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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** 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.
Require Coq.Init.Nat.
Open Scope nat_scope.
Definition eq_S := f_equal S.
Definition f_equal_nat := f_equal (A:=nat).
Hint Resolve f_equal_nat: core.
(** The predecessor function *)
Notation pred := Nat.pred (only parsing).
Definition f_equal_pred := f_equal pred.
Theorem pred_Sn : forall n:nat, n = pred (S n).
Proof.
simpl; reflexivity.
Qed.
(** 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.
Proof.
red; 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 *)
Notation plus := Nat.add (only parsing).
Infix "+" := Nat.add : nat_scope.
Definition f_equal2_plus := f_equal2 plus.
Definition f_equal2_nat := f_equal2 (A1:=nat) (A2:=nat).
Hint Resolve f_equal2_nat: core.
Lemma plus_n_O : forall n:nat, n = n + 0.
Proof.
induction n; simpl; auto.
Qed.
Remove Hints eq_refl : core.
Hint Resolve plus_n_O eq_refl: core. (* We want eq_refl to have higher priority than plus_n_O *)
Lemma plus_O_n : forall n:nat, 0 + n = n.
Proof.
reflexivity.
Qed.
Lemma plus_n_Sm : forall n m:nat, S (n + m) = n + S m.
Proof.
intros n m; induction n; simpl; auto.
Qed.
Hint Resolve plus_n_Sm: core.
Lemma plus_Sn_m : forall n m:nat, S n + m = S (n + m).
Proof.
reflexivity.
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 *)
Notation mult := Nat.mul (only parsing).
Infix "*" := Nat.mul : nat_scope.
Definition f_equal2_mult := f_equal2 mult.
Hint Resolve f_equal2_mult: core.
Lemma mult_n_O : forall n:nat, 0 = n * 0.
Proof.
induction n; simpl; 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; auto.
destruct H; rewrite <- plus_n_Sm; apply eq_S.
pattern m at 1 3; elim m; simpl; 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] *)
Notation minus := Nat.sub (only parsing).
Infix "-" := Nat.sub : 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.
Theorem le_pred : forall n m, n <= m -> pred n <= pred m.
Proof.
induction 1; auto. destruct m; simpl; auto.
Qed.
Theorem le_S_n : forall n m, S n <= S m -> n <= m.
Proof.
intros n m. exact (le_pred (S n) (S m)).
Qed.
Theorem le_0_n : forall n, 0 <= n.
Proof.
induction n; constructor; trivial.
Qed.
Theorem le_n_S : forall n m, n <= m -> S n <= S m.
Proof.
induction 1; constructor; trivial.
Qed.
(** 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; auto.
Qed.
(** Maximum and minimum : definitions and specifications *)
Notation max := Nat.max (only parsing).
Notation min := Nat.min (only parsing).
Lemma max_l n m : m <= n -> Nat.max n m = n.
Proof.
revert m; induction n; destruct m; simpl; trivial.
- inversion 1.
- intros. apply f_equal, IHn, le_S_n; trivial.
Qed.
Lemma max_r n m : n <= m -> Nat.max n m = m.
Proof.
revert m; induction n; destruct m; simpl; trivial.
- inversion 1.
- intros. apply f_equal, IHn, le_S_n; trivial.
Qed.
Lemma min_l n m : n <= m -> Nat.min n m = n.
Proof.
revert m; induction n; destruct m; simpl; trivial.
- inversion 1.
- intros. apply f_equal, IHn, le_S_n; trivial.
Qed.
Lemma min_r n m : m <= n -> Nat.min n m = m.
Proof.
revert m; induction n; destruct m; simpl; trivial.
- inversion 1.
- intros. apply f_equal, IHn, le_S_n; trivial.
Qed.
Lemma nat_rect_succ_r {A} (f: A -> A) (x:A) n :
nat_rect (fun _ => A) x (fun _ => f) (S n) = nat_rect (fun _ => A) (f x) (fun _ => f) n.
Proof.
induction n; intros; simpl; rewrite <- ?IHn; trivial.
Qed.
Theorem nat_rect_plus :
forall (n m:nat) {A} (f:A -> A) (x:A),
nat_rect (fun _ => A) x (fun _ => f) (n + m) =
nat_rect (fun _ => A) (nat_rect (fun _ => A) x (fun _ => f) m) (fun _ => f) n.
Proof.
induction n; intros; simpl; rewrite ?IHn; trivial.
Qed.
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