X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fetc%2Flsuby%2Flsuby.etc;h=e2ee7c7c2605247be0b0a4d2b5c2ee61a5b4ea73;hb=c879284b576409cec07e96c1f08510d9d9ac14f3;hp=6c90b202b9a44388297b107b64f8a35f255eb691;hpb=8a5a354c9ac3ef20ca01dbeb61f6b99902f172a7;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lsuby.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lsuby.etc
index 6c90b202b..e2ee7c7c2 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lsuby.etc
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lsuby.etc
@@ -12,226 +12,225 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "ground_2/ynat/ynat_plus.ma".
-include "basic_2/notation/relations/extlrsubeq_4.ma".
-include "basic_2/relocation/ldrop.ma".
+include "basic_2/notation/relations/lrsubeq_4.ma".
+include "basic_2/substitution/drop.ma".
 
 (* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
 
 inductive lsuby: relation4 ynat ynat lenv lenv ≝
-| lsuby_atom: ∀L,d,e. lsuby d e L (⋆)
+| lsuby_atom: ∀L,l,m. lsuby l m 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) (L1.ⓑ{I1}V) (L2.ⓑ{I2}V)
-| lsuby_succ: ∀I1,I2,L1,L2,V1,V2,d,e.
-              lsuby d e L1 L2 → lsuby (⫯d) e (L1. ⓑ{I1}V1) (L2. ⓑ{I2} V2)
+| lsuby_pair: ∀I1,I2,L1,L2,V,m. lsuby 0 m L1 L2 →
+              lsuby 0 (⫯m) (L1.ⓑ{I1}V) (L2.ⓑ{I2}V)
+| lsuby_succ: ∀I1,I2,L1,L2,V1,V2,l,m.
+              lsuby l m L1 L2 → lsuby (⫯l) m (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
 .
 
 interpretation
   "local environment refinement (extended substitution)"
-  'ExtLRSubEq L1 d e L2 = (lsuby d e L1 L2).
+  'LRSubEq L1 l m L2 = (lsuby l m L1 L2).
 
 (* Basic properties *********************************************************)
 
-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 <(ylt_inv_O1 … He) /2 width=1 by lsuby_pair/
+lemma lsuby_pair_lt: ∀I1,I2,L1,L2,V,m. L1 ⊆[0, ⫰m] L2 → 0 < m →
+                     L1.ⓑ{I1}V ⊆[0, m] L2.ⓑ{I2}V.
+#I1 #I2 #L1 #L2 #V #m #HL12 #Hm <(ylt_inv_O1 … Hm) /2 width=1 by lsuby_pair/
 qed.
 
-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 <(ylt_inv_O1 … Hd) /2 width=1 by lsuby_succ/
+lemma lsuby_succ_lt: ∀I1,I2,L1,L2,V1,V2,l,m. L1 ⊆[⫰l, m] L2 → 0 < l →
+                     L1.ⓑ{I1}V1 ⊆[l, m] L2. ⓑ{I2}V2.
+#I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #Hl <(ylt_inv_O1 … Hl) /2 width=1 by lsuby_succ/
 qed.
 
-lemma lsuby_pair_O_Y: ∀L1,L2. L1 ⊑×[0, ∞] L2 →
-                      ∀I1,I2,V. L1.ⓑ{I1}V ⊑×[0,∞] L2.ⓑ{I2}V.
+lemma lsuby_pair_O_Y: ∀L1,L2. L1 ⊆[0, ∞] L2 →
+                      ∀I1,I2,V. L1.ⓑ{I1}V ⊆[0,∞] L2.ⓑ{I2}V.
 #L1 #L2 #HL12 #I1 #I2 #V lapply (lsuby_pair I1 I2 … V … HL12) -HL12 //
 qed.
 
-lemma lsuby_refl: ∀L,d,e. L ⊑×[d, e] L.
+lemma lsuby_refl: ∀L,l,m. L ⊆[l, m] L.
 #L elim L -L //
-#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/
+#L #I #V #IHL #l elim (ynat_cases … l) [| * #x ]
+#Hl destruct /2 width=1 by lsuby_succ/
+#m elim (ynat_cases … m) [| * #x ]
+#Hm destruct /2 width=1 by lsuby_zero, lsuby_pair/
 qed.
 
