Proof.
Auto.
Qed.
(** Multiplication *)
Fixpoint mult [n:nat] : nat -> nat :=
[m:nat]Cases n of O => O
| (S p) => (plus m (mult p m)) end.
Hint eq_mult : core v62 := Resolve (f_equal2 nat nat nat mult).
V8Infix "*" mult : nat_scope.
Lemma mult_n_O : (n:nat) O=(mult n O).
Proof.
NewInduction n; Simpl; Auto.
Qed.
Hints Resolve mult_n_O : core v62.
Lemma mult_n_Sm : (n,m:nat) (plus (mult n m) n)=(mult n (S m)).
Proof.
Intros; NewInduction n as [|p H]; Simpl; Auto.
NewDestruct H; Rewrite <- plus_n_Sm; Apply (f_equal nat nat S).
Pattern 1 3 m; Elim m; Simpl; Auto.
Qed.
Hints Resolve mult_n_Sm : core v62.
(** Definition of subtraction on [nat] : [m-n] is [0] if [n>=m] *)
Fixpoint minus [n:nat] : nat -> nat :=
[m:nat]Cases n m of
O _ => O
| (S k) O => (S k)
| (S k) (S l) => (minus k l)
end.
V8Infix "-" minus : nat_scope.
(** Definition of the usual orders, the basic properties of [le] and [lt]
can be found in files Le and Lt *)
(** An inductive definition to define the order *)
Inductive le [n:nat] : nat -> Prop
:= le_n : (le n n)
| le_S : (m:nat)(le n m)->(le n (S m)).
V8Infix "<=" le : nat_scope.
Hint constr_le : core v62 := Constructors le.
(*i equivalent to : "Hints Resolve le_n le_S : core v62." i*)
Definition lt := [n,m:nat](le (S n) m).
Hints Unfold lt : core v62.
V8Infix "<" lt : nat_scope.
Definition ge := [n,m:nat](le m n).
Hints Unfold ge : core v62.
V8Infix ">=" ge : nat_scope.
Definition gt := [n,m:nat](lt m n).
Hints Unfold gt : core v62.
V8Infix ">" gt : nat_scope.
V8Notation "x <= y <= z" := (le x y)/\(le y z) : nat_scope.
V8Notation "x <= y < z" := (le x y)/\(lt y z) : nat_scope.
V8Notation "x < y < z" := (lt x y)/\(lt y z) : nat_scope.
V8Notation "x < y <= z" := (lt x y)/\(le y z) : nat_scope.
(** Pattern-Matching on natural numbers *)
Theorem nat_case : (n:nat)(P:nat->Prop)(P O)->((m:nat)(P (S m)))->(P n).
Proof.
NewInduction n ; Auto.
Qed.
(** Principle of double induction *)
Theorem nat_double_ind : (R:nat->nat->Prop)
((n:nat)(R O n)) -> ((n:nat)(R (S n) O))
-> ((n,m:nat)(R n m)->(R (S n) (S m)))
-> (n,m:nat)(R n m).
Proof.
NewInduction n; Auto.
NewDestruct m; Auto.
Qed.
(** Notations *)
V7only[
Syntax constr
level 0:
S [ (S $p) ] -> [$p:"nat_printer":9]
| O [ O ] -> ["(0)"].
].
V7only [
(* For parsing/printing based on scopes *)
Module nat_scope.
Infix 4 "+" plus : nat_scope.
Infix 3 "*" mult : nat_scope.
Infix 4 "-" minus : nat_scope.
Infix NONA 5 "<=" le : nat_scope.
Infix NONA 5 "<" lt : nat_scope.
Infix NONA 5 ">=" ge : nat_scope.
Infix NONA 5 ">" gt : nat_scope.
End nat_scope.
].
|
