X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fsubstitution%2Flift_lift.ma;h=9ab57806f2c2ddeab999d46f15951286da8c096c;hb=c60524dec7ace912c416a90d6b926bee8553250b;hp=06452afe4b69cbfdb2279c9d721c7c8031c3bfa1;hpb=f10cfe417b6b8ec1c7ac85c6ecf5fb1b3fdf37db;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_lift.ma
index 06452afe4..9ab57806f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_lift.ma
@@ -19,50 +19,50 @@ include "basic_2/substitution/lift.ma".
 (* Main properties ***********************************************************)
 
 (* Basic_1: was: lift_inj *)
-theorem lift_inj: ∀d,e,T1,U. ⬆[d,e] T1 ≡ U → ∀T2. ⬆[d,e] T2 ≡ U → T1 = T2.
-#d #e #T1 #U #H elim H -d -e -T1 -U
-[ #k #d #e #X #HX
+theorem lift_inj: ∀l,m,T1,U. ⬆[l,m] T1 ≡ U → ∀T2. ⬆[l,m] T2 ≡ U → T1 = T2.
+#l #m #T1 #U #H elim H -l -m -T1 -U
+[ #k #l #m #X #HX
   lapply (lift_inv_sort2 … HX) -HX //
-| #i #d #e #Hid #X #HX
+| #i #l #m #Hil #X #HX
   lapply (lift_inv_lref2_lt … HX ?) -HX //
-| #i #d #e #Hdi #X #HX
+| #i #l #m #Hli #X #HX
   lapply (lift_inv_lref2_ge … HX ?) -HX /2 width=1 by monotonic_le_plus_l/
-| #p #d #e #X #HX
+| #p #l #m #X #HX
   lapply (lift_inv_gref2 … HX) -HX //
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
   elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
   elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
 ]
 qed-.
 
 (* Basic_1: was: lift_gen_lift *)
-theorem lift_div_le: ∀d1,e1,T1,T. ⬆[d1, e1] T1 ≡ T →
-                     ∀d2,e2,T2. ⬆[d2 + e1, e2] T2 ≡ T →
-                     d1 ≤ d2 →
-                     ∃∃T0. ⬆[d1, e1] T0 ≡ T2 & ⬆[d2, e2] T0 ≡ T1.
-#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
-[ #k #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
+theorem lift_div_le: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
+                     ∀l2,m2,T2. ⬆[l2 + m1, m2] T2 ≡ T →
+                     l1 ≤ l2 →
+                     ∃∃T0. ⬆[l1, m1] T0 ≡ T2 & ⬆[l2, m2] T0 ≡ T1.
+#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
+[ #k #l1 #m1 #l2 #m2 #T2 #Hk #Hl12
   lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct /3 width=3 by lift_sort, ex2_intro/
-| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
-  lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
+| #i #l1 #m1 #Hil1 #l2 #m2 #T2 #Hi #Hl12
+  lapply (lt_to_le_to_lt … Hil1 Hl12) -Hl12 #Hil2
   lapply (lift_inv_lref2_lt … Hi ?) -Hi /3 width=3 by lift_lref_lt, lt_plus_to_minus_r, lt_to_le_to_lt, ex2_intro/
-| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
-  elim (lift_inv_lref2 … Hi) -Hi * #Hid2 #H destruct
-  [ -Hd12 lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3 width=3 by lift_lref_lt, lift_lref_ge, ex2_intro/
-  | -Hid1 >plus_plus_comm_23 in Hid2; #H lapply (le_plus_to_le_r … H) -H #H
-    elim (le_inv_plus_l … H) -H #Hide2 #He2i
-    lapply (transitive_le … Hd12 Hide2) -Hd12 #Hd12
-    >le_plus_minus_comm // >(plus_minus_m_m i e2) in ⊢ (? ? ? %);
+| #i #l1 #m1 #Hil1 #l2 #m2 #T2 #Hi #Hl12
+  elim (lift_inv_lref2 … Hi) -Hi * #Hil2 #H destruct
+  [ -Hl12 lapply (lt_plus_to_lt_l … Hil2) -Hil2 #Hil2 /3 width=3 by lift_lref_lt, lift_lref_ge, ex2_intro/
+  | -Hil1 >plus_plus_comm_23 in Hil2; #H lapply (le_plus_to_le_r … H) -H #H
+    elim (le_inv_plus_l … H) -H #Hilm2 #Hm2i
+    lapply (transitive_le … Hl12 Hilm2) -Hl12 #Hl12
+    >le_plus_minus_comm // >(plus_minus_m_m i m2) in ⊢ (? ? ? %);
     /4 width=3 by lift_lref_ge, ex2_intro/
   ]
-| #p #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
+| #p #l1 #m1 #l2 #m2 #T2 #Hk #Hl12
   lapply (lift_inv_gref2 … Hk) -Hk #Hk destruct /3 width=3 by lift_gref, ex2_intro/
-| #a #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
+| #a #I #W1 #W #U1 #U #l1 #m1 #_ #_ #IHW #IHU #l2 #m2 #T2 #H #Hl12
   lapply (lift_inv_bind2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct
   elim (IHW … HW2) // -IHW -HW2 #W0 #HW2 #HW1
   >plus_plus_comm_23 in HU2; #HU2 elim (IHU … HU2) /3 width=5 by lift_bind, le_S_S, ex2_intro/
-| #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
+| #I #W1 #W #U1 #U #l1 #m1 #_ #_ #IHW #IHU #l2 #m2 #T2 #H #Hl12
   lapply (lift_inv_flat2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct
   elim (IHW … HW2) // -IHW -HW2 #W0 #HW2 #HW1
   elim (IHU … HU2) /3 width=5 by lift_flat, ex2_intro/
@@ -70,74 +70,74 @@ theorem lift_div_le: ∀d1,e1,T1,T. ⬆[d1, e1] T1 ≡ T →
 qed.
 
