# Library Coq.Arith.Minus

``` ```
`minus` (difference between two natural numbers) is defined in `Init/Peano.v` as:
```Fixpoint minus (n m:nat) {struct n} : nat :=
match n, m with
| O, _ => 0
| S k, O => S k
| S k, S l => k - l
end
where "n - m" := (minus n m) : nat_scope.
```

``` Require Import Lt. Require Import Le. Open Local Scope nat_scope. Implicit Types m n p : nat. ```

# 0 is right neutral

``` Lemma minus_n_O : forall n, n = n - 0. Proof.   induction n; simpl in |- *; auto with arith. Qed. Hint Resolve minus_n_O: arith v62. ```

# Permutation with successor

``` Lemma minus_Sn_m : forall n m, m <= n -> S (n - m) = S n - m. Proof.   intros n m Le; pattern m, n in |- *; apply le_elim_rel; simpl in |- *;     auto with arith. Qed. Hint Resolve minus_Sn_m: arith v62. Theorem pred_of_minus : forall n, pred n = n - 1. Proof.   intro x; induction x; simpl in |- *; auto with arith. Qed. ```

# Diagonal

``` Lemma minus_n_n : forall n, 0 = n - n. Proof.   induction n; simpl in |- *; auto with arith. Qed. Hint Resolve minus_n_n: arith v62. ```

# Simplification

``` Lemma minus_plus_simpl_l_reverse : forall n m p, n - m = p + n - (p + m). Proof.   induction p; simpl in |- *; auto with arith. Qed. Hint Resolve minus_plus_simpl_l_reverse: arith v62. ```

# Relation with plus

``` Lemma plus_minus : forall n m p, n = m + p -> p = n - m. Proof.   intros n m p; pattern m, n in |- *; apply nat_double_ind; simpl in |- *;     intros.   replace (n0 - 0) with n0; auto with arith.   absurd (0 = S (n0 + p)); auto with arith.   auto with arith. Qed. Hint Immediate plus_minus: arith v62. Lemma minus_plus : forall n m, n + m - n = m.   symmetry in |- *; auto with arith. Qed. Hint Resolve minus_plus: arith v62. Lemma le_plus_minus : forall n m, n <= m -> m = n + (m - n). Proof.   intros n m Le; pattern n, m in |- *; apply le_elim_rel; simpl in |- *;     auto with arith. Qed. Hint Resolve le_plus_minus: arith v62. Lemma le_plus_minus_r : forall n m, n <= m -> n + (m - n) = m. Proof.   symmetry in |- *; auto with arith. Qed. Hint Resolve le_plus_minus_r: arith v62. ```

# Relation with order

``` Theorem le_minus : forall n m, n - m <= n. Proof.   intros i h; pattern i, h in |- *; apply nat_double_ind;     [ auto       | auto       | intros m n H; simpl in |- *; apply le_trans with (m := m); auto ]. Qed. Lemma lt_minus : forall n m, m <= n -> 0 < m -> n - m < n. Proof.   intros n m Le; pattern m, n in |- *; apply le_elim_rel; simpl in |- *;     auto with arith.   intros; absurd (0 < 0); auto with arith.   intros p q lepq Hp gtp.   elim (le_lt_or_eq 0 p); auto with arith.   auto with arith.   induction 1; elim minus_n_O; auto with arith. Qed. Hint Resolve lt_minus: arith v62. Lemma lt_O_minus_lt : forall n m, 0 < n - m -> m < n. Proof.   intros n m; pattern n, m in |- *; apply nat_double_ind; simpl in |- *;     auto with arith.   intros; absurd (0 < 0); trivial with arith. Qed. Hint Immediate lt_O_minus_lt: arith v62. Theorem not_le_minus_0 : forall n m, ~ m <= n -> n - m = 0. Proof.   intros y x; pattern y, x in |- *; apply nat_double_ind;     [ simpl in |- *; trivial with arith       | intros n H; absurd (0 <= S n); [ assumption | apply le_O_n ]       | simpl in |- *; intros n m H1 H2; apply H1; unfold not in |- *; intros H3;         apply H2; apply le_n_S; assumption ]. Qed. ```