(* Title: HOL/Archimedean_Field.thy
Author: Brian Huffman
*)
section \<open>Archimedean Fields, Floor and Ceiling Functions\<close>
theory Archimedean_Field
imports Main
begin
subsection \<open>Class of Archimedean fields\<close>
text \<open>Archimedean fields have no infinite elements.\<close>
class archimedean_field = linordered_field +
assumes ex_le_of_int: "\<exists>z. x \<le> of_int z"
lemma ex_less_of_int:
fixes x :: "'a::archimedean_field" shows "\<exists>z. x < of_int z"
proof -
from ex_le_of_int obtain z where "x \<le> of_int z" ..
then have "x < of_int (z + 1)" by simp
then show ?thesis ..
qed
lemma ex_of_int_less:
fixes x :: "'a::archimedean_field" shows "\<exists>z. of_int z < x"
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 ex_less_of_nat:
fixes x :: "'a::archimedean_field" shows "\<exists>n. x < of_nat n"
proof -
obtain z where "x < of_int z" using ex_less_of_int ..
also have "\<dots> \<le> of_int (int (nat z))" by simp
also have "\<dots> = of_nat (nat z)" by (simp only: of_int_of_nat_eq)
finally show ?thesis ..
qed
lemma ex_le_of_nat:
fixes x :: "'a::archimedean_field" shows "\<exists>n. x \<le> of_nat n"
proof -
obtain n where "x < of_nat n" using ex_less_of_nat ..
then have "x \<le> of_nat n" by simp
then show ?thesis ..
qed
text \<open>Archimedean fields have no infinitesimal elements.\<close>
lemma ex_inverse_of_nat_Suc_less:
fixes x :: "'a::archimedean_field"
assumes "0 < x" shows "\<exists>n. inverse (of_nat (Suc n)) < x"
proof -
from \<open>0 < x\<close> have "0 < inverse x"
by (rule positive_imp_inverse_positive)
obtain n where "inverse x < of_nat n"
using ex_less_of_nat ..
then obtain m where "inverse x < of_nat (Suc m)"
using \<open>0 < inverse x\<close> by (cases n) (simp_all del: of_nat_Suc)
then have "inverse (of_nat (Suc m)) < inverse (inverse x)"
using \<open>0 < inverse x\<close> by (rule less_imp_inverse_less)
then have "inverse (of_nat (Suc m)) < x"
using \<open>0 < x\<close> 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 "\<exists>n>0. inverse (of_nat n) < x"
using ex_inverse_of_nat_Suc_less [OF \<open>0 < x\<close>] by auto
lemma ex_less_of_nat_mult:
fixes x :: "'a::archimedean_field"
assumes "0 < x" shows "\<exists>n. y < of_nat n * x"
proof -
obtain n where "y / x < of_nat n" using ex_less_of_nat ..
with \<open>0 < x\<close> have "y < of_nat n * x" by (simp add: pos_divide_less_eq)
then show ?thesis ..
qed
subsection \<open>Existence and uniqueness of floor function\<close>
lemma exists_least_lemma:
assumes "\<not> P 0" and "\<exists>n. P n"
shows "\<exists>n. \<not> P n \<and> P (Suc n)"
proof -
from \<open>\<exists>n. P n\<close> have "P (Least P)" by (rule LeastI_ex)
with \<open>\<not> P 0\<close> obtain n where "Least P = Suc n"
by (cases "Least P") auto
then have "n < Least P" by simp
then have "\<not> P n" by (rule not_less_Least)
then have "\<not> P n \<and> P (Suc n)"
using \<open>P (Least P)\<close> \<open>Least P = Suc n\<close> by simp
then show ?thesis ..
qed
lemma floor_exists:
fixes x :: "'a::archimedean_field"
shows "\<exists>z. of_int z \<le> x \<and> x < of_int (z + 1)"
proof (cases)
assume "0 \<le> x"
then have "\<not> x < of_nat 0" by simp
then have "\<exists>n. \<not> x < of_nat n \<and> x < of_nat (Suc n)"
using ex_less_of_nat by (rule exists_least_lemma)
then obtain n where "\<not> x < of_nat n \<and> x < of_nat (Suc n)" ..
then have "of_int (int n) \<le> x \<and> x < of_int (int n + 1)" by simp
then show ?thesis ..
