X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fsubstitution%2Fcpy_lift.ma;h=1d42acd6f5a9f24ea9cac3889a72dbcf7487409b;hb=c60524dec7ace912c416a90d6b926bee8553250b;hp=991d4db29badad333a60668aeb46eb1a1a19058b;hpb=f10cfe417b6b8ec1c7ac85c6ecf5fb1b3fdf37db;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy_lift.ma
index 991d4db29..1d42acd6f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy_lift.ma
@@ -20,59 +20,59 @@ include "basic_2/substitution/cpy.ma".
 (* Properties on relocation *************************************************)
 
 (* Basic_1: was: subst1_lift_lt *)
-lemma cpy_lift_le: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶[dt, et] T2 →
-                   ∀L,U1,U2,s,d,e. ⬇[s, d, e] L ≡ K →
-                   ⬆[d, e] T1 ≡ U1 → ⬆[d, e] T2 ≡ U2 →
-                   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 #s #d #e #_ #H1 #H2 #_
+lemma cpy_lift_le: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 →
+                   ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K →
+                   ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 →
+                   lt + mt ≤ l → ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2.
+#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt
+[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_
   >(lift_mono … H1 … H2) -H1 -H2 //
-| #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #s #d #e #HLK #H #HWU2 #Hdetd
-  lapply (ylt_yle_trans … Hdetd … Hidet) -Hdetd #Hid
-  lapply (ylt_inv_inj … Hid) -Hid #Hid
-  lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct
+| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hlmtl
+  lapply (ylt_yle_trans … Hlmtl … Hilmt) -Hlmtl #Hil
+  lapply (ylt_inv_inj … Hil) -Hil #Hil
+  lapply (lift_inv_lref1_lt … H … Hil) -H #H destruct
   elim (lift_trans_ge … HVW … HWU2) -W // <minus_plus #W #HVW #HWU2
   elim (drop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
-  elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #_ #HVY
+  elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #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 #s #d #e #HLK #H1 #H2 #Hdetd
+| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl
   elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
   elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
   /4 width=7 by cpy_bind, drop_skip, yle_succ/
-| #G #I #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdetd
+| #G #I #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl
   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=7 by cpy_flat/
 ]
 qed-.
 
-lemma cpy_lift_be: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶[dt, et] T2 →
-                   ∀L,U1,U2,s,d,e. ⬇[s, d, e] L ≡ K →
-                   ⬆[d, e] T1 ≡ U1 → ⬆[d, e] T2 ≡ U2 →
-                   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 #s #d #e #_ #H1 #H2 #_ #_
+lemma cpy_lift_be: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 →
+                   ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K →
+                   ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 →
+                   lt ≤ l → l ≤ lt + mt → ⦃G, L⦄ ⊢ U1 ▶[lt, mt + m] U2.
+#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt
+[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_ #_
   >(lift_mono … H1 … H2) -H1 -H2 //
-| #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #s #d #e #HLK #H #HWU2 #Hdtd #_
-  elim (lift_inv_lref1 … H) -H * #Hid #H destruct
-  [ -Hdtd
-    lapply (ylt_yle_trans … (dt+et+e) … Hidet) // -Hidet #Hidete
+| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hltl #_
+  elim (lift_inv_lref1 … H) -H * #Hil #H destruct
+  [ -Hltl
+    lapply (ylt_yle_trans … (lt+mt+m) … Hilmt) // -Hilmt #Hilmtm
     elim (lift_trans_ge … HVW … HWU2) -W // <minus_plus #W #HVW #HWU2
     elim (drop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
-    elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #_ #HVY
+    elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K #Y #_ #HVY
     >(lift_mono … HVY … HVW) -V #H destruct /2 width=5 by cpy_subst/
-  | -Hdti
-    elim (yle_inv_inj2 … Hdtd) -Hdtd #dtt #Hdtd #H destruct
-    lapply (transitive_le … Hdtd Hid) -Hdtd #Hdti
+  | -Hlti
+    elim (yle_inv_inj2 … Hltl) -Hltl #ltt #Hltl #H destruct
+    lapply (transitive_le … Hltl Hil) -Hltl #Hlti
     lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
-    lapply (drop_trans_ge_comm … HLK … HKV ?) -K // -Hid
+    lapply (drop_trans_ge_comm … HLK … HKV ?) -K // -Hil
     /4 width=5 by cpy_subst, drop_inv_gen, monotonic_ylt_plus_dx, yle_plus_dx1_trans, yle_inj/
   ]
-| #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdtd #Hddet
+| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hltl #Hllmt
   elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
   elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
   /4 width=7 by cpy_bind, drop_skip, yle_succ/
-| #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdetd
+| #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl
   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=7 by cpy_flat/
@@ -80,25 +80,25 @@ lemma cpy_lift_be: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶[dt, et] T2 →
 qed-.
 
