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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 include "ground_2/relocation/rtmap_id.ma".
16 include "basic_2/notation/relations/subseteq_4.ma".
17 include "basic_2/syntax/voids_length.ma".
18 include "basic_2/static/frees.ma".
20 (* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
22 inductive fle (T2) (L2) (T1): predicate lenv ≝
23 | fle_intro: ∀f1,f2,L1,n. ⓧ*[n]L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 → L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 →
24 |L1| = |L2| → ⫱*[n]f1 ⊆ f2 → fle T2 L2 T1 (ⓧ*[n]L1)
27 interpretation "free variables inclusion (restricted closure)"
28 'SubSetEq L1 T1 L2 T2 = (fle T2 L2 T1 L1).
30 (* Basic properties *********************************************************)
32 lemma fle_sort: ∀L1,L2. |L1| = |L2| → ∀s1,s2. ⦃L1, ⋆s1⦄ ⊆ ⦃L2, ⋆s2⦄.
33 /3 width=5 by frees_sort, sle_refl, fle_intro/ qed.
35 lemma fle_gref: ∀L1,L2. |L1| = |L2| → ∀l1,l2. ⦃L1, §l1⦄ ⊆ ⦃L2, §l2⦄.
36 /3 width=5 by frees_gref, sle_refl, fle_intro/ qed.
38 (* Basic inversion lemmas ***************************************************)
40 fact fle_inv_voids_sn_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
41 ∀K1,n. |K1| = |L2| → L1 = ⓧ*[n]K1 →
42 ∃∃f1,f2. ⓧ*[n]K1 ⊢ 𝐅*⦃T1⦄ ≡ f1 & L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 & ⫱*[n]f1 ⊆ f2.
43 #L1 #L2 #T1 #T2 * -L1 #f1 #f2 #L1 #n #Hf1 #Hf2 #HL12 #Hf12 #Y #x #HY #H destruct
44 elim (voids_inj_length … H) // -H -HL12 -HY #H1 #H2 destruct
45 /2 width=5 by ex3_2_intro/
48 lemma fle_inv_voids_sn: ∀L1,L2,T1,T2,n. ⦃ⓧ*[n]L1, T1⦄ ⊆ ⦃L2, T2⦄ → |L1| = |L2| →
49 ∃∃f1,f2. ⓧ*[n]L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 & L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 & ⫱*[n]f1 ⊆ f2.
50 /2 width=3 by fle_inv_voids_sn_aux/ qed-.