Library Coq.Init.Peano


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).

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.
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.
Hint Resolve O_S: core.

Theorem n_Sn : forall n:nat, n <> S n.
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.
Hint Resolve plus_n_O: core.

Lemma plus_O_n : forall n:nat, 0 + n = n.

Lemma plus_n_Sm : forall n m:nat, S (n + m) = n + S m.
Hint Resolve plus_n_Sm: core.

Lemma plus_Sn_m : forall n m:nat, S n + m = S (n + m).

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.
Hint Resolve mult_n_O: core.

Lemma mult_n_Sm : forall n m:nat, n * m + n = n * S m.
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.

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.

Theorem le_S_n : forall n m, S n <= S m -> n <= m.

Case analysis

Theorem nat_case :
 forall (n:nat) (P:nat -> Prop), P 0 -> (forall m:nat, P (S m)) -> P n.

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.

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.

Theorem max_r : forall n m : nat, n <= m -> max n m = m.

Theorem min_l : forall n m : nat, n <= m -> min n m = n.

Theorem min_r : forall n m : nat, m <= n -> min n m = m.

nth 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).

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).

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).