From 6907a0f66a3782ce4273a609203c9b574841c7d1 Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Thu, 19 Dec 2013 14:43:18 +0000
Subject: [PATCH] the theory of cpy continues ...

---
 .../lambdadelta/basic_2/relocation/cpy.ma     | 226 ++++++------
 .../basic_2/relocation/cpy_lift.ma            | 328 ++++++++----------
 .../lambdadelta/basic_2/relocation/lsuby.ma   |  28 +-
 3 files changed, 281 insertions(+), 301 deletions(-)

diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma
index 025cd6f65..e1b2afd75 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma
@@ -14,6 +14,7 @@
 
 include "basic_2/notation/relations/extpsubst_6.ma".
 include "basic_2/grammar/genv.ma".
+include "basic_2/grammar/cl_shift.ma".
 include "basic_2/relocation/ldrop_append.ma".
 include "basic_2/relocation/lsuby.ma".
 
@@ -37,12 +38,13 @@ interpretation "context-sensitive extended parallel substritution (term)"
 
 (* Basic properties *********************************************************)
 
-lemma lsuby_cpy_trans: ∀G,d,e.lsub_trans … (cpy d e G) lsuby.
+lemma lsuby_cpy_trans: ∀G,L1,T1,T2,d,e. ⦃G, L1⦄ ⊢ T1 ▶×[d, e] T2 →
+                       ∀L2. L2 ⊑×[d, e] L1 → ⦃G, L2⦄ ⊢ T1 ▶×[d, e] T2.
 #G #d #e #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -d -e
 [ //
 | #I #G #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12
-  elim (lsuby_fwd_ldrop2_bind … HL12 … HLK1) -HL12 -HLK1 /2 width=5 by cpy_subst/
-| /4 width=1 by lsuby_pair, cpy_bind/
+  elim (lsuby_fwd_ldrop2_be … HL12 … HLK1) -HL12 -HLK1 /2 width=5 by cpy_subst/
+| /4 width=1 by lsuby_succ, cpy_bind/
 | /3 width=1 by cpy_flat/
 ]
 qed-.
@@ -117,154 +119,155 @@ lemma cpy_split_up: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ∀i.
   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 cpy_bind, ex2_intro/
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
-  elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
-  -Hdi -Hide /3 width=5/
+  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/
 ]
-qed.
+qed-.
 
-lemma cpy_split_down: ∀L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 →
+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.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
-[ /2 width=3/
-| #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
+#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=8/
+  [ -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=4/
+    >(plus_minus_m_m (d+e) j) in Hide; // -Hjde /3 width=5 by ex2_intro, cpy_subst/
   ]
-| #L #a #I #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/
+| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #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 (cpy_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /3 width=5/
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
-  elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
-  -Hdi -Hide /3 width=5/
+  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/
 ]
-qed.
+qed-.
 
-lemma cpy_append: ∀K,T1,T2,d,e. K⦄ ⊢ T1 ▶×[d, e] T2 →
-                  ∀L. L @@ K⦄ ⊢ T1 ▶×[d, e] T2.
-#K #T1 #T2 #d #e #H elim H -K -T1 -T2 -d -e // /2 width=1/
-#K #K0 #V #W #i #d #e #Hdi #Hide #HK0 #HVW #L
-lapply (ldrop_fwd_ldrop2_length … HK0) #H
-@(cpy_subst … (L@@K0) … HVW) // (**) (* /3/ does not work *)
-@(ldrop_O1_append_sn_le … HK0) /2 width=2/
-qed.
+lemma cpy_append: ∀G,K,T1,T2,d,e. ⦃G, K⦄ ⊢ T1 ▶×[d, e] T2 →
+                  ∀L. ⦃G, L @@ K⦄ ⊢ T1 ▶×[d, e] T2.
+#G #K #T1 #T2 #d #e #H elim H -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
+@(cpy_subst I … (L@@K0) … HVW) // (**) (* /4/ does not work *)
+@(ldrop_O1_append_sn_le … HK0) /2 width=2 by lt_to_le/
+qed-.
 
