From fefe8d334012230f8e8b9d90976d9411a58d4ba8 Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Wed, 10 Aug 2011 12:28:45 +0000
Subject: [PATCH] the refactoring continues ...

---
 .../contribs/lambda-delta/Basic-2/names.txt   |  14 ++
 .../contribs/lambda-delta/Basic-2/notation.ma |  78 ++++++++
 .../contribs/lambda-delta/Ground-2/ground.ma  | 182 ++++++++++++++++++
 .../lambda-delta/Ground-2/xoa.conf.xml        |  34 ++++
 .../contribs/lambda-delta/Ground-2/xoa.ma     | 127 ++++++++++++
 .../lambda-delta/Ground-2/xoa_notation.ma     | 138 +++++++++++++
 .../lambda-delta/Ground-2/xoa_props.ma        |  20 ++
 matita/matita/contribs/lambda-delta/Makefile  |  12 ++
 8 files changed, 605 insertions(+)
 create mode 100644 matita/matita/contribs/lambda-delta/Basic-2/names.txt
 create mode 100644 matita/matita/contribs/lambda-delta/Basic-2/notation.ma
 create mode 100644 matita/matita/contribs/lambda-delta/Ground-2/ground.ma
 create mode 100644 matita/matita/contribs/lambda-delta/Ground-2/xoa.conf.xml
 create mode 100644 matita/matita/contribs/lambda-delta/Ground-2/xoa.ma
 create mode 100644 matita/matita/contribs/lambda-delta/Ground-2/xoa_notation.ma
 create mode 100644 matita/matita/contribs/lambda-delta/Ground-2/xoa_props.ma
 create mode 100644 matita/matita/contribs/lambda-delta/Makefile

diff --git a/matita/matita/contribs/lambda-delta/Basic-2/names.txt b/matita/matita/contribs/lambda-delta/Basic-2/names.txt
new file mode 100644
index 000000000..2a030fb4f
--- /dev/null
+++ b/matita/matita/contribs/lambda-delta/Basic-2/names.txt
@@ -0,0 +1,14 @@
+NAMING CONVENTIONS FOR METAVARIABLES
+
+H      : reserved: transient premise
+IH     : reserved: inductive premise
+I,J    : item
+K,L    : local environment
+T,U,V,W: term
+X,Y,Z  : reserved: transient objet denoted by a capital letter
+
+d      : relocation depth
+e      : relocation height
+h      : sort hierarchy parameter
+i,j    : local reference position index (de Bruijn's)
+k      : sort index
diff --git a/matita/matita/contribs/lambda-delta/Basic-2/notation.ma b/matita/matita/contribs/lambda-delta/Basic-2/notation.ma
new file mode 100644
index 000000000..375d06abc
--- /dev/null
+++ b/matita/matita/contribs/lambda-delta/Basic-2/notation.ma
@@ -0,0 +1,78 @@
+(*
+    ||M||  This file is part of HELM, an Hypertextual, Electronic
+    ||A||  Library of Mathematics, developed at the Computer Science
+    ||T||  Department of the University of Bologna, Italy.
+    ||I||
+    ||T||
+    ||A||  This file is distributed under the terms of the
+    \   /  GNU General Public License Version 2
+     \ /
+      V_______________________________________________________________ *)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+(* Syntax *******************************************************************)
+
+notation "hvbox( ⋆ )"
+ non associative with precedence 90
+ for @{ 'Star }.
+
+notation "hvbox( ⋆ term 90 k )"
+ non associative with precedence 90
+ for @{ 'Star $k }.
+
+notation "hvbox( 𝕚 { I } break (term 90 T1) . break (term 90 T) )"
+ non associative with precedence 90
+ for @{ 'SItem $I $T1 $T }.
+
+notation "hvbox( 𝕓 { I } break (term 90 T1) . break (term 90 T) )"
+ non associative with precedence 90
+ for @{ 'SBind $I $T1 $T }.
+
+notation "hvbox( 𝕗 { I } break (term 90 T1) . break (term 90 T) )"
+ non associative with precedence 90
+ for @{ 'SFlat $I $T1 $T }.
+
+notation "hvbox( T . break 𝕓 { I } break (term 90 T1) )"
+ non associative with precedence 89
+ for @{ 'DBind $T $I $T1 }.
+(*
+notation "hvbox( |  L  | )"
+ non associative with precedence 70
+ for @{ 'Length $L }.