-lemma lsuby_O2: ∀L2,L1,d. |L2| ≤ |L1| → L1 ⊑×[d, yinj 0] L2.
-#L2 elim L2 -L2 // #L2 #I2 #V2 #IHL2 * normalize
-[ #d #H lapply (le_n_O_to_eq … H) -H <plus_n_Sm #H destruct
-| #L1 #I1 #V1 #d #H lapply (le_plus_to_le_r … H) -H #HL12
- elim (ynat_cases d) /3 width=1 by lsuby_zero/
+lemma lsuby_O2: ∀L2,L1,l. |L2| ≤ |L1| → L1 ⊆[l, 0] L2.
+#L2 elim L2 -L2 // #L2 #I2 #V2 #IHL2 *
+[ #l #H elim (ylt_yle_false … H) -H //
+| #L1 #I1 #V1 #l
+ #H lapply (yle_inv_succ … H) -H #HL12
+ elim (ynat_cases l) /3 width=1 by lsuby_zero/
  * /3 width=1 by lsuby_succ/
 ]
 qed.
 
-lemma lsuby_sym: ∀d,e,L1,L2. L1 ⊑×[d, e] L2 → |L1| = |L2| → L2 ⊑×[d, e] L1.
-#d #e #L1 #L2 #H elim H -d -e -L1 -L2
-[ #L1 #d #e #H >(length_inv_zero_dx … H) -L1 //
+lemma lsuby_sym: ∀l,m,L1,L2. L1 ⊆[l, m] L2 → |L1| = |L2| → L2 ⊆[l, m] L1.
+#l #m #L1 #L2 #H elim H -l -m -L1 -L2
+[ #L1 #l #m #H >(length_inv_zero_dx … H) -L1 //
 | /2 width=1 by lsuby_O2/
-| #I1 #I2 #L1 #L2 #V #e #_ #IHL12 #H lapply (injective_plus_l … H)
+| #I1 #I2 #L1 #L2 #V #m #_ #IHL12 #H lapply (ysucc_inv_inj … H) -H
   /3 width=1 by lsuby_pair/
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #H lapply (injective_plus_l … H)
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #H lapply (ysucc_inv_inj … H) -H
   /3 width=1 by lsuby_succ/
 ]
 qed-.
 
 (* Basic inversion lemmas ***************************************************)
 
-fact lsuby_inv_atom1_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 → L1 = ⋆ → L2 = ⋆.
-#L1 #L2 #d #e * -L1 -L2 -d -e //
+fact lsuby_inv_atom1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → L1 = ⋆ → L2 = ⋆.
+#L1 #L2 #l #m * -L1 -L2 -l -m //
 [ #I1 #I2 #L1 #L2 #V1 #V2 #_ #H destruct
-| #I1 #I2 #L1 #L2 #V #e #_ #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #H destruct
+| #I1 #I2 #L1 #L2 #V #m #_ #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #H destruct
 ]
 qed-.
 
-lemma lsuby_inv_atom1: ∀L2,d,e. ⋆ ⊑×[d, e] L2 → L2 = ⋆.
+lemma lsuby_inv_atom1: ∀L2,l,m. ⋆ ⊆[l, m] L2 → L2 = ⋆.
 /2 width=5 by lsuby_inv_atom1_aux/ qed-.
 
-fact lsuby_inv_zero1_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
-                          ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → d = 0 → e = 0 →
+fact lsuby_inv_zero1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+                          ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → l = 0 → m = 0 →
                           L2 = ⋆ ∨
-                          ∃∃J2,K2,W2. K1 ⊑×[0, 0] K2 & L2 = K2.ⓑ{J2}W2.
-#L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
+                          ∃∃J2,K2,W2. K1 ⊆[0, 0] K2 & L2 = K2.ⓑ{J2}W2.
+#L1 #L2 #l #m * -L1 -L2 -l -m /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 #_ #_ #H
+| #I1 #I2 #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #_ #H
   elim (ysucc_inv_O_dx … H)
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W1 #_ #H
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W1 #_ #H
   elim (ysucc_inv_O_dx … H)
 ]
 qed-.
 
-lemma lsuby_inv_zero1: ∀I1,K1,L2,V1. K1.ⓑ{I1}V1 ⊑×[0, 0] L2 →
+lemma lsuby_inv_zero1: ∀I1,K1,L2,V1. K1.ⓑ{I1}V1 ⊆[0, 0] L2 →
                        L2 = ⋆ ∨
-                       ∃∃I2,K2,V2. K1 ⊑×[0, 0] K2 & L2 = K2.ⓑ{I2}V2.
+                       ∃∃I2,K2,V2. K1 ⊆[0, 0] K2 & L2 = K2.ⓑ{I2}V2.
 /2 width=9 by lsuby_inv_zero1_aux/ qed-.
 
