Standard Library | The Coq Proof Assistant

The Coq Proof Assistant

# Library Coq.Arith.Plus

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 (only parsing).
Notation plus_0_r := Nat.add_0_r (only parsing).
Notation plus_comm := Nat.add_comm (only parsing).
Notation plus_assoc := Nat.add_assoc (only parsing).
Notation plus_permute := Nat.add_shuffle3 (only parsing).
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).

# Simplification

Lemma plus_reg_l n m p : p + n = p + m -> n = m.
Lemma plus_le_reg_l n m p : p + n <= p + m -> n <= m.
Lemma plus_lt_reg_l n m p : p + n < p + m -> n < m.

# Compatibility with order

Lemma plus_le_compat_l n m p : n <= m -> p + n <= p + m.
Lemma plus_le_compat_r n m p : n <= m -> n + p <= m + p.
Lemma plus_lt_compat_l n m p : n < m -> p + n < p + m.
Lemma plus_lt_compat_r n m p : n < m -> n + p < m + p.
Lemma plus_le_compat n m p q : n <= m -> p <= q -> n + p <= m + q.
Lemma plus_le_lt_compat n m p q : n <= m -> p < q -> n + p < m + q.
Lemma plus_lt_le_compat n m p q : n < m -> p <= q -> n + p < m + q.
Lemma plus_lt_compat n m p q : n < m -> p < q -> n + p < m + q.
Lemma le_plus_l n m : n <= n + m.
Lemma le_plus_r n m : m <= n + m.
Theorem le_plus_trans n m p : n <= m -> n <= m + p.
Theorem lt_plus_trans n m p : n < m -> n < m + p.

# Inversion lemmas

Lemma plus_is_O n m : n + m = 0 -> n = 0 /\ m = 0.
Definition plus_is_one m n :
m + n = 1 -> {m = 0 /\ n = 1} + {m = 1 /\ n = 0}.

# Derived properties

Notation plus_permute_2_in_4 := Nat.add_shuffle1 (only parsing).

# 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.

# Discrimination

Lemma succ_plus_discr n m : n <> S (m+n).
Lemma n_SSn n : n <> S (S n).
Lemma n_SSSn n : n <> S (S (S n)).
Lemma n_SSSSn n : n <> S (S (S (S n))).

# 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.