+*)
+notation "hvbox( # term 90 x )"
+ non associative with precedence 90
+ for @{ 'Weight $x }.
+
+notation "hvbox( # [ x , break y ] )"
+ non associative with precedence 90
+ for @{ 'Weight $x $y }.
+
+(* Substitution *************************************************************)
+
+notation "hvbox( T1 break [ d , break e ] ≈ break T2 )"
+   non associative with precedence 45
+   for @{ 'Eq $T1 $d $e $T2 }.
+
+notation "hvbox( ↑ [ d , break e ] break T1 ≡ break T2 )"
+   non associative with precedence 45
+   for @{ 'RLift $T1 $d $e $T2 }.
+
+notation "hvbox( ↓ [ d , break e ] break L1 ≡ break L2 )"
+   non associative with precedence 45
+   for @{ 'RDrop $L1 $d $e $L2 }.
+
+notation "hvbox( L ⊢ break term 90 T1 break [ d , break e ] ≫ break T2 )"
+   non associative with precedence 45
+   for @{ 'PSubst $L $T1 $d $e $T2 }.
+
+(* Reduction ****************************************************************)
+
+notation "hvbox( T1 ⇒ break T2 )"
+   non associative with precedence 45
+   for @{ 'PRed $T1 $T2 }.
+
+notation "hvbox( L ⊢ break (term 90 T1) ⇒ break T2 )"
+   non associative with precedence 45
+   for @{ 'PRed $L $T1 $T2 }.
diff --git a/matita/matita/contribs/lambda-delta/Ground-2/ground.ma b/matita/matita/contribs/lambda-delta/Ground-2/ground.ma
new file mode 100644
index 000000000..ed5876107
--- /dev/null
+++ b/matita/matita/contribs/lambda-delta/Ground-2/ground.ma
@@ -0,0 +1,182 @@
+(*
+    ||M||  This file is part of HELM, an Hypertextual, Electronic
+    ||A||  Library of Mathematics, developed at the Computer Science
+    ||T||  Department of the University of Bologna, Italy.
+    ||I||
+    ||T||
+    ||A||  This file is distributed under the terms of the
+    \   /  GNU General Public License Version 2
+     \ /
+      V_______________________________________________________________ *)
+
+include "arithmetics/nat.ma".
+include "lambda-delta/xoa_props.ma".
+
+(* ARITHMETICAL PROPERTIES **************************************************)
+
+lemma plus_S_eq_O_false: ∀n,m. n + S m = 0 → False.
+#n #m <plus_n_Sm #H destruct
+qed.
+
+lemma plus_S_le_to_pos: ∀n,m,p. n + S m ≤ p → 0 < p.
+#n #m #p <plus_n_Sm #H @(lt_to_le_to_lt … H) //
+qed.
+
+lemma minus_le: ∀m,n. m - n ≤ m.
+/2/ qed.
+
+lemma le_O_to_eq_O: ∀n. n ≤ 0 → n = 0.
+/2/ qed.
+
+lemma lt_to_le: ∀a,b. a < b → a ≤ b.
+/2/ qed.
+
+lemma lt_refl_false: ∀n. n < n → False.
+#n #H elim (lt_to_not_eq … H) -H /2/
+qed.
+
+lemma lt_zero_false: ∀n. n < 0 → False.
+#n #H elim (lt_to_not_le … H) -H /2/
+qed.
+
+lemma lt_or_ge: ∀m,n. m < n ∨ n ≤ m.
+#m #n elim (decidable_lt m n) /3/
+qed.
+
+lemma lt_or_eq_or_gt: ∀m,n. ∨∨ m < n | n = m | n < m.
+#m elim m -m
+[ * /2/
+| #m #IHm * [ /2/ ]
+  #n elim (IHm n) -IHm #H 
+  [ @or3_intro0 | @or3_intro1 destruct | @or3_intro2 ] /2/ (**) (* /3/ is slow *)
+  qed. 
+
+lemma le_to_lt_or_eq: ∀m,n. m ≤ n → m < n ∨ m = n.
+#m #n * -n /3/
+qed. 
+
+lemma plus_le_weak: ∀m,n,p. m + n ≤ p → n ≤ p.
+/2/ qed.
+
+lemma plus_lt_false: ∀m,n. m + n < m → False.
