From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Fri, 3 Jan 2014 17:31:49 +0000 (+0000)
Subject: - extended multiple substitutions now uses bounds in ynat (ie. they
X-Git-Tag: make_still_working~1007
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=32bdf7f107be22a121fab8225c5fae4eb6b33633;p=helm.git

- extended multiple substitutions now uses bounds in ynat (ie. they
can be infinite) for use in lleq
- ground_2: additions in ynat
---

diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeqalt_3.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeqalt_3.ma
deleted file mode 100644
index a4d786052..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeqalt_3.ma
+++ /dev/null
@@ -1,19 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||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                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( L1 ⋕ ⋕ break [ term 46 T ] break term 46 L2 )"
-   non associative with precedence 45
-   for @{ 'LazyEqAlt $T $L1 $L2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeqalt_4.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeqalt_4.ma
new file mode 100644
index 000000000..9cd39df3e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeqalt_4.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L1 ⋕ ⋕ break [ term 46 d , break term 46 T ] break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'LazyEqAlt $d $T $L1 $L2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma
index 900bf7d73..41971c43e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma
@@ -12,6 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "ground_2/ynat/ynat_max.ma".
 include "basic_2/notation/relations/extpsubst_6.ma".
 include "basic_2/grammar/genv.ma".
 include "basic_2/grammar/cl_shift.ma".
@@ -21,12 +22,12 @@ include "basic_2/relocation/lsuby.ma".
 (* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
 
 (* activate genv *)
-inductive cpy: nat → nat → relation4 genv lenv term term ≝
+inductive cpy: ynat → ynat → relation4 genv lenv term term ≝
 | cpy_atom : ∀I,G,L,d,e. cpy d e G L (⓪{I}) (⓪{I})
-| cpy_subst: ∀I,G,L,K,V,W,i,d,e. d ≤ i → i < d + e →
-             ⇩[0, i] L ≡ K.ⓑ{I}V → ⇧[0, i + 1] V ≡ W → cpy d e G L (#i) W
+| cpy_subst: ∀I,G,L,K,V,W,i,d,e. d ≤ yinj i → i < d+e →
+             ⇩[0, i] L ≡ K.ⓑ{I}V → ⇧[0, i+1] V ≡ W → cpy d e G L (#i) W
 | cpy_bind : ∀a,I,G,L,V1,V2,T1,T2,d,e.
-             cpy d e G L V1 V2 → cpy (d + 1) e G (L.ⓑ{I}V2) T1 T2 →
+             cpy d e G L V1 V2 → cpy (⫯d) e G (L.ⓑ{I}V2) T1 T2 →
              cpy d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
 | cpy_flat : ∀I,G,L,V1,V2,T1,T2,d,e.
              cpy d e G L V1 V2 → cpy d e G L T1 T2 →
@@ -62,7 +63,7 @@ lemma cpy_full: ∀I,G,K,V,T1,L,d. ⇩[0, d] L ≡ K.ⓑ{I}V →
   destruct
   elim (lift_total V 0 (i+1)) #W #HVW
   elim (lift_split … HVW i i)
-  /3 width=5 by cpy_subst, le_n, ex2_2_intro/
+  /4 width=5 by cpy_subst, ylt_inj, ex2_2_intro/
 | * [ #a ] #J #W1 #U1 #IHW1 #IHU1 #L #d #HLK
   elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
   [ elim (IHU1 (L.ⓑ{J}W2) (d+1)) -IHU1
@@ -76,22 +77,21 @@ qed-.
 lemma cpy_weak: ∀G,L,T1,T2,d1,e1. ⦃G, L⦄ ⊢ T1 ▶×[d1, e1] T2 →
                 ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
                 ⦃G, L⦄ ⊢ T1 ▶×[d2, e2] T2.
-#G #L #T1 #T2 #d1 #e1 #H elim H -G -L -T1 -T2 -d1 -e1
-[ //
-| /3 width=5 by cpy_subst, transitive_le/
-| /4 width=3 by cpy_bind, le_to_lt_to_lt, le_S_S/
+#G #L #T1 #T2 #d1 #e1 #H elim H -G -L -T1 -T2 -d1 -e1 //
+[ /3 width=5 by cpy_subst, ylt_yle_trans, yle_trans/
+| /4 width=3 by cpy_bind, ylt_yle_trans, yle_succ/
 | /3 width=1 by cpy_flat/
 ]
 qed-.
 
 lemma cpy_weak_top: ∀G,L,T1,T2,d,e.
                     ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶×[d, |L| - d] T2.
-#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
-[ //
-| #I #G #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW
+#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e //
+[ #I #G #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW
   lapply (ldrop_fwd_length_lt2 … HLK)
-  /3 width=5 by cpy_subst, lt_to_le_to_lt/
-| normalize /2 width=1 by cpy_bind/
+  /4 width=5 by cpy_subst, ylt_yle_trans, ylt_inj/
+| #a #I #G #L #V1 #V2 normalize in match (|L.ⓑ{I}V2|); (**) (* |?| does not work *)
+  /2 width=1 by cpy_bind/
 | /2 width=1 by cpy_flat/
 ]
 qed-.
@@ -103,53 +103,46 @@ lapply (cpy_weak … HT12 0 (d + e) ? ?) -HT12
 /2 width=2 by cpy_weak_top/
 qed-.
 
-lemma cpy_split_up: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ∀i. d ≤ i → i ≤ d + e →
-                    ∃∃T. ⦃G, L⦄ ⊢ T1 ▶×[d, i - d] T & ⦃G, L⦄ ⊢ T ▶×[i, d + e - i] T2.
+lemma cpy_split_up: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ∀i. i ≤ d + e →
+                    ∃∃T. ⦃G, L⦄ ⊢ T1 ▶×[d, i-d] T & ⦃G, L⦄ ⊢ T ▶×[i, d+e-i] T2.
 #G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
 [ /2 width=3 by ex2_intro/
-| #I #G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
-  elim (lt_or_ge i j) [ -Hide -Hjde | -Hdi -Hdj ]
-  [ >(plus_minus_m_m j d) in ⊢ (%→?);
-    /3 width=5 by cpy_subst, ex2_intro/
-  | #Hij lapply (plus_minus_m_m … Hjde) -Hjde
-    /3 width=9 by cpy_atom, cpy_subst, ex2_intro/
-  ]
-| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
-  elim (IHV12 i) -IHV12 // #V #HV1 #HV2
-  elim (IHT12 (i + 1)) -IHT12 /2 width=1 by le_S_S/
-  -Hdi -Hide >arith_c1x #T #HT1 #HT2
-  lapply (lsuby_cpy_trans … HT1 (L.ⓑ{I} V) ?) -HT1 /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
-| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
-  elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 //
-  -Hdi -Hide /3 width=5 by ex2_intro, cpy_flat/
+| #I #G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hjde
+  elim (ylt_split i j) [ -Hide -Hjde | -Hdi ]
+  /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
+| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
+  elim (IHV12 i) -IHV12 // #V
+  elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hide
+  >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
+  lapply (lsuby_cpy_trans … HT1 (L.ⓑ{I}V) ?) -HT1
+  /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
+| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
+  elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hide
+  /3 width=5 by ex2_intro, cpy_flat/
 ]
 qed-.
 
