src/HOL/RComplete.thy
author huffman
Sun Sep 04 06:56:10 2011 -0700 (2011-09-04)
changeset 44708 37ce74ff4203
parent 44707 487ae6317f7b
child 45966 03ce2b2a29a2
permissions -rw-r--r--
remove unused assumptions from natceiling lemmas
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(*  Title:      HOL/RComplete.thy
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    Author:     Jacques D. Fleuriot, University of Edinburgh
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    Author:     Larry Paulson, University of Cambridge
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    Author:     Jeremy Avigad, Carnegie Mellon University
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    Author:     Florian Zuleger, Johannes Hoelzl, and Simon Funke, TU Muenchen
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*)
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header {* Completeness of the Reals; Floor and Ceiling Functions *}
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theory RComplete
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imports Lubs RealDef
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begin
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lemma real_sum_of_halves: "x/2 + x/2 = (x::real)"
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  by simp
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lemma abs_diff_less_iff:
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  "(\<bar>x - a\<bar> < (r::'a::linordered_idom)) = (a - r < x \<and> x < a + r)"
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  by auto
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subsection {* Completeness of Positive Reals *}
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text {*
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  Supremum property for the set of positive reals
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  Let @{text "P"} be a non-empty set of positive reals, with an upper
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  bound @{text "y"}.  Then @{text "P"} has a least upper bound
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  (written @{text "S"}).
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  FIXME: Can the premise be weakened to @{text "\<forall>x \<in> P. x\<le> y"}?
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*}
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text {* Only used in HOL/Import/HOL4Compat.thy; delete? *}
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lemma posreal_complete:
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  fixes P :: "real set"
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  assumes not_empty_P: "\<exists>x. x \<in> P"
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    and upper_bound_Ex: "\<exists>y. \<forall>x \<in> P. x<y"
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  shows "\<exists>S. \<forall>y. (\<exists>x \<in> P. y < x) = (y < S)"
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proof -
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  from upper_bound_Ex have "\<exists>z. \<forall>x\<in>P. x \<le> z"
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    by (auto intro: less_imp_le)
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  from complete_real [OF not_empty_P this] obtain S
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  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
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  have "\<forall>y. (\<exists>x \<in> P. y < x) = (y < S)"
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  proof
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    fix y show "(\<exists>x\<in>P. y < x) = (y < S)"
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      apply (cases "\<exists>x\<in>P. y < x", simp_all)
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      apply (clarify, drule S1, simp)
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      apply (simp add: not_less S2)
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      done
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  qed
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  thus ?thesis ..
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qed
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text {*
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  \medskip Completeness properties using @{text "isUb"}, @{text "isLub"} etc.
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*}
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lemma real_isLub_unique: "[| isLub R S x; isLub R S y |] ==> x = (y::real)"
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  apply (frule isLub_isUb)
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  apply (frule_tac x = y in isLub_isUb)
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  apply (blast intro!: order_antisym dest!: isLub_le_isUb)
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  done
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text {*
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  \medskip reals Completeness (again!)
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*}
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lemma reals_complete:
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  assumes notempty_S: "\<exists>X. X \<in> S"
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    and exists_Ub: "\<exists>Y. isUb (UNIV::real set) S Y"
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  shows "\<exists>t. isLub (UNIV :: real set) S t"
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proof -
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  from assms have "\<exists>X. X \<in> S" and "\<exists>Y. \<forall>x\<in>S. x \<le> Y"
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    unfolding isUb_def setle_def by simp_all
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  from complete_real [OF this] show ?thesis
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    unfolding isLub_def leastP_def setle_def setge_def Ball_def
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      Collect_def mem_def isUb_def UNIV_def by simp
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qed
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subsection {* The Archimedean Property of the Reals *}
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theorem reals_Archimedean:
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  assumes x_pos: "0 < x"
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  shows "\<exists>n. inverse (real (Suc n)) < x"
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  unfolding real_of_nat_def using x_pos
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  by (rule ex_inverse_of_nat_Suc_less)
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lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"
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  unfolding real_of_nat_def by (rule ex_less_of_nat)
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lemma reals_Archimedean3:
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  assumes x_greater_zero: "0 < x"
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  shows "\<forall>(y::real). \<exists>(n::nat). y < real n * x"
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  unfolding real_of_nat_def using `0 < x`
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  by (auto intro: ex_less_of_nat_mult)
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subsection{*Density of the Rational Reals in the Reals*}
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text{* This density proof is due to Stefan Richter and was ported by TN.  The
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original source is \emph{Real Analysis} by H.L. Royden.
