(*  Title:      HOL/RComplete.thy

     Author:     Jacques D. Fleuriot, University of Edinburgh

     Author:     Larry Paulson, University of Cambridge

     Author:     Jeremy Avigad, Carnegie Mellon University

     Author:     Florian Zuleger, Johannes Hoelzl, and Simon Funke, TU Muenchen

 *)

 header {* Completeness of the Reals; Floor and Ceiling Functions *}

 theory RComplete

 imports Lubs RealDef

 begin

 lemma real_sum_of_halves: "x/2 + x/2 = (x::real)"

   by simp

 lemma abs_diff_less_iff:

   "(\<bar>x - a\<bar> < (r::'a::linordered_idom)) = (a - r < x \<and> x < a + r)"

   by auto

 subsection {* Completeness of Positive Reals *}

 text {*

   Supremum property for the set of positive reals

   Let @{text "P"} be a non-empty set of positive reals, with an upper

   bound @{text "y"}.  Then @{text "P"} has a least upper bound

   (written @{text "S"}).

   FIXME: Can the premise be weakened to @{text "\<forall>x \<in> P. x\<le> y"}?

 *}

 text {* Only used in HOL/Import/HOL4Compat.thy; delete? *}

 lemma posreal_complete:

   fixes P :: "real set"

   assumes not_empty_P: "\<exists>x. x \<in> P"

     and upper_bound_Ex: "\<exists>y. \<forall>x \<in> P. x<y"

   shows "\<exists>S. \<forall>y. (\<exists>x \<in> P. y < x) = (y < S)"

 proof -

   from upper_bound_Ex have "\<exists>z. \<forall>x\<in>P. x \<le> z"

     by (auto intro: less_imp_le)

   from complete_real [OF not_empty_P this] obtain S

   where S1: "\<And>x. x \<in> P \<Longrightarrow> x \<le> S" and S2: "\<And>z. \<forall>x\<in>P. x \<le> z \<Longrightarrow> S \<le> z" by fast

   have "\<forall>y. (\<exists>x \<in> P. y < x) = (y < S)"

   proof

     fix y show "(\<exists>x\<in>P. y < x) = (y < S)"

       apply (cases "\<exists>x\<in>P. y < x", simp_all)

       apply (clarify, drule S1, simp)

       apply (simp add: not_less S2)

       done

   qed

   thus ?thesis ..

 qed

 text {*

   \medskip Completeness properties using @{text "isUb"}, @{text "isLub"} etc.

 *}

 lemma real_isLub_unique: "[| isLub R S x; isLub R S y |] ==> x = (y::real)"

   apply (frule isLub_isUb)

   apply (frule_tac x = y in isLub_isUb)

   apply (blast intro!: order_antisym dest!: isLub_le_isUb)

   done

 text {*

   \medskip reals Completeness (again!)

 *}

 lemma reals_complete:

   assumes notempty_S: "\<exists>X. X \<in> S"

     and exists_Ub: "\<exists>Y. isUb (UNIV::real set) S Y"

   shows "\<exists>t. isLub (UNIV :: real set) S t"

 proof -

   from assms have "\<exists>X. X \<in> S" and "\<exists>Y. \<forall>x\<in>S. x \<le> Y"

     unfolding isUb_def setle_def by simp_all

   from complete_real [OF this] show ?thesis

     by (simp add: isLub_def leastP_def isUb_def setle_def setge_def)

 qed

 subsection {* The Archimedean Property of the Reals *}

 theorem reals_Archimedean:

   assumes x_pos: "0 < x"

   shows "\<exists>n. inverse (real (Suc n)) < x"

   unfolding real_of_nat_def using x_pos

   by (rule ex_inverse_of_nat_Suc_less)

 lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"

   unfolding real_of_nat_def by (rule ex_less_of_nat)

 lemma reals_Archimedean3:

   assumes x_greater_zero: "0 < x"

   shows "\<forall>(y::real). \<exists>(n::nat). y < real n * x"

   unfolding real_of_nat_def using `0 < x`

   by (auto intro: ex_less_of_nat_mult)

 subsection{*Density of the Rational Reals in the Reals*}

 text{* This density proof is due to Stefan Richter and was ported by TN.  The

 original source is \emph{Real Analysis} by H.L. Royden.

