# Theory Word_Lib.More_Divides

```(*
* Copyright Data61, CSIRO (ABN 41 687 119 230)
*
*)

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
using that by (metis div_by_1 abs_ge_zero abs_of_pos int_div_less_self neq_iff
nonneg1_imp_zdiv_pos_iff zabs_less_one_iff)

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"
apply auto
using div_greater_zero_iff apply fastforce
apply (metis One_nat_def div_greater_zero_iff dividend_less_div_times mult.right_neutral mult_Suc mult_numeral_1 numeral_2_eq_2 zero_less_numeral)
done

lemma diff_mod_le:
‹a - a mod b ≤ d - b› if ‹a < d› ‹b dvd d› for a b d :: nat
using that
apply(subst minus_mod_eq_mult_div)
apply(clarsimp simp: dvd_def)
apply(cases ‹b = 0›)
apply simp
apply(subgoal_tac "a div b ≤ k - 1")
prefer 2
apply(subgoal_tac "a div b < k")
apply(subgoal_tac "b * (a div b) < b * ((b * k) div b)")
apply clarsimp
apply(subst div_mult_self1_is_m)
apply arith
apply(rule le_less_trans)
apply simp
apply(subst mult.commute)
apply(rule div_times_less_eq_dividend)
apply assumption
apply clarsimp
apply(subgoal_tac "b * (a div b) ≤ b * (k - 1)")
apply(erule le_trans)
apply simp
done

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
apply safe
apply (erule (1) mod_pos_pos_trivial)
apply (erule_tac [!] subst)
apply auto
done

lemma int_mod_ge': "b < 0 ⟹ 0 < n ⟹ b + n ≤ b mod n"
for b n :: int

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

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

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)›

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_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]

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
apply safe
apply (erule nat_mod_eq')
apply (erule subst)
apply (erule mod_less_divisor)
done

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
apply (rule nat_mod_eq)
apply auto
apply (rule trans)
apply (rule le_mod_geq)
apply simp
apply (rule nat_mod_eq')
apply arith
done

lemma mod_nat_sub: "x < z ⟹ (x - y) mod z = x - y"
for x y :: nat
by (rule nat_mod_eq') arith

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)

(* already have this for naturals, div_mult_self1/2, but not for ints *)
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

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"
proof (subst nat_mult_eq_cancel1 [symmetric])
show "(0::nat) < a ^ n" using av by simp
next
from lt obtain q where mv: "n + q = m"

have "a ^ n * (a ^ m div a ^ n) = a ^ m"
proof (subst mult.commute)
have "a ^ m = (a ^ m div a ^ n) * a ^ n + a ^ m mod a ^ n"
by (rule  div_mult_mod_eq [symmetric])

moreover have "a ^ m mod a ^ n = 0"
by (subst mod_eq_0_iff_dvd, subst dvd_def, rule exI [where x = "a ^ q"],
(subst power_add [symmetric] mv)+, rule refl)

ultimately show "(a ^ m div a ^ n) * a ^ n = a ^ m" by simp
qed

then show "a ^ n * a ^ (m - n) = a ^ n * (a ^ m div a ^ n)" using lt
qed

lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
apply (cut_tac m = q and n = c in mod_less_divisor)
apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
apply (erule_tac P = "%x. lhs < rhs x" for lhs rhs in ssubst)
done

lemma less_two_pow_divD:
"⟦ (x :: nat) < 2 ^ n div 2 ^ m ⟧
⟹ n ≥ m ∧ (x < 2 ^ (n - m))"
apply (rule context_conjI)
apply (rule ccontr)
done

lemma less_two_pow_divI:
"⟦ (x :: nat) < 2 ^ (n - m); m ≤ n ⟧ ⟹ x < 2 ^ n div 2 ^ m"

lemmas m2pths = pos_mod_sign mod_exp_less_eq_exp

lemmas int_mod_eq' = mod_pos_pos_trivial (* FIXME delete *)

lemmas int_mod_le = zmod_le_nonneg_dividend (* FIXME: delete *)

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"
apply -
apply (rule div_less)
apply (rule order_less_le_trans [OF mod_less_divisor]; simp)
done
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"

then have "x mod 2 ^ n = 2 ^ m * (x div 2 ^ m mod 2 ^ q) + x mod 2 ^ m"

then have "?LHS = x div 2 ^ m mod 2 ^ q"

also have "… = ?RHS" using nv
by simp

finally show ?thesis .
qed

lemma mod_mod_power:
fixes k :: nat
shows "k mod 2 ^ m mod 2 ^ n = k mod 2 ^ (min m n)"
proof (cases "m ≤ n")
case True

then have "k mod 2 ^ m mod 2 ^ n = k mod 2 ^ m"
apply -
apply (subst mod_less [where n = "2 ^ n"])
apply (rule order_less_le_trans [OF mod_less_divisor])
apply simp+
done
also have "… = k mod  2 ^ (min m n)" using True by simp
finally show ?thesis .
next
case False
then have "n < m" by simp
then obtain d where md: "m = n + d"
then have "k mod 2 ^ m = 2 ^ n * (k div 2 ^ n mod 2 ^ d) + k mod 2 ^ n"
then have "k mod 2 ^ m mod 2 ^ n = k mod 2 ^ n"
then show ?thesis using False
by simp
qed

lemma mod_div_equality_div_eq:
"a div b * b = (a - (a mod b) :: int)"

lemma zmod_helper:
"n mod m = k ⟹ ((n :: int) + a) mod m = (k + a) mod m"

lemma int_div_sub_1:
"⟦ m ≥ 1 ⟧ ⟹ (n - (1 :: int)) div m = (if m dvd n then (n div m) - 1 else n div m)"
apply (subgoal_tac "m = 0 ∨ (n - (1 :: int)) div m = (if m dvd n then (n div m) - 1 else n div m)")
apply fastforce
apply (subst mult_cancel_right[symmetric])
apply (simp only: left_diff_distrib split: if_split)
apply (simp only: mod_div_equality_div_eq)
apply (clarsimp simp: field_simps)
apply (clarsimp simp: dvd_eq_mod_eq_0)
apply (cases "m = 1")
apply simp
apply (subst mod_diff_eq[symmetric], simp add: zmod_minus1)
apply clarsimp
apply (rule mod_pos_pos_trivial)
apply simp
apply simp
apply (cases "(n - 1) mod m = m - 1")
apply (drule zmod_helper[where a=1])
apply simp
apply (subgoal_tac "1 + (n - 1) mod m ≤ m")
apply simp
apply simp
done

lemma power_minus_is_div:
"b ≤ a ⟹ (2 :: nat) ^ (a - b) = 2 ^ a div 2 ^ b"
apply (induct a arbitrary: b)
apply simp
apply (erule le_SucE)
apply simp
done

lemma two_pow_div_gt_le:
"v < 2 ^ n div (2 ^ m :: nat) ⟹ m ≤ n"
by (clarsimp dest!: less_two_pow_divD)

lemma td_gal_lt:
‹0 < c ⟹ a < b * c ⟷ a div c < b›
for a b c :: nat