-fact lsuby_inv_pair1_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
-                          ∀J1,K1,W. L1 = K1.ⓑ{J1}W → d = 0 → 0 < e →
+fact lsuby_inv_pair1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+                          ∀J1,K1,W. L1 = K1.ⓑ{J1}W → l = 0 → 0 < m →
                           L2 = ⋆ ∨
-                          ∃∃J2,K2. K1 ⊑×[0, ⫰e] K2 & L2 = K2.ⓑ{J2}W.
-#L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
+                          ∃∃J2,K2. K1 ⊆[0, ⫰m] K2 & L2 = K2.ⓑ{J2}W.
+#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/
 [ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W #_ #_ #H
   elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V #e #HL12 #J1 #K1 #W #H #_ #_ destruct
+| #I1 #I2 #L1 #L2 #V #m #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 #_ #H
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #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 →
+lemma lsuby_inv_pair1: ∀I1,K1,L2,V,m. K1.ⓑ{I1}V ⊆[0, m] L2 → 0 < m →
                        L2 = ⋆ ∨
-                       ∃∃I2,K2. K1 ⊑×[0, ⫰e] K2 & L2 = K2.ⓑ{I2}V.
+                       ∃∃I2,K2. K1 ⊆[0, ⫰m] 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 →
+fact lsuby_inv_succ1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+                          ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < l →
                           L2 = ⋆ ∨
-                          ∃∃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/
+                          ∃∃J2,K2,W2. K1 ⊆[⫰l, m] K2 & L2 = K2.ⓑ{J2}W2.
+#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/
 [ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
   elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #H
+| #I1 #I2 #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #H
   elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #J1 #K1 #W1 #H #_ destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J1 #K1 #W1 #H #_ destruct
   /3 width=5 by ex2_3_intro, or_intror/
 ]
 qed-.
 
-lemma lsuby_inv_succ1: ∀I1,K1,L2,V1,d,e. K1.ⓑ{I1}V1 ⊑×[d, e] L2 → 0 < d →
+lemma lsuby_inv_succ1: ∀I1,K1,L2,V1,l,m. K1.ⓑ{I1}V1 ⊆[l, m] L2 → 0 < l →
                        L2 = ⋆ ∨
-                       ∃∃I2,K2,V2. K1 ⊑×[⫰d, e] K2 & L2 = K2.ⓑ{I2}V2.
+                       ∃∃I2,K2,V2. K1 ⊆[⫰l, m] 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 →
-                          ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → d = 0 → e = 0 →
-                          ∃∃J1,K1,W1. K1 ⊑×[0, 0] K2 & L1 = K1.ⓑ{J1}W1.
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ #L1 #d #e #J2 #K2 #W1 #H destruct
+fact lsuby_inv_zero2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+                          ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → l = 0 → m = 0 →
+                          ∃∃J1,K1,W1. K1 ⊆[0, 0] K2 & L1 = K1.ⓑ{J1}W1.
+#L1 #L2 #l #m * -L1 -L2 -l -m
+[ #L1 #l #m #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 #_ #_ #H
+| #I1 #I2 #L1 #L2 #V #m #_ #J2 #K2 #W2 #_ #_ #H
   elim (ysucc_inv_O_dx … H)
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W2 #_ #H
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J2 #K2 #W2 #_ #H
   elim (ysucc_inv_O_dx … H)
 ]
 qed-.
 
-lemma lsuby_inv_zero2: ∀I2,K2,L1,V2. L1 ⊑×[0, 0] K2.ⓑ{I2}V2 →
-                       ∃∃I1,K1,V1. K1 ⊑×[0, 0] K2 & L1 = K1.ⓑ{I1}V1.
+lemma lsuby_inv_zero2: ∀I2,K2,L1,V2. L1 ⊆[0, 0] K2.ⓑ{I2}V2 →
+                       ∃∃I1,K1,V1. K1 ⊆[0, 0] K2 & L1 = K1.ⓑ{I1}V1.
 /2 width=9 by lsuby_inv_zero2_aux/ qed-.
 
-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] K2 & L1 = K1.ⓑ{J1}W.
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ #L1 #d #e #J2 #K2 #W #H destruct
+fact lsuby_inv_pair2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+                          ∀J2,K2,W. L2 = K2.ⓑ{J2}W → l = 0 → 0 < m →
+                          ∃∃J1,K1. K1 ⊆[0, ⫰m] K2 & L1 = K1.ⓑ{J1}W.
+#L1 #L2 #l #m * -L1 -L2 -l -m
+[ #L1 #l #m #J2 #K2 #W #H destruct
 | #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W #_ #_ #H
   elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V #e #HL12 #J2 #K2 #W #H #_ #_ destruct
+| #I1 #I2 #L1 #L2 #V #m #HL12 #J2 #K2 #W #H #_ #_ destruct
   /2 width=4 by ex2_2_intro/
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W #_ #H
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J2 #K2 #W #_ #H
   elim (ysucc_inv_O_dx … H)
 ]
 qed-.
 
