X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambda_delta%2FBasic_2%2Fsubstitution%2Flift.ma;h=b588b29b793e163cfed492afc3981b278e064fbd;hb=d833e40ce45e301a01ddd9ea66c29fb2b34bb685;hp=5f16e9aaf6bfc5071971caa45198e1957576e236;hpb=78f21d7d9014e5c7655f58239e4f1a128ea2c558;p=helm.git

diff --git a/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift.ma b/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift.ma
index 5f16e9aaf..b588b29b7 100644
--- a/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift.ma
+++ b/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift.ma
@@ -13,8 +13,9 @@
 (**************************************************************************)
 
 include "Basic_2/grammar/term_weight.ma".
+include "Basic_2/grammar/term_simple.ma".
 
-(* RELOCATION ***************************************************************)
+(* BASIC TERM RELOCATION ****************************************************)
 
 (* Basic_1: includes:
             lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat
@@ -23,6 +24,7 @@ inductive lift: nat → nat → relation term ≝
 | lift_sort   : ∀k,d,e. lift d e (⋆k) (⋆k)
 | lift_lref_lt: ∀i,d,e. i < d → lift d e (#i) (#i)
 | lift_lref_ge: ∀i,d,e. d ≤ i → lift d e (#i) (#(i + e))
+| lift_gref   : ∀p,d,e. lift d e (§p) (§p)
 | lift_bind   : ∀I,V1,V2,T1,T2,d,e.
                 lift d e V1 V2 → lift (d + 1) e T1 T2 →
                 lift d e (𝕓{I} V1. T1) (𝕓{I} V2. T2)
@@ -33,143 +35,105 @@ inductive lift: nat → nat → relation term ≝
 
 interpretation "relocation" 'RLift d e T1 T2 = (lift d e T1 T2).
 
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: lift_lref_gt *)
-lemma lift_lref_ge_minus: ∀d,e,i. d + e ≤ i → ↑[d, e] #(i - e) ≡ #i.
-#d #e #i #H >(plus_minus_m_m i e) in ⊢ (? ? ? ? %) /3/
-qed.
-
-(* Basic_1: was: lift_r *)
-lemma lift_refl: ∀T,d. ↑[d, 0] T ≡ T.
-#T elim T -T
-[ * #i // #d elim (lt_or_ge i d) /2/
-| * /2/
-]
-qed.
-
-lemma lift_total: ∀T1,d,e. ∃T2. ↑[d,e] T1 ≡ T2.
-#T1 elim T1 -T1
-[ * #i /2/ #d #e elim (lt_or_ge i d) /3/
-| * #I #V1 #T1 #IHV1 #IHT1 #d #e
-  elim (IHV1 d e) -IHV1 #V2 #HV12
-  [ elim (IHT1 (d+1) e) -IHT1 /3/
-  | elim (IHT1 d e) -IHT1 /3/
-  ]
-]
-qed.
-
-(* Basic_1: was: lift_free (right to left) *)
-lemma lift_split: ∀d1,e2,T1,T2. ↑[d1, e2] T1 ≡ T2 → ∀d2,e1.
-                                d1 ≤ d2 → d2 ≤ d1 + e1 → e1 ≤ e2 →
-                                ∃∃T. ↑[d1, e1] T1 ≡ T & ↑[d2, e2 - e1] T ≡ T2.
-#d1 #e2 #T1 #T2 #H elim H -H d1 e2 T1 T2
-[ /3/
-| #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_
-  lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4/
-| #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12
-  lapply (transitive_le …(i+e1) Hd21 ?) /2/ -Hd21 #Hd21
-  <(arith_d1 i e2 e1) // /3/
-| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
-  elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
-  elim (IHT (d2+1) … ? ? He12) /3 width = 5/
-| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
-  elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
-  elim (IHT d2 … ? ? He12) /3 width = 5/
-]
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma tw_lift: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → #[T1] = #[T2].
-#d #e #T1 #T2 #H elim H -d e T1 T2; normalize //
-qed.
-
 (* Basic inversion lemmas ***************************************************)
 
-fact lift_inv_refl_aux: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
-#d #e #T1 #T2 #H elim H -H d e T1 T2 /3/
+fact lift_inv_refl_O2_aux: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
+#d #e #T1 #T2 #H elim H -d -e -T1 -T2 // /3 width=1/
 qed.
 