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It employs the Archimedean property of the reals. *}
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lemma Rats_dense_in_real:
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  fixes x :: real
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  assumes "x < y" shows "\<exists>r\<in>\<rat>. x < r \<and> r < y"
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proof -
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  from `x<y` have "0 < y-x" by simp
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  with reals_Archimedean obtain q::nat 
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    where q: "inverse (real q) < y-x" and "0 < q" by auto
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  def p \<equiv> "ceiling (y * real q) - 1"
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  def r \<equiv> "of_int p / real q"
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  from q have "x < y - inverse (real q)" by simp
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  also have "y - inverse (real q) \<le> r"
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    unfolding r_def p_def
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    by (simp add: le_divide_eq left_diff_distrib le_of_int_ceiling `0 < q`)
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  finally have "x < r" .
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  moreover have "r < y"
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    unfolding r_def p_def
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    by (simp add: divide_less_eq diff_less_eq `0 < q`
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      less_ceiling_iff [symmetric])
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  moreover from r_def have "r \<in> \<rat>" by simp
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  ultimately show ?thesis by fast
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qed
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subsection{*Floor and Ceiling Functions from the Reals to the Integers*}
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lemma number_of_less_real_of_int_iff [simp]:
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     "((number_of n) < real (m::int)) = (number_of n < m)"
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apply auto
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apply (rule real_of_int_less_iff [THEN iffD1])
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apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
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done
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lemma number_of_less_real_of_int_iff2 [simp]:
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     "(real (m::int) < (number_of n)) = (m < number_of n)"
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apply auto
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apply (rule real_of_int_less_iff [THEN iffD1])
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apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
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done
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lemma number_of_le_real_of_int_iff [simp]:
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     "((number_of n) \<le> real (m::int)) = (number_of n \<le> m)"
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by (simp add: linorder_not_less [symmetric])
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lemma number_of_le_real_of_int_iff2 [simp]:
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     "(real (m::int) \<le> (number_of n)) = (m \<le> number_of n)"
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by (simp add: linorder_not_less [symmetric])
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lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n"
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unfolding real_of_nat_def by simp
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lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n"
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unfolding real_of_nat_def by (simp add: floor_minus)
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lemma floor_real_of_int [simp]: "floor (real (n::int)) = n"
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unfolding real_of_int_def by simp
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lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n"
