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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 "arithmetics/nat.ma".
16 include "ground_2/star.ma".
18 (* ARITHMETICAL PROPERTIES **************************************************)
20 (* Equations ****************************************************************)
22 lemma plus_n_2: ∀n. n + 2 = n + 1 + 1.
25 lemma le_plus_minus: ∀m,n,p. p ≤ n → m + n - p = m + (n - p).
26 /2 by plus_minus/ qed.
28 lemma le_plus_minus_comm: ∀n,m,p. p ≤ m → m + n - p = m - p + n.
29 /2 by plus_minus/ qed.
31 lemma arith_b1: ∀a,b,c1. c1 ≤ b → a - c1 - (b - c1) = a - b.
32 #a #b #c1 #H >minus_minus_comm >minus_le_minus_minus_comm //
35 lemma arith_b2: ∀a,b,c1,c2. c1 + c2 ≤ b → a - c1 - c2 - (b - c1 - c2) = a - b.
36 #a #b #c1 #c2 #H >minus_plus >minus_plus >minus_plus /2 width=1/
39 lemma arith_c1x: ∀x,a,b,c1. x + c1 + a - (b + c1) = x + a - b.
40 /3 by monotonic_le_minus_l, le_to_le_to_eq, le_n/ qed.
42 lemma arith_h1: ∀a1,a2,b,c1. c1 ≤ a1 → c1 ≤ b →
43 a1 - c1 + a2 - (b - c1) = a1 + a2 - b.
44 #a1 #a2 #b #c1 #H1 #H2 >plus_minus // /2 width=1/
47 (* Inversion & forward lemmas ***********************************************)
49 axiom eq_nat_dec: ∀n1,n2:nat. Decidable (n1 = n2).
51 axiom lt_dec: ∀n1,n2. Decidable (n1 < n2).
53 lemma lt_or_eq_or_gt: ∀m,n. ∨∨ m < n | n = m | n < m.
54 #m #n elim (lt_or_ge m n) /2 width=1/
55 #H elim H -m /2 width=1/
56 #m #Hm * #H /2 width=1/ /3 width=1/
59 lemma lt_refl_false: ∀n. n < n → ⊥.
60 #n #H elim (lt_to_not_eq … H) -H /2 width=1/
63 lemma lt_zero_false: ∀n. n < 0 → ⊥.
64 #n #H elim (lt_to_not_le … H) -H /2 width=1/
67 lemma false_lt_to_le: ∀x,y. (x < y → ⊥) → y ≤ x.
68 #x #y #H elim (decidable_lt x y) /2 width=1/
72 lemma le_plus_xySz_x_false: ∀y,z,x. x + y + S z ≤ x → ⊥.
74 [ #H lapply (le_n_O_to_eq … H) -H
75 <plus_n_Sm #H destruct
76 | /3 width=1 by le_S_S_to_le/
80 lemma plus_xySz_x_false: ∀z,x,y. x + y + S z = x → ⊥.
81 /2 width=4 by le_plus_xySz_x_false/ qed-.
83 (* Iterators ****************************************************************)
85 (* Note: see also: lib/arithemetcs/bigops.ma *)
86 let rec iter (n:nat) (B:Type[0]) (op: B → B) (nil: B) ≝
89 | S k ⇒ op (iter k B op nil)
92 interpretation "iterated function" 'exp op n = (iter n ? op).
94 lemma iter_SO: ∀B:Type[0]. ∀f:B→B. ∀b,l. f^(l+1) b = f (f^l b).
95 #B #f #b #l >commutative_plus //
98 lemma iter_n_Sm: ∀B:Type[0]. ∀f:B→B. ∀b,l. f^l (f b) = f (f^l b).
99 #B #f #b #l elim l -l normalize //
102 (* Trichotomy operator ******************************************************)
104 (* Note: this is "if eqb n1 n2 then a2 else if leb n1 n2 then a1 else a3" *)
105 let rec tri (A:Type[0]) n1 n2 a1 a2 a3 on n1 : A ≝
107 [ O ⇒ match n2 with [ O ⇒ a2 | S n2 ⇒ a1 ]
108 | S n1 ⇒ match n2 with [ O ⇒ a3 | S n2 ⇒ tri A n1 n2 a1 a2 a3 ]
111 lemma tri_lt: ∀A,a1,a2,a3,n2,n1. n1 < n2 → tri A n1 n2 a1 a2 a3 = a1.
112 #A #a1 #a2 #a3 #n2 elim n2 -n2
113 [ #n1 #H elim (lt_zero_false … H)
114 | #n2 #IH #n1 elim n1 -n1 // /3 width=1/
118 lemma tri_eq: ∀A,a1,a2,a3,n. tri A n n a1 a2 a3 = a2.
119 #A #a1 #a2 #a3 #n elim n -n normalize //
122 lemma tri_gt: ∀A,a1,a2,a3,n1,n2. n2 < n1 → tri A n1 n2 a1 a2 a3 = a3.
123 #A #a1 #a2 #a3 #n1 elim n1 -n1
124 [ #n2 #H elim (lt_zero_false … H)
125 | #n1 #IH #n2 elim n2 -n2 // /3 width=1/