# HG changeset patch
# User hoelzl
# Date 1364296855 -3600
# Node ID 36fa825e0ea7999106841cdc899d14aa96019111
# Parent  e9b361845809b681b0e0f13506dea71ffbfe20d3
merge RComplete into RealDef

diff -r e9b361845809 -r 36fa825e0ea7 src/HOL/Metric_Spaces.thy
--- a/src/HOL/Metric_Spaces.thy	Tue Mar 26 12:20:54 2013 +0100
+++ b/src/HOL/Metric_Spaces.thy	Tue Mar 26 12:20:55 2013 +0100
@@ -6,7 +6,7 @@
 header {* Metric Spaces *}
 
 theory Metric_Spaces
-imports RComplete Topological_Spaces
+imports RealDef Topological_Spaces
 begin
 
 
diff -r e9b361845809 -r 36fa825e0ea7 src/HOL/NSA/NSA.thy
--- a/src/HOL/NSA/NSA.thy	Tue Mar 26 12:20:54 2013 +0100
+++ b/src/HOL/NSA/NSA.thy	Tue Mar 26 12:20:55 2013 +0100
@@ -6,7 +6,7 @@
 header{*Infinite Numbers, Infinitesimals, Infinitely Close Relation*}
 
 theory NSA
-imports HyperDef RComplete
+imports HyperDef
 begin
 
 definition
diff -r e9b361845809 -r 36fa825e0ea7 src/HOL/Quickcheck_Benchmark/Find_Unused_Assms_Examples.thy
--- a/src/HOL/Quickcheck_Benchmark/Find_Unused_Assms_Examples.thy	Tue Mar 26 12:20:54 2013 +0100
+++ b/src/HOL/Quickcheck_Benchmark/Find_Unused_Assms_Examples.thy	Tue Mar 26 12:20:55 2013 +0100
@@ -9,7 +9,6 @@
 find_unused_assms Divides
 find_unused_assms GCD
 find_unused_assms RealDef
-find_unused_assms RComplete
 
 section {* Set Theory *}
 
diff -r e9b361845809 -r 36fa825e0ea7 src/HOL/RComplete.thy
--- a/src/HOL/RComplete.thy	Tue Mar 26 12:20:54 2013 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,489 +0,0 @@
-(*  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 RealDef
-begin
-
-subsection {* Completeness of Positive Reals *}
-
-text {*
-  \medskip Completeness properties using @{text "isUb"}, @{text "isLub"} etc.
-*}
-
-text {*
-  \medskip reals Completeness (again!)
-*}
-
-lemma reals_complete: "\<exists>X. X \<in> S \<Longrightarrow> \<exists>Y. isUb (UNIV::real set) S Y \<Longrightarrow> \<exists>t. isLub (UNIV :: real set) S t"
-  by (intro exI[of _ "Sup S"] isLub_cSup) (auto simp: setle_def isUb_def intro: cSup_upper)
-
-
-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*}
-
-(* FIXME: theorems for negative numerals *)
-lemma numeral_less_real_of_int_iff [simp]:
-     "((numeral n) < real (m::int)) = (numeral 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 numeral_less_real_of_int_iff2 [simp]:
-     "(real (m::int) < (numeral n)) = (m < numeral 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 numeral_le_real_of_int_iff [simp]:
-     "((numeral n) \<le> real (m::int)) = (numeral n \<le> m)"
-by (simp add: linorder_not_less [symmetric])
-
-lemma numeral_le_real_of_int_iff2 [simp]:
-     "(real (m::int) \<le> (numeral n)) = (m \<le> numeral 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 floor_divide_eq_div:
-  "floor (real a / real b) = a div b"
-proof cases
-  assume "b \<noteq> 0 \<or> b dvd a"
-  with real_of_int_div3[of a b] show ?thesis
-    by (auto simp: real_of_int_div[symmetric] intro!: floor_eq2 real_of_int_div4 neq_le_trans)
-       (metis add_left_cancel zero_neq_one real_of_int_div_aux real_of_int_inject
-              real_of_int_zero_cancel right_inverse_eq div_self mod_div_trivial)
-qed (auto simp: real_of_int_div)
-
-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_numeral_eq [simp]: "natfloor (numeral n) = numeral 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_numeral [simp]:
-    "~ neg((numeral n)::int) ==> 0 <= x ==>
-      (numeral n <= natfloor x) = (numeral 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_numeral [simp]:
-    "~neg ((numeral n)::int) ==> 0 <= x ==>
-      natfloor (x + numeral n) = natfloor x + numeral 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 distrib_right)
-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_numeral_eq [simp]: "natceiling (numeral n) = numeral 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_numeral [simp]:
-    "~ neg((numeral n)::int) ==>
-      (natceiling x <= numeral n) = (x <= numeral 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_numeral [simp]:
-    "~ neg ((numeral n)::int) ==> 0 <= x ==>
-      natceiling (x + numeral n) = natceiling x + numeral 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
diff -r e9b361845809 -r 36fa825e0ea7 src/HOL/Real.thy
--- a/src/HOL/Real.thy	Tue Mar 26 12:20:54 2013 +0100
+++ b/src/HOL/Real.thy	Tue Mar 26 12:20:55 2013 +0100
@@ -1,5 +1,5 @@
 theory Real
-imports RComplete RealVector
+imports RealVector
 begin
 