next
assume "\<not> 0 \<le> x"
then have "\<not> - x \<le> of_nat 0" by simp
then have "\<exists>n. \<not> - x \<le> of_nat n \<and> - x \<le> of_nat (Suc n)"
using ex_le_of_nat by (rule exists_least_lemma)
then obtain n where "\<not> - x \<le> of_nat n \<and> - x \<le> of_nat (Suc n)" ..
then have "of_int (- int n - 1) \<le> x \<and> x < of_int (- int n - 1 + 1)" by simp
then show ?thesis ..
qed
lemma floor_exists1:
fixes x :: "'a::archimedean_field"
shows "\<exists>!z. of_int z \<le> x \<and> x < of_int (z + 1)"
proof (rule ex_ex1I)
show "\<exists>z. of_int z \<le> x \<and> x < of_int (z + 1)"
by (rule floor_exists)
next
fix y z assume
"of_int y \<le> x \<and> x < of_int (y + 1)"
"of_int z \<le> x \<and> 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 \<open>Floor function\<close>
class floor_ceiling = archimedean_field +
fixes floor :: "'a \<Rightarrow> int"
assumes floor_correct: "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
notation (xsymbols)
floor ("\<lfloor>_\<rfloor>")
notation (HTML output)
floor ("\<lfloor>_\<rfloor>")
lemma floor_unique: "\<lbrakk>of_int z \<le> x; x < of_int z + 1\<rbrakk> \<Longrightarrow> floor x = z"
using floor_correct [of x] floor_exists1 [of x] by auto
lemma floor_unique_iff:
fixes x :: "'a::floor_ceiling"
shows "\<lfloor>x\<rfloor> = a \<longleftrightarrow> of_int a \<le> x \<and> x < of_int a + 1"
using floor_correct floor_unique by auto
lemma of_int_floor_le: "of_int (floor x) \<le> x"
using floor_correct ..
lemma le_floor_iff: "z \<le> floor x \<longleftrightarrow> of_int z \<le> x"
proof
assume "z \<le> floor x"
then have "(of_int z :: 'a) \<le> of_int (floor x)" by simp
also have "of_int (floor x) \<le> x" by (rule of_int_floor_le)
finally show "of_int z \<le> x" .
next
assume "of_int z \<le> x"
also have "x < of_int (floor x + 1)" using floor_correct ..
finally show "z \<le> floor x" by (simp del: of_int_add)
qed
lemma floor_less_iff: "floor x < z \<longleftrightarrow> x < of_int z"
by (simp add: not_le [symmetric] le_floor_iff)
lemma less_floor_iff: "z < floor x \<longleftrightarrow> of_int z + 1 \<le> x"
using le_floor_iff [of "z + 1" x] by auto
lemma floor_le_iff: "floor x \<le> z \<longleftrightarrow> x < of_int z + 1"
by (simp add: not_less [symmetric] less_floor_iff)
lemma floor_split[arith_split]: "P (floor t) \<longleftrightarrow> (\<forall>i. of_int i \<le> t \<and> t < of_int i + 1 \<longrightarrow> P i)"
by (metis floor_correct floor_unique less_floor_iff not_le order_refl)
lemma floor_mono: assumes "x \<le> y" shows "floor x \<le> floor y"
proof -
have "of_int (floor x) \<le> x" by (rule of_int_floor_le)
also note \<open>x \<le> y\<close>
finally show ?thesis by (simp add: le_floor_iff)
qed
lemma floor_less_cancel: "floor x < floor y \<Longrightarrow> x < y"
by (auto simp add: not_le [symmetric] floor_mono)
lemma floor_of_int [simp]: "floor (of_int z) = z"
by (rule floor_unique) simp_all
lemma floor_of_nat [simp]: "floor (of_nat n) = int n"
using floor_of_int [of "of_nat n"] by simp
lemma le_floor_add: "floor x + floor y \<le> floor (x + y)"
by (simp only: le_floor_iff of_int_add add_mono of_int_floor_le)
