Theory Archimedean_Field

(*  Title:      HOL/Archimedean_Field.thy
    Author:     Brian Huffman
*)

section ‹Archimedean Fields, Floor and Ceiling Functions›

theory Archimedean_Field
imports Main
begin

lemma cInf_abs_ge:
  fixes S :: "'a::{linordered_idom,conditionally_complete_linorder} set"
  assumes "S ≠ {}"
    and bdd: "⋀x. x∈S ⟹ ¦x¦ ≤ a"
  shows "¦Inf S¦ ≤ a"
proof -
  have "Sup (uminus ` S) = - (Inf S)"
  proof (rule antisym)
    have "⋀x. x ∈ S ⟹ bdd_above (uminus ` S)"
      using bdd by (force simp: abs_le_iff bdd_above_def)
  then show "- (Inf S) ≤ Sup (uminus ` S)"
    by (meson cInf_greatest [OF ‹S ≠ {}›] cSUP_upper minus_le_iff)
  next
    have *: "⋀x. x ∈ S ⟹ Inf S ≤ x"
      by (meson abs_le_iff bdd bdd_below_def cInf_lower minus_le_iff)
    show "Sup (uminus ` S) ≤ - Inf S"
      using ‹S ≠ {}› by (force intro: * cSup_least)
  qed
  with cSup_abs_le [of "uminus ` S"] assms show ?thesis
    by fastforce
qed

lemma cSup_asclose:
  fixes S :: "'a::{linordered_idom,conditionally_complete_linorder} set"
  assumes S: "S ≠ {}"
    and b: "∀x∈S. ¦x - l¦ ≤ e"
  shows "¦Sup S - l¦ ≤ e"
proof -
  have *: "¦x - l¦ ≤ e ⟷ l - e ≤ x ∧ x ≤ l + e" for x l e :: 'a
    by arith
  have "bdd_above S"
    using b by (auto intro!: bdd_aboveI[of _ "l + e"])
  with S b show ?thesis
    unfolding * by (auto intro!: cSup_upper2 cSup_least)
qed

lemma cInf_asclose:
  fixes S :: "'a::{linordered_idom,conditionally_complete_linorder} set"
  assumes S: "S ≠ {}"
    and b: "∀x∈S. ¦x - l¦ ≤ e"
  shows "¦Inf S - l¦ ≤ e"
proof -
  have *: "¦x - l¦ ≤ e ⟷ l - e ≤ x ∧ x ≤ l + e" for x l e :: 'a
    by arith
  have "bdd_below S"
    using b by (auto intro!: bdd_belowI[of _ "l - e"])
  with S b show ?thesis
    unfolding * by (auto intro!: cInf_lower2 cInf_greatest)
qed


subsection ‹Class of Archimedean fields›

text ‹Archimedean fields have no infinite elements.›

class archimedean_field = linordered_field +
  assumes ex_le_of_int: "∃z. x ≤ of_int z"

lemma ex_less_of_int: "∃z. x < of_int z"
  for x :: "'a::archimedean_field"
proof -
  from ex_le_of_int obtain z where "x ≤ of_int z" ..
  then have "x < of_int (z + 1)" by simp
  then show ?thesis ..
qed

lemma ex_of_int_less: "∃z. of_int z < x"
  for x :: "'a::archimedean_field"
proof -
  from ex_less_of_int obtain z where "- x < of_int z" ..
  then have "of_int (- z) < x" by simp
  then show ?thesis ..
qed

lemma reals_Archimedean2: "∃n. x < of_nat n"
  for x :: "'a::archimedean_field"
proof -
  obtain z where "x < of_int z"
    using ex_less_of_int ..
  also have "… ≤ of_int (int (nat z))"
    by simp
  also have "… = of_nat (nat z)"
    by (simp only: of_int_of_nat_eq)
  finally show ?thesis ..
qed

lemma real_arch_simple: "∃n. x ≤ of_nat n"
  for x :: "'a::archimedean_field"
proof -
  obtain n where "x < of_nat n"
    using reals_Archimedean2 ..
  then have "x ≤ of_nat n"
    by simp
  then show ?thesis ..
qed

text ‹Archimedean fields have no infinitesimal elements.›

lemma reals_Archimedean:
  fixes x :: "'a::archimedean_field"
  assumes "0 < x"
  shows "∃n. inverse (of_nat (Suc n)) < x"
proof -
  from ‹0 < x› have "0 < inverse x"
    by (rule positive_imp_inverse_positive)
  obtain n where "inverse x < of_nat n"
    using reals_Archimedean2 ..
  then obtain m where "inverse x < of_nat (Suc m)"
    using ‹0 < inverse x› by (cases n) (simp_all del: of_nat_Suc)
  then have "inverse (of_nat (Suc m)) < inverse (inverse x)"
    using ‹0 < inverse x› by (rule less_imp_inverse_less)
  then have "inverse (of_nat (Suc m)) < x"
    using ‹0 < x› by (simp add: nonzero_inverse_inverse_eq)
  then show ?thesis ..
qed

