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14
15 include "basic_2/notation/relations/btsnalt_5.ma".
16 include "basic_2/computation/fpbg_alt.ma".
17 include "basic_2/computation/fsb.ma".
18
19 (* "BIG TREE" STRONGLY NORMALIZING TERMS ************************************)
20
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.
26
27 interpretation
28    "'big tree' strong normalization (closure) alternative"
29    'BTSNAlt h g G L T = (fsba h g G L T).
30
31 (* Basic eliminators ********************************************************)
32
33 theorem fsba_ind_alt: ∀h,g. ∀R: relation3 …. (
34                          ∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⦥⦥[h,g] T1 → (
35                             ∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2
36                          ) → R G1 L1 T1
37                       ) →
38                       ∀G,L,T. ⦃G, L⦄ ⊢ ⦥⦥[h, g] 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/
41 qed-.
42
43 (* Main inversion lemmas ****************************************************)
44
45 theorem fsba_inv_fsb: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⦥⦥[h, g] T → ⦃G, L⦄ ⊢ ⦥[h, g] T.
46 #h #g #G #L #T #H @(fsba_ind_alt … H) -G -L -T
47 /4 width=1 by fsb_intro, fpbc_fpbg/
48 qed-.