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