Coq.Arith.Max

# Module Coq.Arith.Max

``Require` Arith.

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maximum of two natural numbers

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

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Simplifications of `max`

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

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

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`max` and `le`

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

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

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

`Lemma` le_max_r : (n,m:nat)(le m (max n m)).
`Proof`.
NewInduction n; Simpl; Auto with arith.
NewInduction m; Simpl; Auto with arith.
`Qed`.
`Hints` Resolve max_r max_l le_max_l le_max_r: arith v62.

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`max n m` is equal to `n` or `m`

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

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

`Lemma` max_case2 : (n,m:nat)(P:nat->Prop)(P n)->(P m)->(P (max n m)).
`Proof`.
NewInduction n; Simpl; Auto with arith.
NewInduction m; Intros; Simpl; Auto with arith.
Pattern (max n m); Apply IHn ; Auto with arith.
`Qed`.

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Index