 ML_file "Tools/SMT/smt_real.ML"
diff -r e9b361845809 -r 36fa825e0ea7 src/HOL/RealDef.thy
--- a/src/HOL/RealDef.thy	Tue Mar 26 12:20:54 2013 +0100
+++ b/src/HOL/RealDef.thy	Tue Mar 26 12:20:55 2013 +0100
@@ -1,5 +1,8 @@
-(*  Title       : HOL/RealDef.thy
-    Author      : Jacques D. Fleuriot, 1998
+(*  Title:      HOL/RealDef.thy
+    Author:     Jacques D. Fleuriot, University of Edinburgh, 1998
+    Author:     Larry Paulson, University of Cambridge
+    Author:     Jeremy Avigad, Carnegie Mellon University
+    Author:     Florian Zuleger, Johannes Hoelzl, and Simon Funke, TU Muenchen
     Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
     Construction of Cauchy Reals by Brian Huffman, 2010
 *)
@@ -970,6 +973,13 @@
 qed
 end
 
+text {*
+  \medskip Completeness properties using @{text "isUb"}, @{text "isLub"}:
+*}
+
+lemma reals_complete: "\<exists>X. X \<in> S \<Longrightarrow> \<exists>Y. isUb (UNIV::real set) S Y \<Longrightarrow> \<exists>t. isLub (UNIV :: real set) S t"
+  by (intro exI[of _ "Sup S"] isLub_cSup) (auto simp: setle_def isUb_def intro: cSup_upper)
+
 
 subsection {* Hiding implementation details *}
 
@@ -1323,6 +1333,23 @@
 lemma Ints_real_of_nat [simp]: "real (n::nat) \<in> Ints"
 unfolding real_of_nat_def by (rule Ints_of_nat)
 
+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{* Rationals *}
 
@@ -1412,6 +1439,35 @@
   ultimately show ?thesis ..
 qed
 
+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{*Numerals and Arithmetic*}
 
@@ -1576,6 +1632,422 @@
 lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"
 by simp
 
+
+subsection{*Floor and Ceiling Functions from the Reals to the Integers*}
+
+(* FIXME: theorems for negative numerals *)
+lemma numeral_less_real_of_int_iff [simp]:
+     "((numeral n) < real (m::int)) = (numeral 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 numeral_less_real_of_int_iff2 [simp]:
+     "(real (m::int) < (numeral n)) = (m < numeral 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 numeral_le_real_of_int_iff [simp]:
+     "((numeral n) \<le> real (m::int)) = (numeral n \<le> m)"
+by (simp add: linorder_not_less [symmetric])
+
+lemma numeral_le_real_of_int_iff2 [simp]:
+     "(real (m::int) \<le> (numeral n)) = (m \<le> numeral 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 floor_divide_eq_div:
+  "floor (real a / real b) = a div b"
+proof cases
+  assume "b \<noteq> 0 \<or> b dvd a"
+  with real_of_int_div3[of a b] show ?thesis
+    by (auto simp: real_of_int_div[symmetric] intro!: floor_eq2 real_of_int_div4 neq_le_trans)
+       (metis add_left_cancel zero_neq_one real_of_int_div_aux real_of_int_inject
+              real_of_int_zero_cancel right_inverse_eq div_self mod_div_trivial)
+qed (auto simp: real_of_int_div)
+
+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)
+
+
+subsubsection {* 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_numeral_eq [simp]: "natfloor (numeral n) = numeral 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_numeral [simp]:
+    "~ neg((numeral n)::int) ==> 0 <= x ==>
+      (numeral n <= natfloor x) = (numeral 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_numeral [simp]:
+    "~neg ((numeral n)::int) ==> 0 <= x ==>
+      natfloor (x + numeral n) = natfloor x + numeral 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 distrib_right)
+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_numeral_eq [simp]: "natceiling (numeral n) = numeral 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_numeral [simp]:
+    "~ neg((numeral n)::int) ==>
+      (natceiling x <= numeral n) = (x <= numeral 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_numeral [simp]:
+    "~ neg ((numeral n)::int) ==> 0 <= x ==>
+      natceiling (x + numeral n) = natceiling x + numeral 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
+
+
 subsection {* Implementation of rational real numbers *}
 
 text {* Formal constructor *}