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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/notation/relations/predtysnstrong_4.ma".
16 include "static_2/static/reqx.ma".
17 include "basic_2/rt_transition/lpx.ma".
19 (* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
21 definition rsx (h) (G) (T): predicate lenv ≝
22 SN … (lpx h G) (reqx T).
25 "strong normalization for unbound context-sensitive parallel rt-transition on referred entries (local environment)"
26 'PRedTySNStrong h T G L = (rsx h G T L).
28 (* Basic eliminators ********************************************************)
30 (* Basic_2A1: uses: lsx_ind *)
31 lemma rsx_ind (h) (G) (T) (Q:predicate …):
32 (∀L1. G ⊢ ⬈*𝐒[h,T] L1 →
33 (∀L2. ❪G,L1❫ ⊢ ⬈[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
36 ∀L. G ⊢ ⬈*𝐒[h,T] L → Q L.
37 #h #G #T #Q #H0 #L1 #H elim H -L1
38 /5 width=1 by SN_intro/
41 (* Basic properties *********************************************************)
43 (* Basic_2A1: uses: lsx_intro *)
44 lemma rsx_intro (h) (G) (T):
46 (∀L2. ❪G,L1❫ ⊢ ⬈[h] L2 → (L1 ≛[T] L2 → ⊥) → G ⊢ ⬈*𝐒[h,T] L2) →
48 /5 width=1 by SN_intro/ qed.
50 (* Basic forward lemmas *****************************************************)
52 (* Basic_2A1: uses: lsx_fwd_pair_sn lsx_fwd_bind_sn lsx_fwd_flat_sn *)
53 lemma rsx_fwd_pair_sn (h) (G):
54 ∀I,L,V,T. G ⊢ ⬈*𝐒[h,②[I]V.T] L →
57 @(rsx_ind … H) -L #L1 #_ #IHL1
58 @rsx_intro #L2 #HL12 #HnL12
59 /4 width=3 by reqx_fwd_pair_sn/
62 (* Basic_2A1: uses: lsx_fwd_flat_dx *)
63 lemma rsx_fwd_flat_dx (h) (G):
64 ∀I,L,V,T. G ⊢ ⬈*𝐒[h,ⓕ[I]V.T] L →
67 @(rsx_ind … H) -L #L1 #_ #IHL1
68 @rsx_intro #L2 #HL12 #HnL12
69 /4 width=3 by reqx_fwd_flat_dx/
72 fact rsx_fwd_pair_aux (h) (G):
74 ∀I,K,V. L = K.ⓑ[I]V → G ⊢ ⬈*𝐒[h,V] K.
76 @(rsx_ind … H) -L #L1 #_ #IH #I #K1 #V #H destruct
77 /5 width=5 by lpx_pair, rsx_intro, reqx_fwd_zero_pair/
80 lemma rsx_fwd_pair (h) (G):
81 ∀I,K,V. G ⊢ ⬈*𝐒[h,#0] K.ⓑ[I]V → G ⊢ ⬈*𝐒[h,V] K.
82 /2 width=4 by rsx_fwd_pair_aux/ qed-.
84 (* Basic inversion lemmas ***************************************************)
86 (* Basic_2A1: uses: lsx_inv_flat *)
87 lemma rsx_inv_flat (h) (G):
88 ∀I,L,V,T. G ⊢ ⬈*𝐒[h,ⓕ[I]V.T] L →
89 ∧∧ G ⊢ ⬈*𝐒[h,V] L & G ⊢ ⬈*𝐒[h,T] L.
90 /3 width=3 by rsx_fwd_pair_sn, rsx_fwd_flat_dx, conj/ qed-.
92 (* Basic_2A1: removed theorems 9:
94 lsxa_ind lsxa_intro lsxa_lleq_trans lsxa_lpxs_trans lsxa_intro_lpx lsx_lsxa lsxa_inv_lsx