 (* Basic_1: was: subst1_lift_ge *)
-lemma cpy_lift_ge: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶[dt, et] T2 →
-                   ∀L,U1,U2,s,d,e. ⬇[s, d, e] L ≡ K →
-                   ⬆[d, e] T1 ≡ U1 → ⬆[d, e] T2 ≡ U2 →
-                   d ≤ dt → ⦃G, L⦄ ⊢ U1 ▶[dt+e, et] U2.
-#G #K #T1 #T2 #dt #et #H elim H -G -K -T1 -T2 -dt -et
-[ #I #G #K #dt #et #L #U1 #U2 #s #d #e #_ #H1 #H2 #_
+lemma cpy_lift_ge: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 →
+                   ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K →
+                   ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 →
+                   l ≤ lt → ⦃G, L⦄ ⊢ U1 ▶[lt+m, mt] U2.
+#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt
+[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_
   >(lift_mono … H1 … H2) -H1 -H2 //
-| #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #s #d #e #HLK #H #HWU2 #Hddt
-  lapply (yle_trans … Hddt … Hdti) -Hddt #Hid
-  elim (yle_inv_inj2 … Hid) -Hid #dd #Hddi #H0 destruct
-  lapply (lift_inv_lref1_ge … H … Hddi) -H #H destruct
+| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hllt
+  lapply (yle_trans … Hllt … Hlti) -Hllt #Hil
+  elim (yle_inv_inj2 … Hil) -Hil #ll #Hlli #H0 destruct
+  lapply (lift_inv_lref1_ge … H … Hlli) -H #H destruct
   lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
-  lapply (drop_trans_ge_comm … HLK … HKV ?) -K // -Hddi
+  lapply (drop_trans_ge_comm … HLK … HKV ?) -K // -Hlli
   /3 width=5 by cpy_subst, drop_inv_gen, monotonic_ylt_plus_dx, monotonic_yle_plus_dx/
-| #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hddt
+| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hllt
   elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
   elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
   /4 width=6 by cpy_bind, drop_skip, yle_succ/
-| #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hddt
+| #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hllt
   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 by cpy_flat/
@@ -108,28 +108,28 @@ qed-.
 (* Inversion lemmas on relocation *******************************************)
 
 (* Basic_1: was: subst1_gen_lift_lt *)
-lemma cpy_inv_lift1_le: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
-                        ∀K,s,d,e. ⬇[s, d, e] L ≡ K → ∀T1. ⬆[d, e] T1 ≡ U1 →
-                        dt + et ≤ d →
-                        ∃∃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
-[ * #i #G #L #dt #et #K #s #d #e #_ #T1 #H #_
+lemma cpy_inv_lift1_le: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
+                        ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+                        lt + mt ≤ l →
+                        ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
+[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #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 #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #s #d #e #HLK #T1 #H #Hdetd
-  lapply (ylt_yle_trans … Hdetd … Hidet) -Hdetd #Hid
-  lapply (ylt_inv_inj … Hid) -Hid #Hid
-  lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct
+| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hlmtl
+  lapply (ylt_yle_trans … Hlmtl … Hilmt) -Hlmtl #Hil
+  lapply (ylt_inv_inj … Hil) -Hil #Hil
+  lapply (lift_inv_lref2_lt … H … Hil) -H #H destruct
   elim (drop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
-  elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=5 by cpy_subst, ex2_intro/
-| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
+  elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hil /3 width=5 by cpy_subst, ex2_intro/
+| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
   elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
   elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
   elim (IHU12 … HTU1) -IHU12 -HTU1
   /3 width=6 by cpy_bind, yle_succ, drop_skip, lift_bind, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
+| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
   elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
   elim (IHW12 … HLK … HVW1) -W1 //
   elim (IHU12 … HLK … HTU1) -U1 -HLK
@@ -137,38 +137,38 @@ lemma cpy_inv_lift1_le: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
 ]
 qed-.
 