-lemma cpy_split_down: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 →
-                      ∀i. d ≤ i → i ≤ d + e →
-                      ∃∃T. ⦃G, L⦄ ⊢ T1 ▶×[i, d + e - i] T &
-                           ⦃G, L⦄ ⊢ T ▶×[d, i - d] T2.
+lemma cpy_split_down: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ∀i. i ≤ d + e →
+                      ∃∃T. ⦃G, L⦄ ⊢ T1 ▶×[i, d+e-i] T & ⦃G, L⦄ ⊢ T ▶×[d, i-d] T2.
 #G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
 [ /2 width=3 by ex2_intro/
-| #I #G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
-  elim (lt_or_ge i j)
-  [ -Hide -Hjde >(plus_minus_m_m j d) in ⊢ (% → ?); // -Hdj /3 width=9 by ex2_intro, cpy_atom, cpy_subst/
-  | -Hdi -Hdj
-    >(plus_minus_m_m (d+e) j) in Hide; // -Hjde /3 width=5 by ex2_intro, cpy_subst/
-  ]
-| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+| #I #G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hjde
+  elim (ylt_split i j) [ -Hide -Hjde | -Hdi ]
+  /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
+| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
   elim (IHV12 i) -IHV12 // #V
-  elim (IHT12 (i + 1)) -IHT12 /2 width=1 by le_S_S/
-  -Hdi -Hide >arith_c1x #T #HT1 #HT2
-  lapply (lsuby_cpy_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
-| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
-  elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 //
-  -Hdi -Hide /3 width=5 by ex2_intro, cpy_flat/
+  elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hide
+  >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
+  lapply (lsuby_cpy_trans … HT1 (L. ⓑ{I} V) ?) -HT1
+  /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
+| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
+  elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hide
+  /3 width=5 by ex2_intro, cpy_flat/
 ]
 qed-.
 
 lemma cpy_append: ∀G,d,e. l_appendable_sn … (cpy d e G).
-#G #d #e #K #T1 #T2 #H elim H -K -T1 -T2 -d -e
+#G #d #e #K #T1 #T2 #H elim H -G -K -T1 -T2 -d -e
 /2 width=1 by cpy_atom, cpy_bind, cpy_flat/
 #I #G #K #K0 #V #W #i #d #e #Hdi #Hide #HK0 #HVW #L
 lapply (ldrop_fwd_length_lt2 … HK0) #H
@@ -161,7 +154,7 @@ qed-.
 
 fact cpy_inv_atom1_aux: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ∀J. T1 = ⓪{J} →
                         T2 = ⓪{J} ∨
-                        ∃∃I,K,V,i. d ≤ i & i < d + e &
+                        ∃∃I,K,V,i. d ≤ yinj i & i < d + e &
                                    ⇩[O, i] L ≡ K.ⓑ{I}V &
                                    ⇧[O, i + 1] V ≡ T2 &
                                    J = LRef i.
@@ -175,7 +168,7 @@ qed-.
 
 lemma cpy_inv_atom1: ∀I,G,L,T2,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶×[d, e] T2 →
                      T2 = ⓪{I} ∨
-                     ∃∃J,K,V,i. d ≤ i & i < d + e &
+                     ∃∃J,K,V,i. d ≤ yinj i & i < d + e &
                                 ⇩[O, i] L ≡ K.ⓑ{J}V &
                                 ⇧[O, i + 1] V ≡ T2 &
                                 I = LRef i.
@@ -206,7 +199,7 @@ qed-.
 fact cpy_inv_bind1_aux: ∀G,L,U1,U2,d,e. ⦃G, L⦄ ⊢ U1 ▶×[d, e] U2 →
                         ∀a,I,V1,T1. U1 = ⓑ{a,I}V1.T1 →
                         ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶×[d, e] V2 &
-                                 ⦃G, L. ⓑ{I}V2⦄ ⊢ T1 ▶×[d + 1, e] T2 &
+                                 ⦃G, L. ⓑ{I}V2⦄ ⊢ T1 ▶×[⫯d, e] T2 &
                                  U2 = ⓑ{a,I}V2.T2.
 #G #L #U1 #U2 #d #e * -G -L -U1 -U2 -d -e
 [ #I #G #L #d #e #b #J #W1 #U1 #H destruct
@@ -218,7 +211,7 @@ qed-.
 
 lemma cpy_inv_bind1: ∀a,I,G,L,V1,T1,U2,d,e. ⦃G, L⦄ ⊢ ⓑ{a,I} V1. T1 ▶×[d, e] U2 →
                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶×[d, e] V2 &
-                              ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ▶×[d + 1, e] T2 &
+                              ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ▶×[⫯d, e] T2 &
                               U2 = ⓑ{a,I}V2.T2.
 /2 width=3 by cpy_inv_bind1_aux/ qed-.
 
