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. xS  ¦x¦  a"
  shows "¦Inf S¦  a"
proof -
  have "Sup (uminus ` S) = - (Inf S)"
  proof (rule antisym)
    show "- (Inf S)  Sup (uminus ` S)"
      apply (subst minus_le_iff)
      apply (rule cInf_greatest [OF S  {}])
      apply (subst minus_le_iff)
      apply (rule cSup_upper)
       apply force
      using bdd
      apply (force simp: abs_le_iff bdd_above_def)
      done
  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: "xS. ¦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: "xS. ¦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_inta * 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
proof (cases "n = 0")
  case True
  then show ?thesis by simp
next
  case False
  then have *: "of_nat (m mod n) / of_nat n :: 'a = 0"
    by (auto intro: floor_unique)
  have "(of_nat m :: 'a) = of_nat (m div n * n + m mod n)"
    by simp
  also have " = (of_nat (m div n) + of_nat (m mod n) / of_nat n) * of_nat n"
    using False by (simp only: of_nat_add) (simp add: field_simps)
  finally have "(of_nat m / of_nat n :: 'a) =  / of_nat n"
    by simp
  then have "(of_nat m / of_nat n :: 'a) = of_nat (m div n) + of_nat (m mod n) / of_nat n"
    using False by (simp only:) simp
  then have "of_nat m / of_nat n :: 'a = of_nat (m div n) + of_nat (m mod n) / of_nat n :: 'a"
    by simp
  then have "of_nat m / of_nat n :: 'a = of_nat (m mod n) / of_nat n + of_nat (m div n) :: 'a"
    by (simp add: ac_simps)
  moreover have "(of_nat (m div n) :: 'a) = of_int (of_nat (m div n))"
    by simp
  ultimately have "of_nat m / of_nat n :: 'a =
      of_nat (m mod n) / of_nat n :: 'a + of_nat (m div n)"
    by (simp only: floor_add_int)
  with * show ?thesis
    by simp
qed

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  . a  x  x  b}"
proof -
  have "finite {ceiling a..floor b}"
    by simp
  moreover have "{x  . 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  . ¦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: "r0  of_nat (nat r)  r"
  by simp

lemma of_nat_ceiling: "of_nat (nat r)  r"
  by (cases "r0") 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  "
  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  "
  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    y    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    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   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    x  "
proof safe
  assume "frac x  "
  hence "of_int x + frac x  " by auto
  also have "of_int x + frac x = x" by (simp add: frac_def)
  finally show "x  " .
qed (auto simp: frac_def)

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


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