+#m #n #H1 lapply (le_plus_n_r n m) #H2
+lapply (le_to_lt_to_lt … H2 H1) -H2 H1 #H
+elim (lt_refl_false … H)
+qed.
+
+lemma monotonic_lt_minus_l: ∀p,q,n. n ≤ q → q < p → q - n < p - n.
+#p #q #n #H1 #H2
+@lt_plus_to_minus_r <plus_minus_m_m //.
+qed.
+
+lemma plus_le_minus: ∀a,b,c. a + b ≤ c → a ≤ c - b.
+/2/ qed.
+
+lemma lt_plus_minus: ∀i,u,d. u ≤ i → i < d + u → i - u < d.
+/2/ qed.
+
+lemma plus_plus_comm_23: ∀m,n,p. m + n + p = m + p + n.
+// qed.
+
+lemma le_plus_minus_comm: ∀n,m,p. p ≤ m → m + n - p = m - p + n.
+#n #m #p #lepm @plus_to_minus <associative_plus
+>(commutative_plus p) <plus_minus_m_m //
+qed.
+
+lemma minus_le_minus_minus_comm: ∀b,c,a. c ≤ b → a - (b - c) = a + c - b.
+#b elim b -b
+[ #c #a #H >(le_O_to_eq_O … H) -H //
+| #b #IHb #c elim c -c //
+  #c #_ #a #Hcb
+  lapply (le_S_S_to_le … Hcb) -Hcb #Hcb
+  <plus_n_Sm normalize /2/
+]
+qed.
+
+lemma minus_plus_comm: ∀a,b,c. a - b - c = a - (c + b).
+// qed.
+
+lemma minus_minus_comm: ∀a,b,c. a - b - c = a - c - b.
+/3/ qed.
+
+lemma le_plus_minus: ∀a,b,c. c ≤ b → a + b - c = a + (b - c).
+/2/ qed.
+
+lemma plus_minus_m_m_comm: ∀n,m. m ≤ n → n = m + (n - m).
+/2/ qed.
+
+theorem minus_plus_m_m_comm: ∀n,m. n = (m + n) - m.
+/2/ qed.
+
+lemma arith_a2: ∀a,c1,c2. c1 + c2 ≤ a → a - c1 - c2 + (c1 + c2) = a.
+/2/ qed.
+
+lemma arith_b1: ∀a,b,c1. c1 ≤ b → a - c1 - (b - c1) = a - b.
+#a #b #c1 #H >minus_plus @eq_f2 /2/
+qed.
+
+lemma arith_b2: ∀a,b,c1,c2. c1 + c2 ≤ b → a - c1 - c2 - (b - c1 - c2) = a - b.
+#a #b #c1 #c2 #H >minus_plus >minus_plus >minus_plus /2/
+qed.
+
+lemma arith_c1: ∀a,b,c1. a + c1 - (b + c1) = a - b.
+// qed.
+
+lemma arith_c1x: ∀x,a,b,c1. x + c1 + a - (b + c1) = x + a - b.
+#x #a #b #c1 >plus_plus_comm_23 //
+qed.
+
+lemma arith_d1: ∀a,b,c1. c1 ≤ b → a + c1 + (b - c1) = a + b.
+/2/ qed.
+
+lemma arith_e2: ∀a,c1,c2. a ≤ c1 → c1 + c2 - (c1 - a + c2) = a.
+/3/ qed.
+
+lemma arith_f1: ∀a,b,c1. a + b ≤ c1 → c1 - (c1 - a - b) = a + b.
+#a #b #c1 #H >minus_plus <minus_minus //
+qed.
+
+lemma arith_g1: ∀a,b,c1. c1 ≤ b → a - (b - c1) - c1 = a - b.
+/2/ qed.
+
+lemma arith_h1: ∀a1,a2,b,c1. c1 ≤ a1 → c1 ≤ b →
+                a1 - c1 + a2 - (b - c1) = a1 + a2 - b.
+#a1 #a2 #b #c1 #H1 #H2 <le_plus_minus_comm /2/
+qed.
+
+lemma arith_i2: ∀a,c1,c2. c1 + c2 ≤ a → c1 + c2 + (a - c1 - c2) = a.
+/2/ qed.
+
+lemma arith_z1: ∀a,b,c1. a + c1 - b - c1 = a - b.
+// qed.
+
+(* unstable *****************************************************************)
+
+lemma arith1: ∀n,h,m,p. n + h + m ≤ p + h → n + m ≤ p.
