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4 (* ||A|| A project by Andrea Asperti *)
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15 include "ground_2/ynat/ynat_minus.ma".
16 include "basic_2/notation/relations/rminus_3.ma".
17 include "basic_2/multiple/mr2.ma".
19 (* MULTIPLE RELOCATION WITH PAIRS *******************************************)
21 inductive minuss: nat → relation (list2 ynat nat) ≝
22 | minuss_nil: ∀i. minuss i (◊) (◊)
23 | minuss_lt : ∀cs1,cs2,l,m,i. yinj i < l → minuss i cs1 cs2 →
24 minuss i ({l, m} @ cs1) ({l - i, m} @ cs2)
25 | minuss_ge : ∀cs1,cs2,l,m,i. l ≤ yinj i → minuss (m + i) cs1 cs2 →
26 minuss i ({l, m} @ cs1) cs2
29 interpretation "minus (multiple relocation with pairs)"
30 'RMinus cs1 i cs2 = (minuss i cs1 cs2).
32 (* Basic inversion lemmas ***************************************************)
34 fact minuss_inv_nil1_aux: ∀cs1,cs2,i. cs1 ▭ i ≡ cs2 → cs1 = ◊ → cs2 = ◊.
35 #cs1 #cs2 #i * -cs1 -cs2 -i
37 | #cs1 #cs2 #l #m #i #_ #_ #H destruct
38 | #cs1 #cs2 #l #m #i #_ #_ #H destruct
42 lemma minuss_inv_nil1: ∀cs2,i. ◊ ▭ i ≡ cs2 → cs2 = ◊.
43 /2 width=4 by minuss_inv_nil1_aux/ qed-.
45 fact minuss_inv_cons1_aux: ∀cs1,cs2,i. cs1 ▭ i ≡ cs2 →
46 ∀l,m,cs. cs1 = {l, m} @ cs →
47 l ≤ i ∧ cs ▭ m + i ≡ cs2 ∨
48 ∃∃cs0. i < l & cs ▭ i ≡ cs0 &
49 cs2 = {l - i, m} @ cs0.
50 #cs1 #cs2 #i * -cs1 -cs2 -i
51 [ #i #l #m #cs #H destruct
52 | #cs1 #cs #l1 #m1 #i1 #Hil1 #Hcs #l2 #m2 #cs2 #H destruct /3 width=3 by ex3_intro, or_intror/
53 | #cs1 #cs #l1 #m1 #i1 #Hli1 #Hcs #l2 #m2 #cs2 #H destruct /3 width=1 by or_introl, conj/
57 lemma minuss_inv_cons1: ∀cs1,cs2,l,m,i. {l, m} @ cs1 ▭ i ≡ cs2 →
58 l ≤ i ∧ cs1 ▭ m + i ≡ cs2 ∨
59 ∃∃cs. i < l & cs1 ▭ i ≡ cs &
60 cs2 = {l - i, m} @ cs.
61 /2 width=3 by minuss_inv_cons1_aux/ qed-.
63 lemma minuss_inv_cons1_ge: ∀cs1,cs2,l,m,i. {l, m} @ cs1 ▭ i ≡ cs2 →
64 l ≤ i → cs1 ▭ m + i ≡ cs2.
66 elim (minuss_inv_cons1 … H) -H * // #cs #Hil #_ #_ #Hli
67 elim (ylt_yle_false … Hil Hli)
70 lemma minuss_inv_cons1_lt: ∀cs1,cs2,l,m,i. {l, m} @ cs1 ▭ i ≡ cs2 →
72 ∃∃cs. cs1 ▭ i ≡ cs & cs2 = {l - i, m} @ cs.
73 #cs1 #cs2 #l #m #i #H elim (minuss_inv_cons1 … H) -H * /2 width=3 by ex2_intro/
74 #Hli #_ #Hil elim (ylt_yle_false … Hil Hli)