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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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19 (*#**********************************************************************)
21 (* v * The Coq Proof Assistant / The Coq Development Team *)
23 (* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
25 (* \VV/ *************************************************************)
27 (* // * This file is distributed under the terms of the *)
29 (* * GNU Lesser General Public License Version 2.1 *)
31 (*#**********************************************************************)
33 include "Num/LtProps.ma".
35 include "Num/LeAxioms.ma".
37 (*#* Properties of the relation [<=] *)
39 inline procedural "cic:/Coq/Num/LeProps/lt_le.con" as lemma.
41 inline procedural "cic:/Coq/Num/LeProps/eq_le.con" as lemma.
43 (*#* compatibility with equality *)
45 inline procedural "cic:/Coq/Num/LeProps/le_eq_compat.con" as lemma.
48 Hints Resolve le_eq_compat : num.
53 inline procedural "cic:/Coq/Num/LeProps/le_trans.con" as lemma.
56 Hints Resolve le_trans : num.
59 (*#* compatibility with equality, addition and successor *)
61 inline procedural "cic:/Coq/Num/LeProps/le_add_compat_l.con" as lemma.
64 Hints Resolve le_add_compat_l : num.
67 inline procedural "cic:/Coq/Num/LeProps/le_add_compat_r.con" as lemma.
70 Hints Resolve le_add_compat_r : num.
73 inline procedural "cic:/Coq/Num/LeProps/le_add_compat.con" as lemma.
76 Hints Immediate le_add_compat : num.
79 (* compatibility with successor *)
81 inline procedural "cic:/Coq/Num/LeProps/le_S_compat.con" as lemma.
84 Hints Resolve le_S_compat : num.
87 (*#* relating [<=] with [<] *)
89 inline procedural "cic:/Coq/Num/LeProps/le_lt_x_Sy.con" as lemma.
92 Hints Immediate le_lt_x_Sy : num.
95 inline procedural "cic:/Coq/Num/LeProps/le_le_x_Sy.con" as lemma.
98 Hints Immediate le_le_x_Sy : num.
101 inline procedural "cic:/Coq/Num/LeProps/le_Sx_y_lt.con" as lemma.
104 Hints Immediate le_Sx_y_lt : num.
107 (*#* Combined transitivity *)
109 inline procedural "cic:/Coq/Num/LeProps/lt_le_trans.con" as lemma.
111 inline procedural "cic:/Coq/Num/LeProps/le_lt_trans.con" as lemma.
114 Hints Immediate lt_le_trans le_lt_trans : num.
117 (*#* weaker compatibility results involving [<] and [<=] *)
119 inline procedural "cic:/Coq/Num/LeProps/lt_add_compat_weak_l.con" as lemma.
122 Hints Immediate lt_add_compat_weak_l : num.
125 inline procedural "cic:/Coq/Num/LeProps/lt_add_compat_weak_r.con" as lemma.
128 Hints Immediate lt_add_compat_weak_r : num.