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unfolding real_of_int_def by (simp add: floor_minus)
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lemma real_lb_ub_int: " \<exists>n::int. real n \<le> r & r < real (n+1)"
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unfolding real_of_int_def by (rule floor_exists)
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lemma lemma_floor:
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  assumes a1: "real m \<le> r" and a2: "r < real n + 1"
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  shows "m \<le> (n::int)"
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proof -
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  have "real m < real n + 1" using a1 a2 by (rule order_le_less_trans)
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  also have "... = real (n + 1)" by simp
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  finally have "m < n + 1" by (simp only: real_of_int_less_iff)
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  thus ?thesis by arith
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qed
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lemma real_of_int_floor_le [simp]: "real (floor r) \<le> r"
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unfolding real_of_int_def by (rule of_int_floor_le)
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lemma lemma_floor2: "real n < real (x::int) + 1 ==> n \<le> x"
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by (auto intro: lemma_floor)
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lemma real_of_int_floor_cancel [simp]:
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    "(real (floor x) = x) = (\<exists>n::int. x = real n)"
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  using floor_real_of_int by metis
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lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n"
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  unfolding real_of_int_def using floor_unique [of n x] by simp
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lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n"
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  unfolding real_of_int_def by (rule floor_unique)
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lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"
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apply (rule inj_int [THEN injD])
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apply (simp add: real_of_nat_Suc)
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apply (simp add: real_of_nat_Suc floor_eq floor_eq [where n = "int n"])
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done
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lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"
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apply (drule order_le_imp_less_or_eq)
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apply (auto intro: floor_eq3)
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done
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lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)"
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  unfolding real_of_int_def using floor_correct [of r] by simp
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lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real(floor r)"
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  unfolding real_of_int_def using floor_correct [of r] by simp
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lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1"
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  unfolding real_of_int_def using floor_correct [of r] by simp
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lemma real_of_int_floor_add_one_gt [simp]: "r < real(floor r) + 1"
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  unfolding real_of_int_def using floor_correct [of r] by simp
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lemma le_floor: "real a <= x ==> a <= floor x"
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  unfolding real_of_int_def by (simp add: le_floor_iff)
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lemma real_le_floor: "a <= floor x ==> real a <= x"
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  unfolding real_of_int_def by (simp add: le_floor_iff)
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lemma le_floor_eq: "(a <= floor x) = (real a <= x)"
