Theory Word_Lib.More_Divides
section ‹Lemmas on division›
theory More_Divides
imports
"HOL-Library.Word"
begin
declare div_eq_dividend_iff [simp]
lemma int_div_same_is_1 [simp]:
‹a div b = a ⟷ b = 1› if ‹0 < a› for a b :: int
by (metis div_by_1 int_div_less_self less_le_not_le nle_le nonneg1_imp_zdiv_pos_iff that)
lemma int_div_minus_is_minus1 [simp]:
‹a div b = - a ⟷ b = - 1› if ‹0 > a› for a b :: int
using that by (metis div_minus_right equation_minus_iff int_div_same_is_1 neg_0_less_iff_less)
lemma nat_div_eq_Suc_0_iff: "n div m = Suc 0 ⟷ m ≤ n ∧ n < 2 * m" (is "?L=?R")
proof
show "?L⟹?R"
by (metis div_greater_zero_iff div_less_iff_less_mult lessI numeral_2_eq_2)
qed (simp add: div_nat_eqI)
lemma diff_mod_le:
‹a - a mod b ≤ d - b› if ‹a < d› ‹b dvd d› for a b d :: nat
proof(cases ‹b = 0›)
case True
then show ?thesis
by auto
next
case False
then obtain k where k: "d = b * k"
using ‹b dvd d› by blast
then have "a div b < k"
by (metis less_mult_imp_div_less mult.commute that(1))
then have "b * (a div b) ≤ b * (k - 1)"
by auto
then show ?thesis
by (simp add: k minus_mod_eq_mult_div right_diff_distrib')
qed
lemma one_mod_exp_eq_one [simp]:
"1 mod (2 * 2 ^ n) = (1::int)"
using power_gt1 [of 2 n] by (auto intro: mod_pos_pos_trivial)
lemma int_mod_lem: "0 < n ⟹ 0 ≤ b ∧ b < n ⟷ b mod n = b"
for b n :: int
using zmod_trivial_iff by force
lemma int_mod_ge': "b < 0 ⟹ 0 < n ⟹ b + n ≤ b mod n"
for b n :: int
by (metis add_less_same_cancel2 int_mod_ge mod_add_self2)
lemma int_mod_le': "0 ≤ b - n ⟹ b mod n ≤ b - n"
for b n :: int
by (metis minus_mod_self2 zmod_le_nonneg_dividend)
lemma emep1: "even n ⟹ even d ⟹ 0 ≤ d ⟹ (n + 1) mod d = (n mod d) + 1"
for n d :: int
by (auto simp add: pos_zmod_mult_2 add.commute dvd_def)
lemma m1mod2k: "- 1 mod 2 ^ n = (2 ^ n - 1 :: int)"
by (rule zmod_minus1) simp
lemma sb_inc_lem: "a + 2^k < 0 ⟹ a + 2^k + 2^(Suc k) ≤ (a + 2^k) mod 2^(Suc k)"
for a :: int
using int_mod_ge' [where n = "2 ^ (Suc k)" and b = "a + 2 ^ k"]
by simp
lemma sb_inc_lem': "a < - (2^k) ⟹ a + 2^k + 2^(Suc k) ≤ (a + 2^k) mod 2^(Suc k)"
for a :: int
by (rule sb_inc_lem) simp
lemma sb_dec_lem: "0 ≤ - (2 ^ k) + a ⟹ (a + 2 ^ k) mod (2 * 2 ^ k) ≤ - (2 ^ k) + a"
for a :: int
using int_mod_le'[where n = "2 ^ (Suc k)" and b = "a + 2 ^ k"] by simp
lemma sb_dec_lem': "2 ^ k ≤ a ⟹ (a + 2 ^ k) mod (2 * 2 ^ k) ≤ - (2 ^ k) + a"
for a :: int
by (rule sb_dec_lem) simp
lemma mod_2_neq_1_eq_eq_0: "k mod 2 ≠ 1 ⟷ k mod 2 = 0"
for k :: int
by (fact not_mod_2_eq_1_eq_0)
lemma z1pmod2: "(2 * b + 1) mod 2 = (1::int)"
for b :: int
by arith
lemma p1mod22k': "(1 + 2 * b) mod (2 * 2 ^ n) = 1 + 2 * (b mod 2 ^ n)"
for b :: int
by (rule pos_zmod_mult_2) simp
lemma p1mod22k: "(2 * b + 1) mod (2 * 2 ^ n) = 2 * (b mod 2 ^ n) + 1"
for b :: int
by (simp add: p1mod22k' add.commute)
