src/HOL/Real/RComplete.thy
changeset 14641 79b7bd936264
parent 14476 758e7acdea2f
child 15131 c69542757a4d
--- a/src/HOL/Real/RComplete.thy	Thu Apr 22 10:43:06 2004 +0200
+++ b/src/HOL/Real/RComplete.thy	Thu Apr 22 10:45:56 2004 +0200
@@ -2,11 +2,11 @@
     ID          : $Id$
     Author      : Jacques D. Fleuriot
     Copyright   : 1998  University of Cambridge
-    Description : Completeness theorems for positive
-                  reals and reals 
+    Copyright   : 2001,2002  University of Edinburgh
+Converted to Isar and polished by lcp
 *) 
 
-header{*Completeness Theorems for Positive Reals and Reals.*}
+header{*Completeness of the Reals; Floor and Ceiling Functions*}
 
 theory RComplete = Lubs + RealDef:
 
@@ -215,7 +215,322 @@
 val reals_Archimedean3 = thm "reals_Archimedean3";
 *}
 
+
+subsection{*Floor and Ceiling Functions from the Reals to the Integers*}
+
+constdefs
+
+  floor :: "real => int"
+   "floor r == (LEAST n::int. r < real (n+1))"
+
+  ceiling :: "real => int"
+    "ceiling r == - floor (- r)"
+
+syntax (xsymbols)
+  floor :: "real => int"     ("\<lfloor>_\<rfloor>")
+  ceiling :: "real => int"   ("\<lceil>_\<rceil>")
+
+syntax (HTML output)
+  floor :: "real => int"     ("\<lfloor>_\<rfloor>")
+  ceiling :: "real => int"   ("\<lceil>_\<rceil>")
+
+
+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_zero [simp]: "floor 0 = 0"
+apply (simp add: floor_def)
+apply (rule Least_equality, auto)
+done
+
+lemma floor_real_of_nat_zero [simp]: "floor (real (0::nat)) = 0"
+by auto
+
+lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n"
+apply (simp only: floor_def)
+apply (rule Least_equality)
+apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
+apply (drule_tac [2] real_of_int_less_iff [THEN iffD1])
+apply (simp_all add: real_of_int_real_of_nat)
+done
+
+lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n"
+apply (simp only: floor_def)
+apply (rule Least_equality)
+apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
+apply (drule_tac [2] real_of_int_minus [THEN subst])
+apply (drule_tac [2] real_of_int_less_iff [THEN iffD1])
+apply (simp_all add: real_of_int_real_of_nat)
+done
+
+lemma floor_real_of_int [simp]: "floor (real (n::int)) = n"
+apply (simp only: floor_def)
+apply (rule Least_equality)
+apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
+apply (drule_tac [2] real_of_int_less_iff [THEN iffD1], auto)
+done
+
+lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n"
+apply (simp only: floor_def)
+apply (rule Least_equality)
+apply (drule_tac [2] real_of_int_minus [THEN subst])
+apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
+apply (drule_tac [2] real_of_int_less_iff [THEN iffD1], auto)
+done
+
+lemma reals_Archimedean6:
+     "0 \<le> r ==> \<exists>(n::nat). real (n - 1) \<le> r & r < real (n)"
+apply (insert reals_Archimedean2 [of r], safe)
+apply (frule_tac P = "%k. r < real k" and k = n and m = "%x. x"
+       in ex_has_least_nat, auto)
+apply (rule_tac x = x in exI)
+apply (case_tac x, simp)
+apply (rename_tac x')
+apply (drule_tac x = x' in spec, simp)
+done
+
+lemma reals_Archimedean6a: "0 \<le> r ==> \<exists>n. real (n) \<le> r & r < real (Suc n)"
+by (drule reals_Archimedean6, auto)
+
+lemma reals_Archimedean_6b_int:
+     "0 \<le> r ==> \<exists>n::int. real n \<le> r & r < real (n+1)"
+apply (drule reals_Archimedean6a, auto)
+apply (rule_tac x = "int n" in exI)
+apply (simp add: real_of_int_real_of_nat real_of_nat_Suc)
+done
+
+lemma reals_Archimedean_6c_int:
+     "r < 0 ==> \<exists>n::int. real n \<le> r & r < real (n+1)"
+apply (rule reals_Archimedean_6b_int [of "-r", THEN exE], simp, auto)
+apply (rename_tac n)
+apply (drule real_le_imp_less_or_eq, auto)
+apply (rule_tac x = "- n - 1" in exI)
+apply (rule_tac [2] x = "- n" in exI, auto)
+done
+
+lemma real_lb_ub_int: " \<exists>n::int. real n \<le> r & r < real (n+1)"
+apply (case_tac "r < 0")
+apply (blast intro: reals_Archimedean_6c_int)
+apply (simp only: linorder_not_less)
+apply (blast intro: reals_Archimedean_6b_int reals_Archimedean_6c_int)
+done
+
+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" 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"
+apply (simp add: floor_def Least_def)
+apply (insert real_lb_ub_int [of r], safe)