@@ -246,8 +239,7 @@ fact cpy_inv_refl_O2_aux: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 →
 #G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
 [ //
 | #I #G #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct
-  lapply (le_to_lt_to_lt … Hdi … Hide) -Hdi -Hide <plus_n_O #Hdd
-  elim (lt_refl_false … Hdd)
+  elim (ylt_yle_false … Hdi) -Hdi //
 | /3 width=1 by eq_f2/
 | /3 width=1 by eq_f2/
 ]
@@ -275,6 +267,6 @@ lemma cpy_fwd_shift1: ∀G,L1,L,T1,T,d,e. ⦃G, L⦄ ⊢ L1 @@ T1 ▶×[d, e] T
   elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
   >append_length >HL12 -HL12
   @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] (**) (* explicit constructor *)
-  /2 width=3 by trans_eq/ 
+  /2 width=3 by trans_eq/
 ]
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_cpy.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_cpy.ma
index f876a53d7..2acd3705c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_cpy.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_cpy.ma
@@ -53,15 +53,9 @@ theorem cpy_conf_neq: ∀G,L1,T0,T1,d1,e1. ⦃G, L1⦄ ⊢ T0 ▶×[d1, e1] T1 
 | #I1 #G #L1 #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #L2 #T2 #d2 #e2 #H1 #H2
   elim (cpy_inv_lref1 … H1) -H1
   [ #H destruct /3 width=7 by cpy_subst, ex2_intro/
-  | -HLK1 -HVT1 * #I2 #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded
-    [ -Hd1 -Hde2
-      lapply (transitive_le … Hded Hd2) -Hded -Hd2 #H
-      lapply (lt_to_le_to_lt … Hde1 H) -Hde1 -H #H
-      elim (lt_refl_false … H)
-    | -Hd2 -Hde1
-      lapply (transitive_le … Hded Hd1) -Hded -Hd1 #H
-      lapply (lt_to_le_to_lt … Hde2 H) -Hde2 -H #H
-      elim (lt_refl_false … H)
+  | -HLK1 -HVT1 * #I2 #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded [ -Hd1 -Hde2 | -Hd2 -Hde1 ]
+    [ elim (ylt_yle_false … Hde1) -Hde1 /2 width=3 by yle_trans/
+    | elim (ylt_yle_false … Hde2) -Hde2 /2 width=3 by yle_trans/
     ]
   ]
 | #a #I #G #L1 #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
@@ -71,8 +65,7 @@ theorem cpy_conf_neq: ∀G,L1,T0,T1,d1,e1. ⦃G, L1⦄ ⊢ T0 ▶×[d1, e1] T1 
   [ -H #T #HT1 #HT2
     lapply (lsuby_cpy_trans … HT1 (L2.ⓑ{I}V) ?) -HT1 /2 width=1 by lsuby_succ/
     lapply (lsuby_cpy_trans … HT2 (L1.ⓑ{I}V) ?) -HT2 /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
-  | -HV1 -HV2 >plus_plus_comm_23 >plus_plus_comm_23 in ⊢ (? ? %); elim H -H
-    /3 width=1 by monotonic_le_plus_l, or_intror, or_introl/
+  | -HV1 -HV2 elim H -H /3 width=1 by yle_succ, or_introl, or_intror/
   ]
 | #I #G #L1 #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
   elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
@@ -89,11 +82,11 @@ theorem cpy_trans_ge: ∀G,L,T1,T0,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T0 →
   elim (cpy_inv_atom1 … H) -H
   [ #H destruct //
   | * #J #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct
-    lapply (lt_to_le_to_lt … (d+e) Hide2 ?) /2 width=5 by cpy_subst, monotonic_lt_plus_r/
+    lapply (ylt_yle_trans … (d+e) … Hide2) /2 width=5 by cpy_subst, monotonic_yle_plus_dx/
   ]
 | #I #G #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He
-  lapply (cpy_weak … HVT2 0 (i +1) ? ?) -HVT2 /2 width=1 by le_S_S/ #HVT2
-  <(cpy_inv_lift1_eq … HVT2 … HVW) -HVT2 /2 width=5 by cpy_subst/
+  lapply (cpy_weak … HVT2 0 (i+1) ? ?) -HVT2 /3 width=1 by yle_plus_dx2_trans, yle_succ/
+  >yplus_inj #HVT2 <(cpy_inv_lift1_eq … HVW … HVT2) -HVT2 /2 width=5 by cpy_subst/
 | #a #I #G #L #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
   elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct
   lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V0) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
@@ -110,14 +103,14 @@ theorem cpy_trans_down: ∀G,L,T1,T0,d1,e1. ⦃G, L⦄ ⊢ T1 ▶×[d1, e1] T0 
 #G #L #T1 #T0 #d1 #e1 #H elim H -G -L -T1 -T0 -d1 -e1
 [ /2 width=3 by ex2_intro/
 | #I #G #L #K #V #W #i1 #d1 #e1 #Hdi1 #Hide1 #HLK #HVW #T2 #d2 #e2 #HWT2 #Hde2d1
-  lapply (transitive_le … Hde2d1 Hdi1) -Hde2d1 #Hde2i1
-  lapply (cpy_weak … HWT2 0 (i1 + 1) ? ?) -HWT2 normalize /2 width=1 by le_S/ -Hde2i1 #HWT2
-  <(cpy_inv_lift1_eq … HWT2 … HVW) -HWT2 /3 width=9 by cpy_subst, ex2_intro/
+  lapply (yle_trans … Hde2d1 … Hdi1) -Hde2d1 #Hde2i1
+  lapply (cpy_weak … HWT2 0 (i1+1) ? ?) -HWT2 /3 width=1 by yle_succ, yle_pred_sn/ -Hde2i1
+  >yplus_inj #HWT2 <(cpy_inv_lift1_eq … HVW … HWT2) -HWT2 /3 width=9 by cpy_subst, ex2_intro/
 | #a #I #G #L #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
   elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
   lapply (lsuby_cpy_trans … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
-  elim (IHV10 … HV02 ?) -IHV10 -HV02 // #V
-  elim (IHT10 … HT02 ?) -T0 /2 width=1 by le_S_S/ #T #HT1 #HT2
+  elim (IHV10 … HV02) -IHV10 -HV02 // #V
+  elim (IHT10 … HT02) -T0 /2 width=1 by yle_succ/ #T #HT1 #HT2
   lapply (lsuby_cpy_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1 by lsuby_succ/
   lapply (lsuby_cpy_trans … HT2 (L. ⓑ{I} V2) ?) -HT2 /3 width=6 by cpy_bind, lsuby_succ, ex2_intro/
 | #I #G #L #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_lift.ma
index ebb7c9df2..7cb26f477 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_lift.ma
@@ -27,7 +27,8 @@ lemma cpy_lift_le: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶×[dt, et] T2 →
 [ #I #G #K #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
   >(lift_mono … H1 … H2) -H1 -H2 //
 | #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdetd
-  lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
+  lapply (ylt_yle_trans … Hdetd … Hidet) -Hdetd #Hid
+  lapply (ylt_inv_inj … Hid) -Hid #Hid
   lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct
   elim (lift_trans_ge … HVW … HWU2) -W // <minus_plus #W #HVW #HWU2
   elim (ldrop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
@@ -36,7 +37,7 @@ lemma cpy_lift_le: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶×[dt, et] T2 →
 | #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
   elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
   elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
-  /4 width=6 by cpy_bind, ldrop_skip, le_S_S/
+  /4 width=6 by cpy_bind, ldrop_skip, yle_succ/
 | #G #I #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
   elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
   elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
@@ -54,20 +55,22 @@ lemma cpy_lift_be: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶×[dt, et] T2 →
 | #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdtd #_
   elim (lift_inv_lref1 … H) -H * #Hid #H destruct
   [ -Hdtd
-    lapply (lt_to_le_to_lt … (dt+et+e) Hidet ?) // -Hidet #Hidete
+    lapply (ylt_yle_trans … (dt+et+e) … Hidet) // -Hidet #Hidete
     elim (lift_trans_ge … HVW … HWU2) -W // <minus_plus #W #HVW #HWU2
     elim (ldrop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
     elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #_ #HVY
     >(lift_mono … HVY … HVW) -V #H destruct /2 width=5 by cpy_subst/
   | -Hdti
+    elim (yle_inv_inj2 … Hdtd) -Hdtd #dtt #Hdtd #H destruct
     lapply (transitive_le … Hdtd Hid) -Hdtd #Hdti
     lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
-    lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=5 by cpy_subst, lt_minus_to_plus_r, transitive_le/
+    lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid
+    /4 width=5 by cpy_subst, monotonic_ylt_plus_dx, yle_plus_dx1_trans, yle_inj/
   ]
 | #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdtd #Hddet
   elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
   elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
-  /4 width=6 by cpy_bind, ldrop_skip, le_S_S/ (**) (* auto a bit slow *)
+  /4 width=6 by cpy_bind, ldrop_skip, yle_succ/
 | #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
   elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
   elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
@@ -78,19 +81,21 @@ qed-.
 lemma cpy_lift_ge: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶×[dt, et] T2 →
                    ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
                    ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
-                   d ≤ dt → ⦃G, L⦄ ⊢ U1 ▶×[dt + e, et] U2.
+                   d ≤ dt → ⦃G, L⦄ ⊢ U1 ▶×[dt+e, et] U2.
 #G #K #T1 #T2 #dt #et #H elim H -G -K -T1 -T2 -dt -et
 [ #I #G #K #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
   >(lift_mono … H1 … H2) -H1 -H2 //
 | #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hddt
-  lapply (transitive_le … Hddt … Hdti) -Hddt #Hid
-  lapply (lift_inv_lref1_ge … H … Hid) -H #H destruct
+  lapply (yle_trans … Hddt … Hdti) -Hddt #Hid
+  elim (yle_inv_inj2 … Hid) -Hid #dd #Hddi #H0 destruct
+  lapply (lift_inv_lref1_ge … H … Hddi) -H #H destruct
   lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
-  lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=5 by cpy_subst, lt_minus_to_plus_r, monotonic_le_plus_l/
+  lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hddi
+  /3 width=5 by cpy_subst, monotonic_ylt_plus_dx, monotonic_yle_plus_dx/
 | #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
   elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
   elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
-  /4 width=5 by cpy_bind, ldrop_skip, le_minus_to_plus/
+  /4 width=5 by cpy_bind, ldrop_skip, yle_succ/
 | #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
   elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
   elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
@@ -109,15 +114,16 @@ lemma cpy_inv_lift1_le: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U2 
   | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by cpy_atom, lift_gref, ex2_intro/
   ]
 | #I #G #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdetd
-  lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
+  lapply (ylt_yle_trans … Hdetd … Hidet) -Hdetd #Hid
+  lapply (ylt_inv_inj … Hid) -Hid #Hid
   lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct
-  elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
-  elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=5 by cpy_subst, ex2_intro/
+  elim (ldrop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
+  elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=5 by cpy_subst, ex2_intro/
 | #a #I #G #L #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
   elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
   elim (IHV12 … HLK … HWV1) -V1 // #W2 #HW12 #HWV2
   elim (IHU12 … HTU1) -IHU12 -HTU1
-  /3 width=5 by cpy_bind, ldrop_skip, lift_bind, le_S_S, ex2_intro/
+  /3 width=5 by cpy_bind, yle_succ, ldrop_skip, lift_bind, ex2_intro/
 | #I #G #L #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
   elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
   elim (IHV12 … HLK … HWV1) -V1 //
@@ -129,7 +135,7 @@ qed-.
 lemma cpy_inv_lift1_be: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U2 →
                         ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
                         dt ≤ d → d + e ≤ dt + et →
-                        ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[dt, et - e] T2 & ⇧[d, e] T2 ≡ U2.
+                        ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[dt, et-e] T2 & ⇧[d, e] T2 ≡ U2.
 #G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
 [ * #G #L #i #dt #et #K #d #e #_ #T1 #H #_
   [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by cpy_atom, lift_sort, ex2_intro/
@@ -137,30 +143,26 @@ lemma cpy_inv_lift1_be: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U2 
   | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by cpy_atom, lift_gref, ex2_intro/
   ]
 | #I #G #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdtd #Hdedet
-  lapply (le_fwd_plus_plus_ge … Hdtd … Hdedet) #Heet
-  elim (lift_inv_lref2 … H) -H * #Hid #H destruct
-  [ -Hdtd -Hidet
-    lapply (lt_to_le_to_lt … (dt + (et - e)) Hid ?) [ <le_plus_minus /2 width=1 by le_plus_to_minus_r/ ] -Hdedet #Hidete
+  lapply (yle_fwd_plus_ge_inj … Hdtd Hdedet) #Heet
+  elim (lift_inv_lref2 … H) -H * #Hid #H destruct [ -Hdtd -Hidet | -Hdti -Hdedet ]
+  [ lapply (ylt_yle_trans i d (dt+(et-e)) ? ?) /2 width=1 by ylt_inj/
+    [ >yplus_minus_assoc_inj /2 width=1 by yle_plus_to_minus_inj2/ ] -Hdedet #Hidete
     elim (ldrop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
-    elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=5 by cpy_subst, ex2_intro/
-  | -Hdti -Hdedet
-    lapply (transitive_le … (i - e) Hdtd ?) /2 width=1 by le_plus_to_minus_r/ -Hdtd #Hdtie
-    elim (le_inv_plus_l … Hid) #Hdie #Hei
+    elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hid
+    /3 width=5 by cpy_subst, ex2_intro/
+  | elim (le_inv_plus_l … Hid) #Hdie #Hei
+    lapply (yle_trans … Hdtd (i-e) ?) /2 width=1 by yle_inj/ -Hdtd #Hdtie
     lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
     elim (lift_split … HVW d (i-e+1)) -HVW [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hid -Hdie
     #V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H
-    @(ex2_intro … H) @(cpy_subst … Hdtie … HKV HV1) (**) (* explicit constructor *)
-    >commutative_plus >plus_minus /2 width=1 by monotonic_lt_minus_l/
+    @(ex2_intro … H) @(cpy_subst … HKV HV1) // (**) (* explicit constructor *)
+    >yplus_minus_assoc_inj /3 width=1 by monotonic_ylt_minus_dx, yle_inj/
   ]
 | #a #I #G #L #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
   elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
   elim (IHV12 … HLK … HWV1) -V1 // #W2 #HW12 #HWV2
   elim (IHU12 … HTU1) -U1
-  [5: /2 width=2 by ldrop_skip/ |2: skip 
-  |3: >plus_plus_comm_23 >(plus_plus_comm_23 dt) /2 width=1 by le_S_S/
-  |4: /2 width=1 by le_S_S/
-  ]
-  /3 width=5 by cpy_bind, lift_bind, ex2_intro/
+  /3 width=5 by cpy_bind, ldrop_skip, lift_bind, yle_succ, ex2_intro/
 | #I #G #L #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
   elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
   elim (IHV12 … HLK … HWV1) -V1 //
@@ -172,7 +174,7 @@ qed-.
 lemma cpy_inv_lift1_ge: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U2 →
                         ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
                         d + e ≤ dt →
-                        ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[dt - e, et] T2 & ⇧[d, e] T2 ≡ U2.
+                        ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[dt-e, et] T2 & ⇧[d, e] T2 ≡ U2.
 #G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
 [ * #G #L #i #dt #et #K #d #e #_ #T1 #H #_
   [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by cpy_atom, lift_sort, ex2_intro/
@@ -180,21 +182,23 @@ lemma cpy_inv_lift1_ge: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U2 
   | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by cpy_atom, lift_gref, ex2_intro/
   ]
 | #I #G #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdedt
-  lapply (transitive_le … Hdedt … Hdti) #Hdei
-  elim (le_inv_plus_l … Hdedt) -Hdedt #_ #Hedt
-  elim (le_inv_plus_l … Hdei) #Hdie #Hei
-  lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct
-  lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
-  elim (lift_split … HVW d (i-e+1)) -HVW [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hdei -Hdie
-  #V0 #HV10 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H
-  @(ex2_intro … H) @(cpy_subst … HKV HV10) /2 width=1 by monotonic_le_minus_l2/ (**) (* explicit constructor *)
-  >plus_minus /2 width=1 by monotonic_lt_minus_l/
+  lapply (yle_trans … Hdedt … Hdti) #Hdei
+  elim (yle_inv_plus_inj2 … Hdedt) -Hdedt #_ #Hedt
+  elim (yle_inv_plus_inj2 … Hdei) #Hdie #Hei
+  lapply (lift_inv_lref2_ge  … H ?) -H /2 width=1 by yle_inv_inj/ #H destruct
+  lapply (ldrop_conf_ge … HLK … HLKV ?) -L /2 width=1 by yle_inv_inj/ #HKV
+  elim (lift_split … HVW d (i-e+1)) -HVW [2,3,4: /3 width=1 by yle_inv_inj, le_S_S, le_S/ ] -Hdei -Hdie
+  #V0 #HV10 >plus_minus /2 width=1 by yle_inv_inj/ <minus_minus /3 width=1 by yle_inv_inj, le_S/ <minus_n_n <plus_n_O #H
+  @(ex2_intro … H) @(cpy_subst … HKV HV10) (**) (* explicit constructor *)
+  [ /2 width=1 by monotonic_yle_minus_dx/
+  | <yplus_minus_comm_inj /2 width=1 by monotonic_ylt_minus_dx/
+  ]
 | #a #I #G #L #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
   elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
-  elim (le_inv_plus_l … Hdetd) #_ #Hedt
+  elim (yle_inv_plus_inj2 … Hdetd) #_ #Hedt
   elim (IHV12 … HLK … HWV1) -V1 // #W2 #HW12 #HWV2
-  elim (IHU12 … HTU1) -U1 [4: @ldrop_skip // |2: skip |3: /2 width=1 by le_S_S/ ]
-  <plus_minus /3 width=5 by cpy_bind, lift_bind, ex2_intro/
+  elim (IHU12 … HTU1) -U1 [4: @ldrop_skip // |2: skip |3: /2 width=1 by yle_succ/ ]
+  >yminus_succ1_inj /3 width=5 by cpy_bind, lift_bind, ex2_intro/
 | #I #G #L #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
   elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
   elim (IHV12 … HLK … HWV1) -V1 //
@@ -202,23 +206,20 @@ lemma cpy_inv_lift1_ge: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U2 
 ]
 qed-.
 
