1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "basic_2/notation/relations/predtystrong_5.ma".
16 include "basic_2/syntax/tdeq.ma".
17 include "basic_2/rt_transition/cpx.ma".
19 (* STRONGLY NORMALIZING TERMS FOR UNCOUNTED PARALLEL RT-TRANSITION **********)
21 definition csx: ∀h. sd h → relation3 genv lenv term ≝
22 λh,o,G,L. SN … (cpx h G L) (tdeq h o …).
25 "strong normalization for uncounted context-sensitive parallel rt-transition (term)"
26 'PRedTyStrong h o G L T = (csx h o G L T).
28 (* Basic eliminators ********************************************************)
30 lemma csx_ind: ∀h,o,G,L. ∀R:predicate term.
31 (∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
32 (∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛[h, o] T2 → ⊥) → R T2) →
35 ∀T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → R T.
36 #h #o #G #L #R #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,o,G,L,T1.
44 (∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛[h, o] T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄) →
45 ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄.
46 /4 width=1 by SN_intro/ qed.
48 (* Basic forward lemmas *****************************************************)
50 fact csx_fwd_pair_sn_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃U⦄ →
51 ∀I,V,T. U = ②{I}V.T → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄.
52 #h #o #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
53 @csx_intro #V2 #HLV2 #HV2
54 @(IH (②{I}V2.T)) -IH /2 width=3 by cpx_pair_sn/ -HLV2
55 #H elim (tdeq_inv_pair … H) -H /2 width=1 by/
58 (* Basic_1: was just: sn3_gen_head *)
59 lemma csx_fwd_pair_sn: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃②{I}V.T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄.
60 /2 width=5 by csx_fwd_pair_sn_aux/ qed-.
62 fact csx_fwd_bind_dx_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃U⦄ →
63 ∀p,I,V,T. U = ⓑ{p,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
64 #h #o #G #L #U #H elim H -H #U0 #_ #IH #p #I #V #T #H destruct
65 @csx_intro #T2 #HLT2 #HT2
66 @(IH (ⓑ{p,I}V.T2)) -IH /2 width=3 by cpx_bind/ -HLT2
67 #H elim (tdeq_inv_pair … H) -H /2 width=1 by/
70 (* Basic_1: was just: sn3_gen_bind *)
71 lemma csx_fwd_bind_dx: ∀h,o,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓑ{p,I}V.T⦄ → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
72 /2 width=4 by csx_fwd_bind_dx_aux/ qed-.
74 fact csx_fwd_flat_dx_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃U⦄ →
75 ∀I,V,T. U = ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
76 #h #o #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
77 @csx_intro #T2 #HLT2 #HT2
78 @(IH (ⓕ{I}V.T2)) -IH /2 width=3 by cpx_flat/ -HLT2
79 #H elim (tdeq_inv_pair … H) -H /2 width=1 by/
82 (* Basic_1: was just: sn3_gen_flat *)
83 lemma csx_fwd_flat_dx: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓕ{I}V.T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
84 /2 width=5 by csx_fwd_flat_dx_aux/ qed-.
86 lemma csx_fwd_bind: ∀h,o,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓑ{p,I}V.T⦄ →
87 ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
88 /3 width=3 by csx_fwd_pair_sn, csx_fwd_bind_dx, conj/ qed-.
90 lemma csx_fwd_flat: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓕ{I}V.T⦄ →
91 ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ ∧ ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
92 /3 width=3 by csx_fwd_pair_sn, csx_fwd_flat_dx, conj/ qed-.
94 (* Basic_1: removed theorems 14:
96 sn3_gen_cflat sn3_cflat sn3_cpr3_trans sn3_shift sn3_change
97 sn3_appl_cast sn3_appl_beta sn3_appl_lref sn3_appl_abbr
98 sn3_appl_appls sn3_bind sn3_appl_bind sn3_appls_bind
100 (* Basic_2A1: removed theorems 6:
101 csxa_ind csxa_intro csxa_cpxs_trans csxa_intro_cpx