 It employs the Archimedean property of the reals. *}

 lemma Rats_dense_in_real:

   fixes x :: real

   assumes "x < y" shows "\<exists>r\<in>\<rat>. x < r \<and> r < y"

 proof -

   from `x<y` have "0 < y-x" by simp

   with reals_Archimedean obtain q::nat 

     where q: "inverse (real q) < y-x" and "0 < q" by auto

   def p \<equiv> "ceiling (y * real q) - 1"

   def r \<equiv> "of_int p / real q"

   from q have "x < y - inverse (real q)" by simp

   also have "y - inverse (real q) \<le> r"

     unfolding r_def p_def

     by (simp add: le_divide_eq left_diff_distrib le_of_int_ceiling `0 < q`)

   finally have "x < r" .

   moreover have "r < y"

     unfolding r_def p_def

     by (simp add: divide_less_eq diff_less_eq `0 < q`

       less_ceiling_iff [symmetric])

   moreover from r_def have "r \<in> \<rat>" by simp

   ultimately show ?thesis by fast

 qed

 subsection{*Floor and Ceiling Functions from the Reals to the Integers*}

 lemma number_of_less_real_of_int_iff [simp]:

      "((number_of n) < real (m::int)) = (number_of n < m)"

 apply auto

 apply (rule real_of_int_less_iff [THEN iffD1])

 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)

 done

 lemma number_of_less_real_of_int_iff2 [simp]:

      "(real (m::int) < (number_of n)) = (m < number_of n)"

 apply auto

 apply (rule real_of_int_less_iff [THEN iffD1])

 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)

 done

 lemma number_of_le_real_of_int_iff [simp]:

      "((number_of n) \<le> real (m::int)) = (number_of n \<le> m)"

 by (simp add: linorder_not_less [symmetric])

 lemma number_of_le_real_of_int_iff2 [simp]:

      "(real (m::int) \<le> (number_of n)) = (m \<le> number_of n)"

 by (simp add: linorder_not_less [symmetric])

 lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n"

 unfolding real_of_nat_def by simp

 lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n"

 unfolding real_of_nat_def by (simp add: floor_minus)

 lemma floor_real_of_int [simp]: "floor (real (n::int)) = n"

 unfolding real_of_int_def by simp

 lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n"

 unfolding real_of_int_def by (simp add: floor_minus)

 lemma real_lb_ub_int: " \<exists>n::int. real n \<le> r & r < real (n+1)"

 unfolding real_of_int_def by (rule floor_exists)

 lemma lemma_floor:

   assumes a1: "real m \<le> r" and a2: "r < real n + 1"

   shows "m \<le> (n::int)"

 proof -

   have "real m < real n + 1" using a1 a2 by (rule order_le_less_trans)

   also have "... = real (n + 1)" by simp

   finally have "m < n + 1" by (simp only: real_of_int_less_iff)

   thus ?thesis by arith

 qed

 lemma real_of_int_floor_le [simp]: "real (floor r) \<le> r"

 unfolding real_of_int_def by (rule of_int_floor_le)

 lemma lemma_floor2: "real n < real (x::int) + 1 ==> n \<le> x"

 by (auto intro: lemma_floor)

 lemma real_of_int_floor_cancel [simp]:

     "(real (floor x) = x) = (\<exists>n::int. x = real n)"

   using floor_real_of_int by metis

 lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n"

   unfolding real_of_int_def using floor_unique [of n x] by simp

 lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n"

   unfolding real_of_int_def by (rule floor_unique)

 lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"

 apply (rule inj_int [THEN injD])

 apply (simp add: real_of_nat_Suc)

 apply (simp add: real_of_nat_Suc floor_eq floor_eq [where n = "int n"])

 done

 lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"

 apply (drule order_le_imp_less_or_eq)

 apply (auto intro: floor_eq3)

 done

 lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)"

   unfolding real_of_int_def using floor_correct [of r] by simp

 lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real(floor r)"

   unfolding real_of_int_def using floor_correct [of r] by simp

 lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1"

   unfolding real_of_int_def using floor_correct [of r] by simp

 lemma real_of_int_floor_add_one_gt [simp]: "r < real(floor r) + 1"

   unfolding real_of_int_def using floor_correct [of r] by simp

 lemma le_floor: "real a <= x ==> a <= floor x"

   unfolding real_of_int_def by (simp add: le_floor_iff)

 lemma real_le_floor: "a <= floor x ==> real a <= x"