-lemma cpy_inv_lift1_eq: ∀G,L,U1,U2,d,e.
-                        ⦃G, L⦄ ⊢ U1 ▶×[d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
-#G #L #U1 #U2 #d #e #H elim H -G -L -U1 -U2 -d -e
-[ //
-| #I #G #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #T1 #H
-  elim (lift_inv_lref2 … H) -H * #H
-  [ lapply (le_to_lt_to_lt … Hdi … H) -Hdi -H #H
-    elim (lt_refl_false … H)
-  | lapply (lt_to_le_to_lt … Hide … H) -Hide -H #H
-    elim (lt_refl_false … H)
-  ]
-| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
-  elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #H destruct
-  >IHV12 // >IHT12 //
-| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
-  elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #H destruct
-  >IHV12 // >IHT12 //
+lemma cpy_inv_lift1_eq: ∀G,T1,U1,d,e. ⇧[d, e] T1 ≡ U1 →
+                        ∀L,U2. ⦃G, L⦄ ⊢ U1 ▶×[d, e] U2 → U1 = U2.
+#G #T1 #U1 #d #e #H elim H -T1 -U1 -d -e
+[ #k #d #e #L #X #H >(cpy_inv_sort1 … H) -H //
+| #i #d #e #Hid #L #X #H elim (cpy_inv_lref1 … H) -H //
+  * #I #K #V #H elim (ylt_yle_false … H) -H /2 width=1 by ylt_inj/
+| #i #d #e #Hdi #L #X #H elim (cpy_inv_lref1 … H) -H //
+  * #I #K #V #_ #H elim (ylt_yle_false i d)
+  /2 width=2 by ylt_inv_monotonic_plus_dx, yle_inj/
+| #p #d #e #L #X #H >(cpy_inv_gref1 … H) -H //
+| #a #I #V1 #W1 #T1 #U1 #d #e #_ #_ #IHVW1 #IHTU1 #L #X #H elim (cpy_inv_bind1 … H) -H
+  #W2 #U2 #HW12 #HU12 #H destruct /3 width=2 by eq_f2/
+| #I #V1 #W1 #T1 #U1 #d #e #_ #_ #IHVW1 #IHTU1 #L #X #H elim (cpy_inv_flat1 … H) -H
+  #W2 #U2 #HW12 #HU12 #H destruct /3 width=2 by eq_f2/
 ]
 qed-.
 
