From ccf5878f2a2ec7f952f140e162391708a740517b Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Fri, 1 Jan 2021 15:42:16 +0100
Subject: [PATCH] =?utf8?q?arithmetics=20for=20=CE=BB=CE=B4?=
MIME-Version: 1.0
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strict order for non-ngative integers completed
---
 .../contribs/lambdadelta/ground/arith/nat.txt |  6 +--
 .../lambdadelta/ground/arith/nat_le_minus.ma  |  3 ++
 .../lambdadelta/ground/arith/nat_le_pred.ma   | 17 ++++---
 .../lambdadelta/ground/arith/nat_lt.ma        |  3 ++
 .../lambdadelta/ground/arith/nat_lt_minus.ma  | 11 ++++
 .../ground/arith/nat_lt_minus_plus.ma         | 51 +++++++++++++++++++
 .../lambdadelta/ground/arith/nat_lt_pred.ma   |  9 +++-
 .../ground/arith/nat_minus_plus.ma            |  4 +-
 8 files changed, 91 insertions(+), 13 deletions(-)
 create mode 100644 matita/matita/contribs/lambdadelta/ground/arith/nat_lt_minus_plus.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat.txt b/matita/matita/contribs/lambdadelta/ground/arith/nat.txt
index fc8e9e241..2bd0eabca 100644
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat.txt
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat.txt
@@ -111,12 +111,12 @@
 #
 "cic:/matita/arithmetics/nat/minus_pred_pred.con"
 "cic:/matita/arithmetics/nat/monotonic_lt_minus_l.con"
-
-"cic:/matita/arithmetics/nat/lt_minus_to_plus.con"
-"cic:/matita/arithmetics/nat/lt_minus_to_plus_r.con"
+#
 "cic:/matita/arithmetics/nat/lt_plus_to_minus.con"
 "cic:/matita/arithmetics/nat/lt_plus_to_minus_r.con"
 "cic:/matita/arithmetics/nat/lt_inv_plus_l.con"
+"cic:/matita/arithmetics/nat/lt_minus_to_plus.con"
+"cic:/matita/arithmetics/nat/lt_minus_to_plus_r.con"
 
 ####################################
 
diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_le_minus.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_le_minus.ma
index e3da0c922..607a9135e 100644
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_le_minus.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_le_minus.ma
@@ -25,6 +25,9 @@ lemma nle_minus_sn_refl_sn (m) (n): m - n ≤ m.
 #n #IH /2 width=3 by nle_trans/
 qed.
 
+lemma nle_minus_succ_sn (m) (n): ↑n - m ≤ ↑(n - m).
+// qed.
+
 (*** inv_eq_minus_O *)
 lemma nle_eq_minus_O (m) (n): 𝟎 = m - n → m ≤ n.
 #m #n @(nat_ind_2 … m n) //
diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_le_pred.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_le_pred.ma
index da1de97f5..75cd4bf7e 100644
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_le_pred.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_le_pred.ma
@@ -17,8 +17,18 @@ include "ground/arith/nat_le.ma".
 
 (* ORDER FOR NON-NEGATIVE INTEGERS ******************************************)
 
+(* Destructions with npred **************************************************)
+
+lemma nle_inv_pred_sn (m) (n): ↓m ≤ n → m ≤ ↑n.
+#m #n elim m -m
+/2 width=1 by nle_succ_bi/
+qed-.
+
 (* Constructions with npred *************************************************)
 
+lemma nle_succ_pred_dx_refl (m): m ≤ ↑↓m.
+#m @nle_inv_pred_sn // qed.
+
 (*** le_pred_n *)
 lemma nle_pred_sn_refl (m): ↓m ≤ m.
 #m elim m -m //
@@ -34,10 +44,3 @@ lemma nle_pred_sn (m) (n): m ≤ ↑n → ↓m ≤ n.
 #m #n elim m -m //
 /2 width=1 by nle_pred_bi/
 qed-.
-
-(* Destructions with npred **************************************************)
-
-lemma nle_inv_pred_sn (m) (n): ↓m ≤ n → m ≤ ↑n.
-#m #n elim m -m
-/2 width=1 by nle_succ_bi/
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_lt.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt.ma
index f7a868b53..388c8aace 100644
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_lt.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt.ma
@@ -36,6 +36,9 @@ lemma nlt_refl_succ (n): n < ↑n.
 lemma nlt_zero_succ (m): 𝟎 < ↑m.
 /2 width=1 by nle_succ_bi/ qed.
 
