(*  Title:      HOL/Archimedean_Field.thy

     Author:     Brian Huffman

 *)

 header {* Archimedean Fields, Floor and Ceiling Functions *}

 theory Archimedean_Field

 imports Main

 begin

 subsection {* Class of Archimedean fields *}

 text {* Archimedean fields have no infinite elements. *}

 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 {* Archimedean fields have no infinitesimal elements. *}

 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 `0 < x` 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 `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 "\<exists>n>0. inverse (of_nat n) < x"

   using ex_inverse_of_nat_Suc_less [OF `0 < x`] 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 `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 "\<not> P 0" and "\<exists>n. P n"

   shows "\<exists>n. \<not> P n \<and> P (Suc n)"

 proof -

   from `\<exists>n. P n` have "P (Least P)" by (rule LeastI_ex)

   with `\<not> P 0` 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 `P (Least P)` `Least P = Suc n` 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)"

   then have

     "of_int y \<le> x" "x < of_int (y + 1)"

     "of_int z \<le> x" "x < of_int (z + 1)"

     by simp_all

   from le_less_trans [OF `of_int y \<le> x` `x < of_int (z + 1)`]

        le_less_trans [OF `of_int z \<le> x` `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 \<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 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_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 `x \<le> y`

   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 {* Floor with numerals *}

 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 (neg_numeral v) = neg_numeral v"

   using floor_of_int [of "neg_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]:

   "neg_numeral v \<le> floor x \<longleftrightarrow> neg_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]:

   "neg_numeral v < floor x \<longleftrightarrow> neg_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> neg_numeral v \<longleftrightarrow> x < neg_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 < neg_numeral v \<longleftrightarrow> x < neg_numeral v"

   by (simp add: floor_less_iff)

 text {* Addition and subtraction of integers *}

 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_neg_numeral [simp]:

     "floor (x + neg_numeral v) = floor x + neg_numeral v"

   using floor_add_of_int [of x "neg_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_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_neg_numeral [simp]:

   "floor (x - neg_numeral v) = floor x - neg_numeral v"

   using floor_diff_of_int [of x "neg_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

 subsection {* Ceiling function *}

 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 {* Ceiling with numerals *}

 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 (neg_numeral v) = neg_numeral v"

   using ceiling_of_int [of "neg_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> neg_numeral v \<longleftrightarrow> x \<le> neg_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 < neg_numeral v \<longleftrightarrow> x \<le> neg_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]:

   "neg_numeral v \<le> ceiling x \<longleftrightarrow> neg_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]:

   "neg_numeral v < ceiling x \<longleftrightarrow> neg_numeral v < x"

   by (simp add: less_ceiling_iff)

 text {* Addition and subtraction of integers *}

 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_neg_numeral [simp]:

     "ceiling (x + neg_numeral v) = ceiling x + neg_numeral v"

   using ceiling_add_of_int [of x "neg_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_neg_numeral [simp]:

   "ceiling (x - neg_numeral v) = ceiling x - neg_numeral v"

   using ceiling_diff_of_int [of x "neg_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_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 {* Negation *}

 lemma floor_minus: "floor (- x) = - ceiling x"

   unfolding ceiling_def by simp

 lemma ceiling_minus: "ceiling (- x) = - floor x"

   unfolding ceiling_def by simp

 end