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15 include "basic_2/substitution/frsup.ma".
16 include "basic_2/unfold/frsupp.ma".
18 (* STAR-ITERATED RESTRICTED SUPCLOSURE **************************************)
20 definition frsups: bi_relation lenv term ≝ bi_star … frsup.
22 interpretation "star-iterated restricted structural predecessor (closure)"
23 'RestSupTermStar L1 T1 L2 T2 = (frsups L1 T1 L2 T2).
25 (* Basic eliminators ********************************************************)
27 lemma frsups_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_star_ind … IH1 IH2 ? ? H)
34 lemma frsups_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_star_ind_dx … IH1 IH2 ? ? H)
41 (* Basic properties *********************************************************)
43 lemma frsups_refl: bi_reflexive … frsups.
46 lemma frsupp_frsups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄.
49 lemma frsup_frsups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄.
52 lemma frsups_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁* ⦃L, T⦄ → ⦃L, T⦄ ⧁ ⦃L2, T2⦄ →
56 lemma frsups_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁ ⦃L, T⦄ → ⦃L, T⦄ ⧁* ⦃L2, T2⦄ →
60 lemma frsups_frsupp_frsupp: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁* ⦃L, T⦄ →
61 ⦃L, T⦄ ⧁+ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
64 lemma frsupp_frsups_frsupp: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⧁+ ⦃L, T⦄ →
65 ⦃L, T⦄ ⧁* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
68 (* Basic inversion lemmas ***************************************************)
70 lemma frsups_inv_all: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ →
71 (L1 = L2 ∧ T1 = T2) ∨ ⦃L1, T1⦄ ⧁+ ⦃L2, T2⦄.
72 #L1 #L2 #T1 #T2 * /2 width=1/
75 (* Basic forward lemmas *****************************************************)
77 lemma frsups_fwd_fw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → #{L2, T2} ≤ #{L1, T1}.
78 #L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H [ * // ]
79 /3 width=1 by frsupp_fwd_fw, lt_to_le/
82 lemma frsups_fwd_lw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → #{L1} ≤ #{L2}.
83 #L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H [ * // ]
84 /2 width=3 by frsupp_fwd_lw/
87 lemma frsups_fwd_tw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → #{T2} ≤ #{T1}.
88 #L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H [ * // ]
89 /3 width=3 by frsupp_fwd_tw, lt_to_le/
92 lemma frsups_fwd_append: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁* ⦃L2, T2⦄ → ∃L. L2 = L1 @@ L.
93 #L1 #L2 #T1 #T2 #H elim (frsups_inv_all … H) -H
96 | /2 width=3 by frsupp_fwd_append/
99 (* Advanced forward lemmas ***************************************************)
101 lemma lift_frsups_trans: ∀T1,U1,d,e. ⇧[d, e] T1 ≡ U1 →
102 ∀L,K,U2. ⦃L, U1⦄ ⧁* ⦃L @@ K, U2⦄ →
103 ∃T2. ⇧[d + |K|, e] T2 ≡ U2.
104 #T1 #U1 #d #e #HTU1 #L #K #U2 #H elim (frsups_inv_all … H) -H
106 >(append_inv_refl_dx … (sym_eq … H1)) -H1 normalize /2 width=2/
107 | /2 width=5 by lift_frsupp_trans/
111 (* Advanced inversion lemmas for frsupp **************************************)
113 lemma frsupp_inv_atom1_frsups: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ ⧁+ ⦃L2, T2⦄ → ⊥.
114 #J #L1 #L2 #T2 #H @(frsupp_ind … H) -L2 -T2 //
115 #L2 #T2 #H elim (frsup_inv_atom1 … H)
118 lemma frsupp_inv_bind1_frsups: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ ⧁+ ⦃L2, T2⦄ →
119 ⦃L1, W⦄ ⧁* ⦃L2, T2⦄ ∨ ⦃L1.ⓑ{J}W, U⦄ ⧁* ⦃L2, T2⦄.
120 #b #J #L1 #L2 #W #U #T2 #H @(frsupp_ind … H) -L2 -T2
122 elim (frsup_inv_bind1 … H) -H * #H1 #H2 destruct /2 width=1/
123 | #L #T #L2 #T2 #_ #HT2 * /3 width=4/
127 lemma frsupp_inv_flat1_frsups: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ ⧁+ ⦃L2, T2⦄ →
128 ⦃L1, W⦄ ⧁* ⦃L2, T2⦄ ∨ ⦃L1, U⦄ ⧁* ⦃L2, T2⦄.
129 #J #L1 #L2 #W #U #T2 #H @(frsupp_ind … H) -L2 -T2
131 elim (frsup_inv_flat1 … H) -H #H1 * #H2 destruct /2 width=1/
132 | #L #T #L2 #T2 #_ #HT2 * /3 width=4/