text \<open>Floor with numerals\<close>
lemma floor_zero [simp]: "floor 0 = 0"
using floor_of_int [of 0] by simp
lemma floor_one [simp]: "floor 1 = 1"
using floor_of_int [of 1] by simp
lemma floor_numeral [simp]: "floor (numeral v) = numeral v"
using floor_of_int [of "numeral v"] by simp
lemma floor_neg_numeral [simp]: "floor (- numeral v) = - numeral v"
using floor_of_int [of "- numeral v"] by simp
lemma zero_le_floor [simp]: "0 \<le> floor x \<longleftrightarrow> 0 \<le> x"
by (simp add: le_floor_iff)
lemma one_le_floor [simp]: "1 \<le> floor x \<longleftrightarrow> 1 \<le> x"
by (simp add: le_floor_iff)
lemma numeral_le_floor [simp]:
"numeral v \<le> floor x \<longleftrightarrow> numeral v \<le> x"
by (simp add: le_floor_iff)
lemma neg_numeral_le_floor [simp]:
"- numeral v \<le> floor x \<longleftrightarrow> - numeral v \<le> x"
by (simp add: le_floor_iff)
lemma zero_less_floor [simp]: "0 < floor x \<longleftrightarrow> 1 \<le> x"
by (simp add: less_floor_iff)
lemma one_less_floor [simp]: "1 < floor x \<longleftrightarrow> 2 \<le> x"
by (simp add: less_floor_iff)
lemma numeral_less_floor [simp]:
"numeral v < floor x \<longleftrightarrow> numeral v + 1 \<le> x"
by (simp add: less_floor_iff)
lemma neg_numeral_less_floor [simp]:
"- numeral v < floor x \<longleftrightarrow> - numeral v + 1 \<le> x"
by (simp add: less_floor_iff)
lemma floor_le_zero [simp]: "floor x \<le> 0 \<longleftrightarrow> x < 1"
by (simp add: floor_le_iff)
lemma floor_le_one [simp]: "floor x \<le> 1 \<longleftrightarrow> x < 2"
by (simp add: floor_le_iff)
lemma floor_le_numeral [simp]:
"floor x \<le> numeral v \<longleftrightarrow> x < numeral v + 1"
by (simp add: floor_le_iff)
lemma floor_le_neg_numeral [simp]:
"floor x \<le> - numeral v \<longleftrightarrow> x < - numeral v + 1"
by (simp add: floor_le_iff)
lemma floor_less_zero [simp]: "floor x < 0 \<longleftrightarrow> x < 0"
by (simp add: floor_less_iff)
lemma floor_less_one [simp]: "floor x < 1 \<longleftrightarrow> x < 1"
by (simp add: floor_less_iff)
lemma floor_less_numeral [simp]:
"floor x < numeral v \<longleftrightarrow> x < numeral v"
by (simp add: floor_less_iff)
lemma floor_less_neg_numeral [simp]:
"floor x < - numeral v \<longleftrightarrow> x < - numeral v"
by (simp add: floor_less_iff)
text \<open>Addition and subtraction of integers\<close>
lemma floor_add_of_int [simp]: "floor (x + of_int z) = floor x + z"
using floor_correct [of x] by (simp add: floor_unique)
lemma floor_add_numeral [simp]:
"floor (x + numeral v) = floor x + numeral v"
using floor_add_of_int [of x "numeral v"] by simp
lemma floor_add_one [simp]: "floor (x + 1) = floor x + 1"
using floor_add_of_int [of x 1] by simp
lemma floor_diff_of_int [simp]: "floor (x - of_int z) = floor x - z"
using floor_add_of_int [of x "- z"] by (simp add: algebra_simps)
lemma floor_uminus_of_int [simp]: "floor (- (of_int z)) = - z"
by (metis floor_diff_of_int [of 0] diff_0 floor_zero)
lemma floor_diff_numeral [simp]:
"floor (x - numeral v) = floor x - numeral v"
using floor_diff_of_int [of x "numeral v"] by simp
lemma floor_diff_one [simp]: "floor (x - 1) = floor x - 1"