lemma ex_inverse_of_nat_less:
  fixes x :: "'a::archimedean_field"
  assumes "0 < x"
  shows "∃n>0. inverse (of_nat n) < x"
  using reals_Archimedean [OF ‹0 < x›] by auto

lemma ex_less_of_nat_mult:
  fixes x :: "'a::archimedean_field"
  assumes "0 < x"
  shows "∃n. y < of_nat n * x"
proof -
  obtain n where "y / x < of_nat n"
    using reals_Archimedean2 ..
  with ‹0 < x› have "y < of_nat n * x"
    by (simp add: pos_divide_less_eq)
  then show ?thesis ..
qed


subsection ‹Existence and uniqueness of floor function›

lemma exists_least_lemma:
  assumes "¬ P 0" and "∃n. P n"
  shows "∃n. ¬ P n ∧ P (Suc n)"
proof -
  from ‹∃n. P n› have "P (Least P)"
    by (rule LeastI_ex)
  with ‹¬ P 0› obtain n where "Least P = Suc n"
    by (cases "Least P") auto
  then have "n < Least P"
    by simp
  then have "¬ P n"
    by (rule not_less_Least)
  then have "¬ P n ∧ P (Suc n)"
    using ‹P (Least P)› ‹Least P = Suc n› by simp
  then show ?thesis ..
qed

lemma floor_exists:
  fixes x :: "'a::archimedean_field"
  shows "∃z. of_int z ≤ x ∧ x < of_int (z + 1)"
proof (cases "0 ≤ x")
  case True
  then have "¬ x < of_nat 0"
    by simp
  then have "∃n. ¬ x < of_nat n ∧ x < of_nat (Suc n)"
    using reals_Archimedean2 by (rule exists_least_lemma)
  then obtain n where "¬ x < of_nat n ∧ x < of_nat (Suc n)" ..
  then have "of_int (int n) ≤ x ∧ x < of_int (int n + 1)"
    by simp
  then show ?thesis ..
next
  case False
  then have "¬ - x ≤ of_nat 0"
    by simp
  then have "∃n. ¬ - x ≤ of_nat n ∧ - x ≤ of_nat (Suc n)"
    using real_arch_simple by (rule exists_least_lemma)
  then obtain n where "¬ - x ≤ of_nat n ∧ - x ≤ of_nat (Suc n)" ..
  then have "of_int (- int n - 1) ≤ x ∧ x < of_int (- int n - 1 + 1)"
    by simp
  then show ?thesis ..
qed

lemma floor_exists1: "∃!z. of_int z ≤ x ∧ x < of_int (z + 1)"
  for x :: "'a::archimedean_field"
proof (rule ex_ex1I)
  show "∃z. of_int z ≤ x ∧ x < of_int (z + 1)"
    by (rule floor_exists)
next
  fix y z
  assume "of_int y ≤ x ∧ x < of_int (y + 1)"
    and "of_int z ≤ x ∧ x < of_int (z + 1)"
  with le_less_trans [of "of_int y" "x" "of_int (z + 1)"]
       le_less_trans [of "of_int z" "x" "of_int (y + 1)"] show "y = z"
    by (simp del: of_int_add)
qed


subsection ‹Floor function›

class floor_ceiling = archimedean_field +
  fixes floor :: "'a ⇒ int"  ("⌊_⌋")
  assumes floor_correct: "of_int ⌊x⌋ ≤ x ∧ x < of_int (⌊x⌋ + 1)"

lemma floor_unique: "of_int z ≤ x ⟹ x < of_int z + 1 ⟹ ⌊x⌋ = z"
  using floor_correct [of x] floor_exists1 [of x] by auto

lemma floor_eq_iff: "⌊x⌋ = a ⟷ of_int a ≤ x ∧ x < of_int a + 1"
using floor_correct floor_unique by auto

lemma of_int_floor_le [simp]: "of_int ⌊x⌋ ≤ x"
  using floor_correct ..

lemma le_floor_iff: "z ≤ ⌊x⌋ ⟷ of_int z ≤ x"
proof
  assume "z ≤ ⌊x⌋"
  then have "(of_int z :: 'a) ≤ of_int ⌊x⌋" by simp
  also have "of_int ⌊x⌋ ≤ x" by (rule of_int_floor_le)
  finally show "of_int z ≤ x" .
next
  assume "of_int z ≤ x"
  also have "x < of_int (⌊x⌋ + 1)" using floor_correct ..
  finally show "z ≤ ⌊x⌋" by (simp del: of_int_add)
qed

lemma floor_less_iff: "⌊x⌋ < z ⟷ x < of_int z"
  by (simp add: not_le [symmetric] le_floor_iff)

lemma less_floor_iff: "z < ⌊x⌋ ⟷ of_int z + 1 ≤ x"
  using le_floor_iff [of "z + 1" x] by auto

lemma floor_le_iff: "⌊x⌋ ≤ z ⟷ x < of_int z + 1"
  by (simp add: not_less [symmetric] less_floor_iff)

lemma floor_split[linarith_split]: "P ⌊t⌋ ⟷ (∀i. of_int i ≤ t ∧ t < of_int i + 1 ⟶ P i)"
  by (metis floor_correct floor_unique less_floor_iff not_le order_refl)