lemma pos_mod_sign2:
‹0 ≤ a mod 2› for a :: int
by simp
lemma pos_mod_bound2:
‹a mod 2 < 2› for a :: int
by simp
lemma nmod2: "n mod 2 = 0 ∨ n mod 2 = 1"
for n :: int
by arith
lemma eme1p:
"even n ⟹ even d ⟹ 0 ≤ d ⟹ (1 + n) mod d = 1 + n mod d" for n d :: int
using emep1 [of n d] by (simp add: ac_simps)
lemma m1mod22k:
‹- 1 mod (2 * 2 ^ n) = 2 * 2 ^ n - (1::int)›
by (simp add: zmod_minus1)
lemma z1pdiv2: "(2 * b + 1) div 2 = b"
for b :: int
by arith
lemma zdiv_le_dividend:
‹0 ≤ a ⟹ 0 < b ⟹ a div b ≤ a› for a b :: int
by (metis div_by_1 int_one_le_iff_zero_less zdiv_mono2 zero_less_one)
lemma axxmod2: "(1 + x + x) mod 2 = 1 ∧ (0 + x + x) mod 2 = 0"
for x :: int
by arith
lemma axxdiv2: "(1 + x + x) div 2 = x ∧ (0 + x + x) div 2 = x"
for x :: int
by arith
lemmas rdmods =
mod_minus_eq [symmetric]
mod_diff_left_eq [symmetric]
mod_diff_right_eq [symmetric]
mod_add_left_eq [symmetric]
mod_add_right_eq [symmetric]
mod_mult_right_eq [symmetric]
mod_mult_left_eq [symmetric]
lemma mod_plus_right: "(a + x) mod m = (b + x) mod m ⟷ a mod m = b mod m"
for a b m x :: nat
by (induct x) (simp_all add: mod_Suc, arith)
lemma nat_minus_mod: "(n - n mod m) mod m = 0"
for m n :: nat
by (induct n) (simp_all add: mod_Suc)
lemmas nat_minus_mod_plus_right =
trans [OF nat_minus_mod mod_0 [symmetric],
THEN mod_plus_right [THEN iffD2], simplified]
lemmas push_mods' = mod_add_eq
mod_mult_eq mod_diff_eq
mod_minus_eq
lemmas push_mods = push_mods' [THEN eq_reflection]
lemmas pull_mods = push_mods [symmetric] rdmods [THEN eq_reflection]
lemma nat_mod_eq: "b < n ⟹ a mod n = b mod n ⟹ a mod n = b"
for a b n :: nat
by (induct a) auto
lemmas nat_mod_eq' = refl [THEN [2] nat_mod_eq]
lemma nat_mod_lem: "0 < n ⟹ b < n ⟷ b mod n = b"
for b n :: nat
by (metis mod_less_divisor nat_mod_eq')
lemma mod_nat_add: "x < z ⟹ y < z ⟹ (x + y) mod z = (if x + y < z then x + y else x + y - z)"
for x y z :: nat
using mod_if by auto
lemma mod_nat_sub: "x < z ⟹ (x - y) mod z = x - y"
for x y :: nat
by simp
lemma int_mod_eq: "0 ≤ b ⟹ b < n ⟹ a mod n = b mod n ⟹ a mod n = b"
for a b n :: int
by (metis mod_pos_pos_trivial)
lemma zmde:
‹b * (a div b) = a - a mod b› for a b :: ‹'a::{group_add,semiring_modulo}›
using mult_div_mod_eq [of b a] by (simp add: eq_diff_eq)
lemma zdiv_mult_self: "m ≠ 0 ⟹ (a + m * n) div m = a div m + n"
for a m n :: int
by simp
lemma mod_power_lem: "a > 1 ⟹ a ^ n mod a ^ m = (if m ≤ n then 0 else a ^ n)"
for a :: int
by (simp add: mod_eq_0_iff_dvd le_imp_power_dvd)
lemma nonneg_mod_div: "0 ≤ a ⟹ 0 ≤ b ⟹ 0 ≤ (a mod b) ∧ 0 ≤ a div b"
for a b :: int
by (cases "b = 0") (auto intro: pos_imp_zdiv_nonneg_iff [THEN iffD2])
lemma mod_exp_less_eq_exp:
‹a mod 2 ^ n < 2 ^ n› for a :: int
by (rule pos_mod_bound) simp
lemma div_mult_le:
‹a div b * b ≤ a› for a b :: nat
by (fact div_times_less_eq_dividend)
