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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/ynat/ynat_lt.ma".
16 include "basic_2/notation/relations/rat_3.ma".
17 include "basic_2/grammar/term_vector.ma".
19 (* MULTIPLE RELOCATION WITH PAIRS *******************************************)
21 inductive at: list2 ynat nat → relation nat ≝
22 | at_nil: ∀i. at (◊) i i
23 | at_lt : ∀cs,l,m,i1,i2. yinj i1 < l →
24 at cs i1 i2 → at ({l, m} @ cs) i1 i2
25 | at_ge : ∀cs,l,m,i1,i2. l ≤ yinj i1 →
26 at cs (i1 + m) i2 → at ({l, m} @ cs) i1 i2
29 interpretation "application (multiple relocation with pairs)"
30 'RAt i1 cs i2 = (at cs i1 i2).
32 (* Basic inversion lemmas ***************************************************)
34 fact at_inv_nil_aux: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 → cs = ◊ → i1 = i2.
35 #cs #i1 #i2 * -cs -i1 -i2
37 | #cs #l #m #i1 #i2 #_ #_ #H destruct
38 | #cs #l #m #i1 #i2 #_ #_ #H destruct
42 lemma at_inv_nil: ∀i1,i2. @⦃i1, ◊⦄ ≡ i2 → i1 = i2.
43 /2 width=3 by at_inv_nil_aux/ qed-.
45 fact at_inv_cons_aux: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 →
46 ∀l,m,cs0. cs = {l, m} @ cs0 →
47 i1 < l ∧ @⦃i1, cs0⦄ ≡ i2 ∨
48 l ≤ i1 ∧ @⦃i1 + m, cs0⦄ ≡ i2.
49 #cs #i1 #i2 * -cs -i1 -i2
50 [ #i #l #m #cs #H destruct
51 | #cs1 #l1 #m1 #i1 #i2 #Hil1 #Hi12 #l2 #m2 #cs2 #H destruct /3 width=1 by or_introl, conj/
52 | #cs1 #l1 #m1 #i1 #i2 #Hli1 #Hi12 #l2 #m2 #cs2 #H destruct /3 width=1 by or_intror, conj/
56 lemma at_inv_cons: ∀cs,l,m,i1,i2. @⦃i1, {l, m} @ cs⦄ ≡ i2 →
57 i1 < l ∧ @⦃i1, cs⦄ ≡ i2 ∨
58 l ≤ i1 ∧ @⦃i1 + m, cs⦄ ≡ i2.
59 /2 width=3 by at_inv_cons_aux/ qed-.
61 lemma at_inv_cons_lt: ∀cs,l,m,i1,i2. @⦃i1, {l, m} @ cs⦄ ≡ i2 →
62 i1 < l → @⦃i1, cs⦄ ≡ i2.
64 elim (at_inv_cons … H) -H * // #Hli1 #_ #Hi1l
65 elim (ylt_yle_false … Hi1l Hli1)
68 lemma at_inv_cons_ge: ∀cs,l,m,i1,i2. @⦃i1, {l, m} @ cs⦄ ≡ i2 →
69 l ≤ i1 → @⦃i1 + m, cs⦄ ≡ i2.
71 elim (at_inv_cons … H) -H * // #Hi1l #_ #Hli1
72 elim (ylt_yle_false … Hi1l Hli1)