using floor_diff_of_int [of x 1] by simp
lemma le_mult_floor:
assumes "0 \<le> a" and "0 \<le> b"
shows "floor a * floor b \<le> floor (a * b)"
proof -
have "of_int (floor a) \<le> a"
and "of_int (floor b) \<le> b" by (auto intro: of_int_floor_le)
hence "of_int (floor a * floor b) \<le> a * b"
using assms by (auto intro!: mult_mono)
also have "a * b < of_int (floor (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:
fixes k l :: int
shows "\<lfloor>of_int k / of_int l\<rfloor> = k div l"
proof (cases "l = 0")
case True then show ?thesis by simp
next
case False
have *: "\<lfloor>of_int (k mod l) / of_int l :: 'a\<rfloor> = 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
moreover with \<open>l \<noteq> 0\<close> False have "r > 0" by simp
ultimately 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 "\<dots> = (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) = \<dots> / 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 "\<lfloor>of_int k / of_int l :: 'a\<rfloor> = \<lfloor>of_int (k div l) + of_int (k mod l) / of_int l :: 'a\<rfloor>"
by simp
then have "\<lfloor>of_int k / of_int l :: 'a\<rfloor> = \<lfloor>of_int (k mod l) / of_int l + of_int (k div l) :: 'a\<rfloor>"
by (simp add: ac_simps)
then have "\<lfloor>of_int k / of_int l :: 'a\<rfloor> = \<lfloor>of_int (k mod l) / of_int l :: 'a\<rfloor> + k div l"
by simp
with * show ?thesis by simp
qed
lemma floor_divide_of_nat_eq:
fixes m n :: nat
shows "\<lfloor>of_nat m / of_nat n\<rfloor> = of_nat (m div n)"
proof (cases "n = 0")
case True then show ?thesis by simp
next
case False
then have *: "\<lfloor>of_nat (m mod n) / of_nat n :: 'a\<rfloor> = 0"
by (auto intro: floor_unique)
have "(of_nat m :: 'a) = of_nat (m div n * n + m mod n)"
by simp
also have "\<dots> = (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 of_nat_mult)
finally have "(of_nat m / of_nat n :: 'a) = \<dots> / 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 "\<lfloor>of_nat m / of_nat n :: 'a\<rfloor> = \<lfloor>of_nat (m div n) + of_nat (m mod n) / of_nat n :: 'a\<rfloor>"
by simp
then have "\<lfloor>of_nat m / of_nat n :: 'a\<rfloor> = \<lfloor>of_nat (m mod n) / of_nat n + of_nat (m div n) :: 'a\<rfloor>"
by (simp add: ac_simps)
moreover have "(of_nat (m div n) :: 'a) = of_int (of_nat (m div n))"
by simp
ultimately have "\<lfloor>of_nat m / of_nat n :: 'a\<rfloor> = \<lfloor>of_nat (m mod n) / of_nat n :: 'a\<rfloor> + of_nat (m div n)"
by (simp only: floor_add_of_int)
with * show ?thesis by simp
qed
subsection \<open>Ceiling function\<close>
definition
ceiling :: "'a::floor_ceiling \<Rightarrow> int" where
"ceiling x = - floor (- x)"
notation (xsymbols)
ceiling ("\<lceil>_\<rceil>")
notation (HTML output)
ceiling ("\<lceil>_\<rceil>")
lemma ceiling_correct: "of_int (ceiling x) - 1 < x \<and> x \<le> of_int (ceiling x)"
unfolding ceiling_def using floor_correct [of "- x"] by simp
lemma ceiling_unique: "\<lbrakk>of_int z - 1 < x; x \<le> of_int z\<rbrakk> \<Longrightarrow> ceiling x = z"
unfolding ceiling_def using floor_unique [of "- z" "- x"] by simp
lemma le_of_int_ceiling: "x \<le> of_int (ceiling x)"
using ceiling_correct ..
