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15 include "ground_2/notation/relations/rminus_3.ma".
16 include "ground_2/relocation/mr2.ma".
18 (* MULTIPLE RELOCATION WITH PAIRS *******************************************)
20 inductive minuss: nat → relation mr2 ≝
21 | minuss_nil: ∀i. minuss i (◊) (◊)
22 | minuss_lt : ∀cs1,cs2,l,m,i. i < l → minuss i cs1 cs2 →
23 minuss i (❨l, m❩;cs1) (❨l - i, m❩;cs2)
24 | minuss_ge : ∀cs1,cs2,l,m,i. l ≤ i → minuss (m + i) cs1 cs2 →
25 minuss i (❨l, m❩;cs1) cs2
28 interpretation "minus (multiple relocation with pairs)"
29 'RMinus cs1 i cs2 = (minuss i cs1 cs2).
31 (* Basic inversion lemmas ***************************************************)
33 fact minuss_inv_nil1_aux: ∀cs1,cs2,i. cs1 ▭ i ≘ cs2 → cs1 = ◊ → cs2 = ◊.
34 #cs1 #cs2 #i * -cs1 -cs2 -i
36 | #cs1 #cs2 #l #m #i #_ #_ #H destruct
37 | #cs1 #cs2 #l #m #i #_ #_ #H destruct
41 lemma minuss_inv_nil1: ∀cs2,i. ◊ ▭ i ≘ cs2 → cs2 = ◊.
42 /2 width=4 by minuss_inv_nil1_aux/ qed-.
44 fact minuss_inv_cons1_aux: ∀cs1,cs2,i. cs1 ▭ i ≘ cs2 →
45 ∀l,m,cs. cs1 = ❨l, m❩;cs →
46 l ≤ i ∧ cs ▭ m + i ≘ cs2 ∨
47 ∃∃cs0. i < l & cs ▭ i ≘ cs0 &
49 #cs1 #cs2 #i * -cs1 -cs2 -i
50 [ #i #l #m #cs #H destruct
51 | #cs1 #cs #l1 #m1 #i1 #Hil1 #Hcs #l2 #m2 #cs2 #H destruct /3 width=3 by ex3_intro, or_intror/
52 | #cs1 #cs #l1 #m1 #i1 #Hli1 #Hcs #l2 #m2 #cs2 #H destruct /3 width=1 by or_introl, conj/
56 lemma minuss_inv_cons1: ∀cs1,cs2,l,m,i. ❨l, m❩;cs1 ▭ i ≘ cs2 →
57 l ≤ i ∧ cs1 ▭ m + i ≘ cs2 ∨
58 ∃∃cs. i < l & cs1 ▭ i ≘ cs &
60 /2 width=3 by minuss_inv_cons1_aux/ qed-.
62 lemma minuss_inv_cons1_ge: ∀cs1,cs2,l,m,i. ❨l, m❩;cs1 ▭ i ≘ cs2 →
63 l ≤ i → cs1 ▭ m + i ≘ cs2.
65 elim (minuss_inv_cons1 … H) -H * // #cs #Hil #_ #_ #Hli
66 elim (lt_le_false … Hil Hli)
69 lemma minuss_inv_cons1_lt: ∀cs1,cs2,l,m,i. ❨l, m❩;cs1 ▭ i ≘ cs2 →
71 ∃∃cs. cs1 ▭ i ≘ cs & cs2 = ❨l - i, m❩;cs.
72 #cs1 #cs2 #l #m #i #H elim (minuss_inv_cons1 … H) -H * /2 width=3 by ex2_intro/
73 #Hli #_ #Hil elim (lt_le_false … Hil Hli)