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  unfolding real_of_int_def by (rule le_floor_iff)
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lemma floor_less_eq: "(floor x < a) = (x < real a)"
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  unfolding real_of_int_def by (rule floor_less_iff)
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lemma less_floor_eq: "(a < floor x) = (real a + 1 <= x)"
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  unfolding real_of_int_def by (rule less_floor_iff)
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lemma floor_le_eq: "(floor x <= a) = (x < real a + 1)"
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  unfolding real_of_int_def by (rule floor_le_iff)
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lemma floor_add [simp]: "floor (x + real a) = floor x + a"
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  unfolding real_of_int_def by (rule floor_add_of_int)
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lemma floor_subtract [simp]: "floor (x - real a) = floor x - a"
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  unfolding real_of_int_def by (rule floor_diff_of_int)
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lemma le_mult_floor:
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  assumes "0 \<le> (a :: real)" and "0 \<le> b"
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  shows "floor a * floor b \<le> floor (a * b)"
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proof -
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  have "real (floor a) \<le> a"
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    and "real (floor b) \<le> b" by auto
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  hence "real (floor a * floor b) \<le> a * b"
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    using assms by (auto intro!: mult_mono)
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  also have "a * b < real (floor (a * b) + 1)" by auto
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  finally show ?thesis unfolding real_of_int_less_iff by simp
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qed
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lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"
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  unfolding real_of_nat_def by simp
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lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"
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  unfolding real_of_int_def by (rule le_of_int_ceiling)
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lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n"
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  unfolding real_of_int_def by simp
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lemma real_of_int_ceiling_cancel [simp]:
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     "(real (ceiling x) = x) = (\<exists>n::int. x = real n)"
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  using ceiling_real_of_int by metis
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lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1"
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  unfolding real_of_int_def using ceiling_unique [of "n + 1" x] by simp
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lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1"
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  unfolding real_of_int_def using ceiling_unique [of "n + 1" x] by simp
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lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n  |] ==> ceiling x = n"
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  unfolding real_of_int_def using ceiling_unique [of n x] by simp
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lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r"
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  unfolding real_of_int_def using ceiling_correct [of r] by simp
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lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1"
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  unfolding real_of_int_def using ceiling_correct [of r] by simp
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lemma ceiling_le: "x <= real a ==> ceiling x <= a"
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  unfolding real_of_int_def by (simp add: ceiling_le_iff)