   unfolding real_of_int_def by (simp add: le_floor_iff)

 lemma le_floor_eq: "(a <= floor x) = (real a <= x)"

   unfolding real_of_int_def by (rule le_floor_iff)

 lemma floor_less_eq: "(floor x < a) = (x < real a)"

   unfolding real_of_int_def by (rule floor_less_iff)

 lemma less_floor_eq: "(a < floor x) = (real a + 1 <= x)"

   unfolding real_of_int_def by (rule less_floor_iff)

 lemma floor_le_eq: "(floor x <= a) = (x < real a + 1)"

   unfolding real_of_int_def by (rule floor_le_iff)

 lemma floor_add [simp]: "floor (x + real a) = floor x + a"

   unfolding real_of_int_def by (rule floor_add_of_int)

 lemma floor_subtract [simp]: "floor (x - real a) = floor x - a"

   unfolding real_of_int_def by (rule floor_diff_of_int)

 lemma le_mult_floor:

   assumes "0 \<le> (a :: real)" and "0 \<le> b"

   shows "floor a * floor b \<le> floor (a * b)"

 proof -

   have "real (floor a) \<le> a"

     and "real (floor b) \<le> b" by auto

   hence "real (floor a * floor b) \<le> a * b"

     using assms by (auto intro!: mult_mono)

   also have "a * b < real (floor (a * b) + 1)" by auto

   finally show ?thesis unfolding real_of_int_less_iff by simp

 qed

 lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"

   unfolding real_of_nat_def by simp

 lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"

   unfolding real_of_int_def by (rule le_of_int_ceiling)

 lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n"

   unfolding real_of_int_def by simp

 lemma real_of_int_ceiling_cancel [simp]:

      "(real (ceiling x) = x) = (\<exists>n::int. x = real n)"

   using ceiling_real_of_int by metis

 lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1"

   unfolding real_of_int_def using ceiling_unique [of "n + 1" x] by simp

 lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1"

   unfolding real_of_int_def using ceiling_unique [of "n + 1" x] by simp

 lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n  |] ==> ceiling x = n"

   unfolding real_of_int_def using ceiling_unique [of n x] by simp

 lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r"

   unfolding real_of_int_def using ceiling_correct [of r] by simp

 lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1"

   unfolding real_of_int_def using ceiling_correct [of r] by simp

 lemma ceiling_le: "x <= real a ==> ceiling x <= a"

   unfolding real_of_int_def by (simp add: ceiling_le_iff)

 lemma ceiling_le_real: "ceiling x <= a ==> x <= real a"

   unfolding real_of_int_def by (simp add: ceiling_le_iff)

 lemma ceiling_le_eq: "(ceiling x <= a) = (x <= real a)"

   unfolding real_of_int_def by (rule ceiling_le_iff)

 lemma less_ceiling_eq: "(a < ceiling x) = (real a < x)"

   unfolding real_of_int_def by (rule less_ceiling_iff)

 lemma ceiling_less_eq: "(ceiling x < a) = (x <= real a - 1)"

   unfolding real_of_int_def by (rule ceiling_less_iff)

 lemma le_ceiling_eq: "(a <= ceiling x) = (real a - 1 < x)"

   unfolding real_of_int_def by (rule le_ceiling_iff)

 lemma ceiling_add [simp]: "ceiling (x + real a) = ceiling x + a"

   unfolding real_of_int_def by (rule ceiling_add_of_int)

 lemma ceiling_subtract [simp]: "ceiling (x - real a) = ceiling x - a"

   unfolding real_of_int_def by (rule ceiling_diff_of_int)

 subsection {* Versions for the natural numbers *}

 definition

   natfloor :: "real => nat" where

   "natfloor x = nat(floor x)"

 definition

   natceiling :: "real => nat" where

   "natceiling x = nat(ceiling x)"

 lemma natfloor_zero [simp]: "natfloor 0 = 0"

   by (unfold natfloor_def, simp)

 lemma natfloor_one [simp]: "natfloor 1 = 1"

   by (unfold natfloor_def, simp)

 lemma zero_le_natfloor [simp]: "0 <= natfloor x"

   by (unfold natfloor_def, simp)

 lemma natfloor_number_of_eq [simp]: "natfloor (number_of n) = number_of n"

   by (unfold natfloor_def, simp)

 lemma natfloor_real_of_nat [simp]: "natfloor(real n) = n"