-lemma lift_inv_refl: ∀d,T1,T2. ↑[d, 0] T1 ≡ T2 → T1 = T2.
-/2/ qed.
+lemma lift_inv_refl_O2: ∀d,T1,T2. ⇧[d, 0] T1 ≡ T2 → T1 = T2.
+/2 width=4/ qed-.
 
-fact lift_inv_sort1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k.
-#d #e #T1 #T2 * -d e T1 T2 //
+fact lift_inv_sort1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k.
+#d #e #T1 #T2 * -d -e -T1 -T2 //
 [ #i #d #e #_ #k #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
 ]
 qed.
 
-lemma lift_inv_sort1: ∀d,e,T2,k. ↑[d,e] ⋆k ≡ T2 → T2 = ⋆k.
-/2 width=5/ qed.
+lemma lift_inv_sort1: ∀d,e,T2,k. ⇧[d,e] ⋆k ≡ T2 → T2 = ⋆k.
+/2 width=5/ qed-.
 
-fact lift_inv_lref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T1 = #i →
+fact lift_inv_lref1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀i. T1 = #i →
                          (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
-#d #e #T1 #T2 * -d e T1 T2
+#d #e #T1 #T2 * -d -e -T1 -T2
 [ #k #d #e #i #H destruct
-| #j #d #e #Hj #i #Hi destruct /3/
-| #j #d #e #Hj #i #Hi destruct /3/
+| #j #d #e #Hj #i #Hi destruct /3 width=1/
+| #j #d #e #Hj #i #Hi destruct /3 width=1/
+| #p #d #e #i #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
 ]
 qed.
 
-lemma lift_inv_lref1: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 →
+lemma lift_inv_lref1: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 →
                       (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
-/2/ qed.
+/2 width=3/ qed-.
 
-lemma lift_inv_lref1_lt: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → i < d → T2 = #i.
+lemma lift_inv_lref1_lt: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 → i < d → T2 = #i.
 #d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * //
-#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
+#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd
 elim (lt_refl_false … Hdd)
-qed.
+qed-.
 
-lemma lift_inv_lref1_ge: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → d ≤ i → T2 = #(i + e).
+lemma lift_inv_lref1_ge: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 → d ≤ i → T2 = #(i + e).
 #d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * //
-#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
+#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd
 elim (lt_refl_false … Hdd)
+qed-.
+
+fact lift_inv_gref1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀p. T1 = §p → T2 = §p.
+#d #e #T1 #T2 * -d -e -T1 -T2 //
+[ #i #d #e #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+]
 qed.
 
-fact lift_inv_bind1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
+lemma lift_inv_gref1: ∀d,e,T2,p. ⇧[d,e] §p ≡ T2 → T2 = §p.
+/2 width=5/ qed-.
+
+fact lift_inv_bind1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
                          ∀I,V1,U1. T1 = 𝕓{I} V1.U1 →
-                         ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
+                         ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
                                   T2 = 𝕓{I} V2. U2.
-#d #e #T1 #T2 * -d e T1 T2
+#d #e #T1 #T2 * -d -e -T1 -T2
 [ #k #d #e #I #V1 #U1 #H destruct
 | #i #d #e #_ #I #V1 #U1 #H destruct
 | #i #d #e #_ #I #V1 #U1 #H destruct
+| #p #d #e #I #V1 #U1 #H destruct
 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5/
 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct
 ]
 qed.
 
-lemma lift_inv_bind1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕓{I} V1. U1 ≡ T2 →
-                      ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
+lemma lift_inv_bind1: ∀d,e,T2,I,V1,U1. ⇧[d,e] 𝕓{I} V1. U1 ≡ T2 →
+                      ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
                                T2 = 𝕓{I} V2. U2.
-/2/ qed.
+/2 width=3/ qed-.
 