 (* Note: apparently this was missing in basic_1 *)
-theorem lift_div_be: ∀d1,e1,T1,T. ⬆[d1, e1] T1 ≡ T →
-                     ∀e,e2,T2. ⬆[d1 + e, e2] T2 ≡ T →
-                     e ≤ e1 → e1 ≤ e + e2 →
-                     ∃∃T0. ⬆[d1, e] T0 ≡ T2 & ⬆[d1, e + e2 - e1] T0 ≡ T1.
-#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
-[ #k #d1 #e1 #e #e2 #T2 #H >(lift_inv_sort2 … H) -H /2 width=3 by lift_sort, ex2_intro/
-| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2
+theorem lift_div_be: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
+                     ∀m,m2,T2. ⬆[l1 + m, m2] T2 ≡ T →
+                     m ≤ m1 → m1 ≤ m + m2 →
+                     ∃∃T0. ⬆[l1, m] T0 ≡ T2 & ⬆[l1, m + m2 - m1] T0 ≡ T1.
+#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
+[ #k #l1 #m1 #m #m2 #T2 #H >(lift_inv_sort2 … H) -H /2 width=3 by lift_sort, ex2_intro/
+| #i #l1 #m1 #Hil1 #m #m2 #T2 #H #Hm1 #Hm1m2
   >(lift_inv_lref2_lt … H) -H /3 width=3 by lift_lref_lt, lt_plus_to_minus_r, lt_to_le_to_lt, ex2_intro/
-| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2
-  elim (lt_or_ge (i+e1) (d1+e+e2)) #Hie1d1e2
+| #i #l1 #m1 #Hil1 #m #m2 #T2 #H #Hm1 #Hm1m2
+  elim (lt_or_ge (i+m1) (l1+m+m2)) #Him1l1m2
   [ elim (lift_inv_lref2_be … H) -H /2 width=1 by le_plus/
   | >(lift_inv_lref2_ge … H ?) -H //
-    lapply (le_plus_to_minus … Hie1d1e2) #Hd1e21i
-    elim (le_inv_plus_l … Hie1d1e2) -Hie1d1e2 #Hd1e12 #He2ie1
-    @ex2_intro [2: /2 width=1/ | skip ] -Hd1e12
+    lapply (le_plus_to_minus … Him1l1m2) #Hl1m21i
+    elim (le_inv_plus_l … Him1l1m2) -Him1l1m2 #Hl1m12 #Hm2im1
+    @ex2_intro [2: /2 width=1 by lift_lref_ge_minus/ | skip ] -Hl1m12
     @lift_lref_ge_minus_eq [ >plus_minus_associative // | /2 width=1 by minus_le_minus_minus_comm/ ]
   ]
-| #p #d1 #e1 #e #e2 #T2 #H >(lift_inv_gref2 … H) -H /2 width=3 by lift_gref, ex2_intro/
-| #a #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2
+| #p #l1 #m1 #m #m2 #T2 #H >(lift_inv_gref2 … H) -H /2 width=3 by lift_gref, ex2_intro/
+| #a #I #V1 #V #T1 #T #l1 #m1 #_ #_ #IHV1 #IHT1 #m #m2 #X #H #Hm1 #Hm1m2
   elim (lift_inv_bind2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
   elim (IHV1 … HV2) -V // >plus_plus_comm_23 in HT2; #HT2
   elim (IHT1 … HT2) -T /3 width=5 by lift_bind, ex2_intro/
-| #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2
+| #I #V1 #V #T1 #T #l1 #m1 #_ #_ #IHV1 #IHT1 #m #m2 #X #H #Hm1 #Hm1m2
   elim (lift_inv_flat2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
   elim (IHV1 … HV2) -V //
   elim (IHT1 … HT2) -T /3 width=5 by lift_flat, ex2_intro/
 ]
 qed.
 