lemma floor_mono:
  assumes "x ≤ y"
  shows "⌊x⌋ ≤ ⌊y⌋"
proof -
  have "of_int ⌊x⌋ ≤ x" by (rule of_int_floor_le)
  also note ‹x ≤ y›
  finally show ?thesis by (simp add: le_floor_iff)
qed

lemma floor_less_cancel: "⌊x⌋ < ⌊y⌋ ⟹ x < y"
  by (auto simp add: not_le [symmetric] floor_mono)

lemma floor_of_int [simp]: "⌊of_int z⌋ = z"
  by (rule floor_unique) simp_all

lemma floor_of_nat [simp]: "⌊of_nat n⌋ = int n"
  using floor_of_int [of "of_nat n"] by simp

lemma le_floor_add: "⌊x⌋ + ⌊y⌋ ≤ ⌊x + y⌋"
  by (simp only: le_floor_iff of_int_add add_mono of_int_floor_le)


text ‹Floor with numerals.›

lemma floor_zero [simp]: "⌊0⌋ = 0"
  using floor_of_int [of 0] by simp

lemma floor_one [simp]: "⌊1⌋ = 1"
  using floor_of_int [of 1] by simp

lemma floor_numeral [simp]: "⌊numeral v⌋ = numeral v"
  using floor_of_int [of "numeral v"] by simp

lemma floor_neg_numeral [simp]: "⌊- numeral v⌋ = - numeral v"
  using floor_of_int [of "- numeral v"] by simp

lemma zero_le_floor [simp]: "0 ≤ ⌊x⌋ ⟷ 0 ≤ x"
  by (simp add: le_floor_iff)

lemma one_le_floor [simp]: "1 ≤ ⌊x⌋ ⟷ 1 ≤ x"
  by (simp add: le_floor_iff)

lemma numeral_le_floor [simp]: "numeral v ≤ ⌊x⌋ ⟷ numeral v ≤ x"
  by (simp add: le_floor_iff)

lemma neg_numeral_le_floor [simp]: "- numeral v ≤ ⌊x⌋ ⟷ - numeral v ≤ x"
  by (simp add: le_floor_iff)

lemma zero_less_floor [simp]: "0 < ⌊x⌋ ⟷ 1 ≤ x"
  by (simp add: less_floor_iff)

lemma one_less_floor [simp]: "1 < ⌊x⌋ ⟷ 2 ≤ x"
  by (simp add: less_floor_iff)

lemma numeral_less_floor [simp]: "numeral v < ⌊x⌋ ⟷ numeral v + 1 ≤ x"
  by (simp add: less_floor_iff)

lemma neg_numeral_less_floor [simp]: "- numeral v < ⌊x⌋ ⟷ - numeral v + 1 ≤ x"
  by (simp add: less_floor_iff)

lemma floor_le_zero [simp]: "⌊x⌋ ≤ 0 ⟷ x < 1"
  by (simp add: floor_le_iff)

lemma floor_le_one [simp]: "⌊x⌋ ≤ 1 ⟷ x < 2"
  by (simp add: floor_le_iff)

lemma floor_le_numeral [simp]: "⌊x⌋ ≤ numeral v ⟷ x < numeral v + 1"
  by (simp add: floor_le_iff)

lemma floor_le_neg_numeral [simp]: "⌊x⌋ ≤ - numeral v ⟷ x < - numeral v + 1"
  by (simp add: floor_le_iff)

lemma floor_less_zero [simp]: "⌊x⌋ < 0 ⟷ x < 0"
  by (simp add: floor_less_iff)

lemma floor_less_one [simp]: "⌊x⌋ < 1 ⟷ x < 1"
  by (simp add: floor_less_iff)

lemma floor_less_numeral [simp]: "⌊x⌋ < numeral v ⟷ x < numeral v"
  by (simp add: floor_less_iff)

lemma floor_less_neg_numeral [simp]: "⌊x⌋ < - numeral v ⟷ x < - numeral v"
  by (simp add: floor_less_iff)

lemma le_mult_floor_Ints:
  assumes "0 ≤ a" "a ∈ Ints"
  shows "of_int (⌊a⌋ * ⌊b⌋) ≤ (of_int⌊a * b⌋ :: 'a :: linordered_idom)"
  by (metis Ints_cases assms floor_less_iff floor_of_int linorder_not_less mult_left_mono of_int_floor_le of_int_less_iff of_int_mult)


text ‹Addition and subtraction of integers.›

lemma floor_add_int: "⌊x⌋ + z = ⌊x + of_int z⌋"
  using floor_correct [of x] by (simp add: floor_unique[symmetric])

lemma int_add_floor: "z + ⌊x⌋ = ⌊of_int z + x⌋"
  using floor_correct [of x] by (simp add: floor_unique[symmetric])

lemma one_add_floor: "⌊x⌋ + 1 = ⌊x + 1⌋"
  using floor_add_int [of x 1] by simp

lemma floor_diff_of_int [simp]: "⌊x - of_int z⌋ = ⌊x⌋ - z"
  using floor_add_int [of x "- z"] by (simp add: algebra_simps)