lemma power_sub:
fixes a :: nat
assumes lt: "n ≤ m"
and av: "0 < a"
shows "a ^ (m - n) = a ^ m div a ^ n"
using av less_irrefl lt power_diff by blast
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
using mod_less_divisor [where m = q and n = c]
by (smt (verit) div_eq_0_iff add.right_neutral div_mult2_eq div_mult_self3 not_less0 mult.commute)
lemma less_two_pow_divD:
"⟦ (x :: nat) < 2 ^ n div 2 ^ m ⟧ ⟹ n ≥ m ∧ (x < 2 ^ (n - m))"
by (metis One_nat_def div_less lessI not_less0 linorder_not_le numeral_2_eq_2 power_diff power_increasing_iff)
lemma less_two_pow_divI:
"⟦ (x :: nat) < 2 ^ (n - m); m ≤ n ⟧ ⟹ x < 2 ^ n div 2 ^ m"
by (simp add: power_sub)
lemmas m2pths = pos_mod_sign mod_exp_less_eq_exp
lemma power_mod_div:
fixes x :: "nat"
shows "x mod 2 ^ n div 2 ^ m = x div 2 ^ m mod 2 ^ (n - m)" (is "?LHS = ?RHS")
proof (cases "n ≤ m")
case True
then have "?LHS = 0"
by (metis diff_is_0_eq' drop_bit_eq_div drop_bit_take_bit take_bit_0 take_bit_eq_mod)
also have "… = ?RHS" using True
by simp
finally show ?thesis .
next
case False
then have lt: "m < n" by simp
then obtain q where nv: "n = m + q" and "0 < q"
by (auto dest: less_imp_Suc_add)
then have "x mod 2 ^ n = 2 ^ m * (x div 2 ^ m mod 2 ^ q) + x mod 2 ^ m"
by (simp add: power_add mod_mult2_eq)
then have "?LHS = x div 2 ^ m mod 2 ^ q"
by (simp add: div_add1_eq)
also have "… = ?RHS" using nv
by simp
finally show ?thesis .
qed
lemma mod_div_equality_div_eq:
"a div b * b = (a - (a mod b) :: int)"
by (simp add: field_simps)
lemma zmod_helper:
"n mod m = k ⟹ ((n :: int) + a) mod m = (k + a) mod m"
by (metis add.commute mod_add_right_eq)
lemma int_div_sub_1:
assumes "m ≥ 1"
shows "(n - (1 :: int)) div m = (if m dvd n then (n div m) - 1 else n div m)"
proof -
have "(n - 1) div m * m = n div m * m - 1 * m" if "m dvd n"
proof (cases "m = 1")
case False
with that show ?thesis
using assms
unfolding mod_div_equality_div_eq
by (smt (verit, ccfv_SIG) dvd_eq_mod_eq_0 int_mod_ge' mod_diff_eq pos_mod_bound pos_mod_sign)
qed auto
moreover
have "¬ m dvd n ⟹ (n - 1) div m * m = n div m * m"
by (smt (verit, del_insts) assms mod_0_imp_dvd pos_zdiv_mult_2 zdiv_mono1 zdiv_zminus1_eq_if)
ultimately
have "m = 0 ∨ (n - (1 :: int)) div m = (if m dvd n then (n div m) - 1 else n div m)"
by (metis left_diff_distrib' mult.commute nonzero_mult_div_cancel_left)
then show ?thesis
using assms by force
qed
lemma power_minus_is_div:
"b ≤ a ⟹ (2 :: nat) ^ (a - b) = 2 ^ a div 2 ^ b"
by (simp add: power_diff)
lemma two_pow_div_gt_le:
"v < 2 ^ n div (2 ^ m :: nat) ⟹ m ≤ n"
using less_two_pow_divD by blast
lemma td_gal_lt:
‹0 < c ⟹ a < b * c ⟷ a div c < b›
for a b c :: nat
by (simp add: div_less_iff_less_mult)
lemma td_gal:
‹0 < c ⟹ b * c ≤ a ⟷ b ≤ a div c›
for a b c :: nat
by (simp add: less_eq_div_iff_mult_less_eq)
end