commit of the "relocation" component with the new definition of ldrop,

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / relocation / cpy_lift.ma

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 85203f2031091017a97e5073f530aa3d5f5aa25d..38ba768e8c786ba9e527e7312926ad679d815082 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_lift.ma
@@ -20,13 +20,13 @@ include "basic_2/relocation/cpy.ma".
 (* 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 →
+                   ∀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 #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
+| #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
@@ -34,25 +34,25 @@ lemma cpy_lift_le: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶×[dt, et] T2 →
   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, yle_succ/
-| #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 →
+                   ∀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 #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 (ylt_yle_trans … (dt+et+e) … Hidet) // -Hidet #Hidete
@@ -65,68 +65,68 @@ lemma cpy_lift_be: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶×[dt, et] T2 →
     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
-    /4 width=5 by cpy_subst, monotonic_ylt_plus_dx, yle_plus_dx1_trans, yle_inj/
+    /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, yle_succ/
-| #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 →
+                   ∀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 #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
+| #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 // -Hddi
-  /3 width=5 by cpy_subst, monotonic_ylt_plus_dx, monotonic_yle_plus_dx/
-| #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
+  /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, yle_succ/
-| #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-.
 
 (* Inversion lemmas on relocation *******************************************)
 
 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 →
+                        ∀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 #d #e #_ #T1 #H #_
+[ * #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
+| #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 #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+| #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) -W1 // #V2 #HV12 #HVW2
   elim (IHU12 … HTU1) -IHU12 -HTU1
-  /3 width=5 by cpy_bind, yle_succ, ldrop_skip, lift_bind, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+  /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
@@ -135,16 +135,16 @@ 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,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
-                        dt ≤ d → d + e ≤ dt + et →
+                        ∀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 #d #e #_ #T1 #H #_ #_
+[ * #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
+| #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/
@@ -160,12 +160,12 @@ lemma cpy_inv_lift1_be: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U2 
     @(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 #d #e #HLK #X #H #Hdtd #Hdedet
+| #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) -W1 // #V2 #HV12 #HVW2
   elim (IHU12 … HTU1) -U1
-  /3 width=5 by cpy_bind, ldrop_skip, lift_bind, yle_succ, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
+  /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 //
@@ -174,16 +174,16 @@ lemma cpy_inv_lift1_be: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U2 
 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 →
+                        ∀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 #d #e #_ #T1 #H #_
+[ * #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
+| #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
@@ -195,13 +195,13 @@ lemma cpy_inv_lift1_ge: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U2 
   [ /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 #d #e #HLK #X #H #Hdetd
+| #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) -W1 // #V2 #HV12 #HVW2
-  elim (IHU12 … HTU1) -U1 [4: @ldrop_skip // |2: skip |3: /2 width=1 by yle_succ/ ]
+  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 #d #e #HLK #X #H #Hdetd
+| #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/
@@ -211,10 +211,10 @@ 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
+                           ∀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
 lapply (cpy_inv_lift1_eq … HTU1 … HU1) -HU1 #HU1 destruct
@@ -222,20 +222,20 @@ 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 →
+                           ∀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 #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
+#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 →
+                           ∀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 #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde
+#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: >ymax_pre_sn_comm // ] -Hdtd #T #HT1 #HTU