lemma floor_uminus_of_int [simp]: "⌊- (of_int z)⌋ = - z"
  by (metis floor_diff_of_int [of 0] diff_0 floor_zero)

lemma floor_diff_numeral [simp]: "⌊x - numeral v⌋ = ⌊x⌋ - numeral v"
  using floor_diff_of_int [of x "numeral v"] by simp

lemma floor_diff_one [simp]: "⌊x - 1⌋ = ⌊x⌋ - 1"
  using floor_diff_of_int [of x 1] by simp

lemma le_mult_floor:
  assumes "0 ≤ a" and "0 ≤ b"
  shows "⌊a⌋ * ⌊b⌋ ≤ ⌊a * b⌋"
proof -
  have "of_int ⌊a⌋ ≤ a" and "of_int ⌊b⌋ ≤ b"
    by (auto intro: of_int_floor_le)
  then have "of_int (⌊a⌋ * ⌊b⌋) ≤ a * b"
    using assms by (auto intro!: mult_mono)
  also have "a * b < of_int (⌊a * b⌋ + 1)"
    using floor_correct[of "a * b"] by auto
  finally show ?thesis
    unfolding of_int_less_iff by simp
qed

lemma floor_divide_of_int_eq: "⌊of_int k / of_int l⌋ = k div l"
  for k l :: int
proof (cases "l = 0")
  case True
  then show ?thesis by simp
next
  case False
  have *: "⌊of_int (k mod l) / of_int l :: 'a⌋ = 0"
  proof (cases "l > 0")
    case True
    then show ?thesis
      by (auto intro: floor_unique)
  next
    case False
    obtain r where "r = - l"
      by blast
    then have l: "l = - r"
      by simp
    with ‹l ≠ 0› False have "r > 0"
      by simp
    with l show ?thesis
      using pos_mod_bound [of r]
      by (auto simp add: zmod_zminus2_eq_if less_le field_simps intro: floor_unique)
  qed
  have "(of_int k :: 'a) = of_int (k div l * l + k mod l)"
    by simp
  also have "… = (of_int (k div l) + of_int (k mod l) / of_int l) * of_int l"
    using False by (simp only: of_int_add) (simp add: field_simps)
  finally have "(of_int k / of_int l :: 'a) = … / of_int l"
    by simp
  then have "(of_int k / of_int l :: 'a) = of_int (k div l) + of_int (k mod l) / of_int l"
    using False by (simp only:) (simp add: field_simps)
  then have "⌊of_int k / of_int l :: 'a⌋ = ⌊of_int (k div l) + of_int (k mod l) / of_int l :: 'a⌋"
    by simp
  then have "⌊of_int k / of_int l :: 'a⌋ = ⌊of_int (k mod l) / of_int l + of_int (k div l) :: 'a⌋"
    by (simp add: ac_simps)
  then have "⌊of_int k / of_int l :: 'a⌋ = ⌊of_int (k mod l) / of_int l :: 'a⌋ + k div l"
    by (simp add: floor_add_int)
  with * show ?thesis
    by simp
qed

lemma floor_divide_of_nat_eq: "⌊of_nat m / of_nat n⌋ = of_nat (m div n)"
  for m n :: nat
    by (metis floor_divide_of_int_eq of_int_of_nat_eq linordered_euclidean_semiring_class.of_nat_div)

lemma floor_divide_lower:
  fixes q :: "'a::floor_ceiling"
  shows "q > 0 ⟹ of_int ⌊p / q⌋ * q ≤ p"
  using of_int_floor_le pos_le_divide_eq by blast

lemma floor_divide_upper:
  fixes q :: "'a::floor_ceiling"
  shows "q > 0 ⟹ p < (of_int ⌊p / q⌋ + 1) * q"
  by (meson floor_eq_iff pos_divide_less_eq)


subsection ‹Ceiling function›

definition ceiling :: "'a::floor_ceiling ⇒ int"  ("⌈_⌉")
  where "⌈x⌉ = - ⌊- x⌋"

lemma ceiling_correct: "of_int ⌈x⌉ - 1 < x ∧ x ≤ of_int ⌈x⌉"
  unfolding ceiling_def using floor_correct [of "- x"]
  by (simp add: le_minus_iff)

lemma ceiling_unique: "of_int z - 1 < x ⟹ x ≤ of_int z ⟹ ⌈x⌉ = z"
  unfolding ceiling_def using floor_unique [of "- z" "- x"] by simp

lemma ceiling_eq_iff: "⌈x⌉ = a ⟷ of_int a - 1 < x ∧ x ≤ of_int a"
using ceiling_correct ceiling_unique by auto

lemma le_of_int_ceiling [simp]: "x ≤ of_int ⌈x⌉"
  using ceiling_correct ..