+lemma nlt_succ_bi (m) (n): m < n → ↑m < ↑n.
+/2 width=1 by nle_succ_bi/ qed.
+
 (*** le_to_or_lt_eq *)
 lemma nle_lt_eq_e (m) (n): m ≤ n → ∨∨ m < n | m = n.
 #m #n * -n /3 width=1 by nle_succ_bi, or_introl/
diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_minus.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_minus.ma
index 34a16e01f..5ee4b80eb 100644
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_minus.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_minus.ma
@@ -35,3 +35,14 @@ lemma nlt_minus_sn_bi (o) (m) (n): o ≤ m → m < n → m - o < n - o.
 lapply (nle_minus_sn_bi … o Hmn) -Hmn
 <(nminus_succ_sn … Hom) //
 qed.
+
+(* Destructions with nminus *************************************************)
+
+lemma nlt_fwd_minus_dx (o) (m) (n): m < n - o → o < n.
+#o elim o -o
+[ #m #n <nminus_zero_dx
+  /2 width=3 by le_nlt_trans/
+| #o #IH #m #n <nminus_succ_dx_pred_sn #H
+  /3 width=2 by nlt_inv_pred_dx/
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_minus_plus.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_minus_plus.ma
new file mode 100644
index 000000000..0f0106021
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_minus_plus.ma
@@ -0,0 +1,51 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground/arith/nat_le_minus_plus.ma".
+include "ground/arith/nat_lt_minus.ma".
+
+(* STRICT ORDER FOR NON-NEGATIVE INTEGERS ***********************************)
+
+(* Constructions with nminus and nplus **************************************)
+
+(*** lt_plus_to_minus *)
+lemma nlt_minus_sn (o) (m) (n): m ≤ n → n < o + m → n - m < o.
+#o #m #n #Hmn #Ho
+lapply (nle_minus_sn … Ho) -Ho
+<nminus_succ_sn //
+qed.
+
+(*** lt_plus_to_minus_r *)
+lemma nlt_minus_dx (o) (m) (n): m + o < n → m < n - o.
+/2 width=1 by nle_minus_dx/ qed.
+
+(*** lt_inv_plus_l *)
+lemma nlt_minus_dx_full (o) (m) (n): m + o < n → ∧∧ o < n & m < n - o.
+/3 width=3 by nlt_minus_dx, le_nlt_trans, conj/ qed-.
+
+(* Inversions with nminus and nplus *****************************************)
+
+(*** lt_minus_to_plus *)
+lemma nlt_inv_minus_sn (o) (m) (n): m - o < n → m < n + o.
+#o #m #n #Ho
+@nle_inv_minus_sn
+@(nle_trans … Ho) -Ho //
+qed-.
+
+(*** lt_minus_to_plus_r *)
+lemma nlt_inv_minus_dx (o) (m) (n): m < n - o → m + o < n.
+#o #m #n #Ho
+lapply (nle_inv_minus_dx ???? Ho) //
+/3 width=2 by nlt_fwd_minus_dx, nlt_des_le/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_pred.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_pred.ma
index 2b3502259..7ebc811f8 100644
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_pred.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_pred.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "ground/arith/nat_pred_succ.ma".
+include "ground/arith/nat_le_pred.ma".
 include "ground/arith/nat_lt.ma".
 
 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS ***********************************)
@@ -29,3 +29,10 @@ lemma nlt_inv_zero_sn (m): 𝟎 < m → m = ↑↓m.
 #m @(nat_ind … m) -m //
 #H elim (nlt_inv_refl … H)
 qed-.
+
+lemma nlt_inv_pred_dx (m) (n): m < ↓n → ↑m < n.
+#m #n #H >(nlt_inv_zero_sn n)
+[ /2 width=1 by nlt_succ_bi/
+| /3 width=3 by le_nlt_trans, nlt_le_trans/
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_minus_plus.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_minus_plus.ma
index 47e88de49..bf5cf5fee 100644
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_minus_plus.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_minus_plus.ma
@@ -30,14 +30,14 @@ lemma nminus_plus_sn_refl_dx (m) (n): m = n + m - n.
 qed.
 
 (*** minus_plus *)
-theorem nminus_assoc (o) (m) (n): o-m-n = o-(m+n).
+theorem nminus_plus_assoc (o) (m) (n): o-m-n = o-(m+n).
 #o #m #n elim n -n //
 #n #IH <nplus_succ_dx <nminus_succ_dx <nminus_succ_dx //
 qed-.
 
 (*** minus_plus_plus_l *)
 lemma nminus_plus_dx_bi (m) (n) (o): m - n = (m + o) - (n + o).
-#m #n #o <nminus_assoc <nminus_minus_comm //
+#m #n #o <nminus_plus_assoc <nminus_minus_comm //
 qed.
 
 (*** plus_minus_plus_plus_l *) (**)
-- 
2.39.2