X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fsubstitution%2Fcpy.ma;h=8fe2e8d262e967a3e1921e39edd13af43610cc61;hb=c60524dec7ace912c416a90d6b926bee8553250b;hp=439a2cba20ebf53a0b8de0f19e4cbd192e1a9aa9;hpb=f10cfe417b6b8ec1c7ac85c6ecf5fb1b3fdf37db;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy.ma
index 439a2cba2..8fe2e8d26 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy.ma
@@ -21,51 +21,51 @@ include "basic_2/substitution/lsuby.ma".
 
 (* activate genv *)
 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 ≤ yinj i → i < d+e →
-             ⬇[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) e G (L.ⓑ{I}V1) 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 →
-             cpy d e G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+| cpy_atom : ∀I,G,L,l,m. cpy l m G L (⓪{I}) (⓪{I})
+| cpy_subst: ∀I,G,L,K,V,W,i,l,m. l ≤ yinj i → i < l+m →
+             ⬇[i] L ≡ K.ⓑ{I}V → ⬆[0, i+1] V ≡ W → cpy l m G L (#i) W
+| cpy_bind : ∀a,I,G,L,V1,V2,T1,T2,l,m.
+             cpy l m G L V1 V2 → cpy (⫯l) m G (L.ⓑ{I}V1) T1 T2 →
+             cpy l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+| cpy_flat : ∀I,G,L,V1,V2,T1,T2,l,m.
+             cpy l m G L V1 V2 → cpy l m G L T1 T2 →
+             cpy l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
 .
 
 interpretation "context-sensitive extended ordinary substritution (term)"
-   'PSubst G L T1 d e T2 = (cpy d e G L T1 T2).
+   'PSubst G L T1 l m T2 = (cpy l m G L T1 T2).
 
 (* Basic properties *********************************************************)
 
-lemma lsuby_cpy_trans: ∀G,d,e. lsub_trans … (cpy d e G) (lsuby d e).
-#G #d #e #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -d -e
+lemma lsuby_cpy_trans: ∀G,l,m. lsub_trans … (cpy l m G) (lsuby l m).
+#G #l #m #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -l -m
 [ //
-| #I #G #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12
+| #I #G #L1 #K1 #V #W #i #l #m #Hli #Hilm #HLK1 #HVW #L2 #HL12
   elim (lsuby_drop_trans_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-.
 
-lemma cpy_refl: ∀G,T,L,d,e. ⦃G, L⦄ ⊢ T ▶[d, e] T.
+lemma cpy_refl: ∀G,T,L,l,m. ⦃G, L⦄ ⊢ T ▶[l, m] T.
 #G #T elim T -T // * /2 width=1 by cpy_bind, cpy_flat/
 qed.
 
 (* Basic_1: was: subst1_ex *)
-lemma cpy_full: ∀I,G,K,V,T1,L,d. ⬇[d] L ≡ K.ⓑ{I}V →
-                ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ▶[d, 1] T2 & ⬆[d, 1] T ≡ T2.
+lemma cpy_full: ∀I,G,K,V,T1,L,l. ⬇[l] L ≡ K.ⓑ{I}V →
+                ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ▶[l, 1] T2 & ⬆[l, 1] T ≡ T2.
 #I #G #K #V #T1 elim T1 -T1
-[ * #i #L #d #HLK
+[ * #i #L #l #HLK
   /2 width=4 by lift_sort, lift_gref, ex2_2_intro/
-  elim (lt_or_eq_or_gt i d) #Hid
+  elim (lt_or_eq_or_gt i l) #Hil
   /3 width=4 by lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/
   destruct
   elim (lift_total V 0 (i+1)) #W #HVW
   elim (lift_split … HVW i i)
   /4 width=5 by cpy_subst, ylt_inj, ex2_2_intro/
-| * [ #a ] #J #W1 #U1 #IHW1 #IHU1 #L #d #HLK
+| * [ #a ] #J #W1 #U1 #IHW1 #IHU1 #L #l #HLK
   elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
-  [ elim (IHU1 (L.ⓑ{J}W1) (d+1)) -IHU1
+  [ elim (IHU1 (L.ⓑ{J}W1) (l+1)) -IHU1
     /3 width=9 by cpy_bind, drop_drop, lift_bind, ex2_2_intro/
   | elim (IHU1 … HLK) -IHU1 -HLK
     /3 width=8 by cpy_flat, lift_flat, ex2_2_intro/
@@ -73,20 +73,20 @@ lemma cpy_full: ∀I,G,K,V,T1,L,d. ⬇[d] L ≡ K.ⓑ{I}V →
 ]
 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 //
+lemma cpy_weak: ∀G,L,T1,T2,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T2 →
+                ∀l2,m2. l2 ≤ l1 → l1 + m1 ≤ l2 + m2 →
+                ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T2.
+#G #L #T1 #T2 #l1 #m1 #H elim H -G -L -T1 -T2 -l1 -m1 //
 [ /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
+lemma cpy_weak_top: ∀G,L,T1,T2,l,m.
+                    ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶[l, |L| - l] T2.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m //
+[ #I #G #L #K #V #W #i #l #m #Hli #_ #HLK #HVW
   lapply (drop_fwd_length_lt2 … HLK)
   /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 *)
@@ -95,193 +95,193 @@ lemma cpy_weak_top: ∀G,L,T1,T2,d,e.
 ]
 qed-.
 