lemma ceiling_le_iff: "ceiling x \<le> z \<longleftrightarrow> x \<le> of_int z"
unfolding ceiling_def using le_floor_iff [of "- z" "- x"] by auto
lemma less_ceiling_iff: "z < ceiling x \<longleftrightarrow> of_int z < x"
by (simp add: not_le [symmetric] ceiling_le_iff)
lemma ceiling_less_iff: "ceiling x < z \<longleftrightarrow> x \<le> of_int z - 1"
using ceiling_le_iff [of x "z - 1"] by simp
lemma le_ceiling_iff: "z \<le> ceiling x \<longleftrightarrow> of_int z - 1 < x"
by (simp add: not_less [symmetric] ceiling_less_iff)
lemma ceiling_mono: "x \<ge> y \<Longrightarrow> ceiling x \<ge> ceiling y"
unfolding ceiling_def by (simp add: floor_mono)
lemma ceiling_less_cancel: "ceiling x < ceiling y \<Longrightarrow> x < y"
by (auto simp add: not_le [symmetric] ceiling_mono)
lemma ceiling_of_int [simp]: "ceiling (of_int z) = z"
by (rule ceiling_unique) simp_all
lemma ceiling_of_nat [simp]: "ceiling (of_nat n) = int n"
using ceiling_of_int [of "of_nat n"] by simp
lemma ceiling_add_le: "ceiling (x + y) \<le> ceiling x + ceiling y"
by (simp only: ceiling_le_iff of_int_add add_mono le_of_int_ceiling)
text \<open>Ceiling with numerals\<close>
lemma ceiling_zero [simp]: "ceiling 0 = 0"
using ceiling_of_int [of 0] by simp
lemma ceiling_one [simp]: "ceiling 1 = 1"
using ceiling_of_int [of 1] by simp
lemma ceiling_numeral [simp]: "ceiling (numeral v) = numeral v"
using ceiling_of_int [of "numeral v"] by simp
lemma ceiling_neg_numeral [simp]: "ceiling (- numeral v) = - numeral v"
using ceiling_of_int [of "- numeral v"] by simp
lemma ceiling_le_zero [simp]: "ceiling x \<le> 0 \<longleftrightarrow> x \<le> 0"
by (simp add: ceiling_le_iff)
lemma ceiling_le_one [simp]: "ceiling x \<le> 1 \<longleftrightarrow> x \<le> 1"
by (simp add: ceiling_le_iff)
lemma ceiling_le_numeral [simp]:
"ceiling x \<le> numeral v \<longleftrightarrow> x \<le> numeral v"
by (simp add: ceiling_le_iff)
lemma ceiling_le_neg_numeral [simp]:
"ceiling x \<le> - numeral v \<longleftrightarrow> x \<le> - numeral v"
by (simp add: ceiling_le_iff)
lemma ceiling_less_zero [simp]: "ceiling x < 0 \<longleftrightarrow> x \<le> -1"
by (simp add: ceiling_less_iff)
lemma ceiling_less_one [simp]: "ceiling x < 1 \<longleftrightarrow> x \<le> 0"
by (simp add: ceiling_less_iff)
lemma ceiling_less_numeral [simp]:
"ceiling x < numeral v \<longleftrightarrow> x \<le> numeral v - 1"
by (simp add: ceiling_less_iff)
lemma ceiling_less_neg_numeral [simp]:
"ceiling x < - numeral v \<longleftrightarrow> x \<le> - numeral v - 1"
by (simp add: ceiling_less_iff)
lemma zero_le_ceiling [simp]: "0 \<le> ceiling x \<longleftrightarrow> -1 < x"
by (simp add: le_ceiling_iff)
lemma one_le_ceiling [simp]: "1 \<le> ceiling x \<longleftrightarrow> 0 < x"
by (simp add: le_ceiling_iff)
lemma numeral_le_ceiling [simp]:
"numeral v \<le> ceiling x \<longleftrightarrow> numeral v - 1 < x"
by (simp add: le_ceiling_iff)
lemma neg_numeral_le_ceiling [simp]:
"- numeral v \<le> ceiling x \<longleftrightarrow> - numeral v - 1 < x"
by (simp add: le_ceiling_iff)
lemma zero_less_ceiling [simp]: "0 < ceiling x \<longleftrightarrow> 0 < x"
by (simp add: less_ceiling_iff)
lemma one_less_ceiling [simp]: "1 < ceiling x \<longleftrightarrow> 1 < x"
by (simp add: less_ceiling_iff)
lemma numeral_less_ceiling [simp]:
"numeral v < ceiling x \<longleftrightarrow> numeral v < x"
by (simp add: less_ceiling_iff)
lemma neg_numeral_less_ceiling [simp]:
"- numeral v < ceiling x \<longleftrightarrow> - numeral v < x"
by (simp add: less_ceiling_iff)
text \<open>Addition and subtraction of integers\<close>