-lemma cpy_inv_lift1_be: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
-                        ∀K,s,d,e. ⬇[s, d, e] L ≡ K → ∀T1. ⬆[d, e] T1 ≡ U1 →
-                        dt ≤ d → yinj d + e ≤ dt + et →
-                        ∃∃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
-[ * #i #G #L #dt #et #K #s #d #e #_ #T1 #H #_ #_
+lemma cpy_inv_lift1_be: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
+                        ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+                        lt ≤ l → yinj l + m ≤ lt + mt →
+                        ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, mt-m] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
+[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #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 #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #s #d #e #HLK #T1 #H #Hdtd #Hdedet
-  lapply (yle_fwd_plus_ge_inj … Hdtd Hdedet) #Heet
-  elim (lift_inv_lref2 … H) -H * #Hid #H destruct [ -Hdtd -Hidet | -Hdti -Hdedet ]
-  [ lapply (ylt_yle_trans i d (dt+(et-e)) ? ?) /2 width=1 by ylt_inj/
-    [ >yplus_minus_assoc_inj /2 width=1 by yle_plus1_to_minus_inj2/ ] -Hdedet #Hidete
+| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hltl #Hlmlmt
+  lapply (yle_fwd_plus_ge_inj … Hltl Hlmlmt) #Hmmt
+  elim (lift_inv_lref2 … H) -H * #Hil #H destruct [ -Hltl -Hilmt | -Hlti -Hlmlmt ]
+  [ lapply (ylt_yle_trans i l (lt+(mt-m)) ? ?) /2 width=1 by ylt_inj/
+    [ >yplus_minus_assoc_inj /2 width=1 by yle_plus1_to_minus_inj2/ ] -Hlmlmt #Hilmtm
     elim (drop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
-    elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hid
+    elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hil
     /3 width=5 by cpy_subst, ex2_intro/
-  | elim (le_inv_plus_l … Hid) #Hdie #Hei
-    lapply (yle_trans … Hdtd (i-e) ?) /2 width=1 by yle_inj/ -Hdtd #Hdtie
+  | elim (le_inv_plus_l … Hil) #Hlim #Hmi
+    lapply (yle_trans … Hltl (i-m) ?) /2 width=1 by yle_inj/ -Hltl #Hltim
     lapply (drop_conf_ge … HLK … HLKV ?) -L // #HKV
-    elim (lift_split … HVW d (i-e+1)) -HVW [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hid -Hdie
+    elim (lift_split … HVW l (i-m+1)) -HVW [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hil -Hlim
     #V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H
     @(ex2_intro … H) @(cpy_subst … HKV HV1) // (**) (* explicit constructor *)
     >yplus_minus_assoc_inj /3 width=1 by monotonic_ylt_minus_dx, yle_inj/
   ]
-| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdtd #Hdedet
+| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hltl #Hlmlmt
   elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
   elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
   elim (IHU12 … HTU1) -U1
   /3 width=6 by cpy_bind, drop_skip, lift_bind, yle_succ, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdtd #Hdedet
+| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hltl #Hlmlmt
   elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
   elim (IHW12 … HLK … HVW1) -W1 //
   elim (IHU12 … HLK … HTU1) -U1 -HLK //
@@ -177,35 +177,35 @@ lemma cpy_inv_lift1_be: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
 qed-.
 
 (* Basic_1: was: subst1_gen_lift_ge *)
-lemma cpy_inv_lift1_ge: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
-                        ∀K,s,d,e. ⬇[s, d, e] L ≡ K → ∀T1. ⬆[d, e] T1 ≡ U1 →
-                        yinj d + e ≤ dt →
-                        ∃∃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
-[ * #i #G #L #dt #et #K #s #d #e #_ #T1 #H #_
+lemma cpy_inv_lift1_ge: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
+                        ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+                        yinj l + m ≤ lt →
+                        ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt-m, mt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
+[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #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 #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #s #d #e #HLK #T1 #H #Hdedt
-  lapply (yle_trans … Hdedt … Hdti) #Hdei
-  elim (yle_inv_plus_inj2 … Hdedt) -Hdedt #_ #Hedt
-  elim (yle_inv_plus_inj2 … Hdei) #Hdie #Hei
+| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hlmlt
+  lapply (yle_trans … Hlmlt … Hlti) #Hlmi
+  elim (yle_inv_plus_inj2 … Hlmlt) -Hlmlt #_ #Hmlt
+  elim (yle_inv_plus_inj2 … Hlmi) #Hlim #Hmi
   lapply (lift_inv_lref2_ge  … H ?) -H /2 width=1 by yle_inv_inj/ #H destruct
   lapply (drop_conf_ge … HLK … HLKV ?) -L /2 width=1 by yle_inv_inj/ #HKV
-  elim (lift_split … HVW d (i-e+1)) -HVW [2,3,4: /3 width=1 by yle_inv_inj, le_S_S, le_S/ ] -Hdei -Hdie
+  elim (lift_split … HVW l (i-m+1)) -HVW [2,3,4: /3 width=1 by yle_inv_inj, le_S_S, le_S/ ] -Hlmi -Hlim
   #V0 #HV10 >plus_minus /2 width=1 by yle_inv_inj/ <minus_minus /3 width=1 by yle_inv_inj, le_S/ <minus_n_n <plus_n_O #H
   @(ex2_intro … H) @(cpy_subst … HKV HV10) (**) (* explicit constructor *)
   [ /2 width=1 by monotonic_yle_minus_dx/
   | <yplus_minus_comm_inj /2 width=1 by monotonic_ylt_minus_dx/
   ]
-| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
+| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
   elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
-  elim (yle_inv_plus_inj2 … Hdetd) #_ #Hedt
+  elim (yle_inv_plus_inj2 … Hlmtl) #_ #Hmlt
   elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
   elim (IHU12 … HTU1) -U1 [4: @drop_skip // |2,5: skip |3: /2 width=1 by yle_succ/ ]
   >yminus_succ1_inj /3 width=5 by cpy_bind, lift_bind, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
+| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
   elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
   elim (IHW12 … HLK … HVW1) -W1 //
   elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5 by cpy_flat, lift_flat, ex2_intro/
@@ -214,36 +214,36 @@ qed-.
 
