Library Coq.Arith.Minus


Properties of subtraction between natural numbers.
This file is mostly OBSOLETE now, see module PeanoNat.Nat instead.
minus is now an alias for Nat.sub, which is defined in Init/Nat.v as:
Fixpoint sub (n m:nat) : nat :=
  match n, m with
  | S k, S l => k - l
  | _, _ => n
  end
where "n - m" := (sub n m) : nat_scope.

Require Import PeanoNat Lt Le.

Local Open Scope nat_scope.

0 is right neutral


Lemma minus_n_O n : n = n - 0.

Permutation with successor


Lemma minus_Sn_m n m : m <= n -> S (n - m) = S n - m.

Theorem pred_of_minus n : pred n = n - 1.

Diagonal


Notation minus_diag := Nat.sub_diag (compat "8.4").
Lemma minus_diag_reverse n : 0 = n - n.

Notation minus_n_n := minus_diag_reverse.

Simplification


Lemma minus_plus_simpl_l_reverse n m p : n - m = p + n - (p + m).

Relation with plus


Lemma plus_minus n m p : n = m + p -> p = n - m.

Lemma minus_plus n m : n + m - n = m.

Lemma le_plus_minus_r n m : n <= m -> n + (m - n) = m.

Lemma le_plus_minus n m : n <= m -> m = n + (m - n).

Relation with order


Notation minus_le_compat_r :=
  Nat.sub_le_mono_r (compat "8.4").
Notation minus_le_compat_l :=
  Nat.sub_le_mono_l (compat "8.4").
Notation le_minus := Nat.le_sub_l (compat "8.4"). Notation lt_minus := Nat.sub_lt (compat "8.4").
Lemma lt_O_minus_lt n m : 0 < n - m -> m < n.

Theorem not_le_minus_0 n m : ~ m <= n -> n - m = 0.

Hints


Hint Resolve minus_n_O: arith v62.
Hint Resolve minus_Sn_m: arith v62.
Hint Resolve minus_diag_reverse: arith v62.
Hint Resolve minus_plus_simpl_l_reverse: arith v62.
Hint Immediate plus_minus: arith v62.
Hint Resolve minus_plus: arith v62.
Hint Resolve le_plus_minus: arith v62.
Hint Resolve le_plus_minus_r: arith v62.
Hint Resolve lt_minus: arith v62.
Hint Immediate lt_O_minus_lt: arith v62.