- the substitution lemma is proved!

[helm.git] / matita / matita / contribs / lambda-delta / Basic-2 / substitution / lift.ma

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 c6ec4033ffa7e4b2a262b5d52d3e7cadc9f0880e..48e37d63096e4926430f43432d94044b5e37920c 100644 (file)
--- a/matita/matita/contribs/lambda-delta/Basic-2/substitution/lift.ma
+++ b/matita/matita/contribs/lambda-delta/Basic-2/substitution/lift.ma
@@ -1,18 +1,24 @@
-(*
-    ||M||  This file is part of HELM, an Hypertextual, Electronic
-    ||A||  Library of Mathematics, developed at the Computer Science
-    ||T||  Department of the University of Bologna, Italy.
-    ||I||
-    ||T||
-    ||A||  This file is distributed under the terms of the
-    \   /  GNU General Public License Version 2
-     \ /
-      V_______________________________________________________________ *)
-
-include "Basic-2/syntax/term.ma".
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "Basic-2/grammar/term_weight.ma".
 
 (* RELOCATION ***************************************************************)
 
+(* Basic-1: includes:
+            lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat
+*)
 inductive lift: term → nat → nat → term → Prop ≝
 | lift_sort   : ∀k,d,e. lift (⋆k) d e (⋆k)
 | lift_lref_lt: ∀i,d,e. i < d → lift (#i) d e (#i)
@@ -29,22 +35,22 @@ interpretation "relocation" 'RLift T1 d e T2 = (lift T1 d e 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/
-| #I elim I -I /2/
+[ * #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
-[ /2/
-| #i #d #e elim (lt_or_ge i d) /3/
+[ * #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/
@@ -53,6 +59,7 @@ lemma lift_total: ∀T1,d,e. ∃T2. ↑[d,e] T1 ≡ T2.
 ]
 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.
@@ -72,16 +79,22 @@ lemma lift_split: ∀d1,e2,T1,T2. ↑[d1, e2] T1 ≡ T2 → ∀d2,e1.
 ]
 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 ***************************************************)
 
-lemma lift_inv_refl_aux: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
+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/
 qed.
 
 lemma lift_inv_refl: ∀d,T1,T2. ↑[d, 0] T1 ≡ T2 → T1 = T2.
 /2/ qed.
 
-lemma lift_inv_sort1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k.
+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
@@ -92,8 +105,8 @@ qed.
 lemma lift_inv_sort1: ∀d,e,T2,k. ↑[d,e] ⋆k ≡ T2 → T2 = ⋆k.
 /2 width=5/ qed.
 
-lemma lift_inv_lref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T1 = #i →
-                          (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
+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
 [ #k #d #e #i #H destruct
 | #j #d #e #Hj #i #Hi destruct /3/
@@ -119,10 +132,10 @@ lemma lift_inv_lref1_ge: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → d ≤ i → T2 = #(
 elim (lt_refl_false … Hdd)
 qed.
 
-lemma 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 &
-                                   T2 = 𝕓{I} V2. U2.
+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 &
+                                  T2 = 𝕓{I} V2. U2.
 #d #e #T1 #T2 * -d e T1 T2
 [ #k #d #e #I #V1 #U1 #H destruct
 | #i #d #e #_ #I #V1 #U1 #H destruct
@@ -137,10 +150,10 @@ lemma lift_inv_bind1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕓{I} V1. U1 ≡ T2 →
                                T2 = 𝕓{I} V2. U2.
 /2/ qed.
 
-lemma 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 &
-                                   T2 = 𝕗{I} V2. U2.
+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 &
+                                  T2 = 𝕗{I} V2. U2.
 #d #e #T1 #T2 * -d e T1 T2
 [ #k #d #e #I #V1 #U1 #H destruct
 | #i #d #e #_ #I #V1 #U1 #H destruct
@@ -155,7 +168,7 @@ lemma lift_inv_flat1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕗{I} V1. U1 ≡ T2 →
                                T2 = 𝕗{I} V2. U2.
 /2/ qed.
 
-lemma lift_inv_sort2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
+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
@@ -163,11 +176,12 @@ lemma 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_lref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T2 = #i →
-                          (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
+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
 [ #k #d #e #i #H destruct
 | #j #d #e #Hj #i #Hi destruct /3/
@@ -177,26 +191,31 @@ lemma lift_inv_lref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T2 = #i →
 ]
 qed.
 
+(* 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.
 
+(* 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 *)
 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.
 
-lemma 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 &
-                                   T1 = 𝕓{I} V1. U1.
+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 &
+                                  T1 = 𝕓{I} V1. U1.
 #d #e #T1 #T2 * -d e T1 T2
 [ #k #d #e #I #V2 #U2 #H destruct
 | #i #d #e #_ #I #V2 #U2 #H destruct
@@ -206,15 +225,16 @@ lemma lift_inv_bind2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
 ]
 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 &
                                T1 = 𝕓{I} V1. U1.
 /2/ qed.
 
-lemma 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 &
-                                   T1 = 𝕗{I} V1. U1.
+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 &
+                                  T1 = 𝕗{I} V1. U1.
 #d #e #T1 #T2 * -d e T1 T2
 [ #k #d #e #I #V2 #U2 #H destruct
 | #i #d #e #_ #I #V2 #U2 #H destruct
@@ -224,7 +244,14 @@ lemma lift_inv_flat2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
 ]
 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 &
                                T1 = 𝕗{I} V1. U1.
 /2/ qed.
+
+(* 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
+*)