 (* Basic inversion lemmas ***************************************************)
 
-fact cpy_inv_atom1_aux: ∀L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ∀I. T1 = ⓪{I} →
-                        T2 = ⓪{I} ∨
-                        ∃∃K,V,i. d ≤ i & i < d + e &
-                                 ⇩[O, i] L ≡ K. ⓓV &
-                                 ⇧[O, i + 1] V ≡ T2 &
-                                 I = LRef i.
-#L #T1 #T2 #d #e * -L -T1 -T2 -d -e
-[ #L #I #d #e #J #H destruct /2 width=1/
-| #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #I #H destruct /3 width=8/
-| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
+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 &
+                                   ⇩[O, i] L ≡ K.ⓑ{I}V &
+                                   ⇧[O, i + 1] V ≡ T2 &
+                                   J = LRef i.
+#G #L #T1 #T2 #d #e * -G -L -T1 -T2 -d -e
+[ #I #G #L #d #e #J #H destruct /2 width=1 by or_introl/
+| #I #G #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #J #H destruct /3 width=9 by ex5_4_intro, or_intror/
+| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
 ]
-qed.
+qed-.
 
-lemma cpy_inv_atom1: ∀L,T2,I,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶×[d, e] T2 →
+lemma cpy_inv_atom1: ∀I,G,L,T2,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶×[d, e] T2 →
                      T2 = ⓪{I} ∨
-                     ∃∃K,V,i. d ≤ i & i < d + e &
-                              ⇩[O, i] L ≡ K. ⓓV &
-                              ⇧[O, i + 1] V ≡ T2 &
-                              I = LRef i.
-/2 width=3/ qed-.
+                     ∃∃J,K,V,i. d ≤ i & i < d + e &
+                                ⇩[O, i] L ≡ K.ⓑ{J}V &
+                                ⇧[O, i + 1] V ≡ T2 &
+                                I = LRef i.
+/2 width=4 by cpy_inv_atom1_aux/ qed-.
 
-
-(* Basic_1: was: subst1_gen_sort *)
-lemma cpy_inv_sort1: ∀L,T2,k,d,e. ⦃G, L⦄ ⊢ ⋆k ▶×[d, e] T2 → T2 = ⋆k.
-#L #T2 #k #d #e #H
+lemma cpy_inv_sort1: ∀G,L,T2,k,d,e. ⦃G, L⦄ ⊢ ⋆k ▶×[d, e] T2 → T2 = ⋆k.
+#G #L #T2 #k #d #e #H
 elim (cpy_inv_atom1 … H) -H //
-* #K #V #i #_ #_ #_ #_ #H destruct
+* #I #K #V #i #_ #_ #_ #_ #H destruct
 qed-.
 
-(* Basic_1: was: subst1_gen_lref *)
-lemma cpy_inv_lref1: ∀L,T2,i,d,e. ⦃G, L⦄ ⊢ #i ▶×[d, e] T2 →
+lemma cpy_inv_lref1: ∀G,L,T2,i,d,e. ⦃G, L⦄ ⊢ #i ▶×[d, e] T2 →
                      T2 = #i ∨
-                     ∃∃K,V. d ≤ i & i < d + e &
-                            ⇩[O, i] L ≡ K. ⓓV &
-                            ⇧[O, i + 1] V ≡ T2.
-#L #T2 #i #d #e #H
-elim (cpy_inv_atom1 … H) -H /2 width=1/
-* #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=4/
+                     ∃∃I,K,V. d ≤ i & i < d + e &
+                              ⇩[O, i] L ≡ K.ⓑ{I}V &
+                              ⇧[O, i + 1] V ≡ T2.
+#G #L #T2 #i #d #e #H
+elim (cpy_inv_atom1 … H) -H /2 width=1 by or_introl/
+* #I #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=5 by ex4_3_intro, or_intror/
 qed-.
 