lemma ceiling_add_of_int [simp]: "ceiling (x + of_int z) = ceiling x + z"
using ceiling_correct [of x] by (simp add: ceiling_unique)
lemma ceiling_add_numeral [simp]:
"ceiling (x + numeral v) = ceiling x + numeral v"
using ceiling_add_of_int [of x "numeral v"] by simp
lemma ceiling_add_one [simp]: "ceiling (x + 1) = ceiling x + 1"
using ceiling_add_of_int [of x 1] by simp
lemma ceiling_diff_of_int [simp]: "ceiling (x - of_int z) = ceiling x - z"
using ceiling_add_of_int [of x "- z"] by (simp add: algebra_simps)
lemma ceiling_diff_numeral [simp]:
"ceiling (x - numeral v) = ceiling x - numeral v"
using ceiling_diff_of_int [of x "numeral v"] by simp
lemma ceiling_diff_one [simp]: "ceiling (x - 1) = ceiling x - 1"
using ceiling_diff_of_int [of x 1] by simp
lemma ceiling_split[arith_split]: "P (ceiling t) \<longleftrightarrow> (\<forall>i. of_int i - 1 < t \<and> t \<le> of_int i \<longrightarrow> P i)"
by (auto simp add: ceiling_unique ceiling_correct)
lemma ceiling_diff_floor_le_1: "ceiling x - floor x \<le> 1"
proof -
have "of_int \<lceil>x\<rceil> - 1 < x"
using ceiling_correct[of x] by simp
also have "x < of_int \<lfloor>x\<rfloor> + 1"
using floor_correct[of x] by simp_all
finally have "of_int (\<lceil>x\<rceil> - \<lfloor>x\<rfloor>) < (of_int 2::'a)"
by simp
then show ?thesis
unfolding of_int_less_iff by simp
qed
subsection \<open>Negation\<close>
lemma floor_minus: "floor (- x) = - ceiling x"
unfolding ceiling_def by simp
lemma ceiling_minus: "ceiling (- x) = - floor x"
unfolding ceiling_def by simp
subsection \<open>Frac Function\<close>
definition frac :: "'a \<Rightarrow> 'a::floor_ceiling" where
"frac x \<equiv> x - of_int \<lfloor>x\<rfloor>"
lemma frac_lt_1: "frac x < 1"
by (simp add: frac_def) linarith
lemma frac_eq_0_iff [simp]: "frac x = 0 \<longleftrightarrow> x \<in> \<int>"
by (simp add: frac_def) (metis Ints_cases Ints_of_int floor_of_int )
lemma frac_ge_0 [simp]: "frac x \<ge> 0"
unfolding frac_def
by linarith
lemma frac_gt_0_iff [simp]: "frac x > 0 \<longleftrightarrow> x \<notin> \<int>"
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 floor_add: "\<lfloor>x + y\<rfloor> = (if frac x + frac y < 1 then \<lfloor>x\<rfloor> + \<lfloor>y\<rfloor> else (\<lfloor>x\<rfloor> + \<lfloor>y\<rfloor>) + 1)"
proof -
{assume "x + y < 1 + (of_int \<lfloor>x\<rfloor> + of_int \<lfloor>y\<rfloor>)"
then have "\<lfloor>x + y\<rfloor> = \<lfloor>x\<rfloor> + \<lfloor>y\<rfloor>"
by (metis add.commute floor_unique le_floor_add le_floor_iff of_int_add)
}
moreover
{ assume "\<not> x + y < 1 + (of_int \<lfloor>x\<rfloor> + of_int \<lfloor>y\<rfloor>)"
then have "\<lfloor>x + y\<rfloor> = 1 + (\<lfloor>x\<rfloor> + \<lfloor>y\<rfloor>)"
apply (simp add: floor_unique_iff)
apply (auto simp add: algebra_simps)
by linarith
}
ultimately show ?thesis
by (auto simp add: frac_def algebra_simps)
qed
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:
fixes x :: "'a::floor_ceiling"
shows "(frac x) = a \<longleftrightarrow> x - a \<in> \<int> \<and> 0 \<le> a \<and> a < 1"
apply (auto simp: Ints_def frac_def)
apply linarith
apply linarith
by (metis (no_types) add.commute add.left_neutral eq_diff_eq floor_add_of_int floor_unique of_int_0)
lemma frac_eq: "(frac x) = x \<longleftrightarrow> 0 \<le> x \<and> x < 1"
by (simp add: frac_unique_iff)
lemma frac_neg:
fixes x :: "'a::floor_ceiling"
shows "frac (-x) = (if x \<in> \<int> then 0 else 1 - frac x)"
apply (auto simp add: frac_unique_iff)
apply (simp add: frac_def)
by (meson frac_lt_1 less_iff_diff_less_0 not_le not_less_iff_gr_or_eq)
end