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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "basic_2/notation/relations/predtystrong_4.ma".
16 include "static_2/syntax/teqx.ma".
17 include "basic_2/rt_transition/cpx.ma".
19 (* STRONGLY NORMALIZING TERMS FOR UNBOUND PARALLEL RT-TRANSITION ************)
21 definition csx (h) (G) (L): predicate term ≝
22 SN … (cpx h G L) teqx.
25 "strong normalization for unbound context-sensitive parallel rt-transition (term)"
26 'PRedTyStrong h G L T = (csx h G L T).
28 (* Basic eliminators ********************************************************)
30 lemma csx_ind (h) (G) (L) (Q:predicate …):
31 (∀T1. ❪G,L❫ ⊢ ⬈*𝐒[h] T1 →
32 (∀T2. ❪G,L❫ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) →
35 ∀T. ❪G,L❫ ⊢ ⬈*𝐒[h] T → Q T.
36 #h #G #L #Q #H0 #T1 #H elim H -T1
37 /5 width=1 by SN_intro/
40 (* Basic properties *********************************************************)
42 (* Basic_1: was just: sn3_pr2_intro *)
43 lemma csx_intro (h) (G) (L):
44 ∀T1. (∀T2. ❪G,L❫ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → ❪G,L❫ ⊢ ⬈*𝐒[h] T2) →
46 /4 width=1 by SN_intro/ qed.
48 (* Basic forward lemmas *****************************************************)
50 fact csx_fwd_pair_sn_aux (h) (G) (L):
51 ∀U. ❪G,L❫ ⊢ ⬈*𝐒[h] U →
52 ∀I,V,T. U = ②[I]V.T → ❪G,L❫ ⊢ ⬈*𝐒[h] V.
53 #h #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
54 @csx_intro #V2 #HLV2 #HV2
55 @(IH (②[I]V2.T)) -IH /2 width=3 by cpx_pair_sn/ -HLV2
56 #H elim (teqx_inv_pair … H) -H /2 width=1 by/
59 (* Basic_1: was just: sn3_gen_head *)
60 lemma csx_fwd_pair_sn (h) (G) (L):
61 ∀I,V,T. ❪G,L❫ ⊢ ⬈*𝐒[h] ②[I]V.T → ❪G,L❫ ⊢ ⬈*𝐒[h] V.
62 /2 width=5 by csx_fwd_pair_sn_aux/ qed-.
64 fact csx_fwd_bind_dx_aux (h) (G) (L):
65 ∀U. ❪G,L❫ ⊢ ⬈*𝐒[h] U →
66 ∀p,I,V,T. U = ⓑ[p,I]V.T → ❪G,L.ⓑ[I]V❫ ⊢ ⬈*𝐒[h] T.
67 #h #G #L #U #H elim H -H #U0 #_ #IH #p #I #V #T #H destruct
68 @csx_intro #T2 #HLT2 #HT2
69 @(IH (ⓑ[p, I]V.T2)) -IH /2 width=3 by cpx_bind/ -HLT2
70 #H elim (teqx_inv_pair … H) -H /2 width=1 by/
73 (* Basic_1: was just: sn3_gen_bind *)
74 lemma csx_fwd_bind_dx (h) (G) (L):
75 ∀p,I,V,T. ❪G,L❫ ⊢ ⬈*𝐒[h] ⓑ[p,I]V.T → ❪G,L.ⓑ[I]V❫ ⊢ ⬈*𝐒[h] T.
76 /2 width=4 by csx_fwd_bind_dx_aux/ qed-.
78 fact csx_fwd_flat_dx_aux (h) (G) (L):
79 ∀U. ❪G,L❫ ⊢ ⬈*𝐒[h] U →
80 ∀I,V,T. U = ⓕ[I]V.T → ❪G,L❫ ⊢ ⬈*𝐒[h] T.
81 #h #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
82 @csx_intro #T2 #HLT2 #HT2
83 @(IH (ⓕ[I]V.T2)) -IH /2 width=3 by cpx_flat/ -HLT2
84 #H elim (teqx_inv_pair … H) -H /2 width=1 by/
87 (* Basic_1: was just: sn3_gen_flat *)
88 lemma csx_fwd_flat_dx (h) (G) (L):
89 ∀I,V,T. ❪G,L❫ ⊢ ⬈*𝐒[h] ⓕ[I]V.T → ❪G,L❫ ⊢ ⬈*𝐒[h] T.
90 /2 width=5 by csx_fwd_flat_dx_aux/ qed-.
92 lemma csx_fwd_bind (h) (G) (L):
93 ∀p,I,V,T. ❪G,L❫ ⊢ ⬈*𝐒[h] ⓑ[p,I]V.T →
94 ∧∧ ❪G,L❫ ⊢ ⬈*𝐒[h] V & ❪G,L.ⓑ[I]V❫ ⊢ ⬈*𝐒[h] T.
95 /3 width=3 by csx_fwd_pair_sn, csx_fwd_bind_dx, conj/ qed-.
97 lemma csx_fwd_flat (h) (G) (L):
98 ∀I,V,T. ❪G,L❫ ⊢ ⬈*𝐒[h] ⓕ[I]V.T →
99 ∧∧ ❪G,L❫ ⊢ ⬈*𝐒[h] V & ❪G,L❫ ⊢ ⬈*𝐒[h] T.
100 /3 width=3 by csx_fwd_pair_sn, csx_fwd_flat_dx, conj/ qed-.
102 (* Basic_1: removed theorems 14:
104 sn3_gen_cflat sn3_cflat sn3_cpr3_trans sn3_shift sn3_change
105 sn3_appl_cast sn3_appl_beta sn3_appl_lref sn3_appl_abbr
106 sn3_appl_appls sn3_bind sn3_appl_bind sn3_appls_bind
108 (* Basic_2A1: removed theorems 6:
109 csxa_ind csxa_intro csxa_cpxs_trans csxa_intro_cpx