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
.