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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 "basic_2/notation/relations/rat_3.ma".
16 include "basic_2/grammar/term_vector.ma".
18 (* GENERIC RELOCATION WITH PAIRS ********************************************)
20 inductive at: list2 nat nat → relation nat ≝
21 | at_nil: ∀i. at (⟠) i i
22 | at_lt : ∀des,d,e,i1,i2. i1 < d →
23 at des i1 i2 → at ({d, e} @ des) i1 i2
24 | at_ge : ∀des,d,e,i1,i2. d ≤ i1 →
25 at des (i1 + e) i2 → at ({d, e} @ des) i1 i2
28 interpretation "application (generic relocation with pairs)"
29 'RAt i1 des i2 = (at des i1 i2).
31 (* Basic inversion lemmas ***************************************************)
33 fact at_inv_nil_aux: ∀des,i1,i2. @⦃i1, des⦄ ≡ i2 → des = ⟠ → i1 = i2.
34 #des #i1 #i2 * -des -i1 -i2
36 | #des #d #e #i1 #i2 #_ #_ #H destruct
37 | #des #d #e #i1 #i2 #_ #_ #H destruct
41 lemma at_inv_nil: ∀i1,i2. @⦃i1, ⟠⦄ ≡ i2 → i1 = i2.
42 /2 width=3 by at_inv_nil_aux/ qed-.
44 fact at_inv_cons_aux: ∀des,i1,i2. @⦃i1, des⦄ ≡ i2 →
45 ∀d,e,des0. des = {d, e} @ des0 →
46 i1 < d ∧ @⦃i1, des0⦄ ≡ i2 ∨
47 d ≤ i1 ∧ @⦃i1 + e, des0⦄ ≡ i2.
48 #des #i1 #i2 * -des -i1 -i2
49 [ #i #d #e #des #H destruct
50 | #des1 #d1 #e1 #i1 #i2 #Hid1 #Hi12 #d2 #e2 #des2 #H destruct /3 width=1 by or_introl, conj/
51 | #des1 #d1 #e1 #i1 #i2 #Hdi1 #Hi12 #d2 #e2 #des2 #H destruct /3 width=1 by or_intror, conj/
55 lemma at_inv_cons: ∀des,d,e,i1,i2. @⦃i1, {d, e} @ des⦄ ≡ i2 →
56 i1 < d ∧ @⦃i1, des⦄ ≡ i2 ∨
57 d ≤ i1 ∧ @⦃i1 + e, des⦄ ≡ i2.
58 /2 width=3 by at_inv_cons_aux/ qed-.
60 lemma at_inv_cons_lt: ∀des,d,e,i1,i2. @⦃i1, {d, e} @ des⦄ ≡ i2 →
61 i1 < d → @⦃i1, des⦄ ≡ i2.
63 elim (at_inv_cons … H) -H * // #Hdi1 #_ #Hi1d
64 lapply (le_to_lt_to_lt … Hdi1 Hi1d) -Hdi1 -Hi1d #Hd
65 elim (lt_refl_false … Hd)
68 lemma at_inv_cons_ge: ∀des,d,e,i1,i2. @⦃i1, {d, e} @ des⦄ ≡ i2 →
69 d ≤ i1 → @⦃i1 + e, des⦄ ≡ i2.
71 elim (at_inv_cons … H) -H * // #Hi1d #_ #Hdi1
72 lapply (le_to_lt_to_lt … Hdi1 Hi1d) -Hdi1 -Hi1d #Hd
73 elim (lt_refl_false … Hd)