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(* $Id$ *)
(* 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 Export Logic.
Require Export LogicSyntax.
Require Export Datatypes.
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).
Elim (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.
Induction 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).
Lemma plus_n_O : (n:nat) n=(plus n O).
Proof.
Induction n ; Simpl ; Auto.
Qed.
Hints Resolve plus_n_O : core v62.
Lemma plus_n_Sm : (n,m:nat) (S (plus n m))=(plus n (S m)).
Proof.
Intros m n; Elim m; Simpl; Auto.
Qed.
Hints Resolve plus_n_Sm : core v62.
(***************************************)
(* 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).
Lemma mult_n_O : (n:nat) O=(mult n O).
Proof.
Induction 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; Elim n; Simpl; Auto.
Intros p H; Case H; Elim 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 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)).
Hint constr_le : core v62 := Constructors le.
(* equivalent to : "Hints Resolve le_n le_S : core v62." *)
Definition lt := [n,m:nat](le (S n) m).
Hints Unfold lt : core v62.
Definition ge := [n,m:nat](le m n).
Hints Unfold ge : core v62.
Definition gt := [n,m:nat](lt m n).
Hints Unfold gt : core v62.
(*********************************************************)
(* Pattern-Matching on natural numbers *)
(*********************************************************)
Theorem nat_case : (n:nat)(P:nat->Prop)(P O)->((m:nat)(P (S m)))->(P n).
Proof.
Induction 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.
Induction n; Auto.
Induction m; Auto.
Qed.
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