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lemma ceiling_le_real: "ceiling x <= a ==> x <= real a"
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  unfolding real_of_int_def by (simp add: ceiling_le_iff)
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lemma ceiling_le_eq: "(ceiling x <= a) = (x <= real a)"
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  unfolding real_of_int_def by (rule ceiling_le_iff)
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lemma less_ceiling_eq: "(a < ceiling x) = (real a < x)"
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  unfolding real_of_int_def by (rule less_ceiling_iff)
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lemma ceiling_less_eq: "(ceiling x < a) = (x <= real a - 1)"
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  unfolding real_of_int_def by (rule ceiling_less_iff)
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lemma le_ceiling_eq: "(a <= ceiling x) = (real a - 1 < x)"
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  unfolding real_of_int_def by (rule le_ceiling_iff)
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avigad@16819
   301
lemma ceiling_add [simp]: "ceiling (x + real a) = ceiling x + a"
huffman@30097
   302
  unfolding real_of_int_def by (rule ceiling_add_of_int)
avigad@16819
   303
avigad@16819
   304
lemma ceiling_subtract [simp]: "ceiling (x - real a) = ceiling x - a"
huffman@30097
   305
  unfolding real_of_int_def by (rule ceiling_diff_of_int)
avigad@16819
   306
avigad@16819
   307
avigad@16819
   308
subsection {* Versions for the natural numbers *}
avigad@16819
   309
wenzelm@19765
   310
definition
wenzelm@21404
   311
  natfloor :: "real => nat" where
wenzelm@19765
   312
  "natfloor x = nat(floor x)"
wenzelm@21404
   313
wenzelm@21404
   314
definition
wenzelm@21404
   315
  natceiling :: "real => nat" where
wenzelm@19765
   316
  "natceiling x = nat(ceiling x)"
avigad@16819
   317
avigad@16819
   318
lemma natfloor_zero [simp]: "natfloor 0 = 0"
avigad@16819
   319
  by (unfold natfloor_def, simp)
avigad@16819
   320
avigad@16819
   321
lemma natfloor_one [simp]: "natfloor 1 = 1"
avigad@16819
   322
  by (unfold natfloor_def, simp)
avigad@16819
   323
avigad@16819
   324
lemma zero_le_natfloor [simp]: "0 <= natfloor x"
avigad@16819
   325
  by (unfold natfloor_def, simp)
avigad@16819
   326
avigad@16819
   327
lemma natfloor_number_of_eq [simp]: "natfloor (number_of n) = number_of n"
avigad@16819
   328
  by (unfold natfloor_def, simp)
avigad@16819
   329
avigad@16819
   330
lemma natfloor_real_of_nat [simp]: "natfloor(real n) = n"
avigad@16819
   331
  by (unfold natfloor_def, simp)
avigad@16819
   332
avigad@16819
   333
lemma real_natfloor_le: "0 <= x ==> real(natfloor x) <= x"
avigad@16819
   334
  by (unfold natfloor_def, simp)
avigad@16819
   335
avigad@16819
   336
lemma natfloor_neg: "x <= 0 ==> natfloor x = 0"
huffman@44679
   337
  unfolding natfloor_def by simp
huffman@44679
   338
avigad@16819
   339
lemma natfloor_mono: "x <= y ==> natfloor x <= natfloor y"
huffman@44679
   340
  unfolding natfloor_def by (intro nat_mono floor_mono)
avigad@16819
   341
avigad@16819
   342
lemma le_natfloor: "real x <= a ==> x <= natfloor a"
avigad@16819
   343
  apply (unfold natfloor_def)
avigad@16819
   344
  apply (subst nat_int [THEN sym])
huffman@44679
   345
  apply (rule nat_mono)
avigad@16819
   346
  apply (rule le_floor)
avigad@16819
   347
  apply simp
avigad@16819
   348
done
avigad@16819
   349
huffman@44679
   350
lemma natfloor_less_iff: "0 \<le> x \<Longrightarrow> natfloor x < n \<longleftrightarrow> x < real n"
huffman@44679
   351
  unfolding natfloor_def real_of_nat_def
huffman@44679
   352
  by (simp add: nat_less_iff floor_less_iff)
huffman@44679
   353
hoelzl@35578
   354
lemma less_natfloor:
hoelzl@35578
   355
  assumes "0 \<le> x" and "x < real (n :: nat)"
hoelzl@35578
   356
  shows "natfloor x < n"
huffman@44679
   357
  using assms by (simp add: natfloor_less_iff)
hoelzl@35578
   358
avigad@16819
   359
lemma le_natfloor_eq: "0 <= x ==> (a <= natfloor x) = (real a <= x)"
avigad@16819
   360
  apply (rule iffI)
avigad@16819
   361
  apply (rule order_trans)
avigad@16819
   362
  prefer 2
avigad@16819
   363
  apply (erule real_natfloor_le)
avigad@16819
   364
  apply (subst real_of_nat_le_iff)
avigad@16819
   365
  apply assumption
avigad@16819
   366
  apply (erule le_natfloor)
avigad@16819
   367
done
avigad@16819
   368
wenzelm@16893
   369
lemma le_natfloor_eq_number_of [simp]:
avigad@16819
   370
    "~ neg((number_of n)::int) ==> 0 <= x ==>
avigad@16819
   371
      (number_of n <= natfloor x) = (number_of n <= x)"
avigad@16819
   372
  apply (subst le_natfloor_eq, assumption)
avigad@16819
   373
  apply simp
avigad@16819
   374
done
avigad@16819
   375
avigad@16820
   376
lemma le_natfloor_eq_one [simp]: "(1 <= natfloor x) = (1 <= x)"
avigad@16819
   377
  apply (case_tac "0 <= x")
avigad@16819
   378
  apply (subst le_natfloor_eq, assumption, simp)
avigad@16819
   379
  apply (rule iffI)
wenzelm@16893
   380
  apply (subgoal_tac "natfloor x <= natfloor 0")
avigad@16819
   381
  apply simp
avigad@16819
   382
  apply (rule natfloor_mono)
avigad@16819
   383
  apply simp
avigad@16819
   384
  apply simp
avigad@16819
   385
done
avigad@16819
   386
avigad@16819
   387
lemma natfloor_eq: "real n <= x ==> x < real n + 1 ==> natfloor x = n"
huffman@44679
   388
  unfolding natfloor_def by (simp add: floor_eq2 [where n="int n"])
avigad@16819
   389
avigad@16819
   390
lemma real_natfloor_add_one_gt: "x < real(natfloor x) + 1"
avigad@16819
   391
  apply (case_tac "0 <= x")
avigad@16819
   392
  apply (unfold natfloor_def)
avigad@16819
   393
  apply simp
avigad@16819
   394
  apply simp_all
avigad@16819
   395
done
avigad@16819
   396
avigad@16819
   397
lemma real_natfloor_gt_diff_one: "x - 1 < real(natfloor x)"
nipkow@29667
   398
using real_natfloor_add_one_gt by (simp add: algebra_simps)
avigad@16819
   399
avigad@16819
   400
lemma ge_natfloor_plus_one_imp_gt: "natfloor z + 1 <= n ==> z < real n"
avigad@16819
   401
  apply (subgoal_tac "z < real(natfloor z) + 1")
avigad@16819
   402
  apply arith
avigad@16819
   403
  apply (rule real_natfloor_add_one_gt)
avigad@16819
   404
done
avigad@16819
   405
avigad@16819
   406
lemma natfloor_add [simp]: "0 <= x ==> natfloor (x + real a) = natfloor x + a"
huffman@44679
   407
  unfolding natfloor_def
huffman@44679
   408
  unfolding real_of_int_of_nat_eq [symmetric] floor_add
huffman@44679
   409
  by (simp add: nat_add_distrib)
avigad@16819
   410
wenzelm@16893
   411
lemma natfloor_add_number_of [simp]:
wenzelm@16893
   412
    "~neg ((number_of n)::int) ==> 0 <= x ==>
avigad@16819
   413
      natfloor (x + number_of n) = natfloor x + number_of n"
huffman@44679
   414
  by (simp add: natfloor_add [symmetric])
avigad@16819
   415
avigad@16819
   416
lemma natfloor_add_one: "0 <= x ==> natfloor(x + 1) = natfloor x + 1"
huffman@44679
   417
  by (simp add: natfloor_add [symmetric] del: One_nat_def)
avigad@16819
   418
wenzelm@16893
   419
lemma natfloor_subtract [simp]: "real a <= x ==>
avigad@16819
   420
    natfloor(x - real a) = natfloor x - a"
huffman@44679
   421
  unfolding natfloor_def
huffman@44679
   422
  unfolding real_of_int_of_nat_eq [symmetric] floor_subtract
huffman@44679
   423
  by simp
avigad@16819
   424
wenzelm@41550
   425
lemma natfloor_div_nat:
wenzelm@41550
   426
  assumes "1 <= x" and "y > 0"
wenzelm@41550
   427
  shows "natfloor (x / real y) = natfloor x div y"
huffman@44679
   428
proof (rule natfloor_eq)
huffman@44679
   429
  have "(natfloor x) div y * y \<le> natfloor x"
huffman@44679
   430
    by (rule add_leD1 [where k="natfloor x mod y"], simp)
huffman@44679
   431
  thus "real (natfloor x div y) \<le> x / real y"
huffman@44679
   432
    using assms by (simp add: le_divide_eq le_natfloor_eq)
huffman@44679
   433
  have "natfloor x < (natfloor x) div y * y + y"
huffman@44679
   434
    apply (subst mod_div_equality [symmetric])
huffman@44679
   435
    apply (rule add_strict_left_mono)
huffman@44679
   436
    apply (rule mod_less_divisor)
huffman@44679
   437
    apply fact
hoelzl@35578
   438
    done
huffman@44679
   439
  thus "x / real y < real (natfloor x div y) + 1"
huffman@44679
   440
    using assms
huffman@44679
   441
    by (simp add: divide_less_eq natfloor_less_iff left_distrib)
hoelzl@35578
   442
qed
hoelzl@35578
   443
hoelzl@35578
   444
lemma le_mult_natfloor:
hoelzl@35578
   445
  assumes "0 \<le> (a :: real)" and "0 \<le> b"
hoelzl@35578
   446
  shows "natfloor a * natfloor b \<le> natfloor (a * b)"
huffman@44679
   447
  using assms
huffman@44679
   448
  by (simp add: le_natfloor_eq mult_nonneg_nonneg mult_mono' real_natfloor_le)
hoelzl@35578
   449
avigad@16819
   450
lemma natceiling_zero [simp]: "natceiling 0 = 0"
avigad@16819
   451
  by (unfold natceiling_def, simp)
avigad@16819
   452
avigad@16819
   453
lemma natceiling_one [simp]: "natceiling 1 = 1"
avigad@16819
   454
  by (unfold natceiling_def, simp)