   by (unfold natfloor_def, simp)

 lemma real_natfloor_le: "0 <= x ==> real(natfloor x) <= x"

   by (unfold natfloor_def, simp)

 lemma natfloor_neg: "x <= 0 ==> natfloor x = 0"

   unfolding natfloor_def by simp

 lemma natfloor_mono: "x <= y ==> natfloor x <= natfloor y"

   unfolding natfloor_def by (intro nat_mono floor_mono)

 lemma le_natfloor: "real x <= a ==> x <= natfloor a"

   apply (unfold natfloor_def)

   apply (subst nat_int [THEN sym])

   apply (rule nat_mono)

   apply (rule le_floor)

   apply simp

 done

 lemma natfloor_less_iff: "0 \<le> x \<Longrightarrow> natfloor x < n \<longleftrightarrow> x < real n"

   unfolding natfloor_def real_of_nat_def

   by (simp add: nat_less_iff floor_less_iff)

 lemma less_natfloor:

   assumes "0 \<le> x" and "x < real (n :: nat)"

   shows "natfloor x < n"

   using assms by (simp add: natfloor_less_iff)

 lemma le_natfloor_eq: "0 <= x ==> (a <= natfloor x) = (real a <= x)"

   apply (rule iffI)

   apply (rule order_trans)

   prefer 2

   apply (erule real_natfloor_le)

   apply (subst real_of_nat_le_iff)

   apply assumption

   apply (erule le_natfloor)

 done

 lemma le_natfloor_eq_number_of [simp]:

     "~ neg((number_of n)::int) ==> 0 <= x ==>

       (number_of n <= natfloor x) = (number_of n <= x)"

   apply (subst le_natfloor_eq, assumption)

   apply simp

 done

 lemma le_natfloor_eq_one [simp]: "(1 <= natfloor x) = (1 <= x)"

   apply (case_tac "0 <= x")

   apply (subst le_natfloor_eq, assumption, simp)

   apply (rule iffI)

   apply (subgoal_tac "natfloor x <= natfloor 0")

   apply simp

   apply (rule natfloor_mono)

   apply simp

   apply simp

 done

 lemma natfloor_eq: "real n <= x ==> x < real n + 1 ==> natfloor x = n"

   unfolding natfloor_def by (simp add: floor_eq2 [where n="int n"])

 lemma real_natfloor_add_one_gt: "x < real(natfloor x) + 1"

   apply (case_tac "0 <= x")

   apply (unfold natfloor_def)

   apply simp

   apply simp_all

 done

 lemma real_natfloor_gt_diff_one: "x - 1 < real(natfloor x)"

 using real_natfloor_add_one_gt by (simp add: algebra_simps)

 lemma ge_natfloor_plus_one_imp_gt: "natfloor z + 1 <= n ==> z < real n"

   apply (subgoal_tac "z < real(natfloor z) + 1")

   apply arith

   apply (rule real_natfloor_add_one_gt)

 done

 lemma natfloor_add [simp]: "0 <= x ==> natfloor (x + real a) = natfloor x + a"

   unfolding natfloor_def

   unfolding real_of_int_of_nat_eq [symmetric] floor_add

   by (simp add: nat_add_distrib)

 lemma natfloor_add_number_of [simp]:

     "~neg ((number_of n)::int) ==> 0 <= x ==>

       natfloor (x + number_of n) = natfloor x + number_of n"

   by (simp add: natfloor_add [symmetric])

 lemma natfloor_add_one: "0 <= x ==> natfloor(x + 1) = natfloor x + 1"

   by (simp add: natfloor_add [symmetric] del: One_nat_def)

 lemma natfloor_subtract [simp]:

     "natfloor(x - real a) = natfloor x - a"

   unfolding natfloor_def real_of_int_of_nat_eq [symmetric] floor_subtract

   by simp

 lemma natfloor_div_nat:

   assumes "1 <= x" and "y > 0"

   shows "natfloor (x / real y) = natfloor x div y"

 proof (rule natfloor_eq)

   have "(natfloor x) div y * y \<le> natfloor x"

     by (rule add_leD1 [where k="natfloor x mod y"], simp)

   thus "real (natfloor x div y) \<le> x / real y"

     using assms by (simp add: le_divide_eq le_natfloor_eq)

   have "natfloor x < (natfloor x) div y * y + y"

     apply (subst mod_div_equality [symmetric])

     apply (rule add_strict_left_mono)

     apply (rule mod_less_divisor)

     apply fact

     done

   thus "x / real y < real (natfloor x div y) + 1"

     using assms

     by (simp add: divide_less_eq natfloor_less_iff left_distrib)

 qed

 lemma le_mult_natfloor:

   shows "natfloor a * natfloor b \<le> natfloor (a * b)"

   by (cases "0 <= a & 0 <= b")