lemma ceiling_le_iff: "⌈x⌉ ≤ z ⟷ x ≤ of_int z"
  unfolding ceiling_def using le_floor_iff [of "- z" "- x"] by auto

lemma less_ceiling_iff: "z < ⌈x⌉ ⟷ of_int z < x"
  by (simp add: not_le [symmetric] ceiling_le_iff)

lemma ceiling_less_iff: "⌈x⌉ < z ⟷ x ≤ of_int z - 1"
  using ceiling_le_iff [of x "z - 1"] by simp

lemma le_ceiling_iff: "z ≤ ⌈x⌉ ⟷ of_int z - 1 < x"
  by (simp add: not_less [symmetric] ceiling_less_iff)

lemma ceiling_mono: "x ≥ y ⟹ ⌈x⌉ ≥ ⌈y⌉"
  unfolding ceiling_def by (simp add: floor_mono)

lemma ceiling_less_cancel: "⌈x⌉ < ⌈y⌉ ⟹ x < y"
  by (auto simp add: not_le [symmetric] ceiling_mono)

lemma ceiling_of_int [simp]: "⌈of_int z⌉ = z"
  by (rule ceiling_unique) simp_all

lemma ceiling_of_nat [simp]: "⌈of_nat n⌉ = int n"
  using ceiling_of_int [of "of_nat n"] by simp

lemma ceiling_add_le: "⌈x + y⌉ ≤ ⌈x⌉ + ⌈y⌉"
  by (simp only: ceiling_le_iff of_int_add add_mono le_of_int_ceiling)

lemma mult_ceiling_le:
  assumes "0 ≤ a" and "0 ≤ b"
  shows "⌈a * b⌉ ≤ ⌈a⌉ * ⌈b⌉"
  by (metis assms ceiling_le_iff ceiling_mono le_of_int_ceiling mult_mono of_int_mult)

lemma mult_ceiling_le_Ints:
  assumes "0 ≤ a" "a ∈ Ints"
  shows "(of_int ⌈a * b⌉ :: 'a :: linordered_idom) ≤ of_int(⌈a⌉ * ⌈b⌉)"
  by (metis Ints_cases assms ceiling_le_iff ceiling_of_int le_of_int_ceiling mult_left_mono of_int_le_iff of_int_mult)

lemma finite_int_segment:
  fixes a :: "'a::floor_ceiling"
  shows "finite {x ∈ ℤ. a ≤ x ∧ x ≤ b}"
proof -
  have "finite {ceiling a..floor b}"
    by simp
  moreover have "{x ∈ ℤ. a ≤ x ∧ x ≤ b} = of_int ` {ceiling a..floor b}"
    by (auto simp: le_floor_iff ceiling_le_iff elim!: Ints_cases)
  ultimately show ?thesis
    by simp
qed

corollary finite_abs_int_segment:
  fixes a :: "'a::floor_ceiling"
  shows "finite {k ∈ ℤ. ¦k¦ ≤ a}" 
  using finite_int_segment [of "-a" a] by (auto simp add: abs_le_iff conj_commute minus_le_iff)


subsubsection ‹Ceiling with numerals.›

lemma ceiling_zero [simp]: "⌈0⌉ = 0"
  using ceiling_of_int [of 0] by simp

lemma ceiling_one [simp]: "⌈1⌉ = 1"
  using ceiling_of_int [of 1] by simp

lemma ceiling_numeral [simp]: "⌈numeral v⌉ = numeral v"
  using ceiling_of_int [of "numeral v"] by simp

lemma ceiling_neg_numeral [simp]: "⌈- numeral v⌉ = - numeral v"
  using ceiling_of_int [of "- numeral v"] by simp

lemma ceiling_le_zero [simp]: "⌈x⌉ ≤ 0 ⟷ x ≤ 0"
  by (simp add: ceiling_le_iff)

lemma ceiling_le_one [simp]: "⌈x⌉ ≤ 1 ⟷ x ≤ 1"
  by (simp add: ceiling_le_iff)

lemma ceiling_le_numeral [simp]: "⌈x⌉ ≤ numeral v ⟷ x ≤ numeral v"
  by (simp add: ceiling_le_iff)

lemma ceiling_le_neg_numeral [simp]: "⌈x⌉ ≤ - numeral v ⟷ x ≤ - numeral v"
  by (simp add: ceiling_le_iff)

lemma ceiling_less_zero [simp]: "⌈x⌉ < 0 ⟷ x ≤ -1"
  by (simp add: ceiling_less_iff)

lemma ceiling_less_one [simp]: "⌈x⌉ < 1 ⟷ x ≤ 0"
  by (simp add: ceiling_less_iff)

lemma ceiling_less_numeral [simp]: "⌈x⌉ < numeral v ⟷ x ≤ numeral v - 1"
  by (simp add: ceiling_less_iff)

lemma ceiling_less_neg_numeral [simp]: "⌈x⌉ < - numeral v ⟷ x ≤ - numeral v - 1"
  by (simp add: ceiling_less_iff)

lemma zero_le_ceiling [simp]: "0 ≤ ⌈x⌉ ⟷ -1 < x"
  by (simp add: le_ceiling_iff)

lemma one_le_ceiling [simp]: "1 ≤ ⌈x⌉ ⟷ 0 < x"
  by (simp add: le_ceiling_iff)

lemma numeral_le_ceiling [simp]: "numeral v ≤ ⌈x⌉ ⟷ numeral v - 1 < x"
  by (simp add: le_ceiling_iff)

lemma neg_numeral_le_ceiling [simp]: "- numeral v ≤ ⌈x⌉ ⟷ - numeral v - 1 < x"
  by (simp add: le_ceiling_iff)

