X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=inline;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frelocation%2Fcpy.ma;h=829fea318174274d2005d71cc26cdee5fdaeaf4a;hb=2ba2dc23443ad764adab652e06d6f5ed10bd912d;hp=025cd6f65c6ef6edea32ceba4fc9515e1e52bef5;hpb=3e9d72c26091f0e157a024ea9bd6f95a95729860;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma
index 025cd6f65..829fea318 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma
@@ -12,37 +12,39 @@
 (*                                                                        *)
 (**************************************************************************)
 
+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".
 include "basic_2/relocation/ldrop_append.ma".
 include "basic_2/relocation/lsuby.ma".
 
-(* CONTEXT-SENSITIVE EXTENDED PARALLEL SUBSTITUTION FOR TERMS ***************)
+(* 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 →
+             ⇩[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 (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
+             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 d e G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
 .
 
-interpretation "context-sensitive extended parallel substritution (term)"
+interpretation "context-sensitive extended ordinary substritution (term)"
    'ExtPSubst G L T1 d e T2 = (cpy d e G L T1 T2).
 
 (* Basic properties *********************************************************)
 
-lemma lsuby_cpy_trans: ∀G,d,e.lsub_trans … (cpy d e G) lsuby.
+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
 [ //
 | #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-.
@@ -51,7 +53,7 @@ lemma cpy_refl: ∀G,T,L,d,e. ⦃G, L⦄ ⊢ T ▶×[d, e] T.
 #G #T elim T -T // * /2 width=1 by cpy_bind, cpy_flat/
 qed.
 
-lemma cpy_full: ∀I,G,K,V,T1,L,d. ⇩[0, d] L ≡ K.ⓑ{I}V →
+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.
 #I #G #K #V #T1 elim T1 -T1
 [ * #i #L #d #HLK
@@ -61,11 +63,11 @@ 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
-    /3 width=9 by cpy_bind, ldrop_ldrop, lift_bind, ex2_2_intro/
+  [ elim (IHU1 (L.ⓑ{J}W1) (d+1)) -IHU1
+    /3 width=9 by cpy_bind, ldrop_drop, lift_bind, ex2_2_intro/
   | elim (IHU1 … HLK) -IHU1 -HLK
     /3 width=8 by cpy_flat, lift_flat, ex2_2_intro/
   ]
@@ -75,22 +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-.
@@ -102,169 +103,198 @@ 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_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 #_
+  [ 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/
+  | 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
+    @(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
+  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
+  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_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 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/
+| #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 … 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
+  /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 →
-                      ∀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
-  elim (lt_or_ge i j)
-  [ -Hide -Hjde >(plus_minus_m_m j d) in⦄ ⊢ (% → ?); // -Hdj /3 width=8/
-  | -Hdi -Hdj
-    >(plus_minus_m_m (d+e) j) in Hide; // -Hjde /3 width=4/
-  ]
-| #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/
-  -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/
+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 #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 … 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
+  /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,d,e. l_appendable_sn … (cpy d e G).
+#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
+@(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 ≤ yinj i & i < d + e &
+                                   ⇩[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 ≤ yinj i & i < d + e &
+                                ⇩[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 &
+                              ⇩[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}V1⦄ ⊢ 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
+| #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-.
-
-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/
+                              ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶×[⫯d, e] 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 →
+                        ∀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
-  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/
+| #I #G #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct
+  elim (ylt_yle_false … Hdi) -Hdi //
+| /3 width=1 by eq_f2/
+| /3 width=1 by eq_f2/
 ]
-qed.
+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-.
 
-lemma cpy_inv_refl_O2: ∀L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▶×[d, 0] T2 → T1 = T2.
-/2 width=6/ qed-.
+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_up … HU12 … HTU1) -HU12 -HTU1
+/2 width=4 by cpy_inv_refl_O2/
+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 +303,7 @@ 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
-*)