-theorem lift_mono: ∀d,e,T,U1. ⬆[d,e] T ≡ U1 → ∀U2. ⬆[d,e] T ≡ U2 → U1 = U2.
-#d #e #T #U1 #H elim H -d -e -T -U1
-[ #k #d #e #X #HX
+theorem lift_mono: ∀l,m,T,U1. ⬆[l,m] T ≡ U1 → ∀U2. ⬆[l,m] T ≡ U2 → U1 = U2.
+#l #m #T #U1 #H elim H -l -m -T -U1
+[ #k #l #m #X #HX
   lapply (lift_inv_sort1 … HX) -HX //
-| #i #d #e #Hid #X #HX
+| #i #l #m #Hil #X #HX
   lapply (lift_inv_lref1_lt … HX ?) -HX //
-| #i #d #e #Hdi #X #HX
+| #i #l #m #Hli #X #HX
   lapply (lift_inv_lref1_ge … HX ?) -HX //
-| #p #d #e #X #HX
+| #p #l #m #X #HX
   lapply (lift_inv_gref1 … HX) -HX //
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
   elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
   elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
 ]
 qed-.
 
 (* Basic_1: was: lift_free (left to right) *)
-theorem lift_trans_be: ∀d1,e1,T1,T. ⬆[d1, e1] T1 ≡ T →
-                       ∀d2,e2,T2. ⬆[d2, e2] T ≡ T2 →
-                       d1 ≤ d2 → d2 ≤ d1 + e1 → ⬆[d1, e1 + e2] T1 ≡ T2.
-#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
-[ #k #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
+theorem lift_trans_be: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
+                       ∀l2,m2,T2. ⬆[l2, m2] T ≡ T2 →
+                       l1 ≤ l2 → l2 ≤ l1 + m1 → ⬆[l1, m1 + m2] T1 ≡ T2.
+#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
+[ #k #l1 #m1 #l2 #m2 #T2 #HT2 #_ #_
   >(lift_inv_sort1 … HT2) -HT2 //
-| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #Hd12 #_
-  lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
-  lapply (lift_inv_lref1_lt … HT2 Hid2) /2 width=1 by lift_lref_lt/
-| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #_ #Hd21
+| #i #l1 #m1 #Hil1 #l2 #m2 #T2 #HT2 #Hl12 #_
+  lapply (lt_to_le_to_lt … Hil1 Hl12) -Hl12 #Hil2
+  lapply (lift_inv_lref1_lt … HT2 Hil2) /2 width=1 by lift_lref_lt/
+| #i #l1 #m1 #Hil1 #l2 #m2 #T2 #HT2 #_ #Hl21
   lapply (lift_inv_lref1_ge … HT2 ?) -HT2
-  [ @(transitive_le … Hd21 ?) -Hd21 /2 width=1 by monotonic_le_plus_l/
-  | -Hd21 /2 width=1 by lift_lref_ge/
+  [ @(transitive_le … Hl21 ?) -Hl21 /2 width=1 by monotonic_le_plus_l/
+  | -Hl21 /2 width=1 by lift_lref_ge/
   ]
-| #p #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
+| #p #l1 #m1 #l2 #m2 #T2 #HT2 #_ #_
   >(lift_inv_gref1 … HT2) -HT2 //
-| #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
+| #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl12 #Hl21
   elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct 
   lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
   lapply (IHT12 … HT20 ? ?) /2 width=1 by lift_bind, le_S_S/ (**) (* full auto a bit slow *)
-| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
+| #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl12 #Hl21
   elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
   lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
   lapply (IHT12 … HT20 ? ?) /2 width=1 by lift_flat/ (**) (* full auto a bit slow *)
@@ -145,26 +145,26 @@ theorem lift_trans_be: ∀d1,e1,T1,T. ⬆[d1, e1] T1 ≡ T →
 qed.
 
