Module Coq.Init.Peano

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 Logic.
Require LogicSyntax.
Require 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.
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.


Index