@@ -228,17 +229,18 @@ lemma cpy_inv_lift1_ge_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U
                            ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[d, dt + et - (d + e)] T2 & ⇧[d, e] T2 ≡ U2.
 #G #L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
 elim (cpy_split_up … HU12 (d + e)) -HU12 // -Hdedet #U #HU1 #HU2
-lapply (cpy_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1 by le_minus_to_plus_r/ ] -Hddt -Hdtde #HU1
-lapply (cpy_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
-elim (cpy_inv_lift1_ge … HU2 … HLK … HTU1) -U -L // <minus_plus_m_m /2 width=3 by ex2_intro/
+lapply (cpy_weak … HU1 d e ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hddt -Hdtde #HU1
+lapply (cpy_inv_lift1_eq … HTU1 … HU1) -HU1 #HU1 destruct
+elim (cpy_inv_lift1_ge … HU2 … HLK … HTU1) -U -L /2 width=3 by ex2_intro/
 qed-.
 
 lemma cpy_inv_lift1_be_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U2 →
                            ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
                            dt ≤ d → dt + et ≤ d + e →
-                           ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
+                           ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[dt, d-dt] T2 & ⇧[d, e] T2 ≡ U2.
 #G #L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
-lapply (cpy_weak … HU12 dt (d + e - dt) ? ?) -HU12 /2 width=3 by transitive_le, le_n/ -Hdetde #HU12
+lapply (cpy_weak … HU12 dt (d+e-dt) ? ?) -HU12 //
+[ >ymax_pre_sn_comm /2 width=1 by yle_plus_dx1_trans/ ] -Hdetde #HU12
 elim (cpy_inv_lift1_be … HU12 … HLK … HTU1) -U1 -L /2 width=3 by ex2_intro/
 qed-.
 
@@ -248,7 +250,9 @@ lemma cpy_inv_lift1_le_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U
                            ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
 #G #L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde
 elim (cpy_split_up … HU12 d) -HU12 // #U #HU1 #HU2
-elim (cpy_inv_lift1_le … HU1 … HLK … HTU1) -U1 [2: >commutative_plus /2 width=1 by le_minus_to_plus_r/ ] -Hdtd #T #HT1 #HTU
-lapply (cpy_weak … HU2 d e ? ?) -HU2 // [ >commutative_plus <plus_minus_m_m // ] -Hddet -Hdetde #HU2
-lapply (cpy_inv_lift1_eq … HU2 … HTU) -L #H destruct /2 width=3 by ex2_intro/
+elim (cpy_inv_lift1_le … HU1 … HLK … HTU1) -U1
+[2: >ymax_pre_sn_comm // ] -Hdtd #T #HT1 #HTU
+lapply (cpy_weak … HU2 d e ? ?) -HU2 //
+[ >ymax_pre_sn_comm // ] -Hddet -Hdetde #HU2
+lapply (cpy_inv_lift1_eq … HTU … HU2) -L #H destruct /2 width=3 by ex2_intro/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/ldrop_leq.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/ldrop_leq.ma
index 2c58473a3..47ac4653d 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/ldrop_leq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/ldrop_leq.ma
@@ -66,8 +66,8 @@ lemma ldrop_leq_conf_be: ∀L1,L2,d,e. L1 ≃[d, e] L2 →
     >yminus_succ /3 width=5 by ldrop_ldrop_lt, ex2_3_intro/
   ]
 | #I #L1 #L2 #V #d #e #_ #IHL12 #J1 #K1 #W1 #i #H #Hdi lapply (ylt_yle_trans 0 … Hdi ?) //
-  #Hi <(ylt_inv_O1 … Hi) >yplus_succ1 >yminus_succ lapply (yle_fwd_succ1 … Hdi) -Hdi
-  #Hdi #Hide lapply (ylt_inv_succ … Hide)
+  #Hi <(ylt_inv_O1 … Hi) >yplus_succ1 >yminus_succ elim (yle_inv_succ1 … Hdi) -Hdi
+  #Hdi #_ #Hide lapply (ylt_inv_succ … Hide)
   #Hide lapply (ylt_inv_inj … Hi) -Hi
   #Hi lapply (ldrop_inv_ldrop1_lt … H ?) -H //
   #H elim (IHL12 … H) -IHL12 -H
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lsuby.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lsuby.ma
index 3c89103c3..a8567a4b2 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lsuby.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lsuby.ma
@@ -12,19 +12,20 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "ground_2/ynat/ynat_plus.ma".
 include "basic_2/notation/relations/extlrsubeq_4.ma".
 include "basic_2/relocation/ldrop.ma".
 
 (* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
 
-inductive lsuby: relation4 nat nat lenv lenv ≝
+inductive lsuby: relation4 ynat ynat lenv lenv ≝
 | lsuby_atom: ∀L,d,e. lsuby d e L (⋆)
 | lsuby_zero: ∀I1,I2,L1,L2,V1,V2.
               lsuby 0 0 L1 L2 → lsuby 0 0 (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
 | lsuby_pair: ∀I1,I2,L1,L2,V,e. lsuby 0 e L1 L2 →
-              lsuby 0 (e + 1) (L1.ⓑ{I1}V) (L2.ⓑ{I2}V)
+              lsuby 0 (⫯e) (L1.ⓑ{I1}V) (L2.ⓑ{I2}V)
 | lsuby_succ: ∀I1,I2,L1,L2,V1,V2,d,e.
-              lsuby d e L1 L2 → lsuby (d + 1) e (L1. ⓑ{I1}V1) (L2. ⓑ{I2} V2)
+              lsuby d e L1 L2 → lsuby (⫯d) e (L1. ⓑ{I1}V1) (L2. ⓑ{I2} V2)
 .
 
 interpretation
@@ -33,23 +34,25 @@ interpretation
 
 (* Basic properties *********************************************************)
 
-lemma lsuby_pair_lt: ∀I1,I2,L1,L2,V,e. L1 ⊑×[0, e-1] L2 → 0 < e →
+lemma lsuby_pair_lt: ∀I1,I2,L1,L2,V,e. L1 ⊑×[0, ⫰e] L2 → 0 < e →
                      L1.ⓑ{I1}V ⊑×[0, e] L2.ⓑ{I2}V.
-#I1 #I2 #L1 #L2 #V #e #HL12 #He >(plus_minus_m_m e 1) /2 width=1 by lsuby_pair/
+#I1 #I2 #L1 #L2 #V #e #HL12 #He <(ylt_inv_O1 … He) /2 width=1 by lsuby_pair/
 qed.
 
-lemma lsuby_succ_lt: ∀I1,I2,L1,L2,V1,V2,d,e. L1 ⊑×[d-1, e] L2 → 0 < d →
+lemma lsuby_succ_lt: ∀I1,I2,L1,L2,V1,V2,d,e. L1 ⊑×[⫰d, e] L2 → 0 < d →
                      L1.ⓑ{I1}V1 ⊑×[d, e] L2. ⓑ{I2}V2.
-#I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #Hd >(plus_minus_m_m d 1) /2 width=1 by lsuby_succ/
+#I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #Hd <(ylt_inv_O1 … Hd) /2 width=1 by lsuby_succ/
 qed.
 
 lemma lsuby_refl: ∀L,d,e. L ⊑×[d, e] L.
 #L elim L -L //
-#L #I #V #IHL #d @(nat_ind_plus … d) -d /2 width=1 by lsuby_succ/
-#e @(nat_ind_plus … e) -e /2 width=2 by lsuby_pair, lsuby_zero/
+#L #I #V #IHL #d elim (ynat_cases … d) [| * #x ]
+#Hd destruct /2 width=1 by lsuby_succ/
+#e elim (ynat_cases … e) [| * #x ]
+#He destruct /2 width=1 by lsuby_zero, lsuby_pair/
 qed.
 
-lemma lsuby_length: ∀L1,L2. |L2| ≤ |L1| → L1 ⊑×[0, 0] L2.
+lemma lsuby_length: ∀L1,L2. |L2| ≤ |L1| → L1 ⊑×[yinj 0, yinj 0] L2.
 #L1 elim L1 -L1
 [ #X #H lapply (le_n_O_to_eq … H) -H
   #H lapply (length_inv_zero_sn … H) #H destruct /2 width=1 by lsuby_atom/  
@@ -78,10 +81,10 @@ fact lsuby_inv_zero1_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
 #L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
 [ #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct
   /3 width=5 by ex2_3_intro, or_intror/
-| #I1 #I2 #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #_
-  <plus_n_Sm #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W1 #_
-  <plus_n_Sm #H destruct
+| #I1 #I2 #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #_ #H
+  elim (ysucc_inv_O_dx … H)
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W1 #_ #H
+  elim (ysucc_inv_O_dx … H)
 ]
 qed-.
 