-lemma cpy_inv_gref1: ∀L,T2,p,d,e. ⦃G, L⦄ ⊢ §p ▶×[d, e] T2 → T2 = §p.
-#L #T2 #p #d #e #H
+lemma cpy_inv_gref1: ∀G,L,T2,p,d,e. ⦃G, L⦄ ⊢ §p ▶×[d, e] T2 → T2 = §p.
+#G #L #T2 #p #d #e #H
 elim (cpy_inv_atom1 … H) -H //
-* #K #V #i #_ #_ #_ #_ #H destruct
+* #I #K #V #i #_ #_ #_ #_ #H destruct
 qed-.
 
-fact cpy_inv_bind1_aux: ∀d,e,L,U1,U2. ⦃G, L⦄ ⊢ U1 ▶×[d, e] U2 →
-                        ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 →
+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 &
-                                 L. ⓑ{I} V2⦄ ⊢ T1 ▶×[d + 1, e] T2 &
-                                 U2 = ⓑ{a,I} V2. T2.
-#d #e #L #U1 #U2 * -d -e -L -U1 -U2
-[ #L #k #d #e #a #I #V1 #T1 #H destruct
-| #L #K #V #W #i #d #e #_ #_ #_ #_ #a #I #V1 #T1 #H destruct
-| #L #b #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #a #I #V #T #H destruct /2 width=5/
-| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #a #I #V #T #H destruct
+                                 ⦃G, L. ⓑ{I}V2⦄ ⊢ T1 ▶×[d + 1, 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
+| #I #G #L #K #V #W #i #d #e #_ #_ #_ #_ #b #J #W1 #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #b #J #W1 #U1 #H destruct
 ]
-qed.
+qed-.
 
-lemma cpy_inv_bind1: ∀d,e,L,a,I,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I} V1. T1 ▶×[d, e] U2 →
+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 &
-                              L. ⓑ{I} V2⦄ ⊢ T1 ▶×[d + 1, e] T2 &
-                              U2 = ⓑ{a,I} V2. T2.
-/2 width=3/ qed-.
+                              ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ▶×[d + 1, e] T2 &
+                              U2 = ⓑ{a,I}V2.T2.
+/2 width=3 by cpy_inv_bind1_aux/ qed-.
 
-fact cpy_inv_flat1_aux: ∀d,e,L,U1,U2. ⦃G, L⦄ ⊢ U1 ▶×[d, e] U2 →
-                        ∀I,V1,T1. U1 = ⓕ{I} V1. T1 →
-                        ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶×[d, e] V2 & ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 &
-                                 U2 =  ⓕ{I} V2. T2.
-#d #e #L #U1 #U2 * -d -e -L -U1 -U2
-[ #L #k #d #e #I #V1 #T1 #H destruct
-| #L #K #V #W #i #d #e #_ #_ #_ #_ #I #V1 #T1 #H destruct
-| #L #a #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
-| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
+fact cpy_inv_flat1_aux: ∀G,L,U1,U2,d,e. ⦃G, L⦄ ⊢ U1 ▶×[d, e] U2 →
+                        ∀I,V1,T1. U1 = ⓕ{I}V1.T1 →
+                        ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶×[d, e] V2 &
+                                 ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 &
+                                 U2 = ⓕ{I}V2.T2.
+#G #L #U1 #U2 #d #e * -G -L -U1 -U2 -d -e
+[ #I #G #L #d #e #J #W1 #U1 #H destruct
+| #I #G #L #K #V #W #i #d #e #_ #_ #_ #_ #J #W1 #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #J #W1 #U1 #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
 ]
-qed.
+qed-.
+
+lemma cpy_inv_flat1: ∀I,G,L,V1,T1,U2,d,e. ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ▶×[d, e] U2 →
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶×[d, e] V2 &
+                              ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 &
+                              U2 = ⓕ{I}V2.T2.
+/2 width=3 by cpy_inv_flat1_aux/ qed-.
 
