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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 *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "basics/star.ma".
16 include "basics/lists/lstar.ma".
17 include "arithmetics/exp.ma".
19 include "xoa_notation.ma".
24 (* Note: For some reason this cannot be in the standard library *)
25 interpretation "logical false" 'false = False.
28 non associative with precedence 90
33 lemma lt_refl_false: ∀n. n < n → ⊥.
34 #n #H elim (lt_to_not_eq … H) -H /2 width=1/
37 lemma lt_zero_false: ∀n. n < 0 → ⊥.
38 #n #H elim (lt_to_not_le … H) -H /2 width=1/
41 lemma plus_lt_false: ∀m,n. m + n < m → ⊥.
42 #m #n #H elim (lt_to_not_le … H) -H /2 width=1/
45 lemma lt_or_eq_or_gt: ∀m,n. ∨∨ m < n | n = m | n < m.
46 #m #n elim (lt_or_ge m n) /2 width=1/
47 #H elim H -m /2 width=1/
48 #m #Hm * #H /2 width=1/ /3 width=1/
51 (* trichotomy operator *)
53 (* Note: this is "if eqb n1 n2 then a2 else if leb n1 n2 then a1 else a3" *)
54 let rec tri (A:Type[0]) n1 n2 a1 a2 a3 on n1 : A ≝
56 [ O ⇒ match n2 with [ O ⇒ a2 | S n2 ⇒ a1 ]
57 | S n1 ⇒ match n2 with [ O ⇒ a3 | S n2 ⇒ tri A n1 n2 a1 a2 a3 ]
60 lemma tri_lt: ∀A,a1,a2,a3,n2,n1. n1 < n2 → tri A n1 n2 a1 a2 a3 = a1.
61 #A #a1 #a2 #a3 #n2 elim n2 -n2
62 [ #n1 #H elim (lt_zero_false … H)
63 | #n2 #IH #n1 elim n1 -n1 // /3 width=1/
67 lemma tri_eq: ∀A,a1,a2,a3,n. tri A n n a1 a2 a3 = a2.
68 #A #a1 #a2 #a3 #n elim n -n normalize //
71 lemma tri_gt: ∀A,a1,a2,a3,n1,n2. n2 < n1 → tri A n1 n2 a1 a2 a3 = a3.
72 #A #a1 #a2 #a3 #n1 elim n1 -n1
73 [ #n2 #H elim (lt_zero_false … H)
74 | #n1 #IH #n2 elim n2 -n2 // /3 width=1/
80 (* Note: notation for nil not involving brackets *)
82 non associative with precedence 90
85 definition map_cons: ∀A. A → list (list A) → list (list A) ≝ λA,a.
88 interpretation "map_cons" 'ho_cons a l = (map_cons ? a l).
90 notation "hvbox(a ::: break l)"
91 right associative with precedence 47
92 for @{'ho_cons $a $l}.
96 (* Note: this cannot be in lib because of the missing xoa quantifier *)
97 lemma lstar_inv_pos: ∀A,B,R,l,b1,b2. lstar A B R l b1 b2 → 0 < |l| →
98 ∃∃a,ll,b. a::ll = l & R a b1 b & lstar A B R ll b b2.
99 #A #B #R #l #b1 #b2 #H @(lstar_ind_l ????????? H) -b1
100 [ #H elim (lt_refl_false … H)
101 | #a #ll #b1 #b #Hb1 #Hb2 #_ #_ /2 width=6/ (**) (* auto fail if we do not remove the inductive premise *)