-fact lift_inv_flat1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
+fact lift_inv_flat1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
                          ∀I,V1,U1. T1 = 𝕗{I} V1.U1 →
-                         ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
+                         ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
                                   T2 = 𝕗{I} V2. U2.
-#d #e #T1 #T2 * -d e T1 T2
+#d #e #T1 #T2 * -d -e -T1 -T2
 [ #k #d #e #I #V1 #U1 #H destruct
 | #i #d #e #_ #I #V1 #U1 #H destruct
 | #i #d #e #_ #I #V1 #U1 #H destruct
+| #p #d #e #I #V1 #U1 #H destruct
 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct
 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5/
 ]
 qed.
 
-lemma lift_inv_flat1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕗{I} V1. U1 ≡ T2 →
-                      ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
+lemma lift_inv_flat1: ∀d,e,T2,I,V1,U1. ⇧[d,e] 𝕗{I} V1. U1 ≡ T2 →
+                      ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
                                T2 = 𝕗{I} V2. U2.
-/2/ qed.
+/2 width=3/ qed-.
 
-fact lift_inv_sort2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
-#d #e #T1 #T2 * -d e T1 T2 //
+fact lift_inv_sort2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
+#d #e #T1 #T2 * -d -e -T1 -T2 //
 [ #i #d #e #_ #k #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
@@ -177,78 +141,233 @@ fact lift_inv_sort2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k 
 qed.
 
 (* Basic_1: was: lift_gen_sort *)
-lemma lift_inv_sort2: ∀d,e,T1,k. ↑[d,e] T1 ≡ ⋆k → T1 = ⋆k.
-/2 width=5/ qed.
+lemma lift_inv_sort2: ∀d,e,T1,k. ⇧[d,e] T1 ≡ ⋆k → T1 = ⋆k.
+/2 width=5/ qed-.
 
-fact lift_inv_lref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T2 = #i →
+fact lift_inv_lref2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀i. T2 = #i →
                          (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
-#d #e #T1 #T2 * -d e T1 T2
+#d #e #T1 #T2 * -d -e -T1 -T2
 [ #k #d #e #i #H destruct
-| #j #d #e #Hj #i #Hi destruct /3/
-| #j #d #e #Hj #i #Hi destruct <minus_plus_m_m /4/
+| #j #d #e #Hj #i #Hi destruct /3 width=1/
+| #j #d #e #Hj #i #Hi destruct <minus_plus_m_m /4 width=1/
+| #p #d #e #i #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
 ]
 qed.
 
 (* Basic_1: was: lift_gen_lref *)
-lemma lift_inv_lref2: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i →
+lemma lift_inv_lref2: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i →
                       (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
-/2/ qed.
+/2 width=3/ qed-.
 
 (* Basic_1: was: lift_gen_lref_lt *)
-lemma lift_inv_lref2_lt: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i → i < d → T1 = #i.
+lemma lift_inv_lref2_lt: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i → i < d → T1 = #i.
 #d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
-#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
-elim (plus_lt_false … Hdd)
-qed.
+#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd
+elim (lt_inv_plus_l … Hdd) -Hdd #Hdd
+elim (lt_refl_false … Hdd)
+qed-.
 
 (* Basic_1: was: lift_gen_lref_false *)
+lemma lift_inv_lref2_be: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i →
+                         d ≤ i → i < d + e → False.
+#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H *
+[ #H1 #_ #H2 #_ | #H2 #_ #_ #H1 ]
+lapply (le_to_lt_to_lt … H2 H1) -H2 -H1 #H
+elim (lt_refl_false … H)
+qed-.
 
 (* Basic_1: was: lift_gen_lref_ge *)
-lemma lift_inv_lref2_ge: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i → d + e ≤ i → T1 = #(i - e).
+lemma lift_inv_lref2_ge: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i → d + e ≤ i → T1 = #(i - e).
 #d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
-#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
-elim (plus_lt_false … Hdd)
+#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd
+elim (lt_inv_plus_l … Hdd) -Hdd #Hdd
+elim (lt_refl_false … Hdd)
+qed-.
+
+fact lift_inv_gref2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀p. T2 = §p → T1 = §p.
+#d #e #T1 #T2 * -d -e -T1 -T2 //
+[ #i #d #e #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+]
 qed.
 