-lemma cpy_inv_flat1: ∀d,e,L,I,V1,T1,U2. ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ▶×[d, e] U2 →
-                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶×[d, e] V2 & ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 &
-                              U2 =  ⓕ{I} V2. T2.
-/2 width=3/ qed-.
 
-fact cpy_inv_refl_O2_aux: ∀L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → e = 0 → T1 = T2.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
+fact cpy_inv_refl_O2_aux: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → e = 0 → T1 = T2.
+#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
 [ //
-| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct
+| #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)
-| /3 width=1/
-| /3 width=1/
+| /3 width=1 by eq_f2/
+| /3 width=1 by eq_f2/
 ]
-qed.
+qed-.
 
-lemma cpy_inv_refl_O2: ∀L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▶×[d, 0] T2 → T1 = T2.
-/2 width=6/ qed-.
+lemma cpy_inv_refl_O2: ∀G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▶×[d, 0] T2 → T1 = T2.
+/2 width=6 by cpy_inv_refl_O2_aux/ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma cpy_fwd_tw: ∀L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ♯{T1} ≤ ♯{T2}.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e normalize
-/3 by monotonic_le_plus_l, le_plus/ (**) (* just /3 width=1/ is too slow *)
+lemma cpy_fwd_tw: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ♯{T1} ≤ ♯{T2}.
+#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e normalize
+/3 width=1 by monotonic_le_plus_l, le_plus/
 qed-.
 
-lemma cpy_fwd_shift1: ∀L1,L,T1,T,d,e. ⦃G, L⦄ ⊢ L1 @@ T1 ▶[d, e] T →
+lemma cpy_fwd_shift1: ∀G,L1,L,T1,T,d,e. ⦃G, L⦄ ⊢ L1 @@ T1 ▶×[d, e] T →
                       ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#L1 @(lenv_ind_dx … L1) -L1 normalize
+#G #L1 @(lenv_ind_dx … L1) -L1 normalize
 [ #L #T1 #T #d #e #HT1
   @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
 | #I #L1 #V1 #IH #L #T1 #X #d #e
@@ -273,16 +276,7 @@ lemma cpy_fwd_shift1: ∀L1,L,T1,T,d,e. ⦃G, L⦄ ⊢ L1 @@ T1 ▶[d, e] T →
   #V0 #T0 #_ #HT10 #H destruct
   elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
   >append_length >HL12 -HL12
-  @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
+  @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] (**) (* explicit constructor *)
+  /2 width=3 by trans_eq/ 
 ]
 qed-.
-
-(* Basic_1: removed theorems 25:
-            subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
-            subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
-            subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
-            subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
-            subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
-            subst0_confluence_lift subst0_tlt
-            subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift
-*)
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 1d65d8d63..161b21b01 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_lift.ma
@@ -12,283 +12,243 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/substitution/ldrop_ldrop.ma".
-include "basic_2/substitution/tps.ma".
+include "basic_2/relocation/ldrop_ldrop.ma".
+include "basic_2/relocation/cpy.ma".
 
-(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-fact tps_inv_S2_aux: ∀L,T1,T2,d,e1. L ⊢ T1 ▶ [d, e1] T2 → ∀e2. e1 = e2 + 1 →
-                     ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶ [d + 1, e2] T2.
-#L #T1 #T2 #d #e1 #H elim H -L -T1 -T2 -d -e1
-[ //
-| #L #K0 #V0 #W #i #d #e1 #Hdi #Hide1 #HLK0 #HV0 #e2 #He12 #K #V #HLK destruct
-  elim (lt_or_ge i (d+1)) #HiSd
-  [ -Hide1 -HV0
-    lapply (le_to_le_to_eq … Hdi ?) /2 width=1/ #H destruct
-    lapply (ldrop_mono … HLK0 … HLK) #H destruct
-  | -V -Hdi /2 width=4/
-  ]
-| /4 width=3/
-| /3 width=3/
-]
-qed.
-
-lemma tps_inv_S2: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e + 1] T2 →
-                  ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶ [d + 1, e] T2.
-/2 width=3/ qed-.
-
-lemma tps_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 ▶ [d, 1] T2 →
-                        ∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2.
-#L #T1 #T2 #d #HT12 #K #V #HLK
-lapply (tps_inv_S2 … T1 T2 … 0 … HLK) -K // -HT12 #HT12
-lapply (tps_inv_refl_O2 … HT12) -HT12 //
-qed-.
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL SUBSTITUTION FOR TERMS ***************)
 