     (auto simp add: le_natfloor_eq mult_nonneg_nonneg mult_mono' real_natfloor_le natfloor_neg)

 lemma natceiling_zero [simp]: "natceiling 0 = 0"

   by (unfold natceiling_def, simp)

 lemma natceiling_one [simp]: "natceiling 1 = 1"

   by (unfold natceiling_def, simp)

 lemma zero_le_natceiling [simp]: "0 <= natceiling x"

   by (unfold natceiling_def, simp)

 lemma natceiling_number_of_eq [simp]: "natceiling (number_of n) = number_of n"

   by (unfold natceiling_def, simp)

 lemma natceiling_real_of_nat [simp]: "natceiling(real n) = n"

   by (unfold natceiling_def, simp)

 lemma real_natceiling_ge: "x <= real(natceiling x)"

   unfolding natceiling_def by (cases "x < 0", simp_all)

 lemma natceiling_neg: "x <= 0 ==> natceiling x = 0"

   unfolding natceiling_def by simp

 lemma natceiling_mono: "x <= y ==> natceiling x <= natceiling y"

   unfolding natceiling_def by (intro nat_mono ceiling_mono)

 lemma natceiling_le: "x <= real a ==> natceiling x <= a"

   unfolding natceiling_def real_of_nat_def

   by (simp add: nat_le_iff ceiling_le_iff)

 lemma natceiling_le_eq: "(natceiling x <= a) = (x <= real a)"

   unfolding natceiling_def real_of_nat_def

   by (simp add: nat_le_iff ceiling_le_iff)

 lemma natceiling_le_eq_number_of [simp]:

     "~ neg((number_of n)::int) ==>

       (natceiling x <= number_of n) = (x <= number_of n)"

   by (simp add: natceiling_le_eq)

 lemma natceiling_le_eq_one: "(natceiling x <= 1) = (x <= 1)"

   unfolding natceiling_def

   by (simp add: nat_le_iff ceiling_le_iff)

 lemma natceiling_eq: "real n < x ==> x <= real n + 1 ==> natceiling x = n + 1"

   unfolding natceiling_def

   by (simp add: ceiling_eq2 [where n="int n"])

 lemma natceiling_add [simp]: "0 <= x ==>

     natceiling (x + real a) = natceiling x + a"

   unfolding natceiling_def

   unfolding real_of_int_of_nat_eq [symmetric] ceiling_add

   by (simp add: nat_add_distrib)

 lemma natceiling_add_number_of [simp]:

     "~ neg ((number_of n)::int) ==> 0 <= x ==>

       natceiling (x + number_of n) = natceiling x + number_of n"

   by (simp add: natceiling_add [symmetric])

 lemma natceiling_add_one: "0 <= x ==> natceiling(x + 1) = natceiling x + 1"

   by (simp add: natceiling_add [symmetric] del: One_nat_def)

 lemma natceiling_subtract [simp]: "natceiling(x - real a) = natceiling x - a"

   unfolding natceiling_def real_of_int_of_nat_eq [symmetric] ceiling_subtract

   by simp

 subsection {* Exponentiation with floor *}

 lemma floor_power:

   assumes "x = real (floor x)"

   shows "floor (x ^ n) = floor x ^ n"

 proof -

   have *: "x ^ n = real (floor x ^ n)"

     using assms by (induct n arbitrary: x) simp_all

   show ?thesis unfolding real_of_int_inject[symmetric]

     unfolding * floor_real_of_int ..

 qed

 lemma natfloor_power:

   assumes "x = real (natfloor x)"

   shows "natfloor (x ^ n) = natfloor x ^ n"

 proof -

   from assms have "0 \<le> floor x" by auto

   note assms[unfolded natfloor_def real_nat_eq_real[OF `0 \<le> floor x`]]

   from floor_power[OF this]

   show ?thesis unfolding natfloor_def nat_power_eq[OF `0 \<le> floor x`, symmetric]

     by simp

 qed

 end