Standard Library | The Coq Proof Assistant

The Coq Proof Assistant

# Library Coq.Arith.Gt

Theorems about gt in nat.
This file is DEPRECATED now, see module PeanoNat.Nat instead, which favor lt over gt.
gt is defined in Init/Peano.v as:

    Definition gt (n m:nat) := m < n.

Require Import PeanoNat Le Lt Plus.
Local Open Scope nat_scope.

# Order and successor

Theorem gt_Sn_O n : S n > 0.
Theorem gt_Sn_n n : S n > n.
Theorem gt_n_S n m : n > m -> S n > S m.
Lemma gt_S_n n m : S m > S n -> m > n.
Theorem gt_S n m : S n > m -> n > m \/ m = n.
Lemma gt_pred n m : m > S n -> pred m > n.

# Irreflexivity

Lemma gt_irrefl n : ~ n > n.

# Asymmetry

Lemma gt_asym n m : n > m -> ~ m > n.

# Relating strict and large orders

Lemma le_not_gt n m : n <= m -> ~ n > m.
Lemma gt_not_le n m : n > m -> ~ n <= m.
Theorem le_S_gt n m : S n <= m -> m > n.
Lemma gt_S_le n m : S m > n -> n <= m.
Lemma gt_le_S n m : m > n -> S n <= m.
Lemma le_gt_S n m : n <= m -> S m > n.

# Transitivity

Theorem le_gt_trans n m p : m <= n -> m > p -> n > p.
Theorem gt_le_trans n m p : n > m -> p <= m -> n > p.
Lemma gt_trans n m p : n > m -> m > p -> n > p.
Theorem gt_trans_S n m p : S n > m -> m > p -> n > p.

# Comparison to 0

Theorem gt_0_eq n : n > 0 \/ 0 = n.

# Simplification and compatibility

Lemma plus_gt_reg_l n m p : p + n > p + m -> n > m.
Lemma plus_gt_compat_l n m p : n > m -> p + n > p + m.

# Hints

Hint Resolve gt_Sn_O gt_Sn_n gt_n_S : arith.
Hint Immediate gt_S_n gt_pred : arith.
Hint Resolve gt_irrefl gt_asym : arith.
Hint Resolve le_not_gt gt_not_le : arith.
Hint Immediate le_S_gt gt_S_le : arith.
Hint Resolve gt_le_S le_gt_S : arith.
Hint Resolve gt_trans_S le_gt_trans gt_le_trans: arith.
Hint Resolve plus_gt_compat_l: arith.