 (* Relocation properties ****************************************************)
 
-(* Basic_1: was: subst1_lift_lt *)
-lemma tps_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 →
+lemma cpy_lift_le: ∀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 →
-                   dt + et ≤ d →
-                   L ⊢ U1 ▶ [dt, et] U2.
-#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
-[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
+                   dt + et ≤ d → ⦃G, L⦄ ⊢ U1 ▶×[dt, 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 //
-| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdetd
+| #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 (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/ #X #HLK #H
-  elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K #Y #_ #HVY
-  >(lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=4/
-| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
+  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) -Y -HVW #H destruct /2 width=5 by cpy_subst/
+| #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
-  @tps_bind [ /2 width=6/ | @IHT12 /2 width=6/ ] (**) (* /3 width=6/ is too slow, arith3 needed to avoid crash *)
-| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
+  /4 width=6 by cpy_bind, ldrop_skip, le_S_S/
+| #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 /3 width=6/
+  elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+  /3 width=6 by cpy_flat/
 ]
-qed.
+qed-.
 
-lemma tps_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 →
+lemma cpy_lift_be: ∀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 →
-                   dt ≤ d → d ≤ dt + et →
-                   L ⊢ U1 ▶ [dt, et + e] U2.
-#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
-[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_ #_
+                   dt ≤ d → d ≤ dt + et → ⦃G, L⦄ ⊢ U1 ▶×[dt, et + e] 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 //
-| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdtd #_
+| #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
-    elim (lift_trans_ge … HVW … HWU2 ?) -W // <minus_plus #W #HVW #HWU2
-    elim (ldrop_trans_le … HLK … HKV ?) -K /2 width=2/ #X #HLK #H
-    elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K #Y #_ #HVY
-    >(lift_mono … HVY … HVW) -V #H destruct /2 width=4/
+    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
     lapply (transitive_le … Hdtd Hid) -Hdtd #Hdti
-    lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2
-    lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/
+    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/
   ]
-| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdtd #Hddet
+| #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
-  @tps_bind [ /2 width=6/ | @IHT12 [3,4: // | skip |5,6: /2 width=1/ | /2 width=1/ ]
-            ] (**) (* /3 width=6/ is too slow, simplification like tps_lift_le is too slow *)
-| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
+  /4 width=6 by cpy_bind, ldrop_skip, le_S_S/ (**) (* auto a bit slow *)
+| #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 /3 width=6/
+  elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+  /3 width=6 by cpy_flat/
 ]
-qed.
+qed-.
 
-(* Basic_1: was: subst1_lift_ge *)
-lemma tps_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 ▶ [dt, et] T2 →
+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 →
-                   L ⊢ U1 ▶ [dt + e, et] U2.
-#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
-[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
+                   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 //
-| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hddt
+| #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 (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2
-  lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/
-| #K #a #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
+  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/
+| #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
-  @tps_bind [ /2 width=5/ | /3 width=5/ ] (**) (* explicit constructor *)
-| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
+  /4 width=5 by cpy_bind, ldrop_skip, le_minus_to_plus/
+| #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 /3 width=5/
+  elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+  /3 width=5 by cpy_flat/
 ]
-qed.
+qed-.
 
