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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** Properties of addition.
This file is mostly OBSOLETE now, see module [PeanoNat.Nat] instead.
[Nat.add] is defined in [Init/Nat.v] as:
<<
Fixpoint add (n m:nat) : nat :=
match n with
| O => m
| S p => S (p + m)
end
where "n + m" := (add n m) : nat_scope.
>>
*)
Require Import PeanoNat.
Local Open Scope nat_scope.
(** * Neutrality of 0, commutativity, associativity *)
Notation plus_0_l := Nat.add_0_l (compat "8.4").
Notation plus_0_r := Nat.add_0_r (compat "8.4").
Notation plus_comm := Nat.add_comm (compat "8.4").
Notation plus_assoc := Nat.add_assoc (compat "8.4").
Notation plus_permute := Nat.add_shuffle3 (compat "8.4").
Definition plus_Snm_nSm : forall n m, S n + m = n + S m :=
Peano.plus_n_Sm.
Lemma plus_assoc_reverse n m p : n + m + p = n + (m + p).
Proof.
symmetry. apply Nat.add_assoc.
Qed.
(** * Simplification *)
Lemma plus_reg_l n m p : p + n = p + m -> n = m.
Proof.
apply Nat.add_cancel_l.
Qed.
Lemma plus_le_reg_l n m p : p + n <= p + m -> n <= m.
Proof.
apply Nat.add_le_mono_l.
Qed.
Lemma plus_lt_reg_l n m p : p + n < p + m -> n < m.
Proof.
apply Nat.add_lt_mono_l.
Qed.
(** * Compatibility with order *)
Lemma plus_le_compat_l n m p : n <= m -> p + n <= p + m.
Proof.
apply Nat.add_le_mono_l.
Qed.
Lemma plus_le_compat_r n m p : n <= m -> n + p <= m + p.
Proof.
apply Nat.add_le_mono_r.
Qed.
Lemma plus_lt_compat_l n m p : n < m -> p + n < p + m.
Proof.
apply Nat.add_lt_mono_l.
Qed.
Lemma plus_lt_compat_r n m p : n < m -> n + p < m + p.
Proof.
apply Nat.add_lt_mono_r.
Qed.
Lemma plus_le_compat n m p q : n <= m -> p <= q -> n + p <= m + q.
Proof.
apply Nat.add_le_mono.
Qed.
Lemma plus_le_lt_compat n m p q : n <= m -> p < q -> n + p < m + q.
Proof.
apply Nat.add_le_lt_mono.
Qed.
Lemma plus_lt_le_compat n m p q : n < m -> p <= q -> n + p < m + q.
Proof.
apply Nat.add_lt_le_mono.
Qed.
Lemma plus_lt_compat n m p q : n < m -> p < q -> n + p < m + q.
Proof.
apply Nat.add_lt_mono.
Qed.
Lemma le_plus_l n m : n <= n + m.
Proof.
apply Nat.le_add_r.
Qed.
Lemma le_plus_r n m : m <= n + m.
Proof.
rewrite Nat.add_comm. apply Nat.le_add_r.
Qed.
Theorem le_plus_trans n m p : n <= m -> n <= m + p.
Proof.
intros. now rewrite <- Nat.le_add_r.
Qed.
Theorem lt_plus_trans n m p : n < m -> n < m + p.
Proof.
intros. apply Nat.lt_le_trans with m. trivial. apply Nat.le_add_r.
Qed.
(** * Inversion lemmas *)
Lemma plus_is_O n m : n + m = 0 -> n = 0 /\ m = 0.
Proof.
destruct n; now split.
Qed.
Definition plus_is_one m n :
m + n = 1 -> {m = 0 /\ n = 1} + {m = 1 /\ n = 0}.
Proof.
destruct m as [| m]; auto.
destruct m; auto.
discriminate.
Defined.
(** * Derived properties *)
Notation plus_permute_2_in_4 := Nat.add_shuffle1 (compat "8.4").
(** * Tail-recursive plus *)
(** [tail_plus] is an alternative definition for [plus] which is
tail-recursive, whereas [plus] is not. This can be useful
when extracting programs. *)
Fixpoint tail_plus n m : nat :=
match n with
| O => m
| S n => tail_plus n (S m)
end.
Lemma plus_tail_plus : forall n m, n + m = tail_plus n m.
Proof.
induction n as [| n IHn]; simpl; auto.
intro m; rewrite <- IHn; simpl; auto.
Qed.
(** * Discrimination *)
Lemma succ_plus_discr n m : n <> S (m+n).
Proof.
apply Nat.succ_add_discr.
Qed.
Lemma n_SSn n : n <> S (S n).
Proof (succ_plus_discr n 1).
Lemma n_SSSn n : n <> S (S (S n)).
Proof (succ_plus_discr n 2).
Lemma n_SSSSn n : n <> S (S (S (S n))).
Proof (succ_plus_discr n 3).
(** * Compatibility Hints *)
Hint Immediate plus_comm : arith.
Hint Resolve plus_assoc plus_assoc_reverse : arith.
Hint Resolve plus_le_compat_l plus_le_compat_r : arith.
Hint Resolve le_plus_l le_plus_r le_plus_trans : arith.
Hint Immediate lt_plus_trans : arith.
Hint Resolve plus_lt_compat_l plus_lt_compat_r : arith.
(** For compatibility, we "Require" the same files as before *)
Require Import Le Lt.
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