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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_5.ma".
16 include "static_2/static/rdeq.ma".
17 include "basic_2/rt_transition/lpx.ma".
19 (* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
21 definition rdsx (h) (o) (G) (T): predicate lenv ≝
22 SN … (lpx h G) (rdeq h o T).
25 "strong normalization for unbound context-sensitive parallel rt-transition on referred entries (local environment)"
26 'PRedTySNStrong h o T G L = (rdsx h o G T L).
28 (* Basic eliminators ********************************************************)
30 (* Basic_2A1: uses: lsx_ind *)
31 lemma rdsx_ind (h) (o) (G) (T):
33 (∀L1. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
34 (∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[h, o, T] L2 → ⊥) → Q L2) →
37 ∀L. G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄ → Q L.
38 #h #o #G #T #Q #H0 #L1 #H elim H -L1
39 /5 width=1 by SN_intro/
42 (* Basic properties *********************************************************)
44 (* Basic_2A1: uses: lsx_intro *)
45 lemma rdsx_intro (h) (o) (G) (T):
47 (∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[h, o, T] L2 → ⊥) → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄) →
48 G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄.
49 /5 width=1 by SN_intro/ qed.
51 (* Basic forward lemmas *****************************************************)
53 (* Basic_2A1: uses: lsx_fwd_pair_sn lsx_fwd_bind_sn lsx_fwd_flat_sn *)
54 lemma rdsx_fwd_pair_sn (h) (o) (G):
55 ∀I,L,V,T. G ⊢ ⬈*[h, o, ②{I}V.T] 𝐒⦃L⦄ →
57 #h #o #G #I #L #V #T #H
58 @(rdsx_ind … H) -L #L1 #_ #IHL1
59 @rdsx_intro #L2 #HL12 #HnL12
60 /4 width=3 by rdeq_fwd_pair_sn/
63 (* Basic_2A1: uses: lsx_fwd_flat_dx *)
64 lemma rdsx_fwd_flat_dx (h) (o) (G):
65 ∀I,L,V,T. G ⊢ ⬈*[h, o, ⓕ{I}V.T] 𝐒⦃L⦄ →
67 #h #o #G #I #L #V #T #H
68 @(rdsx_ind … H) -L #L1 #_ #IHL1
69 @rdsx_intro #L2 #HL12 #HnL12
70 /4 width=3 by rdeq_fwd_flat_dx/
73 fact rdsx_fwd_pair_aux (h) (o) (G): ∀L. G ⊢ ⬈*[h, o, #0] 𝐒⦃L⦄ →
74 ∀I,K,V. L = K.ⓑ{I}V → G ⊢ ⬈*[h, o, V] 𝐒⦃K⦄.
76 @(rdsx_ind … H) -L #L1 #_ #IH #I #K1 #V #H destruct
77 /5 width=5 by lpx_pair, rdsx_intro, rdeq_fwd_zero_pair/
80 lemma rdsx_fwd_pair (h) (o) (G):
81 ∀I,K,V. G ⊢ ⬈*[h, o, #0] 𝐒⦃K.ⓑ{I}V⦄ → G ⊢ ⬈*[h, o, V] 𝐒⦃K⦄.
82 /2 width=4 by rdsx_fwd_pair_aux/ qed-.
84 (* Basic inversion lemmas ***************************************************)
86 (* Basic_2A1: uses: lsx_inv_flat *)
87 lemma rdsx_inv_flat (h) (o) (G): ∀I,L,V,T. G ⊢ ⬈*[h, o, ⓕ{I}V.T] 𝐒⦃L⦄ →
88 ∧∧ G ⊢ ⬈*[h, o, V] 𝐒⦃L⦄ & G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄.
89 /3 width=3 by rdsx_fwd_pair_sn, rdsx_fwd_flat_dx, conj/ qed-.
91 (* Basic_2A1: removed theorems 9:
93 lsxa_ind lsxa_intro lsxa_lleq_trans lsxa_lpxs_trans lsxa_intro_lpx lsx_lsxa lsxa_inv_lsx