lemma zero_less_ceiling [simp]: "0 < ⌈x⌉ ⟷ 0 < x"
  by (simp add: less_ceiling_iff)

lemma one_less_ceiling [simp]: "1 < ⌈x⌉ ⟷ 1 < x"
  by (simp add: less_ceiling_iff)

lemma numeral_less_ceiling [simp]: "numeral v < ⌈x⌉ ⟷ numeral v < x"
  by (simp add: less_ceiling_iff)

lemma neg_numeral_less_ceiling [simp]: "- numeral v < ⌈x⌉ ⟷ - numeral v < x"
  by (simp add: less_ceiling_iff)

lemma ceiling_altdef: "⌈x⌉ = (if x = of_int ⌊x⌋ then ⌊x⌋ else ⌊x⌋ + 1)"
  by (intro ceiling_unique; simp, linarith?)

lemma floor_le_ceiling [simp]: "⌊x⌋ ≤ ⌈x⌉"
  by (simp add: ceiling_altdef)


subsubsection ‹Addition and subtraction of integers.›

lemma ceiling_add_of_int [simp]: "⌈x + of_int z⌉ = ⌈x⌉ + z"
  using ceiling_correct [of x] by (simp add: ceiling_def)

lemma ceiling_add_numeral [simp]: "⌈x + numeral v⌉ = ⌈x⌉ + numeral v"
  using ceiling_add_of_int [of x "numeral v"] by simp

lemma ceiling_add_one [simp]: "⌈x + 1⌉ = ⌈x⌉ + 1"
  using ceiling_add_of_int [of x 1] by simp

lemma ceiling_diff_of_int [simp]: "⌈x - of_int z⌉ = ⌈x⌉ - z"
  using ceiling_add_of_int [of x "- z"] by (simp add: algebra_simps)

lemma ceiling_diff_numeral [simp]: "⌈x - numeral v⌉ = ⌈x⌉ - numeral v"
  using ceiling_diff_of_int [of x "numeral v"] by simp

lemma ceiling_diff_one [simp]: "⌈x - 1⌉ = ⌈x⌉ - 1"
  using ceiling_diff_of_int [of x 1] by simp

lemma ceiling_split[linarith_split]: "P ⌈t⌉ ⟷ (∀i. of_int i - 1 < t ∧ t ≤ of_int i ⟶ P i)"
  by (auto simp add: ceiling_unique ceiling_correct)

lemma ceiling_diff_floor_le_1: "⌈x⌉ - ⌊x⌋ ≤ 1"
proof -
  have "of_int ⌈x⌉ - 1 < x"
    using ceiling_correct[of x] by simp
  also have "x < of_int ⌊x⌋ + 1"
    using floor_correct[of x] by simp_all
  finally have "of_int (⌈x⌉ - ⌊x⌋) < (of_int 2::'a)"
    by simp
  then show ?thesis
    unfolding of_int_less_iff by simp
qed

lemma nat_approx_posE:
  fixes e:: "'a::{archimedean_field,floor_ceiling}"
  assumes "0 < e"
  obtains n :: nat where "1 / of_nat(Suc n) < e"
proof 
  have "(1::'a) / of_nat (Suc (nat ⌈1/e⌉)) < 1 / of_int (⌈1/e⌉)"
  proof (rule divide_strict_left_mono)
    show "(of_int ⌈1 / e⌉::'a) < of_nat (Suc (nat ⌈1 / e⌉))"
      using assms by (simp add: field_simps)
    show "(0::'a) < of_nat (Suc (nat ⌈1 / e⌉)) * of_int ⌈1 / e⌉"
      using assms by (auto simp: zero_less_mult_iff pos_add_strict)
  qed auto
  also have "1 / of_int (⌈1/e⌉) ≤ 1 / (1/e)"
    by (rule divide_left_mono) (auto simp: ‹0 < e› ceiling_correct)
  also have "… = e" by simp
  finally show  "1 / of_nat (Suc (nat ⌈1 / e⌉)) < e"
    by metis 
qed

lemma ceiling_divide_upper:
  fixes q :: "'a::floor_ceiling"
  shows "q > 0 ⟹ p ≤ of_int (ceiling (p / q)) * q"
  by (meson divide_le_eq le_of_int_ceiling)

lemma ceiling_divide_lower:
  fixes q :: "'a::floor_ceiling"
  shows "q > 0 ⟹ (of_int ⌈p / q⌉ - 1) * q < p"
  by (meson ceiling_eq_iff pos_less_divide_eq)

subsection ‹Negation›

lemma floor_minus: "⌊- x⌋ = - ⌈x⌉"
  unfolding ceiling_def by simp

lemma ceiling_minus: "⌈- x⌉ = - ⌊x⌋"
  unfolding ceiling_def by simp

subsection ‹Natural numbers›

lemma of_nat_floor: "r≥0 ⟹ of_nat (nat ⌊r⌋) ≤ r"
  by simp

lemma of_nat_ceiling: "of_nat (nat ⌈r⌉) ≥ r"
  by (cases "r≥0") auto

lemma of_nat_int_floor [simp]: "x≥0 ⟹ of_nat (nat⌊x⌋) = of_int ⌊x⌋"
  by auto

lemma of_nat_int_ceiling [simp]: "x≥0 ⟹ of_nat (nat ⌈x⌉) = of_int ⌈x⌉"
  by auto

subsection ‹Frac Function›

definition frac :: "'a ⇒ 'a::floor_ceiling"
  where "frac x ≡ x - of_int ⌊x⌋"

