X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frelocation%2Fcpy_lift.ma;h=aa0b2416a01d2ef5a1ef6f1025a30e681c25f2de;hb=944b1f7b762774a6f8d99a2c2846f865b6788712;hp=161b21b01d86092d0758e4b37d595e280830219d;hpb=6907a0f66a3782ce4273a609203c9b574841c7d1;p=helm.git

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 161b21b01..aa0b2416a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_lift.ma
@@ -15,240 +15,235 @@
 include "basic_2/relocation/ldrop_ldrop.ma".
 include "basic_2/relocation/cpy.ma".
 
-(* CONTEXT-SENSITIVE EXTENDED PARALLEL SUBSTITUTION FOR TERMS ***************)
+(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
 
-(* Relocation properties ****************************************************)
+(* Properties on relocation *************************************************)
 
-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 →
+(* 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.
+                   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 #_
+[ #I #G #K #dt #et #L #U1 #U2 #s #d #e #_ #H1 #H2 #_
   >(lift_mono … H1 … H2) -H1 -H2 //
-| #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
+| #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
   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
+| #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #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
-  /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
+  /4 width=7 by cpy_bind, ldrop_skip, yle_succ/
+| #G #I #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #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 by cpy_flat/
+  /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,d,e. ⇩[d, e] L ≡ K →
+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.
+                   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 #_ #_
+[ #I #G #K #dt #et #L #U1 #U2 #s #d #e #_ #H1 #H2 #_ #_
   >(lift_mono … H1 … H2) -H1 -H2 //
-| #I #G #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 #s #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
+    lapply (ylt_yle_trans … (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 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
+    elim (yle_inv_inj2 … Hdtd) -Hdtd #dtt #Hdtd #H destruct
     lapply (transitive_le … Hdtd Hid) -Hdtd #Hdti
     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/
+    lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid
+    /4 width=5 by cpy_subst, ldrop_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 #d #e #HLK #H1 #H2 #Hdtd #Hddet
+| #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #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
-  /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
+  /4 width=7 by cpy_bind, ldrop_skip, yle_succ/
+| #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #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 by cpy_flat/
+  /3 width=7 by cpy_flat/
 ]
 qed-.
 
-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 →
+(* 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.
+                   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 #_
+[ #I #G #K #dt #et #L #U1 #U2 #s #d #e #_ #H1 #H2 #_
   >(lift_mono … H1 … H2) -H1 -H2 //
-| #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
+| #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
   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
+  lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hddi
+  /3 width=5 by cpy_subst, ldrop_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
   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=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
+  /4 width=6 by cpy_bind, ldrop_skip, yle_succ/
+| #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #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 by cpy_flat/
+  /3 width=6 by cpy_flat/
 ]
 qed-.
 
-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 →
+(* 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.
+                        ∃∃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/
+[ * #i #G #L #dt #et #K #s #d #e #_ #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/
+  | 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 #d #e #HLK #T1 #H #Hdetd
-  lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
+| #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
-  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/
-| #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 (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/
+| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
+  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=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 //
+  /3 width=6 by cpy_bind, yle_succ, ldrop_skip, lift_bind, ex2_intro/
+| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
+  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/
 ]
 qed-.
 
-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. ⦃G, K⦄ ⊢ T1 ▶×[dt, et - e] T2 & ⇧[d, e] T2 ≡ U2.
+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
-[ * #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/
+[ * #i #G #L #dt #et #K #s #d #e #_ #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/
+  | 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 #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 by le_plus_to_minus_r/ ] -Hdedet #Hidete
+| #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_plus_to_minus_inj2/ ] -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 by le_plus_to_minus_r/ -Hdtd #Hdtie
-    elim (le_inv_plus_l … Hid) #Hdie #Hei
+    elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hid
+    /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
     lapply (ldrop_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
     #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/
+    @(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 #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
+| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdtd #Hdedet
+  elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+  elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
   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 //
+  /3 width=6 by cpy_bind, ldrop_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
+  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/
 ]
 qed-.
 
-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. ⦃G, K⦄ ⊢ T1 ▶×[dt - e, et] T2 & ⇧[d, e] T2 ≡ U2.
+(* 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
-[ * #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/
+[ * #i #G #L #dt #et #K #s #d #e #_ #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/
+  | 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 #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 [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 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 //
+| #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
+  lapply (lift_inv_lref2_ge  … H ?) -H /2 width=1 by yle_inv_inj/ #H destruct
+  lapply (ldrop_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
+  #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
+  elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+  elim (yle_inv_plus_inj2 … Hdetd) #_ #Hedt
+  elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
+  elim (IHU12 … HTU1) -U1 [4: @ldrop_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
+  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/
 ]
 qed-.
 
-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
-[ //
-| #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)
-  ]
-| #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 //
-| #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-.
+(* Advancd inversion lemmas on relocation ***********************************)
 
-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. ⦃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
+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 // [ >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/
+lapply (cpy_weak … HU1 d e ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hddt -Hdtde #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,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
-                           dt ≤ d → dt + et ≤ d + e →
-                           ∃∃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
+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
 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,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
-                           dt ≤ d → d ≤ dt + et → dt + et ≤ d + e →
-                           ∃∃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
+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
-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/
+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
+lapply (cpy_inv_lift1_eq … HTU … HU2) -L #H destruct /2 width=3 by ex2_intro/
 qed-.