-(* Basic_1: was: subst1_gen_lift_lt *)
-lemma tps_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+lemma cpy_inv_lift1_le: ∀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 + et ≤ d →
-                        ∃∃T2. K ⊢ T1 ▶ [dt, et] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
-[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
-  [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
-  | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
-  | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
+                        ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[dt, 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/
+  | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3 by cpy_atom, lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+  | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by cpy_atom, lift_gref, ex2_intro/
   ]
-| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdetd
+| #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 (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=4/
-| #L #a #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+  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: /2 width=1/ |4: @ldrop_skip // |2: skip ] -HLK -Hdetd (**) (* /3 width=5/ is too slow *)
-  /3 width=5/
-| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+  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/
+| #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 //
-  elim (IHU12 … HLK … HTU1 ?) -U1 -HLK // /3 width=5/
+  elim (IHV12 … HLK … HWV1) -V1 //
+  elim (IHU12 … HLK … HTU1) -U1 -HLK
+  /3 width=5 by cpy_flat, lift_flat, ex2_intro/
 ]
-qed.
+qed-.
 
-lemma tps_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+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. K ⊢ T1 ▶ [dt, et - e] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
-[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
-  [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
-  | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
-  | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
+                        ∃∃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/
+  | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3 by cpy_atom, lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+  | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by cpy_atom, lift_gref, ex2_intro/
   ]
-| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdtd #Hdedet
+| #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/ ] -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=4/
+    lapply (lt_to_le_to_lt … (dt + (et - e)) Hid ?) [ <le_plus_minus /2 width=1 by le_plus_to_minus_r/ ] -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/ -Hdtd #Hdtie
+    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
     lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
-    elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hid -Hdie
-    #V1 #HV1 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
-    @ex2_intro [3: @H | skip | @tps_subst [3,5,6: // |1,2: skip | >commutative_plus >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
+    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/
   ]
-| #L #a #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
+| #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: @ldrop_skip // |2: skip |3: >plus_plus_comm_23 >(plus_plus_comm_23 dt) /2 width=1/ |4: /2 width=1/ ] (**) (* 29s without the rewrites *)
-  /3 width=5/
-| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
+  elim (IHV12 … HLK … HWV1) -V1 // #W2 #HW12 #HWV2
+  elim (IHU12 … HTU1) -U1
+  [5: @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/
+  ] (**) (* 29s without the rewrites *)
+  /3 width=5 by _//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 #Hdtd #Hdedet
   elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
-  elim (IHV12 … HLK … HWV1 ? ?) -V1 //
-  elim (IHU12 … HLK … HTU1 ? ?) -U1 -HLK // /3 width=5/
+  elim (IHV12 … HLK … HWV1) -V1 //
+  elim (IHU12 … HLK … HTU1) -U1 -HLK //
+  /3 width=5 by cpy_flat, lift_flat, ex2_intro/
 ]
-qed.
+qed-.
 
-(* Basic_1: was: subst1_gen_lift_ge *)
-lemma tps_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+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. K ⊢ T1 ▶ [dt - e, et] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
-[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
-  [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
-  | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
-  | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
+                        ∃∃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/
+  | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3 by cpy_atom, lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+  | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by cpy_atom, lift_gref, ex2_intro/
   ]
-| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdedt
+| #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 [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hdei -Hdie
-  #V0 #HV10 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
-  @ex2_intro [3: @H | skip | @tps_subst [5,6: // |1,2: skip | /2 width=1/ | >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
-| #L #a #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+  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/
+| #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 (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2
-  elim (IHU12 … HTU1 ?) -U1 [4: @ldrop_skip // |2: skip |3: /2 width=1/ ]
-  <plus_minus // /3 width=5/
-| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+  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/
+| #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 //
-  elim (IHU12 … HLK … HTU1 ?) -U1 -HLK // /3 width=5/
+  elim (IHV12 … HLK … HWV1) -V1 //
+  elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5 by cpy_flat, lift_flat, ex2_intro/
 ]
-qed.
+qed-.
 