-lemma cpy_weak_full: ∀G,L,T1,T2,d,e.
-                     ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶[0, |L|] T2.
-#G #L #T1 #T2 #d #e #HT12
-lapply (cpy_weak … HT12 0 (d + e) ? ?) -HT12
+lemma cpy_weak_full: ∀G,L,T1,T2,l,m.
+                     ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶[0, |L|] T2.
+#G #L #T1 #T2 #l #m #HT12
+lapply (cpy_weak … HT12 0 (l + m) ? ?) -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. 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
+lemma cpy_split_up: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀i. i ≤ l + m →
+                    ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l, i-l] T & ⦃G, L⦄ ⊢ T ▶[i, l+m-i] T2.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
 [ /2 width=3 by ex2_intro/
-| #I #G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hjde
-  elim (ylt_split i j) [ -Hide -Hjde | -Hdi ]
+| #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #j #Hjlm
+  elim (ylt_split i j) [ -Hilm -Hjlm | -Hli ]
   /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
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
   elim (IHV12 i) -IHV12 // #V
-  elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hide
+  elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hilm
   >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
   lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2
   /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
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
+  elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hilm
   /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. 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
+lemma cpy_split_down: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀i. i ≤ l + m →
+                      ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[i, l+m-i] T & ⦃G, L⦄ ⊢ T ▶[l, i-l] T2.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
 [ /2 width=3 by ex2_intro/
-| #I #G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hjde
-  elim (ylt_split i j) [ -Hide -Hjde | -Hdi ]
+| #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #j #Hjlm
+  elim (ylt_split i j) [ -Hilm -Hjlm | -Hli ]
   /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
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
   elim (IHV12 i) -IHV12 // #V
-  elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hide
+  elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hilm
   >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
   lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2
   /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
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
+  elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hilm
   /3 width=5 by ex2_intro, cpy_flat/
 ]
 qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma cpy_fwd_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
-                  ∀T1,d,e. ⬆[d, e] T1 ≡ U1 →
-                  d ≤ dt → d + e ≤ dt + et →
-                  ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶[d+e, dt+et-(d+e)] U2 & ⬆[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
-[ * #i #G #L #dt #et #T1 #d #e #H #_
+lemma cpy_fwd_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
+                  ∀T1,l,m. ⬆[l, m] T1 ≡ U1 →
+                  l ≤ lt → l + m ≤ lt + mt →
+                  ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶[l+m, lt+mt-(l+m)] U2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
+[ * #i #G #L #lt #mt #T1 #l #m #H #_
   [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
-  | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+  | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
   | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
   ]
-| #I #G #L #K #V #W #i #dt #et #Hdti #Hidet #HLK #HVW #T1 #d #e #H #Hddt #Hdedet
-  elim (lift_inv_lref2 … H) -H * #Hid #H destruct [ -V -Hidet -Hdedet | -Hdti -Hddt ]
-  [ elim (ylt_yle_false … Hddt) -Hddt /3 width=3 by yle_ylt_trans, ylt_inj/
-  | elim (le_inv_plus_l … Hid) #Hdie #Hei
-    elim (lift_split … HVW d (i-e+1) ? ? ?) [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hdie
-    #T2 #_ >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H -Hei
+| #I #G #L #K #V #W #i #lt #mt #Hlti #Hilmt #HLK #HVW #T1 #l #m #H #Hllt #Hlmlmt
+  elim (lift_inv_lref2 … H) -H * #Hil #H destruct [ -V -Hilmt -Hlmlmt | -Hlti -Hllt ]
+  [ elim (ylt_yle_false … Hllt) -Hllt /3 width=3 by yle_ylt_trans, ylt_inj/
+  | elim (le_inv_plus_l … Hil) #Hlim #Hmi
+    elim (lift_split … HVW l (i-m+1) ? ? ?) [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hlim
+    #T2 #_ >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H -Hmi
     @(ex2_intro … H) -H @(cpy_subst … HLK HVW) /2 width=1 by yle_inj/ >ymax_pre_sn_comm // (**) (* explicit constructor *)
   ]
-| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #X #d #e #H #Hddt #Hdedet
+| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt
   elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
   elim (IHW12 … HVW1) -V1 -IHW12 //
   elim (IHU12 … HTU1) -T1 -IHU12 /2 width=1 by yle_succ/
   <yplus_inj >yplus_SO2 >yplus_succ1 >yplus_succ1
   /3 width=2 by cpy_bind, lift_bind, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #X #d #e #H #Hddt #Hdedet
+| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt
   elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
   elim (IHW12 … HVW1) -V1 -IHW12 // elim (IHU12 … HTU1) -T1 -IHU12
   /3 width=2 by cpy_flat, lift_flat, ex2_intro/
 ]
 qed-.
 