lemma frac_lt_1: "frac x < 1"
  by (simp add: frac_def) linarith

lemma frac_eq_0_iff [simp]: "frac x = 0 ⟷ x ∈ ℤ"
  by (simp add: frac_def) (metis Ints_cases Ints_of_int floor_of_int )

lemma frac_ge_0 [simp]: "frac x ≥ 0"
  unfolding frac_def by linarith

lemma frac_gt_0_iff [simp]: "frac x > 0 ⟷ x ∉ ℤ"
  by (metis frac_eq_0_iff frac_ge_0 le_less less_irrefl)

lemma frac_of_int [simp]: "frac (of_int z) = 0"
  by (simp add: frac_def)

lemma frac_frac [simp]: "frac (frac x) = frac x"
  by (simp add: frac_def)

lemma floor_add: "⌊x + y⌋ = (if frac x + frac y < 1 then ⌊x⌋ + ⌊y⌋ else (⌊x⌋ + ⌊y⌋) + 1)"
proof -
  have "x + y < 1 + (of_int ⌊x⌋ + of_int ⌊y⌋) ⟹ ⌊x + y⌋ = ⌊x⌋ + ⌊y⌋"
    by (metis add.commute floor_unique le_floor_add le_floor_iff of_int_add)
  moreover
  have "¬ x + y < 1 + (of_int ⌊x⌋ + of_int ⌊y⌋) ⟹ ⌊x + y⌋ = 1 + (⌊x⌋ + ⌊y⌋)"
    apply (simp add: floor_eq_iff)
    apply (auto simp add: algebra_simps)
    apply linarith
    done
  ultimately show ?thesis by (auto simp add: frac_def algebra_simps)
qed

lemma floor_add2[simp]: "x ∈ ℤ ∨ y ∈ ℤ ⟹ ⌊x + y⌋ = ⌊x⌋ + ⌊y⌋"
by (metis add.commute add.left_neutral frac_lt_1 floor_add frac_eq_0_iff)

lemma frac_add:
  "frac (x + y) = (if frac x + frac y < 1 then frac x + frac y else (frac x + frac y) - 1)"
  by (simp add: frac_def floor_add)

lemma frac_unique_iff: "frac x = a ⟷ x - a ∈ ℤ ∧ 0 ≤ a ∧ a < 1"
  for x :: "'a::floor_ceiling"
  apply (auto simp: Ints_def frac_def algebra_simps floor_unique)
   apply linarith+
  done

lemma frac_eq: "frac x = x ⟷ 0 ≤ x ∧ x < 1"
  by (simp add: frac_unique_iff)

lemma frac_neg: "frac (- x) = (if x ∈ ℤ then 0 else 1 - frac x)"
  for x :: "'a::floor_ceiling"
  apply (auto simp add: frac_unique_iff)
   apply (simp add: frac_def)
  apply (meson frac_lt_1 less_iff_diff_less_0 not_le not_less_iff_gr_or_eq)
  done

lemma frac_in_Ints_iff [simp]: "frac x ∈ ℤ ⟷ x ∈ ℤ"
proof safe
  assume "frac x ∈ ℤ"
  hence "of_int ⌊x⌋ + frac x ∈ ℤ" by auto
  also have "of_int ⌊x⌋ + frac x = x" by (simp add: frac_def)
  finally show "x ∈ ℤ" .
qed (auto simp: frac_def)

lemma frac_1_eq: "frac (x+1) = frac x"
  by (simp add: frac_def)


subsection ‹Fractional part arithmetic›
text ‹Many thanks to Stepan Holub›

lemma frac_non_zero: "frac x ≠ 0 ⟹ frac (-x) =  1 - frac x"
  using frac_eq_0_iff frac_neg by metis

lemma frac_add_simps [simp]: 
   "frac (frac a + b) = frac (a + b)"
   "frac (a + frac b) = frac (a + b)"      
  by (simp_all add: frac_add)

lemma frac_neg_frac:  "frac (- frac x) = frac (-x)"
  unfolding frac_neg frac_frac by force

lemma frac_diff_simp: "frac (y - frac x) = frac (y - x)"
    unfolding diff_conv_add_uminus frac_add frac_neg_frac..

lemma frac_diff: "frac (a - b) = frac (frac a + (- frac b))" 
  unfolding frac_add_simps(1) 
  unfolding ab_group_add_class.ab_diff_conv_add_uminus[symmetric] frac_diff_simp..