-(* Basic_1: was: subst1_gen_lift_eq *)
-lemma tps_inv_lift1_eq: ∀L,U1,U2,d,e.
-                        L ⊢ U1 ▶ [d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
-#L #U1 #U2 #d #e #H elim H -L -U1 -U2 -d -e
+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
 [ //
-| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #T1 #H
+| #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)
   ]
-| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+| #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 //
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+| #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 //
 ]
-qed.
-(*
-      Theorem subst0_gen_lift_rev_ge: (t1,v,u2,i,h,d:?)
-                                      (subst0 i v t1 (lift h d u2)) ->
-                                      (le (plus d h) i) ->
-                                      (EX u1 | (subst0 (minus i h) v u1 u2) &
-                                               t1 = (lift h d u1)
-                                      ).
-
+qed-.
 
-      Theorem subst0_gen_lift_rev_lelt: (t1,v,u2,i,h,d:?)
-                                        (subst0 i v t1 (lift h d u2)) ->
-                                        (le d i) -> (lt i (plus d h)) ->
-                                        (EX u1 | t1 = (lift (minus (plus d h) (S i)) (S i) u1)).
-*)
-lemma tps_inv_lift1_ge_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+lemma cpy_inv_lift1_ge_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 →
                            d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
-                           ∃∃T2. K ⊢ T1 ▶ [d, dt + et - (d + e)] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
-elim (tps_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
-lapply (tps_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1
-lapply (tps_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
-elim (tps_inv_lift1_ge … HU2 … HLK … HTU1 ?) -U -L // <minus_plus_m_m /2 width=3/
-qed.
+                           ∃∃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/
+qed-.
 
-lemma tps_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+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. K ⊢ T1 ▶ [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
-lapply (tps_weak … HU12 dt (d + e - dt) ? ?) -HU12 // /2 width=3/ -Hdetde #HU12
-elim (tps_inv_lift1_be … HU12 … HLK … HTU1 ? ?) -U1 -L // /2 width=3/
-qed.
+                           ∃∃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
+elim (cpy_inv_lift1_be … HU12 … HLK … HTU1) -U1 -L /2 width=3 by ex2_intro/
+qed-.
 
-lemma tps_inv_lift1_le_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶ [dt, et] U2 →
+lemma cpy_inv_lift1_le_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 → d ≤ dt + et → dt + et ≤ d + e →
-                           ∃∃T2. K ⊢ T1 ▶ [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde
-elim (tps_split_up … HU12 d ? ?) -HU12 // #U #HU1 #HU2
-elim (tps_inv_lift1_le … HU1 … HLK … HTU1 ?) -U1 [2: >commutative_plus /2 width=1/ ] -Hdtd #T #HT1 #HTU
-lapply (tps_weak … HU2 d e ? ?) -HU2 // [ >commutative_plus <plus_minus_m_m // ] -Hddet -Hdetde #HU2
-lapply (tps_inv_lift1_eq … HU2 … HTU) -L #H destruct /2 width=3/
-qed.
+                           ∃∃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/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lsuby.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lsuby.ma
index 367afba27..d2aa16144 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lsuby.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lsuby.ma
@@ -13,7 +13,7 @@
 (**************************************************************************)
 
 include "basic_2/notation/relations/extlrsubeq_4.ma".
-include "basic_2/grammar/lenv_length.ma".
+include "basic_2/relocation/ldrop.ma".
 
 (* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
 
@@ -196,3 +196,29 @@ lemma lsuby_inv_succ2: ∀I2,K2,L1,V2,d,e. L1 ⊑×[d, e] K2.ⓑ{I2}V2 → 0 < d
 lemma lsuby_fwd_length: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 → |L2| ≤ |L1|.
 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize /2 width=1 by le_S_S/
 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.
+#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/
+  ]
+| #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/
+]
+qed-.
-- 
2.39.2