-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
+lemma cpy_fwd_tw: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ♯{T1} ≤ ♯{T2}.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m normalize
 /3 width=1 by monotonic_le_plus_l, le_plus/
 qed-.
 
 (* Basic inversion lemmas ***************************************************)
 
-fact cpy_inv_atom1_aux: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ∀J. T1 = ⓪{J} →
+fact cpy_inv_atom1_aux: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀J. T1 = ⓪{J} →
                         T2 = ⓪{J} ∨
-                        ∃∃I,K,V,i. d ≤ yinj i & i < d + e &
+                        ∃∃I,K,V,i. l ≤ yinj i & i < l + m &
                                    ⬇[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
+#G #L #T1 #T2 #l #m * -G -L -T1 -T2 -l -m
+[ #I #G #L #l #m #J #H destruct /2 width=1 by or_introl/
+| #I #G #L #K #V #T2 #i #l #m #Hli #Hilm #HLK #HVT2 #J #H destruct /3 width=9 by ex5_4_intro, or_intror/
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct
 ]
 qed-.
 
-lemma cpy_inv_atom1: ∀I,G,L,T2,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶[d, e] T2 →
+lemma cpy_inv_atom1: ∀I,G,L,T2,l,m. ⦃G, L⦄ ⊢ ⓪{I} ▶[l, m] T2 →
                      T2 = ⓪{I} ∨
-                     ∃∃J,K,V,i. d ≤ yinj i & i < d + e &
+                     ∃∃J,K,V,i. l ≤ yinj i & i < l + m &
                                 ⬇[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: ∀G,L,T2,k,d,e. ⦃G, L⦄ ⊢ ⋆k ▶[d, e] T2 → T2 = ⋆k.
-#G #L #T2 #k #d #e #H
+lemma cpy_inv_sort1: ∀G,L,T2,k,l,m. ⦃G, L⦄ ⊢ ⋆k ▶[l, m] T2 → T2 = ⋆k.
+#G #L #T2 #k #l #m #H
 elim (cpy_inv_atom1 … H) -H //
 * #I #K #V #i #_ #_ #_ #_ #H destruct
 qed-.
 
