Coq.Arith.Gt

# Module Coq.Arith.Gt

``Require` Le.
`Require` Lt.
`Require` Plus.

`Theorem` gt_Sn_O : (n:nat)(gt (S n) O).
`Proof`.
Auto with arith.
`Qed`.
`Hints` Resolve gt_Sn_O : arith v62.

`Theorem` gt_Sn_n : (n:nat)(gt (S n) n).
`Proof`.
Auto with arith.
`Qed`.
`Hints` Resolve gt_Sn_n : arith v62.

`Theorem` le_S_gt : (n,m:nat)(le (S n) m)->(gt m n).
`Proof`.
Auto with arith.
`Qed`.
`Hints` `Immediate` le_S_gt : arith v62.

`Theorem` gt_n_S : (n,m:nat)(gt n m)->(gt (S n) (S m)).
`Proof`.
Auto with arith.
`Qed`.
`Hints` Resolve gt_n_S : arith v62.

`Theorem` gt_trans_S : (n,m,p:nat)(gt (S n) m)->(gt m p)->(gt n p).
`Proof`.
Red; Intros; Apply lt_le_trans with m; Auto with arith.
`Qed`.

`Theorem` le_gt_trans : (n,m,p:nat)(le m n)->(gt m p)->(gt n p).
`Proof`.
Red; Intros; Apply lt_le_trans with m; Auto with arith.
`Qed`.

`Theorem` gt_le_trans : (n,m,p:nat)(gt n m)->(le p m)->(gt n p).
`Proof`.
Red; Intros; Apply le_lt_trans with m; Auto with arith.
`Qed`.

`Hints` Resolve gt_trans_S le_gt_trans gt_le_trans : arith v62.

`Lemma` le_not_gt : (n,m:nat)(le n m) -> ~(gt n m).
`Proof` le_not_lt.
`Hints` Resolve le_not_gt : arith v62.

`Lemma` gt_antirefl : (n:nat)~(gt n n).
`Proof` lt_n_n.
`Hints` Resolve gt_antirefl : arith v62.

`Lemma` gt_not_sym : (n,m:nat)(gt n m) -> ~(gt m n).
`Proof` [n,m:nat](lt_not_sym m n).

`Lemma` gt_not_le : (n,m:nat)(gt n m) -> ~(le n m).
`Proof`.
Auto with arith.
`Qed`.
`Hints` Resolve gt_not_sym gt_not_le : arith v62.

`Lemma` gt_trans : (n,m,p:nat)(gt n m)->(gt m p)->(gt n p).
`Proof`.
Red; Intros n m p H1 H2.
Apply lt_trans with m; Auto with arith.
`Qed`.

`Lemma` gt_S_n : (n,p:nat)(gt (S p) (S n))->(gt p n).
`Proof`.
Auto with arith.
`Qed`.
`Hints` `Immediate` gt_S_n : arith v62.

`Lemma` gt_S_le : (n,p:nat)(gt (S p) n)->(le n p).
`Proof`.
Intros n p; Exact (lt_n_Sm_le n p).
`Qed`.
`Hints` `Immediate` gt_S_le : arith v62.

`Lemma` gt_le_S : (n,p:nat)(gt p n)->(le (S n) p).
`Proof`.
Auto with arith.
`Qed`.
`Hints` Resolve gt_le_S : arith v62.

`Lemma` le_gt_S : (n,p:nat)(le n p)->(gt (S p) n).
`Proof`.
Auto with arith.
`Qed`.
`Hints` Resolve le_gt_S : arith v62.

`Lemma` gt_pred : (n,p:nat)(gt p (S n))->(gt (pred p) n).
`Proof`.
Auto with arith.
`Qed`.
`Hints` `Immediate` gt_pred : arith v62.

`Theorem` gt_S : (n,m:nat)(gt (S n) m)->((gt n m)\/(<nat>m=n)).
`Proof`.
Intros n m H; Unfold gt; Apply le_lt_or_eq; Auto with arith.
`Qed`.

`Theorem` gt_O_eq : (n:nat)((gt n O)\/(<nat>O=n)).
`Proof`.
Intro n ; Apply gt_S ; Auto with arith.
`Qed`.

`Lemma` simpl_gt_plus_l : (n,m,p:nat)(gt (plus p n) (plus p m))->(gt n m).
`Proof`.
Red; Intros n m p H; Apply simpl_lt_plus_l with p; Auto with arith.
`Qed`.

`Lemma` gt_reg_l : (n,m,p:nat)(gt n m)->(gt (plus p n) (plus p m)).
`Proof`.
Auto with arith.
`Qed`.
`Hints` Resolve gt_reg_l : arith v62.

`

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