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15 include "basic_2/unfold/frsupp.ma".
17 (* STAR-ITERATED RESTRICTED SUPCLOSURE **************************************)
19 definition frsups: bi_relation lenv term ≝ bi_star … frsup.
21 interpretation "star-iterated restricted structural predecessor (closure)"
22 'RestSupTermStar L1 T1 L2 T2 = (frsups L1 T1 L2 T2).
24 (* Basic eliminators ********************************************************)
26 lemma frsups_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 →
27 (∀L,L2,T,T2. ⦃L1, T1⦄ ⧁* ⦃L, T⦄ → ⦃L, T⦄ ⧁ ⦃L2, T2⦄ → R L T → R L2 T2) →
28 ∀L2,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → R L2 T2.
29 #L1 #T1 #R #IH1 #IH2 #L2 #T2 #H
30 @(bi_star_ind … IH1 IH2 ? ? H)
33 lemma frsups_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 →
34 (∀L1,L,T1,T. ⦃L1, T1⦄ ⧁ ⦃L, T⦄ → ⦃L, T⦄ ⧁* ⦃L2, T2⦄ → R L T → R L1 T1) →
35 ∀L1,T1. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → R L1 T1.
36 #L2 #T2 #R #IH1 #IH2 #L1 #T1 #H
37 @(bi_star_ind_dx … IH1 IH2 ? ? H)
40 (* Basic properties *********************************************************)
42 lemma frsups_refl: bi_reflexive … frsups.
45 lemma frsupp_frsups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄.
48 lemma frsup_frsups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄.
51 lemma frsups_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁* ⦃L, T⦄ → ⦃L, T⦄ ⧁ ⦃L2, T2⦄ →
55 lemma frsups_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁ ⦃L, T⦄ → ⦃L, T⦄ ⧁* ⦃L2, T2⦄ →
59 lemma frsups_frsupp_frsupp: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁* ⦃L, T⦄ →
60 ⦃L, T⦄ ⧁+ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
63 lemma frsupp_frsups_frsupp: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁+ ⦃L, T⦄ →
64 ⦃L, T⦄ ⧁* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
67 (* Basic inversion lemmas ***************************************************)
69 lemma frsups_inv_all: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ →
70 (L1 = L2 ∧ T1 = T2) ∨ ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
71 #L1 #L2 #T1 #T2 * /2 width=1/
74 (* Basic forward lemmas *****************************************************)
76 lemma frsups_fwd_fw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → #{L2, T2} ≤ #{L1, T1}.
77 #L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H [ * // ]
78 /3 width=1 by frsupp_fwd_fw, lt_to_le/
81 lemma frsups_fwd_lw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → #{L1} ≤ #{L2}.
82 #L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H [ * // ]
83 /2 width=3 by frsupp_fwd_lw/
86 lemma frsups_fwd_tw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → #{T2} ≤ #{T1}.
87 #L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H [ * // ]
88 /3 width=3 by frsupp_fwd_tw, lt_to_le/
91 lemma frsups_fwd_append: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → ∃L. L2 = L1 @@ L.
92 #L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H
95 | /2 width=3 by frsupp_fwd_append/
98 (* Advanced forward lemmas ***************************************************)
100 lemma lift_frsups_trans: ∀T1,U1,d,e. ⇧[d, e] T1 ≡ U1 →
101 ∀L,K,U2. ⦃L, U1⦄ ⧁* ⦃L @@ K, U2⦄ →
102 ∃T2. ⇧[d + |K|, e] T2 ≡ U2.
103 #T1 #U1 #d #e #HTU1 #L #K #U2 #H elim (frsups_inv_all … H) -H
105 >(append_inv_refl_dx … (sym_eq … H1)) -H1 normalize /2 width=2/
106 | /2 width=5 by lift_frsupp_trans/
110 (* Advanced inversion lemmas for frsupp **************************************)
112 lemma frsupp_inv_atom1_frsups: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ ⧁+ ⦃L2, T2⦄ → ⊥.
113 #J #L1 #L2 #T2 #H @(frsupp_ind … H) -L2 -T2 //
114 #L2 #T2 #H elim (frsup_inv_atom1 … H)
117 lemma frsupp_inv_bind1_frsups: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ ⧁+ ⦃L2, T2⦄ →
118 ⦃L1, W⦄ ⧁* ⦃L2, T2⦄ ∨ ⦃L1.ⓑ{J}W, U⦄ ⧁* ⦃L2, T2⦄.
119 #b #J #L1 #L2 #W #U #T2 #H @(frsupp_ind … H) -L2 -T2
121 elim (frsup_inv_bind1 … H) -H * #H1 #H2 destruct /2 width=1/
122 | #L #T #L2 #T2 #_ #HT2 * /3 width=4/
126 lemma frsupp_inv_flat1_frsups: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ ⧁+ ⦃L2, T2⦄ →
127 ⦃L1, W⦄ ⧁* ⦃L2, T2⦄ ∨ ⦃L1, U⦄ ⧁* ⦃L2, T2⦄.
128 #J #L1 #L2 #W #U #T2 #H @(frsupp_ind … H) -L2 -T2
130 elim (frsup_inv_flat1 … H) -H #H1 * #H2 destruct /2 width=1/
131 | #L #T #L2 #T2 #_ #HT2 * /3 width=4/