 (* Basic_1: was: lift_d (right to left) *)
-theorem lift_trans_le: ∀d1,e1,T1,T. ⬆[d1, e1] T1 ≡ T →
-                       ∀d2,e2,T2. ⬆[d2, e2] T ≡ T2 → d2 ≤ d1 →
-                       ∃∃T0. ⬆[d2, e2] T1 ≡ T0 & ⬆[d1 + e2, e1] T0 ≡ T2.
-#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
-[ #k #d1 #e1 #d2 #e2 #X #HX #_
+theorem lift_trans_le: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
+                       ∀l2,m2,T2. ⬆[l2, m2] T ≡ T2 → l2 ≤ l1 →
+                       ∃∃T0. ⬆[l2, m2] T1 ≡ T0 & ⬆[l1 + m2, m1] T0 ≡ T2.
+#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
+[ #k #l1 #m1 #l2 #m2 #X #HX #_
   >(lift_inv_sort1 … HX) -HX /2 width=3 by lift_sort, ex2_intro/
-| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
-  lapply (lt_to_le_to_lt … (d1+e2) Hid1 ?) // #Hie2
-  elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct /4 width=3 by lift_lref_ge_minus, lift_lref_lt, lt_minus_to_plus, monotonic_le_plus_l, ex2_intro/
-| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hd21
-  lapply (transitive_le … Hd21 Hid1) -Hd21 #Hid2
+| #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #_
+  lapply (lt_to_le_to_lt … (l1+m2) Hil1 ?) // #Him2
+  elim (lift_inv_lref1 … HX) -HX * #Hil2 #HX destruct /4 width=3 by lift_lref_ge_minus, lift_lref_lt, lt_minus_to_plus, monotonic_le_plus_l, ex2_intro/
+| #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #Hl21
+  lapply (transitive_le … Hl21 Hil1) -Hl21 #Hil2
   lapply (lift_inv_lref1_ge … HX ?) -HX /2 width=3 by transitive_le/ #HX destruct
   >plus_plus_comm_23 /4 width=3 by lift_lref_ge_minus, lift_lref_ge, monotonic_le_plus_l, ex2_intro/
-| #p #d1 #e1 #d2 #e2 #X #HX #_
+| #p #l1 #m1 #l2 #m2 #X #HX #_
   >(lift_inv_gref1 … HX) -HX /2 width=3 by lift_gref, ex2_intro/
-| #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
+| #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl21
   elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
   elim (IHV12 … HV20) -IHV12 -HV20 //
   elim (IHT12 … HT20) -IHT12 -HT20 /3 width=5 by lift_bind, le_S_S, ex2_intro/
-| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
+| #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl21
   elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
   elim (IHV12 … HV20) -IHV12 -HV20 //
   elim (IHT12 … HT20) -IHT12 -HT20 /3 width=5 by lift_flat, ex2_intro/
@@ -172,27 +172,27 @@ theorem lift_trans_le: ∀d1,e1,T1,T. ⬆[d1, e1] T1 ≡ T →
 qed.
 
