- the relocation properties of cpr are closed!

[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 cccd3b03c5c98c35aa1bbc1d2d181f20ab0e4a3a..1934c37bcce9305b589f020d1b547211e9e8dcc4 100644 (file)
--- 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)
@@ -35,12 +36,12 @@ interpretation "relocation" 'RLift d e T1 T2 = (lift d e T1 T2).
 
 (* Basic properties *********************************************************)
 
-(* Basic-1: was: lift_lref_gt *)
+(* 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 *)
+(* 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/
@@ -59,7 +60,7 @@ lemma lift_total: ∀T1,d,e. ∃T2. ↑[d,e] T1 ≡ T2.
 ]
 qed.
 
-(* Basic-1: was: lift_free (right to left) *)
+(* 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.
@@ -70,6 +71,7 @@ lemma lift_split: ∀d1,e2,T1,T2. ↑[d1, e2] T1 ≡ T2 → ∀d2,e1.
 | #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/
@@ -111,6 +113,7 @@ 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
 ]
@@ -132,6 +135,17 @@ lemma lift_inv_lref1_ge: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → d ≤ i → T2 = #(
 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 +154,7 @@ 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
 ]
@@ -158,6 +173,7 @@ 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/
 ]
@@ -176,7 +192,7 @@ 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.
 
@@ -186,32 +202,44 @@ 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.
 
-(* 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_false *)
 
-(* Basic-1: was: lift_gen_lref_ge *)
+(* 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 &
@@ -220,12 +248,13 @@ 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.
@@ -239,18 +268,19 @@ 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.
 
-(* 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