@@ -93,32 +96,31 @@ lemma lsuby_inv_zero1: ∀I1,K1,L2,V1. K1.ⓑ{I1}V1 ⊑×[0, 0] L2 →
 fact lsuby_inv_pair1_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
                           ∀J1,K1,W. L1 = K1.ⓑ{J1}W → d = 0 → 0 < e →
                           L2 = ⋆ ∨
-                          ∃∃J2,K2. K1 ⊑×[0, e-1] K2 & L2 = K2.ⓑ{J2}W.
+                          ∃∃J2,K2. K1 ⊑×[0, ⫰e] K2 & L2 = K2.ⓑ{J2}W.
 #L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
 [ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W #_ #_ #H
-  elim (lt_zero_false … H)
+  elim (ylt_yle_false … H) //
 | #I1 #I2 #L1 #L2 #V #e #HL12 #J1 #K1 #W #H #_ #_ destruct
   /3 width=4 by ex2_2_intro, or_intror/
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W #_
-  <plus_n_Sm #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W #_ #H
+  elim (ysucc_inv_O_dx … H)
 ]
 qed-.
 
 lemma lsuby_inv_pair1: ∀I1,K1,L2,V,e. K1.ⓑ{I1}V ⊑×[0, e] L2 → 0 < e →
                        L2 = ⋆ ∨
-                       ∃∃I2,K2. K1 ⊑×[0, e-1] K2 & L2 = K2.ⓑ{I2}V.
+                       ∃∃I2,K2. K1 ⊑×[0, ⫰e] K2 & L2 = K2.ⓑ{I2}V.
 /2 width=6 by lsuby_inv_pair1_aux/ qed-.
 
-
 fact lsuby_inv_succ1_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
                           ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < d →
                           L2 = ⋆ ∨
-                          ∃∃J2,K2,W2. K1 ⊑×[d-1, e] K2 & L2 = K2.ⓑ{J2}W2.
+                          ∃∃J2,K2,W2. K1 ⊑×[⫰d, e] K2 & L2 = K2.ⓑ{J2}W2.
 #L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
 [ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
-  elim (lt_zero_false … H)
+  elim (ylt_yle_false … H) //
 | #I1 #I2 #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #H
-  elim (lt_zero_false … H)
+  elim (ylt_yle_false … H) //
 | #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #J1 #K1 #W1 #H #_ destruct
   /3 width=5 by ex2_3_intro, or_intror/
 ]
@@ -126,7 +128,7 @@ qed-.
 
 lemma lsuby_inv_succ1: ∀I1,K1,L2,V1,d,e. K1.ⓑ{I1}V1 ⊑×[d, e] L2 → 0 < d →
                        L2 = ⋆ ∨
-                       ∃∃I2,K2,V2. K1 ⊑×[d - 1, e] K2 & L2 = K2.ⓑ{I2}V2.
+                       ∃∃I2,K2,V2. K1 ⊑×[⫰d, e] K2 & L2 = K2.ⓑ{I2}V2.
 /2 width=5 by lsuby_inv_succ1_aux/ qed-.
 
 fact lsuby_inv_zero2_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
@@ -136,10 +138,10 @@ fact lsuby_inv_zero2_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
 [ #L1 #d #e #J2 #K2 #W1 #H destruct
 | #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J2 #K2 #W2 #H #_ #_ destruct
   /2 width=5 by ex2_3_intro/
-| #I1 #I2 #L1 #L2 #V #e #_ #J2 #K2 #W2 #_ #_
-  <plus_n_Sm #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W2 #_
-  <plus_n_Sm #H destruct
+| #I1 #I2 #L1 #L2 #V #e #_ #J2 #K2 #W2 #_ #_ #H
+  elim (ysucc_inv_O_dx … H)
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W2 #_ #H
+  elim (ysucc_inv_O_dx … H)
 ]
 qed-.
 
@@ -149,15 +151,15 @@ lemma lsuby_inv_zero2: ∀I2,K2,L1,V2. L1 ⊑×[0, 0] K2.ⓑ{I2}V2 →
 
 fact lsuby_inv_pair2_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
                           ∀J2,K2,W. L2 = K2.ⓑ{J2}W → d = 0 → 0 < e →
-                          ∃∃J1,K1. K1 ⊑×[0, e-1] K2 & L1 = K1.ⓑ{J1}W.
+                          ∃∃J1,K1. K1 ⊑×[0, ⫰e] K2 & L1 = K1.ⓑ{J1}W.
 #L1 #L2 #d #e * -L1 -L2 -d -e
 [ #L1 #d #e #J2 #K2 #W #H destruct
 | #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W #_ #_ #H
-  elim (lt_zero_false … H)
+  elim (ylt_yle_false … H) //
 | #I1 #I2 #L1 #L2 #V #e #HL12 #J2 #K2 #W #H #_ #_ destruct
   /2 width=4 by ex2_2_intro/
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W #_
-  <plus_n_Sm #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W #_ #H
+  elim (ysucc_inv_O_dx … H)
 ]
 qed-.
 
@@ -167,20 +169,20 @@ lemma lsuby_inv_pair2: ∀I2,K2,L1,V,e. L1 ⊑×[0, e] K2.ⓑ{I2}V → 0 < e →
 
 fact lsuby_inv_succ2_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
                           ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → 0 < d →
-                          ∃∃J1,K1,W1. K1 ⊑×[d-1, e] K2 & L1 = K1.ⓑ{J1}W1.
+                          ∃∃J1,K1,W1. K1 ⊑×[⫰d, e] K2 & L1 = K1.ⓑ{J1}W1.
 #L1 #L2 #d #e * -L1 -L2 -d -e
 [ #L1 #d #e #J2 #K2 #W2 #H destruct
 | #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W2 #_ #H
-  elim (lt_zero_false … H)
+  elim (ylt_yle_false … H) //
 | #I1 #I2 #L1 #L2 #V #e #_ #J2 #K1 #W2 #_ #H
-  elim (lt_zero_false … H)
+  elim (ylt_yle_false … H) //
 | #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #J2 #K2 #W2 #H #_ destruct
   /2 width=5 by ex2_3_intro/
 ]
 qed-.
 
 lemma lsuby_inv_succ2: ∀I2,K2,L1,V2,d,e. L1 ⊑×[d, e] K2.ⓑ{I2}V2 → 0 < d →
-                       ∃∃I1,K1,V1. K1 ⊑×[d-1, e] K2 & L1 = K1.ⓑ{I1}V1.
+                       ∃∃I1,K1,V1. K1 ⊑×[⫰d, e] K2 & L1 = K1.ⓑ{I1}V1.
 /2 width=5 by lsuby_inv_succ2_aux/ qed-.
 
 (* Basic forward lemmas *****************************************************)
@@ -192,25 +194,26 @@ qed-.
 lemma lsuby_fwd_ldrop2_be: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
                            ∀I2,K2,W,i. ⇩[0, i] L2 ≡ K2.ⓑ{I2}W →
                            d ≤ i → i < d + e →
-                           ∃∃I1,K1. K1 ⊑×[0, d+e-i-1] K2 & ⇩[0, i] L1 ≡ K1.ⓑ{I1}W.
+                           ∃∃I1,K1. K1 ⊑×[0, ⫰(d+e-i)] K2 & ⇩[0, i] L1 ≡ K1.ⓑ{I1}W.
 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e
 [ #L1 #d #e #J2 #K2 #W #i #H
   elim (ldrop_inv_atom1 … H) -H #H destruct
 | #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #i #_ #_ #H
-  elim (lt_zero_false … H)
-| #I1 #I2 #L1 #L2 #V #e #HL12 #IHL12 #J2 #K2 #W #i #H #_ #Hie
-  elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK1
-  [ -IHL12 -Hie destruct normalize -I2
-    <minus_n_O <minus_plus_m_m /2 width=4 by ldrop_pair, ex2_2_intro/
-  | -HL12
-    elim (IHL12 … HLK1) -IHL12 -HLK1 // [2: /2 width=1 by lt_plus_to_minus/ ] -Hie normalize 
-    >minus_minus_comm >arith_b1 /3 width=4 by ldrop_ldrop_lt, ex2_2_intro/
+  elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V #e #HL12 #IHL12 #J2 #K2 #W #i #H #_ >yplus_O_sn
+  elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
+  [ #_ destruct -I2 >ypred_succ
+    /2 width=4 by ldrop_pair, ex2_2_intro/
+  | lapply (ylt_inv_O1 i ?) /2 width=1 by ylt_inj/
+    #H <H -H #H lapply (ylt_inv_succ … H) -H
+    #Hie elim (IHL12 … HLK1) -IHL12 -HLK1 // -Hie
+    >yminus_succ <yminus_inj /3 width=4 by ldrop_ldrop_lt, ex2_2_intro/
   ]
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #J2 #K2 #W #i #H #Hdi >plus_plus_comm_23 #Hide
-  elim (le_inv_plus_l … Hdi) #_ #Hi
-  lapply (ldrop_inv_ldrop1_lt … H ?) -H // #HLK1
-  elim (IHL12 … HLK1) -IHL12 -HLK1
-  [2,3: /2 width=1 by lt_plus_to_minus, monotonic_pred/ ] -Hdi -Hide
-  >minus_minus_comm >arith_b1 /3 width=4 by ldrop_ldrop_lt, ex2_2_intro/
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #J2 #K2 #W #i #HLK2 #Hdi
+  elim (yle_inv_succ1 … Hdi) -Hdi
+  #Hdi #Hi <Hi >yplus_succ1 #H lapply (ylt_inv_succ … H) -H
+  #Hide lapply (ldrop_inv_ldrop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/
+  #HLK1 elim (IHL12 … HLK1) -IHL12 -HLK1 <yminus_inj >yminus_SO2
+  /4 width=4 by ylt_O, ldrop_ldrop_lt, ex2_2_intro/
 ]
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys.ma
index 91f9f35aa..033034d53 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys.ma
@@ -17,7 +17,7 @@ include "basic_2/relocation/cpy.ma".
 