-fact lift_inv_bind2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
+lemma lift_inv_gref2: ∀d,e,T1,p. ⇧[d,e] T1 ≡ §p → T1 = §p.
+/2 width=5/ qed-.
+
+fact lift_inv_bind2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
                          ∀I,V2,U2. T2 = 𝕓{I} V2.U2 →
-                         ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
+                         ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
                                   T1 = 𝕓{I} V1. U1.
-#d #e #T1 #T2 * -d e T1 T2
+#d #e #T1 #T2 * -d -e -T1 -T2
 [ #k #d #e #I #V2 #U2 #H destruct
 | #i #d #e #_ #I #V2 #U2 #H destruct
 | #i #d #e #_ #I #V2 #U2 #H destruct
+| #p #d #e #I #V2 #U2 #H destruct
 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width=5/
 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct
 ]
 qed.
 
 (* Basic_1: was: lift_gen_bind *)
-lemma lift_inv_bind2: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡  𝕓{I} V2. U2 →
-                      ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
+lemma lift_inv_bind2: ∀d,e,T1,I,V2,U2. ⇧[d,e] T1 ≡  𝕓{I} V2. U2 →
+                      ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
                                T1 = 𝕓{I} V1. U1.
-/2/ qed.
+/2 width=3/ qed-.
 
-fact lift_inv_flat2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
+fact lift_inv_flat2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
                          ∀I,V2,U2. T2 = 𝕗{I} V2.U2 →
-                         ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
+                         ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
                                   T1 = 𝕗{I} V1. U1.
-#d #e #T1 #T2 * -d e T1 T2
+#d #e #T1 #T2 * -d -e -T1 -T2
 [ #k #d #e #I #V2 #U2 #H destruct
 | #i #d #e #_ #I #V2 #U2 #H destruct
 | #i #d #e #_ #I #V2 #U2 #H destruct
+| #p #d #e #I #V2 #U2 #H destruct
 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct
-| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width = 5/
+| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width=5/
 ]
 qed.
 
