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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 "arithmetics/nat.ma".
16 include "Ground_2/star.ma".
18 (* ARITHMETICAL PROPERTIES **************************************************)
20 axiom eq_nat_dec: ∀n1,n2:nat. Decidable (n1 = n2).
22 axiom lt_dec: ∀n1,n2. Decidable (n1 < n2).
24 lemma plus_S_eq_O_false: ∀n,m. n + S m = 0 → False.
25 #n #m <plus_n_Sm #H destruct
28 lemma plus_S_le_to_pos: ∀n,m,p. n + S m ≤ p → 0 < p.
29 #n #m #p <plus_n_Sm #H @(lt_to_le_to_lt … H) //
32 lemma minus_le: ∀m,n. m - n ≤ m.
35 lemma le_O_to_eq_O: ∀n. n ≤ 0 → n = 0.
38 lemma lt_to_le: ∀a,b. a < b → a ≤ b.
41 lemma lt_refl_false: ∀n. n < n → False.
42 #n #H elim (lt_to_not_eq … H) -H /2/
45 lemma lt_zero_false: ∀n. n < 0 → False.
46 #n #H elim (lt_to_not_le … H) -H /2/
49 lemma lt_or_ge: ∀m,n. m < n ∨ n ≤ m.
50 #m #n elim (decidable_lt m n) /3/
53 lemma lt_or_eq_or_gt: ∀m,n. ∨∨ m < n | n = m | n < m.
57 #n elim (IHm n) -IHm #H
58 [ @or3_intro0 | @or3_intro1 destruct | @or3_intro2 ] /2/ (**) (* /3/ is slow *)
61 lemma le_to_lt_or_eq: ∀m,n. m ≤ n → m < n ∨ m = n.
62 /2/ qed. (**) (* REMOVE: this is le_to_or_lt_eq *)
64 lemma plus_le_weak: ∀m,n,p. m + n ≤ p → n ≤ p.
67 lemma plus_lt_false: ∀m,n. m + n < m → False.
68 #m #n #H1 lapply (le_plus_n_r n m) #H2
69 lapply (le_to_lt_to_lt … H2 H1) -H2 H1 #H
70 elim (lt_refl_false … H)
73 lemma le_fwd_plus_plus_ge: ∀m1,m2. m2 ≤ m1 → ∀n1,n2. m1 + n1 ≤ m2 + n2 → n1 ≤ n2.
74 #m1 #m2 #H elim H -H m1
76 | #m1 #_ #IHm1 #n1 #n2 #H @IHm1 /2/
80 lemma monotonic_lt_minus_l: ∀p,q,n. n ≤ q → q < p → q - n < p - n.
82 @lt_plus_to_minus_r <plus_minus_m_m //.
85 lemma plus_le_minus: ∀a,b,c. a + b ≤ c → a ≤ c - b.
88 lemma lt_plus_minus: ∀i,u,d. u ≤ i → i < d + u → i - u < d.
91 lemma plus_plus_comm_23: ∀m,n,p. m + n + p = m + p + n.
94 lemma le_plus_minus_comm: ∀n,m,p. p ≤ m → m + n - p = m - p + n.
95 #n #m #p #lepm @plus_to_minus <associative_plus
96 >(commutative_plus p) <plus_minus_m_m //
99 lemma minus_le_minus_minus_comm: ∀b,c,a. c ≤ b → a - (b - c) = a + c - b.
101 [ #c #a #H >(le_O_to_eq_O … H) -H //
102 | #b #IHb #c elim c -c //
104 lapply (le_S_S_to_le … Hcb) -Hcb #Hcb
105 <plus_n_Sm normalize /2/
109 lemma minus_plus_comm: ∀a,b,c. a - b - c = a - (c + b).
112 lemma minus_minus_comm: ∀a,b,c. a - b - c = a - c - b.
115 lemma le_plus_minus: ∀a,b,c. c ≤ b → a + b - c = a + (b - c).
118 lemma plus_minus_m_m_comm: ∀n,m. m ≤ n → n = m + (n - m).
121 lemma minus_plus_m_m_comm: ∀n,m. n = (m + n) - m.
124 lemma arith_a2: ∀a,c1,c2. c1 + c2 ≤ a → a - c1 - c2 + (c1 + c2) = a.
127 lemma arith_b1: ∀a,b,c1. c1 ≤ b → a - c1 - (b - c1) = a - b.
128 #a #b #c1 #H >minus_plus @eq_f2 /2/
131 lemma arith_b2: ∀a,b,c1,c2. c1 + c2 ≤ b → a - c1 - c2 - (b - c1 - c2) = a - b.
132 #a #b #c1 #c2 #H >minus_plus >minus_plus >minus_plus /2/
135 lemma arith_c1: ∀a,b,c1. a + c1 - (b + c1) = a - b.
138 lemma arith_c1x: ∀x,a,b,c1. x + c1 + a - (b + c1) = x + a - b.
139 #x #a #b #c1 >plus_plus_comm_23 //
142 lemma arith_d1: ∀a,b,c1. c1 ≤ b → a + c1 + (b - c1) = a + b.
145 lemma arith_e2: ∀a,c1,c2. a ≤ c1 → c1 + c2 - (c1 - a + c2) = a.
148 lemma arith_f1: ∀a,b,c1. a + b ≤ c1 → c1 - (c1 - a - b) = a + b.
149 #a #b #c1 #H >minus_plus <minus_minus //
152 lemma arith_g1: ∀a,b,c1. c1 ≤ b → a - (b - c1) - c1 = a - b.
155 lemma arith_h1: ∀a1,a2,b,c1. c1 ≤ a1 → c1 ≤ b →
156 a1 - c1 + a2 - (b - c1) = a1 + a2 - b.
157 #a1 #a2 #b #c1 #H1 #H2 <le_plus_minus_comm /2/
160 lemma arith_i2: ∀a,c1,c2. c1 + c2 ≤ a → c1 + c2 + (a - c1 - c2) = a.
163 lemma arith_z1: ∀a,b,c1. a + c1 - b - c1 = a - b.
166 (* unstable *****************************************************************)
168 lemma arith1: ∀n,h,m,p. n + h + m ≤ p + h → n + m ≤ p.
171 lemma arith2: ∀j,i,e,d. d + e ≤ i → d ≤ i - e + j.
172 #j #i #e #d #H lapply (plus_le_minus … H) -H /2/
175 lemma arith3: ∀a1,a2,b,c1. a1 + a2 ≤ b → a1 + c1 + a2 ≤ b + c1.
178 lemma arith4: ∀h,d,e1,e2. d ≤ e1 + e2 → d + h ≤ e1 + h + e2.
181 lemma arith5: ∀a,b1,b2,c1. c1 ≤ b1 → c1 ≤ a → a < b1 + b2 → a - c1 < b1 - c1 + b2.
182 #a #b1 #b2 #c1 #H1 #H2 #H3
183 <le_plus_minus_comm // @monotonic_lt_minus_l //
186 lemma arith8: ∀a,b. a < a + b + 1.
189 lemma arith9: ∀a,b,c. c < a + (b + c + 1) + 1.
192 lemma arith10: ∀a,b,c,d,e. a ≤ b → c + (a - d - e) ≤ c + (b - d - e).
194 >minus_plus >minus_plus @monotonic_le_plus_r @monotonic_le_minus_l //