lemma frac_diff_pos: "frac x ≤ frac y ⟹ frac (y - x) = frac y - frac x"
  unfolding diff_conv_add_uminus frac_add frac_neg
  using frac_lt_1 by force 
   
lemma frac_diff_neg: assumes "frac y < frac x" 
  shows "frac (y - x) = frac y + 1 - frac x"
proof-
  have "x ∉ ℤ"
    unfolding frac_gt_0_iff[symmetric]
    using assms frac_ge_0[of y] by order
  have "frac y + (1 + - frac x) < 1"
    using frac_lt_1[of x] assms by fastforce
  show ?thesis
    unfolding diff_conv_add_uminus frac_add frac_neg 
    if_not_P[OF ‹x ∉ ℤ›] if_P[OF ‹frac y + (1 + - frac x) < 1›]
    by simp
qed

lemma frac_diff_eq: assumes "frac y = frac x" 
  shows "frac (y - x) = 0"
  by (simp add: assms frac_diff_pos)

lemma frac_diff_zero: assumes "frac (x - y) = 0"
  shows "frac x = frac y"
  using frac_add_simps(1)[of "x - y" y, symmetric]
  unfolding assms add.group_left_neutral diff_add_cancel.

lemma frac_neg_eq_iff: "frac (-x) = frac (-y) ⟷ frac x = frac y"
  using add.inverse_inverse frac_neg_frac by metis

subsection ‹Rounding to the nearest integer›

definition round :: "'a::floor_ceiling ⇒ int"
  where "round x = ⌊x + 1/2⌋"

lemma of_int_round_ge: "of_int (round x) ≥ x - 1/2"
  and of_int_round_le: "of_int (round x) ≤ x + 1/2"
  and of_int_round_abs_le: "¦of_int (round x) - x¦ ≤ 1/2"
  and of_int_round_gt: "of_int (round x) > x - 1/2"
proof -
  from floor_correct[of "x + 1/2"] have "x + 1/2 < of_int (round x) + 1"
    by (simp add: round_def)
  from add_strict_right_mono[OF this, of "-1"] show A: "of_int (round x) > x - 1/2"
    by simp
  then show "of_int (round x) ≥ x - 1/2"
    by simp
  from floor_correct[of "x + 1/2"] show "of_int (round x) ≤ x + 1/2"
    by (simp add: round_def)
  with A show "¦of_int (round x) - x¦ ≤ 1/2"
    by linarith
qed

lemma round_of_int [simp]: "round (of_int n) = n"
  unfolding round_def by (subst floor_eq_iff) force

lemma round_0 [simp]: "round 0 = 0"
  using round_of_int[of 0] by simp

lemma round_1 [simp]: "round 1 = 1"
  using round_of_int[of 1] by simp

lemma round_numeral [simp]: "round (numeral n) = numeral n"
  using round_of_int[of "numeral n"] by simp

lemma round_neg_numeral [simp]: "round (-numeral n) = -numeral n"
  using round_of_int[of "-numeral n"] by simp

lemma round_of_nat [simp]: "round (of_nat n) = of_nat n"
  using round_of_int[of "int n"] by simp

lemma round_mono: "x ≤ y ⟹ round x ≤ round y"
  unfolding round_def by (intro floor_mono) simp

lemma round_unique: "of_int y > x - 1/2 ⟹ of_int y ≤ x + 1/2 ⟹ round x = y"
  unfolding round_def
proof (rule floor_unique)
  assume "x - 1 / 2 < of_int y"
  from add_strict_left_mono[OF this, of 1] show "x + 1 / 2 < of_int y + 1"
    by simp
qed

lemma round_unique': "¦x - of_int n¦ < 1/2 ⟹ round x = n"
  by (subst (asm) abs_less_iff, rule round_unique) (simp_all add: field_simps)

lemma round_altdef: "round x = (if frac x ≥ 1/2 then ⌈x⌉ else ⌊x⌋)"
  by (cases "frac x ≥ 1/2")
    (rule round_unique, ((simp add: frac_def field_simps ceiling_altdef; linarith)+)[2])+

lemma floor_le_round: "⌊x⌋ ≤ round x"
  unfolding round_def by (intro floor_mono) simp

lemma ceiling_ge_round: "⌈x⌉ ≥ round x"
  unfolding round_altdef by simp

lemma round_diff_minimal: "¦z - of_int (round z)¦ ≤ ¦z - of_int m¦"
  for z :: "'a::floor_ceiling"
proof (cases "of_int m ≥ z")
  case True
  then have "¦z - of_int (round z)¦ ≤ ¦of_int ⌈z⌉ - z¦"
    unfolding round_altdef by (simp add: field_simps ceiling_altdef frac_def) linarith
  also have "of_int ⌈z⌉ - z ≥ 0"
    by linarith
  with True have "¦of_int ⌈z⌉ - z¦ ≤ ¦z - of_int m¦"
    by (simp add: ceiling_le_iff)
  finally show ?thesis .
next
  case False
  then have "¦z - of_int (round z)¦ ≤ ¦of_int ⌊z⌋ - z¦"
    unfolding round_altdef by (simp add: field_simps ceiling_altdef frac_def) linarith
  also have "z - of_int ⌊z⌋ ≥ 0"
    by linarith
  with False have "¦of_int ⌊z⌋ - z¦ ≤ ¦z - of_int m¦"
    by (simp add: le_floor_iff)
  finally show ?thesis .
qed

end