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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 "ground_2/xoa/ex_4_3.ma".
16 include "basic_2/notation/relations/topredtysnstrong_4.ma".
17 include "basic_2/rt_computation/rsx.ma".
19 (* COMPATIBILITY OF STRONG NORMALIZATION FOR UNBOUND RT-TRANSITION **********)
21 (* Note: this should be an instance of a more general sex *)
22 (* Basic_2A1: uses: lcosx *)
23 inductive jsx (h) (G): relation lenv ≝
24 | jsx_atom: jsx h G (⋆) (⋆)
25 | jsx_bind: ∀I,K1,K2. jsx h G K1 K2 →
26 jsx h G (K1.ⓘ[I]) (K2.ⓘ[I])
27 | jsx_pair: ∀I,K1,K2,V. jsx h G K1 K2 →
28 G ⊢ ⬈*[h,V] 𝐒❪K2❫ → jsx h G (K1.ⓑ[I]V) (K2.ⓧ)
32 "strong normalization for unbound parallel rt-transition (compatibility)"
33 'ToPRedTySNStrong h G L1 L2 = (jsx h G L1 L2).
35 (* Basic inversion lemmas ***************************************************)
37 fact jsx_inv_atom_sn_aux (h) (G):
38 ∀L1,L2. G ⊢ L1 ⊒[h] L2 → L1 = ⋆ → L2 = ⋆.
39 #h #G #L1 #L2 * -L1 -L2
41 | #I #K1 #K2 #_ #H destruct
42 | #I #K1 #K2 #V #_ #_ #H destruct
46 lemma jsx_inv_atom_sn (h) (G): ∀L2. G ⊢ ⋆ ⊒[h] L2 → L2 = ⋆.
47 /2 width=5 by jsx_inv_atom_sn_aux/ qed-.
49 fact jsx_inv_bind_sn_aux (h) (G):
50 ∀L1,L2. G ⊢ L1 ⊒[h] L2 →
52 ∨∨ ∃∃K2. G ⊢ K1 ⊒[h] K2 & L2 = K2.ⓘ[I]
53 | ∃∃J,K2,V. G ⊢ K1 ⊒[h] K2 & G ⊢ ⬈*[h,V] 𝐒❪K2❫ & I = BPair J V & L2 = K2.ⓧ.
54 #h #G #L1 #L2 * -L1 -L2
56 | #I #K1 #K2 #HK12 #J #L1 #H destruct /3 width=3 by ex2_intro, or_introl/
57 | #I #K1 #K2 #V #HK12 #HV #J #L1 #H destruct /3 width=7 by ex4_3_intro, or_intror/
61 lemma jsx_inv_bind_sn (h) (G):
62 ∀I,K1,L2. G ⊢ K1.ⓘ[I] ⊒[h] L2 →
63 ∨∨ ∃∃K2. G ⊢ K1 ⊒[h] K2 & L2 = K2.ⓘ[I]
64 | ∃∃J,K2,V. G ⊢ K1 ⊒[h] K2 & G ⊢ ⬈*[h,V] 𝐒❪K2❫ & I = BPair J V & L2 = K2.ⓧ.
65 /2 width=3 by jsx_inv_bind_sn_aux/ qed-.
67 (* Advanced inversion lemmas ************************************************)
69 (* Basic_2A1: uses: lcosx_inv_pair *)
70 lemma jsx_inv_pair_sn (h) (G):
71 ∀I,K1,L2,V. G ⊢ K1.ⓑ[I]V ⊒[h] L2 →
72 ∨∨ ∃∃K2. G ⊢ K1 ⊒[h] K2 & L2 = K2.ⓑ[I]V
73 | ∃∃K2. G ⊢ K1 ⊒[h] K2 & G ⊢ ⬈*[h,V] 𝐒❪K2❫ & L2 = K2.ⓧ.
74 #h #G #I #K1 #L2 #V #H elim (jsx_inv_bind_sn … H) -H *
75 [ /3 width=3 by ex2_intro, or_introl/
76 | #J #K2 #X #HK12 #HX #H1 #H2 destruct /3 width=4 by ex3_intro, or_intror/
80 lemma jsx_inv_void_sn (h) (G):
81 ∀K1,L2. G ⊢ K1.ⓧ ⊒[h] L2 →
82 ∃∃K2. G ⊢ K1 ⊒[h] K2 & L2 = K2.ⓧ.
83 #h #G #K1 #L2 #H elim (jsx_inv_bind_sn … H) -H *
84 /2 width=3 by ex2_intro/
87 (* Advanced forward lemmas **************************************************)
89 lemma jsx_fwd_bind_sn (h) (G):
90 ∀I1,K1,L2. G ⊢ K1.ⓘ[I1] ⊒[h] L2 →
91 ∃∃I2,K2. G ⊢ K1 ⊒[h] K2 & L2 = K2.ⓘ[I2].
92 #h #G #I1 #K1 #L2 #H elim (jsx_inv_bind_sn … H) -H *
93 /2 width=4 by ex2_2_intro/
96 (* Advanced properties ******************************************************)
98 (* Basic_2A1: uses: lcosx_O *)
99 lemma jsx_refl (h) (G): reflexive … (jsx h G).
100 #h #G #L elim L -L /2 width=1 by jsx_atom, jsx_bind/
103 (* Basic_2A1: removed theorems 2:
104 lcosx_drop_trans_lt lcosx_inv_succ