+/2/ qed.
+
+lemma arith2: ∀j,i,e,d. d + e ≤ i → d ≤ i - e + j.
+#j #i #e #d #H lapply (plus_le_minus … H) -H /2/
+qed.
+
+lemma arith3: ∀a1,a2,b,c1. a1 + a2 ≤ b → a1 + c1 + a2 ≤ b + c1.
+/2/ qed.
+
+lemma arith4: ∀h,d,e1,e2. d ≤ e1 + e2 → d + h ≤ e1 + h + e2.
+/2/ qed.
+
+lemma arith5: ∀a,b1,b2,c1. c1 ≤ b1 → c1 ≤ a → a < b1 + b2 → a - c1 < b1 - c1 + b2.
+#a #b1 #b2 #c1 #H1 #H2 #H3
+<le_plus_minus_comm // @monotonic_lt_minus_l //
+qed.
+
+lemma arith8: ∀a,b. a < a + b + 1.
+// qed.
+
+lemma arith9: ∀a,b,c. c < a + (b + c + 1) + 1.
+// qed.
+
+lemma arith10: ∀a,b,c,d,e. a ≤ b → c + (a - d - e) ≤ c + (b - d - e).
+#a #b #c #d #e #H
+>minus_plus >minus_plus @monotonic_le_plus_r @monotonic_le_minus_l //
+qed.
diff --git a/matita/matita/contribs/lambda-delta/Ground-2/xoa.conf.xml b/matita/matita/contribs/lambda-delta/Ground-2/xoa.conf.xml
new file mode 100644
index 000000000..d25c5dcca
--- /dev/null
+++ b/matita/matita/contribs/lambda-delta/Ground-2/xoa.conf.xml
@@ -0,0 +1,34 @@
+<?xml version="1.0" encoding="utf-8"?>
+<helm_registry>
+  <section name="matita">
+    <key name="rt_base_dir">$(MATITA_RT_BASE_DIR)</key>
+<!--
+    <key name="system">false</key>
+    <key name="map_unicode_to_tex">false</key>
+    <key name="do_heavy_checks">true</key>
+    <key name="include_path">lib</key>
+-->
+  </section>
+  <section name="xoa">
+    <key name="output_dir">lib/lambda-delta</key>
+    <key name="objects">xoa</key>    
+    <key name="notations">xoa_notation</key>
+    <key name="include">basics/pts.ma</key>
+    <key name="ex">2 1</key>
+    <key name="ex">2 2</key>
+    <key name="ex">3 1</key>
+    <key name="ex">3 2</key>
+    <key name="ex">3 3</key>
+    <key name="ex">4 3</key>
+    <key name="ex">4 4</key>
+    <key name="ex">5 3</key>
+    <key name="ex">5 4</key>
+    <key name="ex">6 6</key>
+    <key name="ex">7 6</key>
+    <key name="or">3</key>
+    <key name="or">4</key>
+<!--    
+    <key name="and">3</key>
+-->
+  </section>
+</helm_registry>
diff --git a/matita/matita/contribs/lambda-delta/Ground-2/xoa.ma b/matita/matita/contribs/lambda-delta/Ground-2/xoa.ma
new file mode 100644
index 000000000..68a0776dc
--- /dev/null
+++ b/matita/matita/contribs/lambda-delta/Ground-2/xoa.ma
@@ -0,0 +1,127 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was generated by xoa.native: do not edit *********************)
+
+include "basics/pts.ma".
+
+(* multiple existental quantifier (2, 1) *)
+
+inductive ex2_1 (A0:Type[0]) (P0,P1:A0→Prop) : Prop ≝
+   | ex2_1_intro: ∀x0. P0 x0 → P1 x0 → ex2_1 ? ? ?
+.
+
+interpretation "multiple existental quantifier (2, 1)" 'Ex P0 P1 = (ex2_1 ? P0 P1).
+
+(* multiple existental quantifier (2, 2) *)
+
+inductive ex2_2 (A0,A1:Type[0]) (P0,P1:A0→A1→Prop) : Prop ≝
+   | ex2_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → ex2_2 ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (2, 2)" 'Ex P0 P1 = (ex2_2 ? ? P0 P1).
