Coq.Arith.Min

# Module Coq.Arith.Min

``Require` Arith.

`

minimum of two natural numbers

`

`Fixpoint` min [n:nat] : nat -> nat :=
[m:nat]Cases n m of
O _ => O
| (S n') O => O
| (S n') (S m') => (S (min n' m'))
end.

`

Simplifications of `min`

`

`Lemma` min_SS : (n,m:nat)((S (min n m))=(min (S n) (S m))).
`Proof`.
Auto with arith.
`Qed`.

`Lemma` min_sym : (n,m:nat)(min n m)=(min m n).
`Proof`.
NewInduction n;NewInduction m;Simpl;Auto with arith.
`Qed`.

`

`min` and `le`

`

`Lemma` min_l : (n,m:nat)(le n m)->(min n m)=n.
`Proof`.
NewInduction n;NewInduction m;Simpl;Auto with arith.
`Qed`.

`Lemma` min_r : (n,m:nat)(le m n)->(min n m)=m.
`Proof`.
NewInduction n;NewInduction m;Simpl;Auto with arith.
`Qed`.

`Lemma` le_min_l : (n,m:nat)(le (min n m) n).
`Proof`.
NewInduction n; Intros; Simpl; Auto with arith.
Elim m; Intros; Simpl; Auto with arith.
`Qed`.

`Lemma` le_min_r : (n,m:nat)(le (min n m) m).
`Proof`.
NewInduction n; Simpl; Auto with arith.
NewInduction m; Simpl; Auto with arith.
`Qed`.
`Hints` Resolve min_l min_r le_min_l le_min_r : arith v62.

`

`min n m` is equal to `n` or `m`

`

`Lemma` min_dec : (n,m:nat){(min n m)=n}+{(min n m)=m}.
`Proof`.
NewInduction n;NewInduction m;Simpl;Auto with arith.
Elim (IHn m);Intro H;Elim H;Auto.
`Qed`.

`Lemma` min_case : (n,m:nat)(P:nat->Set)(P n)->(P m)->(P (min n m)).
`Proof`.
NewInduction n; Simpl; Auto with arith.
NewInduction m; Intros; Simpl; Auto with arith.
Pattern (min n m); Apply IHn ; Auto with arith.
`Qed`.

`Lemma` min_case2 : (n,m:nat)(P:nat->Prop)(P n)->(P m)->(P (min n m)).
`Proof`.
NewInduction n; Simpl; Auto with arith.
NewInduction m; Intros; Simpl; Auto with arith.
Pattern (min n m); Apply IHn ; Auto with arith.
`Qed`.

`

Index