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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_2A/notation/relations/btsnalt_5.ma".
16 include "basic_2A/computation/fpbg_fpbs.ma".
17 include "basic_2A/computation/fsb.ma".
19 (* "QRST" STRONGLY NORMALIZING CLOSURES *************************************)
21 (* Note: alternative definition of fsb *)
22 inductive fsba (h) (g): relation3 genv lenv term ≝
23 | fsba_intro: ∀G1,L1,T1. (
24 ∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → fsba h g G2 L2 T2
25 ) → fsba h g G1 L1 T1.
28 "'big tree' strong normalization (closure) alternative"
29 'BTSNAlt h g G L T = (fsba h g G L T).
31 (* Basic eliminators ********************************************************)
33 lemma fsba_ind_alt: ∀h,g. ∀R: relation3 …. (
34 ∀G1,L1,T1. ⦥⦥[h,g] ⦃G1, L1, T1⦄ → (
35 ∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2
38 ∀G,L,T. ⦥⦥[h, g] ⦃G, L, T⦄ → R G L T.
39 #h #g #R #IH #G #L #T #H elim H -G -L -T
40 /4 width=1 by fsba_intro/
43 (* Basic properties *********************************************************)
45 lemma fsba_fpbs_trans: ∀h,g,G1,L1,T1. ⦥⦥[h, g] ⦃G1, L1, T1⦄ →
46 ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦥⦥[h, g] ⦃G2, L2, T2⦄.
47 #h #g #G1 #L1 #T1 #H @(fsba_ind_alt … H) -G1 -L1 -T1
48 /4 width=5 by fsba_intro, fpbs_fpbg_trans/
51 (* Main properties **********************************************************)
53 theorem fsb_fsba: ∀h,g,G,L,T. ⦥[h, g] ⦃G, L, T⦄ → ⦥⦥[h, g] ⦃G, L, T⦄.
54 #h #g #G #L #T #H @(fsb_ind_alt … H) -G -L -T
55 #G1 #L1 #T1 #_ #IH @fsba_intro
56 #G2 #L2 #T2 * /3 width=5 by fsba_fpbs_trans/
59 (* Main inversion lemmas ****************************************************)
61 theorem fsba_inv_fsb: ∀h,g,G,L,T. ⦥⦥[h, g] ⦃G, L, T⦄ → ⦥[h, g] ⦃G, L, T⦄.
62 #h #g #G #L #T #H @(fsba_ind_alt … H) -G -L -T
63 /4 width=1 by fsb_intro, fpb_fpbg/
66 (* Advanced properties ******************************************************)
68 lemma fsb_fpbs_trans: ∀h,g,G1,L1,T1. ⦥[h, g] ⦃G1, L1, T1⦄ →
69 ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦥[h, g] ⦃G2, L2, T2⦄.
70 /4 width=5 by fsba_inv_fsb, fsb_fsba, fsba_fpbs_trans/ qed-.
72 (* Advanced eliminators *****************************************************)
74 lemma fsb_ind_fpbg: ∀h,g. ∀R:relation3 genv lenv term.
75 (∀G1,L1,T1. ⦥[h, g] ⦃G1, L1, T1⦄ →
76 (∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
79 ∀G1,L1,T1. ⦥[h, g] ⦃G1, L1, T1⦄ → R G1 L1 T1.
80 #h #g #R #IH #G1 #L1 #T1 #H @(fsba_ind_alt h g … G1 L1 T1)
81 /3 width=1 by fsba_inv_fsb, fsb_fsba/