+
+(* multiple existental quantifier (3, 1) *)
+
+inductive ex3_1 (A0:Type[0]) (P0,P1,P2:A0→Prop) : Prop ≝
+   | ex3_1_intro: ∀x0. P0 x0 → P1 x0 → P2 x0 → ex3_1 ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (3, 1)" 'Ex P0 P1 P2 = (ex3_1 ? P0 P1 P2).
+
+(* multiple existental quantifier (3, 2) *)
+
+inductive ex3_2 (A0,A1:Type[0]) (P0,P1,P2:A0→A1→Prop) : Prop ≝
+   | ex3_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → P2 x0 x1 → ex3_2 ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (3, 2)" 'Ex P0 P1 P2 = (ex3_2 ? ? P0 P1 P2).
+
+(* multiple existental quantifier (3, 3) *)
+
+inductive ex3_3 (A0,A1,A2:Type[0]) (P0,P1,P2:A0→A1→A2→Prop) : Prop ≝
+   | ex3_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → ex3_3 ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (3, 3)" 'Ex P0 P1 P2 = (ex3_3 ? ? ? P0 P1 P2).
+
+(* multiple existental quantifier (4, 3) *)
+
+inductive ex4_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3:A0→A1→A2→Prop) : Prop ≝
+   | ex4_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → ex4_3 ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (4, 3)" 'Ex P0 P1 P2 P3 = (ex4_3 ? ? ? P0 P1 P2 P3).
+
+(* multiple existental quantifier (4, 4) *)
+
+inductive ex4_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3:A0→A1→A2→A3→Prop) : Prop ≝
+   | ex4_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → ex4_4 ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (4, 4)" 'Ex P0 P1 P2 P3 = (ex4_4 ? ? ? ? P0 P1 P2 P3).
+
+(* multiple existental quantifier (5, 3) *)
+
+inductive ex5_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→Prop) : Prop ≝
+   | ex5_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → P4 x0 x1 x2 → ex5_3 ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (5, 3)" 'Ex P0 P1 P2 P3 P4 = (ex5_3 ? ? ? P0 P1 P2 P3 P4).
+
+(* multiple existental quantifier (5, 4) *)
+
+inductive ex5_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→A3→Prop) : Prop ≝
+   | ex5_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → ex5_4 ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (5, 4)" 'Ex P0 P1 P2 P3 P4 = (ex5_4 ? ? ? ? P0 P1 P2 P3 P4).
+
+(* multiple existental quantifier (6, 6) *)
+
+inductive ex6_6 (A0,A1,A2,A3,A4,A5:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→A4→A5→Prop) : Prop ≝
+   | ex6_6_intro: ∀x0,x1,x2,x3,x4,x5. P0 x0 x1 x2 x3 x4 x5 → P1 x0 x1 x2 x3 x4 x5 → P2 x0 x1 x2 x3 x4 x5 → P3 x0 x1 x2 x3 x4 x5 → P4 x0 x1 x2 x3 x4 x5 → P5 x0 x1 x2 x3 x4 x5 → ex6_6 ? ? ? ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (6, 6)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_6 ? ? ? ? ? ? P0 P1 P2 P3 P4 P5).
+
+(* multiple existental quantifier (7, 6) *)
+
+inductive ex7_6 (A0,A1,A2,A3,A4,A5:Type[0]) (P0,P1,P2,P3,P4,P5,P6:A0→A1→A2→A3→A4→A5→Prop) : Prop ≝
+   | ex7_6_intro: ∀x0,x1,x2,x3,x4,x5. P0 x0 x1 x2 x3 x4 x5 → P1 x0 x1 x2 x3 x4 x5 → P2 x0 x1 x2 x3 x4 x5 → P3 x0 x1 x2 x3 x4 x5 → P4 x0 x1 x2 x3 x4 x5 → P5 x0 x1 x2 x3 x4 x5 → P6 x0 x1 x2 x3 x4 x5 → ex7_6 ? ? ? ? ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (7, 6)" 'Ex P0 P1 P2 P3 P4 P5 P6 = (ex7_6 ? ? ? ? ? ? P0 P1 P2 P3 P4 P5 P6).
+
+(* multiple disjunction connective (3) *)
+
+inductive or3 (P0,P1,P2:Prop) : Prop ≝
+   | or3_intro0: P0 → or3 ? ? ?
+   | or3_intro1: P1 → or3 ? ? ?
+   | or3_intro2: P2 → or3 ? ? ?
+.
