X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambda_delta%2FBasic_2%2Fsubstitution%2Flift.ma;h=7d0f43c8188b42d29f392aa4775808fe440f37a3;hb=9581b03be2b2bc830820b93992920aaa2c021cc9;hp=cccd3b03c5c98c35aa1bbc1d2d181f20ab0e4a3a;hpb=55dc00c1c44cc21c7ae179cb9df03e7446002c46;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 cccd3b03c..7d0f43c81 100644
--- a/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift.ma
+++ b/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift.ma
@@ -12,17 +12,18 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "Basic-2/grammar/term_weight.ma".
+include "Basic_2/grammar/term_weight.ma".
 
 (* RELOCATION ***************************************************************)
 
-(* Basic-1: includes:
+(* Basic_1: includes:
             lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat
 *)
 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)
@@ -33,58 +34,6 @@ 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.
@@ -92,7 +41,7 @@ fact lift_inv_refl_aux: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
 qed.
 
 lemma lift_inv_refl: ∀d,T1,T2. ↑[d, 0] T1 ≡ T2 → T1 = T2.
-/2/ qed.
+/2/ 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 //
@@ -103,7 +52,7 @@ fact lift_inv_sort1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k 
 qed.
 
 lemma lift_inv_sort1: ∀d,e,T2,k. ↑[d,e] ⋆k ≡ T2 → T2 = ⋆k.
-/2 width=5/ qed.
+/2 width=5/ qed-.
 
 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)).
@@ -111,6 +60,7 @@ fact lift_inv_lref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T1 = #i →
 [ #k #d #e #i #H destruct
 | #j #d #e #Hj #i #Hi destruct /3/
 | #j #d #e #Hj #i #Hi destruct /3/
+| #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
 ]
@@ -118,20 +68,31 @@ qed.
 
 lemma lift_inv_lref1: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 →
                       (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
-/2/ qed.
+/2/ qed-.
 
 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
 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).
 #d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * //
 #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.
 
+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 &
@@ -140,6 +101,7 @@ fact lift_inv_bind1_aux: ∀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
 ]
@@ -148,7 +110,7 @@ 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 &
                                T2 = 𝕓{I} V2. U2.
-/2/ qed.
+/2/ qed-.
 
 fact lift_inv_flat1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
                          ∀I,V1,U1. T1 = 𝕗{I} V1.U1 →
@@ -158,6 +120,7 @@ fact lift_inv_flat1_aux: ∀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/
 ]
@@ -166,7 +129,7 @@ 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 &
                                T2 = 𝕗{I} V2. U2.
-/2/ qed.
+/2/ 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 //
@@ -176,9 +139,9 @@ fact lift_inv_sort2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k 
 ]
 qed.
 
-(* Basic-1: was: lift_gen_sort *)
+(* Basic_1: was: lift_gen_sort *)
 lemma lift_inv_sort2: ∀d,e,T1,k. ↑[d,e] T1 ≡ ⋆k → T1 = ⋆k.
-/2 width=5/ qed.
+/2 width=5/ qed-.
 
 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)).
@@ -186,32 +149,51 @@ fact lift_inv_lref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T2 = #i →
 [ #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/
+| #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 *)
+(* Basic_1: was: lift_gen_lref *)
 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/ qed-.
 
-(* Basic-1: was: lift_gen_lref_lt *)
+(* Basic_1: was: lift_gen_lref_lt *)
 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.
-
-(* Basic-1: was: lift_gen_lref_false *)
-
-(* Basic-1: was: lift_gen_lref_ge *)
+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).
 #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)
+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.
 
+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 &
@@ -220,16 +202,17 @@ fact lift_inv_bind2_aux: ∀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 *)
+(* 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 &
                                T1 = 𝕓{I} V1. U1.
-/2/ qed.
+/2/ qed-.
 
 fact lift_inv_flat2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
                          ∀I,V2,U2. T2 = 𝕗{I} V2.U2 →
@@ -239,18 +222,136 @@ fact lift_inv_flat2_aux: ∀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/
 ]
 qed.
 
-(* Basic-1: was: lift_gen_flat *)
+(* 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 &
                                T1 = 𝕗{I} V1. U1.
-/2/ qed.
+/2/ 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 -J T W1 /2/
+  | elim (lift_inv_flat2 … H) -H #W1 #U1 #HW12 #_ #H destruct -J T W1 /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 -J U1 W1 /2/
+  | elim (lift_inv_flat2 … H) -H #W1 #U1 #_ #HU12 #H destruct -J U1 W1 /2/
+  ]
+]
+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 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 width=2/
+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/
+| /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_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/
+      | -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:
+(* Basic_1: removed theorems 7:
             lift_head lift_gen_head
             lift_weight_map lift_weight lift_weight_add lift_weight_add_O
             lift_tlt_dx