 (* Basic_1: was: lift_d (left to right) *)
-theorem lift_trans_ge: ∀d1,e1,T1,T. ⬆[d1, e1] T1 ≡ T →
-                       ∀d2,e2,T2. ⬆[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 →
-                       ∃∃T0. ⬆[d2 - e1, e2] T1 ≡ T0 & ⬆[d1, e1] T0 ≡ T2.
-#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
-[ #k #d1 #e1 #d2 #e2 #X #HX #_
+theorem lift_trans_ge: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
+                       ∀l2,m2,T2. ⬆[l2, m2] T ≡ T2 → l1 + m1 ≤ l2 →
+                       ∃∃T0. ⬆[l2 - m1, m2] T1 ≡ T0 & ⬆[l1, m1] T0 ≡ T2.
+#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
+[ #k #l1 #m1 #l2 #m2 #X #HX #_
   >(lift_inv_sort1 … HX) -HX /2 width=3 by lift_sort, ex2_intro/
-| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hded
-  lapply (lt_to_le_to_lt … (d1+e1) Hid1 ?) // #Hid1e
-  lapply (lt_to_le_to_lt … (d2-e1) Hid1 ?) /2 width=1 by le_plus_to_minus_r/ #Hid2e
-  lapply (lt_to_le_to_lt … Hid1e Hded) -Hid1e -Hded #Hid2
+| #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #Hlml
+  lapply (lt_to_le_to_lt … (l1+m1) Hil1 ?) // #Hil1m
+  lapply (lt_to_le_to_lt … (l2-m1) Hil1 ?) /2 width=1 by le_plus_to_minus_r/ #Hil2m
+  lapply (lt_to_le_to_lt … Hil1m Hlml) -Hil1m -Hlml #Hil2
   lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct /3 width=3 by lift_lref_lt, ex2_intro/
-| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
-  elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct /4 width=3 by lift_lref_lt, lift_lref_ge, monotonic_le_minus_l, lt_plus_to_minus_r, transitive_le, ex2_intro/
-| #p #d1 #e1 #d2 #e2 #X #HX #_
+| #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #_
+  elim (lift_inv_lref1 … HX) -HX * #Himl #HX destruct /4 width=3 by lift_lref_lt, lift_lref_ge, monotonic_le_minus_l, lt_plus_to_minus_r, transitive_le, ex2_intro/
+| #p #l1 #m1 #l2 #m2 #X #HX #_
   >(lift_inv_gref1 … HX) -HX /2 width=3 by lift_gref, ex2_intro/
-| #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
+| #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hlml
   elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
   elim (IHV12 … HV20) -IHV12 -HV20 //
   elim (IHT12 … HT20) -IHT12 -HT20 /2 width=1 by le_S_S/ #T
   <plus_minus /3 width=5 by lift_bind, le_plus_to_minus_r, le_plus_b, ex2_intro/
-| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
+| #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hlml
   elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
   elim (IHV12 … HV20) -IHV12 -HV20 //
   elim (IHT12 … HT20) -IHT12 -HT20 /3 width=5 by lift_flat, ex2_intro/
@@ -201,17 +201,17 @@ qed.
 
 (* Advanced properties ******************************************************)
 
-lemma lift_conf_O1: ∀T,T1,d1,e1. ⬆[d1, e1] T ≡ T1 → ∀T2,e2. ⬆[0, e2] T ≡ T2 →
-                    ∃∃T0. ⬆[0, e2] T1 ≡ T0 & ⬆[d1 + e2, e1] T2 ≡ T0.
-#T #T1 #d1 #e1 #HT1 #T2 #e2 #HT2
-elim (lift_total T1 0 e2) #T0 #HT10
+lemma lift_conf_O1: ∀T,T1,l1,m1. ⬆[l1, m1] T ≡ T1 → ∀T2,m2. ⬆[0, m2] T ≡ T2 →
+                    ∃∃T0. ⬆[0, m2] T1 ≡ T0 & ⬆[l1 + m2, m1] T2 ≡ T0.
+#T #T1 #l1 #m1 #HT1 #T2 #m2 #HT2
+elim (lift_total T1 0 m2) #T0 #HT10
 elim (lift_trans_le … HT1 … HT10) -HT1 // #X #HTX #HT20
 lapply (lift_mono … HTX … HT2) -T #H destruct /2 width=3 by ex2_intro/
 qed.
 
-lemma lift_conf_be: ∀T,T1,d,e1. ⬆[d, e1] T ≡ T1 → ∀T2,e2. ⬆[d, e2] T ≡ T2 →
-                    e1 ≤ e2 → ⬆[d + e1, e2 - e1] T1 ≡ T2.
-#T #T1 #d #e1 #HT1 #T2 #e2 #HT2 #He12
-elim (lift_split … HT2 (d+e1) e1) -HT2 // #X #H
+lemma lift_conf_be: ∀T,T1,l,m1. ⬆[l, m1] T ≡ T1 → ∀T2,m2. ⬆[l, m2] T ≡ T2 →
+                    m1 ≤ m2 → ⬆[l + m1, m2 - m1] T1 ≡ T2.
+#T #T1 #l #m1 #HT1 #T2 #m2 #HT2 #Hm12
+elim (lift_split … HT2 (l+m1) m1) -HT2 // #X #H
 >(lift_mono … H … HT1) -T //
 qed.