 (* Basic_1: was: subst1_gen_lref *)
-lemma cpy_inv_lref1: ∀G,L,T2,i,d,e. ⦃G, L⦄ ⊢ #i ▶[d, e] T2 →
+lemma cpy_inv_lref1: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶[l, m] T2 →
                      T2 = #i ∨
-                     ∃∃I,K,V. d ≤ i & i < d + e &
+                     ∃∃I,K,V. l ≤ i & i < l + m &
                               ⬇[i] L ≡ K.ⓑ{I}V &
                               ⬆[O, i+1] V ≡ T2.
-#G #L #T2 #i #d #e #H
+#G #L #T2 #i #l #m #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/
+* #I #K #V #j #Hlj #Hjlm #HLK #HVT2 #H destruct /3 width=5 by ex4_3_intro, or_intror/
 qed-.
 
-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
+lemma cpy_inv_gref1: ∀G,L,T2,p,l,m. ⦃G, L⦄ ⊢ §p ▶[l, m] T2 → T2 = §p.
+#G #L #T2 #p #l #m #H
 elim (cpy_inv_atom1 … H) -H //
 * #I #K #V #i #_ #_ #_ #_ #H destruct
 qed-.
 
-fact cpy_inv_bind1_aux: ∀G,L,U1,U2,d,e. ⦃G, L⦄ ⊢ U1 ▶[d, e] U2 →
+fact cpy_inv_bind1_aux: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 →
                         ∀a,I,V1,T1. U1 = ⓑ{a,I}V1.T1 →
-                        ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[d, e] V2 &
-                                 ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ▶[⫯d, e] T2 &
+                        ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
+                                 ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ▶[⫯l, m] 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
+#G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m
+[ #I #G #L #l #m #b #J #W1 #U1 #H destruct
+| #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #b #J #W1 #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #b #J #W1 #U1 #H destruct
 ]
 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}V1⦄ ⊢ T1 ▶[⫯d, e] T2 &
+lemma cpy_inv_bind1: ∀a,I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓑ{a,I} V1. T1 ▶[l, m] U2 →
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
+                              ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶[⫯l, m] T2 &
                               U2 = ⓑ{a,I}V2.T2.
 /2 width=3 by cpy_inv_bind1_aux/ qed-.
 
-fact cpy_inv_flat1_aux: ∀G,L,U1,U2,d,e. ⦃G, L⦄ ⊢ U1 ▶[d, e] U2 →
+fact cpy_inv_flat1_aux: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 →
                         ∀I,V1,T1. U1 = ⓕ{I}V1.T1 →
-                        ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[d, e] V2 &
-                                 ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 &
+                        ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
+                                 ⦃G, L⦄ ⊢ T1 ▶[l, m] 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/
+#G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m
+[ #I #G #L #l #m #J #W1 #U1 #H destruct
+| #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #J #W1 #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #W1 #U1 #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
 ]
 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 &
+lemma cpy_inv_flat1: ∀I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ▶[l, m] U2 →
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
+                              ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 &
                               U2 = ⓕ{I}V2.T2.
 /2 width=3 by cpy_inv_flat1_aux/ qed-.
 
 
-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
+fact cpy_inv_refl_O2_aux: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → m = 0 → T1 = T2.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
 [ //
-| #I #G #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct
-  elim (ylt_yle_false … Hdi) -Hdi //
+| #I #G #L #K #V #W #i #l #m #Hli #Hilm #_ #_ #H destruct
+  elim (ylt_yle_false … Hli) -Hli //
 | /3 width=1 by eq_f2/
 | /3 width=1 by eq_f2/
 ]
 qed-.
 
-lemma cpy_inv_refl_O2: ∀G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▶[d, 0] T2 → T1 = T2.
+lemma cpy_inv_refl_O2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶[l, 0] T2 → T1 = T2.
 /2 width=6 by cpy_inv_refl_O2_aux/ qed-.
 
 (* Basic_1: was: subst1_gen_lift_eq *)
-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 #HTU1 #L #U2 #HU12 elim (cpy_fwd_up … HU12 … HTU1) -HU12 -HTU1
+lemma cpy_inv_lift1_eq: ∀G,T1,U1,l,m. ⬆[l, m] T1 ≡ U1 →
+                        ∀L,U2. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 → U1 = U2.
+#G #T1 #U1 #l #m #HTU1 #L #U2 #HU12 elim (cpy_fwd_up … HU12 … HTU1) -HU12 -HTU1
 /2 width=4 by cpy_inv_refl_O2/
 qed-.