diff options
Diffstat (limited to 'theories/Arith/Lt.v')
| -rw-r--r-- | theories/Arith/Lt.v | 39 |
1 files changed, 23 insertions, 16 deletions
diff --git a/theories/Arith/Lt.v b/theories/Arith/Lt.v index bfc2b91a..0c7515c6 100644 --- a/theories/Arith/Lt.v +++ b/theories/Arith/Lt.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Strict order on natural numbers. @@ -23,7 +25,7 @@ Local Open Scope nat_scope. (** * Irreflexivity *) -Notation lt_irrefl := Nat.lt_irrefl (compat "8.4"). (* ~ x < x *) +Notation lt_irrefl := Nat.lt_irrefl (only parsing). (* ~ x < x *) Hint Resolve lt_irrefl: arith. @@ -62,12 +64,12 @@ Hint Immediate le_not_lt lt_not_le: arith. (** * Asymmetry *) -Notation lt_asym := Nat.lt_asymm (compat "8.4"). (* n<m -> ~m<n *) +Notation lt_asym := Nat.lt_asymm (only parsing). (* n<m -> ~m<n *) (** * Order and 0 *) -Notation lt_0_Sn := Nat.lt_0_succ (compat "8.4"). (* 0 < S n *) -Notation lt_n_0 := Nat.nlt_0_r (compat "8.4"). (* ~ n < 0 *) +Notation lt_0_Sn := Nat.lt_0_succ (only parsing). (* 0 < S n *) +Notation lt_n_0 := Nat.nlt_0_r (only parsing). (* ~ n < 0 *) Theorem neq_0_lt n : 0 <> n -> 0 < n. Proof. @@ -84,8 +86,8 @@ Hint Immediate neq_0_lt lt_0_neq: arith. (** * Order and successor *) -Notation lt_n_Sn := Nat.lt_succ_diag_r (compat "8.4"). (* n < S n *) -Notation lt_S := Nat.lt_lt_succ_r (compat "8.4"). (* n < m -> n < S m *) +Notation lt_n_Sn := Nat.lt_succ_diag_r (only parsing). (* n < S n *) +Notation lt_S := Nat.lt_lt_succ_r (only parsing). (* n < m -> n < S m *) Theorem lt_n_S n m : n < m -> S n < S m. Proof. @@ -107,6 +109,11 @@ Proof. intros. symmetry. now apply Nat.lt_succ_pred with m. Qed. +Lemma S_pred_pos n: O < n -> n = S (pred n). +Proof. + apply S_pred. +Qed. + Lemma lt_pred n m : S n < m -> n < pred m. Proof. apply Nat.lt_succ_lt_pred. @@ -122,28 +129,28 @@ Hint Resolve lt_pred_n_n: arith. (** * Transitivity properties *) -Notation lt_trans := Nat.lt_trans (compat "8.4"). -Notation lt_le_trans := Nat.lt_le_trans (compat "8.4"). -Notation le_lt_trans := Nat.le_lt_trans (compat "8.4"). +Notation lt_trans := Nat.lt_trans (only parsing). +Notation lt_le_trans := Nat.lt_le_trans (only parsing). +Notation le_lt_trans := Nat.le_lt_trans (only parsing). Hint Resolve lt_trans lt_le_trans le_lt_trans: arith. (** * Large = strict or equal *) -Notation le_lt_or_eq_iff := Nat.lt_eq_cases (compat "8.4"). +Notation le_lt_or_eq_iff := Nat.lt_eq_cases (only parsing). Theorem le_lt_or_eq n m : n <= m -> n < m \/ n = m. Proof. apply Nat.lt_eq_cases. Qed. -Notation lt_le_weak := Nat.lt_le_incl (compat "8.4"). +Notation lt_le_weak := Nat.lt_le_incl (only parsing). Hint Immediate lt_le_weak: arith. (** * Dichotomy *) -Notation le_or_lt := Nat.le_gt_cases (compat "8.4"). (* n <= m \/ m < n *) +Notation le_or_lt := Nat.le_gt_cases (only parsing). (* n <= m \/ m < n *) Theorem nat_total_order n m : n <> m -> n < m \/ m < n. Proof. |