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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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11 (* v GNU General Public License Version 2 *)
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15 include "Basic_2/unfold/tpss.ma".
17 (* DELIFT ON TERMS **********************************************************)
19 definition delift: nat → nat → lenv → relation term ≝
20 λd,e,L,T1,T2. ∃∃T. L ⊢ T1 [d, e] ▶* T & ⇧[d, e] T2 ≡ T.
22 interpretation "delift (term)"
23 'TSubst L T1 d e T2 = (delift d e L T1 T2).
25 (* Basic properties *********************************************************)
27 lemma delift_lsubs_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≡ T2 →
28 ∀L2. L1 [d, e] ≼ L2 → L2 ⊢ T1 [d, e] ≡ T2.
29 #L1 #T1 #T2 #d #e * /3 width=3/
32 lemma delift_bind: ∀I,L,V1,V2,T1,T2,d,e.
33 L ⊢ V1 [d, e] ≡ V2 → L. 𝕓{I} V2 ⊢ T1 [d+1, e] ≡ T2 →
34 L ⊢ 𝕓{I} V1. T1 [d, e] ≡ 𝕓{I} V2. T2.
35 #I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * #T #HT1 #HT2
36 lapply (tpss_lsubs_conf … HT1 (L. 𝕓{I} V) ?) -HT1 /2 width=1/ /3 width=5/
39 lemma delift_flat: ∀I,L,V1,V2,T1,T2,d,e.
40 L ⊢ V1 [d, e] ≡ V2 → L ⊢ T1 [d, e] ≡ T2 →
41 L ⊢ 𝕗{I} V1. T1 [d, e] ≡ 𝕗{I} V2. T2.
42 #I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * /3 width=5/
45 (* Basic forward lemmas *****************************************************)
47 lemma delift_fwd_sort1: ∀L,U2,d,e,k. L ⊢ ⋆k [d, e] ≡ U2 → U2 = ⋆k.
48 #L #U2 #d #e #k * #U #HU
49 >(tpss_inv_sort1 … HU) -HU #HU2
50 >(lift_inv_sort2 … HU2) -HU2 //
53 lemma delift_fwd_gref1: ∀L,U2,d,e,p. L ⊢ §p [d, e] ≡ U2 → U2 = §p.
54 #L #U #d #e #p * #U #HU
55 >(tpss_inv_gref1 … HU) -HU #HU2
56 >(lift_inv_gref2 … HU2) -HU2 //
59 lemma delift_fwd_bind1: ∀I,L,V1,T1,U2,d,e. L ⊢ 𝕓{I} V1. T1 [d, e] ≡ U2 →
60 ∃∃V2,T2. L ⊢ V1 [d, e] ≡ V2 &
61 L. 𝕓{I} V2 ⊢ T1 [d+1, e] ≡ T2 &
63 #I #L #V1 #T1 #U2 #d #e * #U #HU #HU2
64 elim (tpss_inv_bind1 … HU) -HU #V #T #HV1 #HT1 #X destruct
65 elim (lift_inv_bind2 … HU2) -HU2 #V2 #T2 #HV2 #HT2
66 lapply (tpss_lsubs_conf … HT1 (L. 𝕓{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
69 lemma delift_fwd_flat1: ∀I,L,V1,T1,U2,d,e. L ⊢ 𝕗{I} V1. T1 [d, e] ≡ U2 →
70 ∃∃V2,T2. L ⊢ V1 [d, e] ≡ V2 &
73 #I #L #V1 #T1 #U2 #d #e * #U #HU #HU2
74 elim (tpss_inv_flat1 … HU) -HU #V #T #HV1 #HT1 #X destruct
75 elim (lift_inv_flat2 … HU2) -HU2 /3 width=5/
78 (* Basic Inversion lemmas ***************************************************)
80 lemma delift_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 [d, 0] ≡ T2 → T1 = T2.
81 #L #T1 #T2 #d * #T #HT1
82 >(tpss_inv_refl_O2 … HT1) -HT1 #HT2
83 >(lift_inv_refl_O2 … HT2) -HT2 //