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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "ground_2/notation/relations/rat_3.ma".
16 include "ground_2/relocation/mr2.ma".
18 (* MULTIPLE RELOCATION WITH PAIRS *******************************************)
20 inductive at: mr2 → relation nat ≝
21 | at_nil: ∀i. at (◊) i i
22 | at_lt : ∀cs,l,m,i1,i2. i1 < l →
23 at cs i1 i2 → at ({l, m};cs) i1 i2
24 | at_ge : ∀cs,l,m,i1,i2. l ≤ i1 →
25 at cs (i1 + m) i2 → at ({l, m};cs) i1 i2
28 interpretation "application (multiple relocation with pairs)"
29 'RAt i1 cs i2 = (at cs i1 i2).
31 (* Basic inversion lemmas ***************************************************)
33 fact at_inv_nil_aux: ∀cs,i1,i2. @⦃i1, cs⦄ ≘ i2 → cs = ◊ → i1 = i2.
34 #cs #i1 #i2 * -cs -i1 -i2
36 | #cs #l #m #i1 #i2 #_ #_ #H destruct
37 | #cs #l #m #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: ∀cs,i1,i2. @⦃i1, cs⦄ ≘ i2 →
45 ∀l,m,cs0. cs = {l, m};cs0 →
46 i1 < l ∧ @⦃i1, cs0⦄ ≘ i2 ∨
47 l ≤ i1 ∧ @⦃i1 + m, cs0⦄ ≘ i2.
48 #cs #i1 #i2 * -cs -i1 -i2
49 [ #i #l #m #cs #H destruct
50 | #cs1 #l1 #m1 #i1 #i2 #Hil1 #Hi12 #l2 #m2 #cs2 #H destruct /3 width=1 by or_introl, conj/
51 | #cs1 #l1 #m1 #i1 #i2 #Hli1 #Hi12 #l2 #m2 #cs2 #H destruct /3 width=1 by or_intror, conj/
55 lemma at_inv_cons: ∀cs,l,m,i1,i2. @⦃i1, {l, m};cs⦄ ≘ i2 →
56 i1 < l ∧ @⦃i1, cs⦄ ≘ i2 ∨
57 l ≤ i1 ∧ @⦃i1 + m, cs⦄ ≘ i2.
58 /2 width=3 by at_inv_cons_aux/ qed-.
60 lemma at_inv_cons_lt: ∀cs,l,m,i1,i2. @⦃i1, {l, m};cs⦄ ≘ i2 →
61 i1 < l → @⦃i1, cs⦄ ≘ i2.
63 elim (at_inv_cons … H) -H * // #Hli1 #_ #Hi1l
64 elim (lt_le_false … Hi1l Hli1)
67 lemma at_inv_cons_ge: ∀cs,l,m,i1,i2. @⦃i1, {l, m};cs⦄ ≘ i2 →
68 l ≤ i1 → @⦃i1 + m, cs⦄ ≘ i2.
70 elim (at_inv_cons … H) -H * // #Hi1l #_ #Hli1
71 elim (lt_le_false … Hi1l Hli1)
74 (* Main properties **********************************************************)
76 theorem at_mono: ∀cs,i,i1. @⦃i, cs⦄ ≘ i1 → ∀i2. @⦃i, cs⦄ ≘ i2 → i1 = i2.
77 #cs #i #i1 #H elim H -cs -i -i1
78 [ #i #x #H <(at_inv_nil … H) -x //
79 | #cs #l #m #i #i1 #Hil #_ #IHi1 #x #H
80 lapply (at_inv_cons_lt … H Hil) -H -Hil /2 width=1 by/
81 | #cs #l #m #i #i1 #Hli #_ #IHi1 #x #H
82 lapply (at_inv_cons_ge … H Hli) -H -Hli /2 width=1 by/