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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "ground_2/notation/functions/basic_2.ma".
16 include "ground_2/relocation/rtmap_at.ma".
18 (* RELOCATION MAP ***********************************************************)
20 definition basic: nat → nat → rtmap ≝ λm,n. ⫯*[m] 𝐔❨n❩.
22 interpretation "basic relocation (rtmap)"
23 'Basic m n = (basic m n).
25 (* Prioerties with application **********************************************)
27 lemma at_basic_lt: ∀m,n,i. i < m → @❪i, 𝐁❨m,n❩❫ ≘ i.
28 #m elim m -m [ #n #i #H elim (lt_zero_false … H) ]
29 #m #IH #n * [ /2 width=2 by refl, at_refl/ ]
30 #i #H lapply (lt_S_S_to_lt … H) -H /3 width=7 by refl, at_push/
33 lemma at_basic_ge: ∀m,n,i. m ≤ i → @❪i, 𝐁❨m,n❩❫ ≘ n+i.
36 elim (le_inv_S1 … H) -H #i #Hmi #H destruct
37 /3 width=7 by refl, at_push/
40 (* Inversion lemmas with application ****************************************)
42 lemma at_basic_inv_lt: ∀m,n,i,j. i < m → @❪i, 𝐁❨m,n❩❫ ≘ j → i = j.
43 /3 width=4 by at_basic_lt, at_mono/ qed-.
45 lemma at_basic_inv_ge: ∀m,n,i,j. m ≤ i → @❪i, 𝐁❨m,n❩❫ ≘ j → n+i = j.
46 /3 width=4 by at_basic_ge, at_mono/ qed-.