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15 include "basic_2/computation/fpbs_fleq.ma".
16 include "basic_2/computation/fpbg_fpbg.ma".
17 include "basic_2/computation/fsb.ma".
19 (* "BIG TREE" STRONGLY NORMALIZING TERMS ************************************)
21 axiom fleq_fpbc_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⋕[h, g] ⦃G, L, T⦄ →
22 ⦃G, L, T⦄ ≻[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄.
24 axiom fleq_fpbg_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⋕[h, g] ⦃G, L, T⦄ →
25 ⦃G, L, T⦄ >[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
27 (* Properties ******************************************************)
29 lemma fsb_fleq_trans: ∀h,g,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⦥[h, g] T1 →
30 ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⋕[h, g] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⦥[h, g] T2.
31 #h #g #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1
32 /4 width=5 by fsb_intro, fleq_fpbc_trans/
35 (* Advanced eliminators *****************************************************)
37 lemma fsb_ind_fpbg_fpbs: ∀h,g. ∀R:relation3 genv lenv term.
38 (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⦥[h, g] T1 →
39 (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
42 ∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⦥[h, g] T1 →
43 ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2.
44 #h #g #R #IH #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1
45 #G1 #L1 #T1 #H1 #IH1 #G2 #L2 #T2 #H12 elim (fpbs_fwd_fpb_sn … H12) -H12
46 [ #H12 @IH -IH /2 width=5 by fsb_fleq_trans/
47 -H1 #G0 #L0 #T0 #H20 lapply (fleq_fpbg_trans … H12 H20) -G2 -L2 -T2
48 #H10 elim (fpbg_fwd_fpb_sn … H10) -H10 /2 width=5 by/
49 | -H1 -IH * /2 width=5 by/