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15 include "basic_2/notation/relations/suptermstar_4.ma".
16 include "basic_2/relocation/fsupq.ma".
18 (* STAR-ITERATED SUPCLOSURE *************************************************)
20 definition fsups: bi_relation lenv term ≝ bi_TC … fsupq.
22 interpretation "star-iterated structural successor (closure)"
23 'SupTermStar L1 T1 L2 T2 = (fsups L1 T1 L2 T2).
25 (* Basic eliminators ********************************************************)
27 lemma fsups_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 →
28 (∀L,L2,T,T2. ⦃L1, T1⦄ ⊃* ⦃L, T⦄ → ⦃L, T⦄ ⊃⸮ ⦃L2, T2⦄ → R L T → R L2 T2) →
29 ∀L2,T2. ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄ → R L2 T2.
30 #L1 #T1 #R #IH1 #IH2 #L2 #T2 #H
31 @(bi_TC_star_ind … IH1 IH2 ? ? H) //
34 lemma fsups_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 →
35 (∀L1,L,T1,T. ⦃L1, T1⦄ ⊃⸮ ⦃L, T⦄ → ⦃L, T⦄ ⊃* ⦃L2, T2⦄ → R L T → R L1 T1) →
36 ∀L1,T1. ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄ → R L1 T1.
37 #L2 #T2 #R #IH1 #IH2 #L1 #T1 #H
38 @(bi_TC_star_ind_dx … IH1 IH2 ? ? H) //
41 (* Basic properties *********************************************************)
43 lemma fsups_refl: bi_reflexive … fsups.
46 lemma fsupq_fsups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃⸮ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄.
49 lemma fsups_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⊃* ⦃L, T⦄ → ⦃L, T⦄ ⊃⸮ ⦃L2, T2⦄ →
53 lemma fsups_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⊃⸮ ⦃L, T⦄ → ⦃L, T⦄ ⊃* ⦃L2, T2⦄ →
57 (* Basic forward lemmas *****************************************************)
59 lemma fsups_fwd_fw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄ → ♯{L2, T2} ≤ ♯{L1, T1}.
60 #L1 #L2 #T1 #T2 #H @(fsups_ind … H) -L2 -T2 //
61 /3 width=3 by fsupq_fwd_fw, transitive_le/ (**) (* slow even with trace *)
64 (* Advanced inversion lemmas on plus-iterated supclosure ********************)
66 lamma fsupp_inv_bind1_fsups: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ ⊃+ ⦃L2, T2⦄ →
67 ⦃L1, W⦄ ⊃* ⦃L2, T2⦄ ∨ ⦃L1.ⓑ{J}W, U⦄ ⊃* ⦃L2, T2⦄.
68 #b #J #L1 #L2 #W #U #T2 #H @(fsupp_ind … H) -L2 -T2
70 elim (fsup_inv_bind1 … H) -H * #H1 #H2 destruct /2 width=1/
71 | #L #T #L2 #T2 #_ #HT2 * /3 width=4/
75 lamma fsupp_inv_flat1_fsups: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ ⊃+ ⦃L2, T2⦄ →
76 ⦃L1, W⦄ ⊃* ⦃L2, T2⦄ ∨ ⦃L1, U⦄ ⊃* ⦃L2, T2⦄.
77 #J #L1 #L2 #W #U #T2 #H @(fsupp_ind … H) -L2 -T2
79 elim (fsup_inv_flat1 … H) -H #H1 * #H2 destruct /2 width=1/
80 | #L #T #L2 #T2 #_ #HT2 * /3 width=4/