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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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15 include "basic_2/rt_equivalence/cpcs_cprs.ma".
16 include "basic_2/dynamic/nta.ma".
18 (* NATIVE TYPE ASSIGNMENT FOR TERMS *****************************************)
20 (* Properties with r-equivalence for terms **********************************)
22 lemma nta_conv_cnv (a) (h) (G) (L) (T):
23 ∀U1. ⦃G,L⦄ ⊢ T :[a,h] U1 →
24 ∀U2. ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 → ⦃G,L⦄ ⊢ U2 ![a,h] → ⦃G,L⦄ ⊢ T :[a,h] U2.
25 #a #h #G #L #T #U1 #H1 #U2 #HU12 #HU2
26 elim (cnv_inv_cast … H1) -H1 #X1 #HU1 #HT #HUX1 #HTX1
27 lapply (cpcs_cprs_conf … HUX1 … HU12) -U1 #H
28 elim (cpcs_inv_cprs … H) -H #X2 #HX12 #HU12
29 /3 width=5 by cnv_cast, cpms_cprs_trans/
32 (* Basic_1: was by definition: ty3_conv *)
33 (* Basic_2A1: was by definition: nta_conv ntaa_conv *)
34 lemma nta_conv (a) (h) (G) (L) (T):
35 ∀U1. ⦃G,L⦄ ⊢ T :[a,h] U1 →
36 ∀U2. ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 →
37 ∀W2. ⦃G,L⦄ ⊢ U2 :[a,h] W2 → ⦃G,L⦄ ⊢ T :[a,h] U2.
38 #a #h #G #L #T #U1 #H1 #U2 #HU12 #W2 #H2
39 /3 width=3 by nta_conv_cnv, nta_fwd_cnv_sn/
42 (* Inversion lemmas with r-equivalence for terms ***************************)
44 (* Basic_1: was: ty3_gen_sort *)
45 (* Basic_2A1: was: nta_inv_sort1 *)
46 lemma nta_inv_sort_sn (a) (h) (G) (L) (X2):
47 ∀s. ⦃G,L⦄ ⊢ ⋆s :[a,h] X2 →
48 ∧∧ ⦃G,L⦄ ⊢ ⋆(next h s) ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h].
50 elim (cnv_inv_cast … H) -H #X1 #HX2 #_ #HX21 #H
51 lapply (cpms_inv_sort1 … H) -H #H destruct
52 /3 width=1 by cpcs_cprs_sn, conj/
55 lemma nta_inv_ldec_sn_cnv (a) (h) (G) (K) (V):
56 ∀X2. ⦃G,K.ⓛV⦄ ⊢ #0 :[a,h] X2 →
57 ∃∃U. ⦃G,K⦄ ⊢ V ![a,h] & ⬆*[1] V ≘ U & ⦃G,K.ⓛV⦄ ⊢ U ⬌*[h] X2 & ⦃G,K.ⓛV⦄ ⊢ X2 ![a,h].
59 elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
60 elim (cnv_inv_zero … H1) -H1 #Z #K #V #HV #H destruct
61 elim (cpms_inv_ell_sn … H2) -H2 *
63 | #m #W #HVW #HWX1 #H destruct
64 elim (lifts_total V (𝐔❴1❵)) #U #HVU
65 lapply (cpms_lifts_bi … HVW (Ⓣ) … (K.ⓛV) … HVU … HWX1) -W
66 [ /3 width=1 by drops_refl, drops_drop/ ] #HUX1
67 /3 width=5 by cprs_div, ex4_intro/