 (* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
 
-definition cpys: nat → nat → relation4 genv lenv term term ≝
+definition cpys: ynat → ynat → relation4 genv lenv term term ≝
                  λd,e,G. LTC … (cpy d e G).
 
 interpretation "context-sensitive extended multiple substritution (term)"
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_alt.ma
index cf0b10e76..045aa4490 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_alt.ma
@@ -18,9 +18,9 @@ include "basic_2/substitution/cpys_lift.ma".
 (* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
 
 (* alternative definition of cpys *)
-inductive cpysa: nat → nat → relation4 genv lenv term term ≝
+inductive cpysa: ynat → ynat → relation4 genv lenv term term ≝
 | cpysa_atom : ∀I,G,L,d,e. cpysa d e G L (⓪{I}) (⓪{I})
-| cpysa_subst: ∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
+| cpysa_subst: ∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → i < d+e →
                ⇩[0, i] L ≡ K.ⓑ{I}V1 → cpysa 0 (d+e-i-1) G K V1 V2 →
                ⇧[0, i+1] V2 ≡ W2 → cpysa d e G L (#i) W2
 | cpysa_bind : ∀a,I,G,L,V1,V2,T1,T2,d,e.
@@ -60,7 +60,8 @@ lemma cpysa_cpy_trans: ∀G,L,T1,T,d,e. ⦃G, L⦄ ⊢ T1 ▶▶*×[d, e] T →
 | #I #G #L #K #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK #_ #HVW2 #IHV12 #T2 #H
   lapply (ldrop_fwd_ldrop2 … HLK) #H0LK
   lapply (cpy_weak … H 0 (d+e) ? ?) -H // #H
-  elim (cpy_inv_lift1_be … H … H0LK … HVW2) -H -H0LK -HVW2 /3 width=7 by cpysa_subst/
+  elim (cpy_inv_lift1_be … H … H0LK … HVW2) -H -H0LK -HVW2
+  /3 width=7 by cpysa_subst, ylt_fwd_le_succ/
 | #a #I #G #L #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
   elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
   lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /2 width=1 by lsuby_succ/ #HT2
@@ -82,9 +83,9 @@ lemma cpysa_cpys: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶▶*×[d, e] T2 → ⦃
 /2 width=7 by cpys_subst, cpys_flat, cpys_bind, cpy_cpys/
 qed-.
 
-lemma cpys_ind_alt: ∀R:nat→nat→relation4 genv lenv term term.
+lemma cpys_ind_alt: ∀R:ynat→ynat→relation4 genv lenv term term.
                     (∀I,G,L,d,e. R d e G L (⓪{I}) (⓪{I})) →
-                    (∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
+                    (∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → i < d + e →
                      ⇩[O, i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*×[O, d+e-i-1] V2 →
                      ⇧[O, i + 1] V2 ≡ W2 → R O (d+e-i-1) G K V1 V2 → R d e G L (#i) W2
                     ) →
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_cpys.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_cpys.ma
index f84916b9b..41ab4081d 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_cpys.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_cpys.ma
@@ -52,9 +52,9 @@ lemma cpys_split_up: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*×[d, e] T2 →
 #G #L #T1 #T2 #d #e #H #i #Hdi #Hide @(cpys_ind … H) -T2
 [ /2 width=3 by ex2_intro/
 | #T #T2 #_ #HT12 * #T3 #HT13 #HT3
-  elim (cpy_split_up … HT12 … Hdi Hide) -HT12 -Hide #T0 #HT0 #HT02
-  elim (cpys_strap1_down … HT3 … HT0 ?) -T /3 width=5 by cpys_strap1, ex2_intro/
-  >commutative_plus /2 width=1 by le_minus_to_plus_r/
+  elim (cpy_split_up … HT12 … Hide) -HT12 -Hide #T0 #HT0 #HT02
+  elim (cpys_strap1_down … HT3 … HT0) -T /3 width=5 by cpys_strap1, ex2_intro/
+  >ymax_pre_sn_comm //
 ]
 qed-.
 
@@ -65,9 +65,10 @@ lemma cpys_inv_lift1_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2
                                ⇧[d, e] T2 ≡ U2.
 #G #L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
 elim (cpys_split_up … HU12 (d + e)) -HU12 // -Hdedet #U #HU1 #HU2
-lapply (cpys_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1 by le_minus_to_plus_r/ ] -Hddt -Hdtde #HU1
+lapply (cpys_weak … HU1 d e ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hddt -Hdtde #HU1
 lapply (cpys_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
-elim (cpys_inv_lift1_ge … HU2 … HLK … HTU1) -HU2 -HLK -HTU1 // <minus_plus_m_m /2 width=3 by ex2_intro/
+elim (cpys_inv_lift1_ge … HU2 … HLK … HTU1) -HU2 -HLK -HTU1 //
+>yplus_minus_inj /2 width=3 by ex2_intro/
 qed-.
 
 (* Main properties **********************************************************)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_lift.ma
index 631ab202d..3cf6f8b31 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_lift.ma
@@ -20,39 +20,41 @@ include "basic_2/substitution/cpys.ma".
 (* Advanced properties ******************************************************)
 
 lemma cpys_subst: ∀I,G,L,K,V,U1,i,d,e.
-                  d ≤ i → i < d + e →
-                  ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*×[0, d+e-i-1] U1 →
-                  ∀U2. ⇧[0, i + 1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*×[d, e] U2.
+                  d ≤ yinj i → i < d + e →
+                  ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*×[0, ⫰(d+e-i)] U1 →
+                  ∀U2. ⇧[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*×[d, e] U2.
 #I #G #L #K #V #U1 #i #d #e #Hdi #Hide #HLK #H @(cpys_ind … H) -U1
 [ /3 width=5 by cpy_cpys, cpy_subst/
 | #U #U1 #_ #HU1 #IHU #U2 #HU12
   elim (lift_total U 0 (i+1)) #U0 #HU0
   lapply (IHU … HU0) -IHU #H
   lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
-  lapply (cpy_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // normalize #HU02
-  lapply (cpy_weak … HU02 d e ? ?) -HU02 [2,3: /2 width=3 by cpys_strap1, le_S/ ]
-  >minus_plus >commutative_plus /2 width=1 by le_minus_to_plus_r/
+  lapply (cpy_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // #HU02
+  lapply (cpy_weak … HU02 d e ? ?) -HU02
+  [2,3: /2 width=3 by cpys_strap1, yle_succ_dx/ ]
+  >yplus_O_sn <yplus_inj >ymax_pre_sn_comm /2 width=1 by ylt_fwd_le_succ/
 ]
 qed.
 