avigad@16819
   455
avigad@16819
   456
lemma zero_le_natceiling [simp]: "0 <= natceiling x"
avigad@16819
   457
  by (unfold natceiling_def, simp)
avigad@16819
   458
avigad@16819
   459
lemma natceiling_number_of_eq [simp]: "natceiling (number_of n) = number_of n"
avigad@16819
   460
  by (unfold natceiling_def, simp)
avigad@16819
   461
avigad@16819
   462
lemma natceiling_real_of_nat [simp]: "natceiling(real n) = n"
avigad@16819
   463
  by (unfold natceiling_def, simp)
avigad@16819
   464
avigad@16819
   465
lemma real_natceiling_ge: "x <= real(natceiling x)"
huffman@44679
   466
  unfolding natceiling_def by (cases "x < 0", simp_all)
avigad@16819
   467
avigad@16819
   468
lemma natceiling_neg: "x <= 0 ==> natceiling x = 0"
huffman@44679
   469
  unfolding natceiling_def by simp
avigad@16819
   470
avigad@16819
   471
lemma natceiling_mono: "x <= y ==> natceiling x <= natceiling y"
huffman@44679
   472
  unfolding natceiling_def by (intro nat_mono ceiling_mono)
huffman@44679
   473
avigad@16819
   474
lemma natceiling_le: "x <= real a ==> natceiling x <= a"
huffman@44679
   475
  unfolding natceiling_def real_of_nat_def
huffman@44679
   476
  by (simp add: nat_le_iff ceiling_le_iff)
avigad@16819
   477
huffman@44708
   478
lemma natceiling_le_eq: "(natceiling x <= a) = (x <= real a)"
huffman@44708
   479
  unfolding natceiling_def real_of_nat_def
huffman@44679
   480
  by (simp add: nat_le_iff ceiling_le_iff)
avigad@16819
   481
wenzelm@16893
   482
lemma natceiling_le_eq_number_of [simp]:
huffman@44708
   483
    "~ neg((number_of n)::int) ==>
avigad@16820
   484
      (natceiling x <= number_of n) = (x <= number_of n)"
huffman@44679
   485
  by (simp add: natceiling_le_eq)
avigad@16819
   486
avigad@16820
   487
lemma natceiling_le_eq_one: "(natceiling x <= 1) = (x <= 1)"
huffman@44679
   488
  unfolding natceiling_def
huffman@44679
   489
  by (simp add: nat_le_iff ceiling_le_iff)
avigad@16819
   490
avigad@16819
   491
lemma natceiling_eq: "real n < x ==> x <= real n + 1 ==> natceiling x = n + 1"
huffman@44679
   492
  unfolding natceiling_def
huffman@44679
   493
  by (simp add: ceiling_eq2 [where n="int n"])
avigad@16819
   494
wenzelm@16893
   495
lemma natceiling_add [simp]: "0 <= x ==>
avigad@16819
   496
    natceiling (x + real a) = natceiling x + a"
huffman@44679
   497
  unfolding natceiling_def
huffman@44679
   498
  unfolding real_of_int_of_nat_eq [symmetric] ceiling_add
huffman@44679
   499
  by (simp add: nat_add_distrib)
avigad@16819
   500
wenzelm@16893
   501
lemma natceiling_add_number_of [simp]:
wenzelm@16893
   502
    "~ neg ((number_of n)::int) ==> 0 <= x ==>
avigad@16820
   503
      natceiling (x + number_of n) = natceiling x + number_of n"
huffman@44679
   504
  by (simp add: natceiling_add [symmetric])
avigad@16819
   505
avigad@16819
   506
lemma natceiling_add_one: "0 <= x ==> natceiling(x + 1) = natceiling x + 1"
huffman@44679
   507
  by (simp add: natceiling_add [symmetric] del: One_nat_def)
avigad@16819
   508
wenzelm@16893
   509
lemma natceiling_subtract [simp]: "real a <= x ==>
avigad@16819
   510
    natceiling(x - real a) = natceiling x - a"
huffman@44679
   511
  unfolding natceiling_def
huffman@44679
   512
  unfolding real_of_int_of_nat_eq [symmetric] ceiling_subtract
huffman@44679
   513
  by simp
avigad@16819
   514
huffman@36826
   515
subsection {* Exponentiation with floor *}
huffman@36826
   516
huffman@36826
   517
lemma floor_power:
huffman@36826
   518
  assumes "x = real (floor x)"
huffman@36826
   519
  shows "floor (x ^ n) = floor x ^ n"
huffman@36826
   520
proof -
huffman@36826
   521
  have *: "x ^ n = real (floor x ^ n)"
huffman@36826
   522
    using assms by (induct n arbitrary: x) simp_all
huffman@36826
   523
  show ?thesis unfolding real_of_int_inject[symmetric]
huffman@36826
   524
    unfolding * floor_real_of_int ..
huffman@36826
   525
qed
huffman@36826
   526
huffman@36826
   527
lemma natfloor_power:
huffman@36826
   528
  assumes "x = real (natfloor x)"
huffman@36826
   529
  shows "natfloor (x ^ n) = natfloor x ^ n"
huffman@36826
   530
proof -
huffman@36826
   531
  from assms have "0 \<le> floor x" by auto
huffman@36826
   532
  note assms[unfolded natfloor_def real_nat_eq_real[OF `0 \<le> floor x`]]
huffman@36826
   533
  from floor_power[OF this]
huffman@36826
   534
  show ?thesis unfolding natfloor_def nat_power_eq[OF `0 \<le> floor x`, symmetric]
huffman@36826
   535
    by simp
huffman@36826
   536
qed
avigad@16819
   537
paulson@14365
   538
end