+
+interpretation "multiple disjunction connective (3)" 'Or P0 P1 P2 = (or3 P0 P1 P2).
+
+(* multiple disjunction connective (4) *)
+
+inductive or4 (P0,P1,P2,P3:Prop) : Prop ≝
+   | or4_intro0: P0 → or4 ? ? ? ?
+   | or4_intro1: P1 → or4 ? ? ? ?
+   | or4_intro2: P2 → or4 ? ? ? ?
+   | or4_intro3: P3 → or4 ? ? ? ?
+.
+
+interpretation "multiple disjunction connective (4)" 'Or P0 P1 P2 P3 = (or4 P0 P1 P2 P3).
+
diff --git a/matita/matita/contribs/lambda-delta/Ground-2/xoa_notation.ma b/matita/matita/contribs/lambda-delta/Ground-2/xoa_notation.ma
new file mode 100644
index 000000000..b8270b660
--- /dev/null
+++ b/matita/matita/contribs/lambda-delta/Ground-2/xoa_notation.ma
@@ -0,0 +1,138 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was generated by xoa.native: do not edit *********************)
+
+(* multiple existental quantifier (2, 1) *)
+
+notation > "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.$P0) (λ${ident x0}.$P1) }.
+
+notation < "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.$P0) (λ${ident x0}:$T0.$P1) }.
+
+(* multiple existental quantifier (2, 2) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) }.
+
+(* multiple existental quantifier (3, 1) *)
+
+notation > "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.$P0) (λ${ident x0}.$P1) (λ${ident x0}.$P2) }.
+
+notation < "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.$P0) (λ${ident x0}:$T0.$P1) (λ${ident x0}:$T0.$P2) }.
+
+(* multiple existental quantifier (3, 2) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) (λ${ident x0}.λ${ident x1}.$P2) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P2) }.
+
+(* multiple existental quantifier (3, 3) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) }.
+
+(* multiple existental quantifier (4, 3) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P3) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) }.
+
+(* multiple existental quantifier (4, 4) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) }.
+
+(* multiple existental quantifier (5, 3) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P4) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P4) }.
+
+(* multiple existental quantifier (5, 4) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P4) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) }.
+
+(* multiple existental quantifier (6, 6) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P5) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P5) }.
+
+(* multiple existental quantifier (7, 6) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P5) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P6) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P6) }.
+
+(* multiple disjunction connective (3) *)
+
+notation "hvbox(∨∨ term 29 P0 break | term 29 P1 break | term 29 P2)"
+ non associative with precedence 30
+ for @{ 'Or $P0 $P1 $P2 }.
+
+(* multiple disjunction connective (4) *)
+
+notation "hvbox(∨∨ term 29 P0 break | term 29 P1 break | term 29 P2 break | term 29 P3)"
+ non associative with precedence 30
+ for @{ 'Or $P0 $P1 $P2 $P3 }.
+
diff --git a/matita/matita/contribs/lambda-delta/Ground-2/xoa_props.ma b/matita/matita/contribs/lambda-delta/Ground-2/xoa_props.ma
new file mode 100644
index 000000000..f07bc89be
--- /dev/null
+++ b/matita/matita/contribs/lambda-delta/Ground-2/xoa_props.ma
@@ -0,0 +1,20 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "lambda-delta/xoa_notation.ma".
+include "lambda-delta/xoa.ma".
+
+lemma ex2_1_comm: ∀A0. ∀P0,P1:A0→Prop. (∃∃x0. P0 x0 & P1 x0) → ∃∃x0. P1 x0 & P0 x0.
+#A0 #P0 #P1 * /2/
+qed.
diff --git a/matita/matita/contribs/lambda-delta/Makefile b/matita/matita/contribs/lambda-delta/Makefile
new file mode 100644
index 000000000..8feb040db
--- /dev/null
+++ b/matita/matita/contribs/lambda-delta/Makefile
@@ -0,0 +1,12 @@
+H       = @
+XOA_DIR = ../../../components/binaries/xoa
+XOA     = xoa.native
+
+CONF    = xoa.conf.xml
+TARGETS = xoa_natation.ma xoa.ma
+
+all: $(TARGETS)
+
+$(TARGETS): $(CONF)
+	@echo "  EXEC $(XOA) $(CONF)"
+	$(H)MATITA_RT_BASE_DIR=../.. $(XOA_DIR)/$(XOA) $(CONF)
-- 
2.39.2