-lemma lsuby_inv_pair2: ∀I2,K2,L1,V,e. L1 ⊑×[0, e] K2.ⓑ{I2}V → 0 < e →
-                       ∃∃I1,K1. K1 ⊑×[0, ⫰e] K2 & L1 = K1.ⓑ{I1}V.
+lemma lsuby_inv_pair2: ∀I2,K2,L1,V,m. L1 ⊆[0, m] K2.ⓑ{I2}V → 0 < m →
+                       ∃∃I1,K1. K1 ⊆[0, ⫰m] K2 & L1 = K1.ⓑ{I1}V.
 /2 width=6 by lsuby_inv_pair2_aux/ qed-.
 
-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, e] K2 & L1 = K1.ⓑ{J1}W1.
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ #L1 #d #e #J2 #K2 #W2 #H destruct
+fact lsuby_inv_succ2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+                          ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → 0 < l →
+                          ∃∃J1,K1,W1. K1 ⊆[⫰l, m] K2 & L1 = K1.ⓑ{J1}W1.
+#L1 #L2 #l #m * -L1 -L2 -l -m
+[ #L1 #l #m #J2 #K2 #W2 #H destruct
 | #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W2 #_ #H
   elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V #e #_ #J2 #K1 #W2 #_ #H
+| #I1 #I2 #L1 #L2 #V #m #_ #J2 #K1 #W2 #_ #H
   elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #J2 #K2 #W2 #H #_ destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #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, e] K2 & L1 = K1.ⓑ{I1}V1.
+lemma lsuby_inv_succ2: ∀I2,K2,L1,V2,l,m. L1 ⊆[l, m] K2.ⓑ{I2}V2 → 0 < l →
+                       ∃∃I1,K1,V1. K1 ⊆[⫰l, m] K2 & L1 = K1.ⓑ{I1}V1.
 /2 width=5 by lsuby_inv_succ2_aux/ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-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/
+lemma lsuby_fwd_length: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → |L2| ≤ |L1|.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m /2 width=1 by yle_succ/
 qed-.
 
 (* Properties on basic slicing **********************************************)
 
-lemma lsuby_ldrop_trans_be: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
-                            ∀I2,K2,W,s,i. ⇩[s, 0, i] L2 ≡ K2.ⓑ{I2}W →
-                            d ≤ i → i < d + e →
-                            ∃∃I1,K1. K1 ⊑×[0, ⫰(d+e-i)] K2 & ⇩[s, 0, i] L1 ≡ K1.ⓑ{I1}W.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
-[ #L1 #d #e #J2 #K2 #W #s #i #H
-  elim (ldrop_inv_atom1 … H) -H #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #s #i #_ #_ #H
-  elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V #e #HL12 #IHL12 #J2 #K2 #W #s #i #H #_ >yplus_O1
-  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_drop_lt, ex2_2_intro/
+lemma lsuby_drop_trans_be: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+                           ∀I2,K2,W,s,i. ⬇[s, 0, i] L2 ≡ K2.ⓑ{I2}W →
+                           l ≤ i → ∀m0. i + ⫯m0 = l + m →
+                           ∃∃I1,K1. K1 ⊆[0, m0] K2 & ⬇[s, 0, i] L1 ≡ K1.ⓑ{I1}W.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m
+[ #L1 #l #m #J2 #K2 #W #s #i #H
+  elim (drop_inv_atom1 … H) -H #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #s #i #_ #_ #m0
+  >yplus_O2 >yplus_succ2 #H elim (ysucc_inv_O_dx … H)
+| #I1 #I2 #L1 #L2 #V #m #HL12 #IHL12 #J2 #K2 #W #s #i #H #_ #m0
+  >yplus_succ2 >yplus_succ2 #H0 lapply (ysucc_inv_inj … H0) -H0
+  elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
+  [ destruct -I2 /2 width=4 by drop_pair, ex2_2_intro/
+  | lapply (ylt_inv_O1 … Hi)
+    #H <H -H <yplus_succ_swap #Him elim (IHL12 … HLK1 … Him) -IHL12 -HLK1 -Him
+    /3 width=4 by drop_drop_lt, ex2_2_intro/
   ]
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #J2 #K2 #W #s #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_drop1_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_drop_lt, ex2_2_intro/
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #J2 #K2 #W #s #i #HLK2 #Hli #m0
+  elim (yle_inv_succ1 … Hli) -Hli #Hli #Hi
+  lapply (drop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O1/ #HLK2
+  >yplus_succ1 >yplus_succ2 #H lapply (ysucc_inv_inj … H) -H
+  <Hi <yplus_succ_swap #H elim (IHL12 … HLK2 … H) -IHL12 -HLK2 -H
+  /3 width=4 by drop_drop, ex2_2_intro/
 ]
 qed-.