 (* Advanced inversion lemmas on relocation ***********************************)
 
-lemma cpy_inv_lift1_ge_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
-                           ∀K,s,d,e. ⬇[s, d, e] L ≡ K → ∀T1. ⬆[d, e] T1 ≡ U1 →
-                           d ≤ dt → dt ≤ yinj d + e → yinj d + e ≤ dt + et →
-                           ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[d, dt + et - (yinj d + e)] T2 & ⬆[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #HU12 #K #s #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 // [ >ymax_pre_sn_comm // ] -Hddt -Hdtde #HU1
+lemma cpy_inv_lift1_ge_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
+                           ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+                           l ≤ lt → lt ≤ yinj l + m → yinj l + m ≤ lt + mt →
+                           ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[l, lt + mt - (yinj l + m)] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt
+elim (cpy_split_up … HU12 (l + m)) -HU12 // -Hlmlmt #U #HU1 #HU2
+lapply (cpy_weak … HU1 l m ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hllt -Hltlm #HU1
 lapply (cpy_inv_lift1_eq … HTU1 … HU1) -HU1 #HU1 destruct
 elim (cpy_inv_lift1_ge … HU2 … HLK … HTU1) -U -L /2 width=3 by ex2_intro/
 qed-.
 
-lemma cpy_inv_lift1_be_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
-                           ∀K,s,d,e. ⬇[s, d, e] L ≡ K → ∀T1. ⬆[d, e] T1 ≡ U1 →
-                           dt ≤ d → dt + et ≤ yinj d + e →
-                           ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[dt, d-dt] T2 & ⬆[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #HU12 #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
-lapply (cpy_weak … HU12 dt (d+e-dt) ? ?) -HU12 //
-[ >ymax_pre_sn_comm /2 width=1 by yle_plus_dx1_trans/ ] -Hdetde #HU12
+lemma cpy_inv_lift1_be_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
+                           ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+                           lt ≤ l → lt + mt ≤ yinj l + m →
+                           ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, l-lt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmtlm
+lapply (cpy_weak … HU12 lt (l+m-lt) ? ?) -HU12 //
+[ >ymax_pre_sn_comm /2 width=1 by yle_plus_dx1_trans/ ] -Hlmtlm #HU12
 elim (cpy_inv_lift1_be … HU12 … HLK … HTU1) -U1 -L /2 width=3 by ex2_intro/
 qed-.
 
-lemma cpy_inv_lift1_le_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
-                           ∀K,s,d,e. ⬇[s, d, e] L ≡ K → ∀T1. ⬆[d, e] T1 ≡ U1 →
-                           dt ≤ d → d ≤ dt + et → dt + et ≤ yinj d + e →
-                           ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[dt, d - dt] T2 & ⬆[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #HU12 #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde
-elim (cpy_split_up … HU12 d) -HU12 // #U #HU1 #HU2
+lemma cpy_inv_lift1_le_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
+                           ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+                           lt ≤ l → l ≤ lt + mt → lt + mt ≤ yinj l + m →
+                           ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, l - lt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hllmt #Hlmtlm
+elim (cpy_split_up … HU12 l) -HU12 // #U #HU1 #HU2
 elim (cpy_inv_lift1_le … HU1 … HLK … HTU1) -U1
-[2: >ymax_pre_sn_comm // ] -Hdtd #T #HT1 #HTU
-lapply (cpy_weak … HU2 d e ? ?) -HU2 //
-[ >ymax_pre_sn_comm // ] -Hddet -Hdetde #HU2
+[2: >ymax_pre_sn_comm // ] -Hltl #T #HT1 #HTU
+lapply (cpy_weak … HU2 l m ? ?) -HU2 //
+[ >ymax_pre_sn_comm // ] -Hllmt -Hlmtlm #HU2
 lapply (cpy_inv_lift1_eq … HTU … HU2) -L #H destruct /2 width=3 by ex2_intro/
 qed-.