 (* Basic_1: was: lift_gen_flat *)
-lemma lift_inv_flat2: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡  𝕗{I} V2. U2 →
-                      ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
+lemma lift_inv_flat2: ∀d,e,T1,I,V2,U2. ⇧[d,e] T1 ≡  𝕗{I} V2. U2 →
+                      ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
                                T1 = 𝕗{I} V1. U1.
-/2/ qed.
+/2 width=3/ qed-.
+
+lemma lift_inv_pair_xy_x: ∀d,e,I,V,T. ⇧[d, e] 𝕔{I} V. T ≡ V → False.
+#d #e #J #V elim V -V
+[ * #i #T #H
+  [ lapply (lift_inv_sort2 … H) -H #H destruct
+  | elim (lift_inv_lref2 … H) -H * #_ #H destruct
+  | lapply (lift_inv_gref2 … H) -H #H destruct
+  ]
+| * #I #W2 #U2 #IHW2 #_ #T #H
+  [ elim (lift_inv_bind2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2/
+  | elim (lift_inv_flat2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2/
+  ]
+]
+qed-.
+
+lemma lift_inv_pair_xy_y: ∀I,T,V,d,e. ⇧[d, e] 𝕔{I} V. T ≡ T → False.
+#J #T elim T -T
+[ * #i #V #d #e #H
+  [ lapply (lift_inv_sort2 … H) -H #H destruct
+  | elim (lift_inv_lref2 … H) -H * #_ #H destruct
+  | lapply (lift_inv_gref2 … H) -H #H destruct
+  ]
+| * #I #W2 #U2 #_ #IHU2 #V #d #e #H
+  [ elim (lift_inv_bind2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4/
+  | elim (lift_inv_flat2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4/
+  ]
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma tw_lift: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → #[T1] = #[T2].
+#d #e #T1 #T2 #H elim H -d -e -T1 -T2 normalize //
+qed-.
+
+lemma lift_simple_dx: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → 𝕊[T1] → 𝕊[T2].
+#d #e #T1 #T2 #H elim H -d -e -T1 -T2 //
+#I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #H
+elim (simple_inv_bind … H)
+qed-.
+
+lemma lift_simple_sn: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → 𝕊[T2] → 𝕊[T1].
+#d #e #T1 #T2 #H elim H -d -e -T1 -T2 //
+#I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #H
+elim (simple_inv_bind … H)
+qed-. 
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: lift_lref_gt *)
+lemma lift_lref_ge_minus: ∀d,e,i. d + e ≤ i → ⇧[d, e] #(i - e) ≡ #i.
+#d #e #i #H >(plus_minus_m_m i e) in ⊢ (? ? ? ? %); /2 width=2/ /3 width=2/
+qed.
+
+lemma lift_lref_ge_minus_eq: ∀d,e,i,j. d + e ≤ i → j = i - e → ⇧[d, e] #j ≡ #i.
+/2 width=1/ qed-.
+
+(* Basic_1: was: lift_r *)
+lemma lift_refl: ∀T,d. ⇧[d, 0] T ≡ T.
+#T elim T -T
+[ * #i // #d elim (lt_or_ge i d) /2 width=1/
+| * /2 width=1/
+]
+qed.
+
+lemma lift_total: ∀T1,d,e. ∃T2. ⇧[d,e] T1 ≡ T2.
+#T1 elim T1 -T1
+[ * #i /2 width=2/ #d #e elim (lt_or_ge i d) /3 width=2/
+| * #I #V1 #T1 #IHV1 #IHT1 #d #e
+  elim (IHV1 d e) -IHV1 #V2 #HV12
+  [ elim (IHT1 (d+1) e) -IHT1 /3 width=2/
+  | elim (IHT1 d e) -IHT1 /3 width=2/
+  ]
+]
+qed.
+
+(* Basic_1: was: lift_free (right to left) *)
+lemma lift_split: ∀d1,e2,T1,T2. ⇧[d1, e2] T1 ≡ T2 →
+                  ∀d2,e1. d1 ≤ d2 → d2 ≤ d1 + e1 → e1 ≤ e2 →
+                  ∃∃T. ⇧[d1, e1] T1 ≡ T & ⇧[d2, e2 - e1] T ≡ T2.
+#d1 #e2 #T1 #T2 #H elim H -d1 -e2 -T1 -T2
+[ /3 width=3/
+| #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_
+  lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4 width=3/
+| #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12
+  lapply (transitive_le … (i+e1) Hd21 ?) /2 width=1/ -Hd21 #Hd21
+  >(plus_minus_m_m e2 e1 ?) // /3 width=3/
+| /3 width=3/
+| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
+  elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
+  elim (IHT (d2+1) … ? ? He12) /2 width=1/ /3 width=5/
+| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
+  elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
+  elim (IHT d2 … ? ? He12) // /3 width=5/
+]
+qed.
+
+(* Basic_1: was only: dnf_dec2 dnf_dec *)
+lemma is_lift_dec: ∀T2,d,e. Decidable (∃T1. ⇧[d,e] T1 ≡ T2).
+#T1 elim T1 -T1
+[ * [1,3: /3 width=2/ ] #i #d #e
+  elim (lt_dec i d) #Hid
+  [ /4 width=2/
+  | lapply (false_lt_to_le … Hid) -Hid #Hid
+    elim (lt_dec i (d + e)) #Hide
+    [ @or_intror * #T1 #H
+      elim (lift_inv_lref2_be … H Hid Hide)
+    | lapply (false_lt_to_le … Hide) -Hide /4 width=2/
+    ]
+  ]
+| * #I #V2 #T2 #IHV2 #IHT2 #d #e
+  [ elim (IHV2 d e) -IHV2
+    [ * #V1 #HV12 elim (IHT2 (d+1) e) -IHT2
+      [ * #T1 #HT12 @or_introl /3 width=2/
+      | -V1 #HT2 @or_intror * #X #H
+        elim (lift_inv_bind2 … H) -H /3 width=2/
+      ]
+    | -IHT2 #HV2 @or_intror * #X #H
+      elim (lift_inv_bind2 … H) -H /3 width=2/
+    ]
+  | elim (IHV2 d e) -IHV2
+    [ * #V1 #HV12 elim (IHT2 d e) -IHT2
+      [ * #T1 #HT12 /4 width=2/
+      | -V1 #HT2 @or_intror * #X #H
+        elim (lift_inv_flat2 … H) -H /3 width=2/
+      ]
+    | -IHT2 #HV2 @or_intror * #X #H
+      elim (lift_inv_flat2 … H) -H /3 width=2/
+    ]
+  ]
+]
+qed.
 
 (* Basic_1: removed theorems 7:
             lift_head lift_gen_head