 (* Advanced inverion lemmas *************************************************)
 
-lemma cpys_inv_atom1: ∀G,L,T2,I,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶*×[d, e] T2 →
+lemma cpys_inv_atom1: ∀I,G,L,T2,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶*×[d, e] T2 →
                       T2 = ⓪{I} ∨
-                      ∃∃J,K,V1,V2,i. d ≤ i & i < d + e &
+                      ∃∃J,K,V1,V2,i. d ≤ yinj i & i < d + e &
                                     ⇩[O, i] L ≡ K.ⓑ{J}V1 &
                                      ⦃G, K⦄ ⊢ V1 ▶*×[0, d+e-i-1] V2 &
-                                     ⇧[O, i + 1] V2 ≡ T2 &
+                                     ⇧[O, i+1] V2 ≡ T2 &
                                      I = LRef i.
-#G #L #T2 #I #d #e #H @(cpys_ind … H) -T2
+#I #G #L #T2 #d #e #H @(cpys_ind … H) -T2
 [ /2 width=1 by or_introl/
 | #T #T2 #_ #HT2 *
   [ #H destruct
     elim (cpy_inv_atom1 … HT2) -HT2 [ /2 width=1 by or_introl/ | * /3 width=11 by ex6_5_intro, or_intror/ ]
   | * #J #K #V1 #V #i #Hdi #Hide #HLK #HV1 #HVT #HI
     lapply (ldrop_fwd_ldrop2 … HLK) #H
-    elim (cpy_inv_lift1_ge_up … HT2 … H … HVT) normalize -HT2 -H -HVT [2,3,4: /2 width=1 by le_S/ ]
-    <minus_plus /4 width=11 by cpys_strap1, ex6_5_intro, or_intror/
+    elim (cpy_inv_lift1_ge_up … HT2 … H … HVT) -HT2 -H -HVT
+    [2,3,4: /2 width=1 by ylt_fwd_le_succ, yle_succ_dx/ ]
+    /4 width=11 by cpys_strap1, ex6_5_intro, or_intror/
   ]
 ]
 qed-.
@@ -61,8 +63,8 @@ lemma cpys_inv_lref1: ∀G,L,T2,i,d,e. ⦃G, L⦄ ⊢ #i ▶*×[d, e] T2 →
                       T2 = #i ∨
                       ∃∃I,K,V1,V2. d ≤ i & i < d + e &
                                    ⇩[O, i] L ≡ K.ⓑ{I}V1 &
-                                   ⦃G, K⦄ ⊢ V1 ▶*×[0, d + e - i - 1] V2 &
-                                   ⇧[O, i + 1] V2 ≡ T2.
+                                   ⦃G, K⦄ ⊢ V1 ▶*×[0, d+e-i-1] V2 &
+                                   ⇧[O, i+1] V2 ≡ T2.
 #G #L #T2 #i #d #e #H elim (cpys_inv_atom1 … H) -H /2 width=1 by or_introl/
 * #I #K #V1 #V2 #j #Hdj #Hjde #HLK #HV12 #HVT2 #H destruct /3 width=7 by ex5_4_intro, or_intror/
 qed-.
@@ -70,7 +72,7 @@ qed-.
 (* Relocation properties ****************************************************)
 
 lemma cpys_lift_le: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 →
-                    ∀L,U1,d,e. dt + et ≤ d → ⇩[d, e] L ≡ K →
+                    ∀L,U1,d,e. dt + et ≤ yinj d → ⇩[d, e] L ≡ K →
                     ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
                     ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2.
 #G #K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdetd #HLK #HTU1 @(cpys_ind … H) -T2
@@ -83,7 +85,7 @@ lemma cpys_lift_le: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 →
 qed-.
 
 lemma cpys_lift_be: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 →
-                    ∀L,U1,d,e. dt ≤ d → d ≤ dt + et →
+                    ∀L,U1,d,e. dt ≤ yinj d → d ≤ dt + et →
                     ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
                     ∀U2. ⇧[d, e] T2 ≡ U2 → ⦃G, L⦄ ⊢ U1 ▶*×[dt, et + e] U2.
 #G #K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1 @(cpys_ind … H) -T2
@@ -96,9 +98,9 @@ lemma cpys_lift_be: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 →
 qed-.
 
 lemma cpys_lift_ge: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 →
-                    ∀L,U1,d,e. d ≤ dt → ⇩[d, e] L ≡ K →
+                    ∀L,U1,d,e. yinj d ≤ dt → ⇩[d, e] L ≡ K →
                     ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
-                    ⦃G, L⦄ ⊢ U1 ▶*×[dt + e, et] U2.
+                    ⦃G, L⦄ ⊢ U1 ▶*×[dt+e, et] U2.
 #G #K #T1 #T2 #dt #et #H #L #U1 #d #e #Hddt #HLK #HTU1 @(cpys_ind … H) -T2
 [ #U2 #H >(lift_mono … HTU1 … H) -H //
 | -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
@@ -137,15 +139,15 @@ lemma cpys_inv_lift1_ge: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2
 #G #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdedt @(cpys_ind … H) -U2
 [ /2 width=3 by ex2_intro/
 | -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
-  elim (cpy_inv_lift1_ge … HU2 … HLK … HTU ?) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+  elim (cpy_inv_lift1_ge … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
 ]
 qed-.
 
-lemma cpys_inv_lift1_eq: ∀G,L,U1,U2,d,e.
+lemma cpys_inv_lift1_eq: ∀G,L,U1,U2. ∀d,e:nat.
                          ⦃G, L⦄ ⊢ U1 ▶*×[d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
 #G #L #U1 #U2 #d #e #H #T1 #HTU1 @(cpys_ind … H) -U2 //
 #U #U2 #_ #HU2 #IHU destruct
-<(cpy_inv_lift1_eq … HU2 … HTU1) -HU2 -HTU1 //
+<(cpy_inv_lift1_eq … HTU1 … HU2) -HU2 -HTU1 //
 qed-.
 
 lemma cpys_inv_lift1_ge_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 →
diff --git a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_lt.ma b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_lt.ma
index 265572bae..6cc21d5bc 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_lt.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_lt.ma
@@ -26,10 +26,14 @@ interpretation "ynat 'less than'" 'lt x y = (ylt x y).
 
 (* Basic forward lemmas *****************************************************)
 
-lemma ylt_inv_gen: ∀x,y. x < y → ∃m. x = yinj m.
+lemma ylt_fwd_gen: ∀x,y. x < y → ∃m. x = yinj m.
 #x #y * -x -y /2 width=2 by ex_intro/
 qed-.
 
+lemma ylt_fwd_le_succ: ∀x,y. x < y → ⫯x ≤ y.
+#x #y * -x -y /2 width=1 by yle_inj/
+qed-.
+
 (* Basic inversion lemmas ***************************************************)
 
 fact ylt_inv_inj2_aux: ∀x,y. x < y → ∀n. y = yinj n →
@@ -51,7 +55,7 @@ lemma ylt_inv_inj: ∀m,n. yinj m < yinj n → m < n.
 qed-.
 
 lemma ylt_inv_Y1: ∀n. ∞ < n → ⊥.
-#n #H elim (ylt_inv_gen … H) -H
+#n #H elim (ylt_fwd_gen … H) -H
 #y #H destruct
 qed-.
 
diff --git a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_minus.ma b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_minus.ma
index a8ede2637..10a841cbd 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_minus.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_minus.ma
@@ -55,7 +55,7 @@ lemma yminus_succ: ∀n,m. ⫯m - ⫯n = m - n.
 #m >yminus_inj //
 qed.
 
-lemma yminus_succ1_inj: ∀n:nat. ∀m:ynat. n ≤ m →  ⫯m - n = ⫯(m - n).
+lemma yminus_succ1_inj: ∀n:nat. ∀m:ynat. n ≤ m → ⫯m - n = ⫯(m - n).
 #n *
 [ #m #Hmn >yminus_inj >yminus_inj
   /4 width=1 by yle_inv_inj, plus_minus, eq_f/
@@ -63,6 +63,10 @@ lemma yminus_succ1_inj: ∀n:nat. ∀m:ynat. n ≤ m →  ⫯m - n = ⫯(m - n).
 ]
 qed-.
 
+lemma yminus_succ2: ∀y,x. x - ⫯y = ⫰(x-y).
+* //
+qed.
+
 (* Properties on order ******************************************************)
 
 lemma yle_minus_sn: ∀n,m. m - n ≤ m.