+apply (rule theI2, auto)
+done
+
+lemma floor_le: "x < y ==> floor x \<le> floor y"
+apply (simp add: floor_def Least_def)
+apply (insert real_lb_ub_int [of x])
+apply (insert real_lb_ub_int [of y], safe)
+apply (rule theI2)
+apply (rule_tac [3] theI2, auto)
+done
+
+lemma floor_le2: "x \<le> y ==> floor x \<le> floor y"
+by (auto dest: real_le_imp_less_or_eq simp add: 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)"
+apply (simp add: floor_def Least_def)
+apply (insert real_lb_ub_int [of x], erule exE)
+apply (rule theI2)
+apply (auto intro: lemma_floor)
+done
+
+lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n"
+apply (simp add: floor_def)
+apply (rule Least_equality)
+apply (auto intro: lemma_floor)
+done
+
+lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n"
+apply (simp add: floor_def)
+apply (rule Least_equality)
+apply (auto intro: lemma_floor)
+done
+
+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 = "of_nat 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 floor_number_of_eq [simp]:
+     "floor(number_of n :: real) = (number_of n :: int)"
+apply (subst real_number_of [symmetric])
+apply (rule floor_real_of_int)
+done
+
+lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)"
+apply (simp add: floor_def Least_def)
+apply (insert real_lb_ub_int [of r], safe)
+apply (rule theI2)
+apply (auto intro: lemma_floor)
+done
+
+lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1"
+apply (insert real_of_int_floor_ge_diff_one [of r])
+apply (auto simp del: real_of_int_floor_ge_diff_one)
+done
+
+
+subsection{*Ceiling Function for Positive Reals*}
+
+lemma ceiling_zero [simp]: "ceiling 0 = 0"
+by (simp add: ceiling_def)
+
+lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"
+by (simp add: ceiling_def)
+
+lemma ceiling_real_of_nat_zero [simp]: "ceiling (real (0::nat)) = 0"
+by auto
+
+lemma ceiling_floor [simp]: "ceiling (real (floor r)) = floor r"
+by (simp add: ceiling_def)
+
+lemma floor_ceiling [simp]: "floor (real (ceiling r)) = ceiling r"
+by (simp add: ceiling_def)
+
+lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"
+apply (simp add: ceiling_def)
+apply (subst le_minus_iff, simp)
+done
+
+lemma ceiling_le: "x < y ==> ceiling x \<le> ceiling y"
+by (simp add: floor_le ceiling_def)
+
+lemma ceiling_le2: "x \<le> y ==> ceiling x \<le> ceiling y"
+by (simp add: floor_le2 ceiling_def)
+
+lemma real_of_int_ceiling_cancel [simp]:
+     "(real (ceiling x) = x) = (\<exists>n::int. x = real n)"
+apply (auto simp add: ceiling_def)
+apply (drule arg_cong [where f = uminus], auto)
+apply (rule_tac x = "-n" in exI, auto)
+done
+
+lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1"
+apply (simp add: ceiling_def)
+apply (rule minus_equation_iff [THEN iffD1])
+apply (simp add: floor_eq [where n = "-(n+1)"])
+done
+
+lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1"
+by (simp add: ceiling_def floor_eq2 [where n = "-(n+1)"])
+
+lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n  |] ==> ceiling x = n"
+by (simp add: ceiling_def floor_eq2 [where n = "-n"])
+
+lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n"
+by (simp add: ceiling_def)
+
+lemma ceiling_number_of_eq [simp]:
+     "ceiling (number_of n :: real) = (number_of n)"
+apply (subst real_number_of [symmetric])
+apply (rule ceiling_real_of_int)
+done
+
+lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r"
+apply (rule neg_le_iff_le [THEN iffD1])
+apply (simp add: ceiling_def diff_minus)
+done
+
+lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1"
+apply (insert real_of_int_ceiling_diff_one_le [of r])
+apply (simp del: real_of_int_ceiling_diff_one_le)
+done
+
+ML
+{*
+val number_of_less_real_of_int_iff = thm "number_of_less_real_of_int_iff";
+val number_of_less_real_of_int_iff2 = thm "number_of_less_real_of_int_iff2";
+val number_of_le_real_of_int_iff = thm "number_of_le_real_of_int_iff";
+val number_of_le_real_of_int_iff2 = thm "number_of_le_real_of_int_iff2";
+val floor_zero = thm "floor_zero";
+val floor_real_of_nat_zero = thm "floor_real_of_nat_zero";
+val floor_real_of_nat = thm "floor_real_of_nat";
+val floor_minus_real_of_nat = thm "floor_minus_real_of_nat";
+val floor_real_of_int = thm "floor_real_of_int";
+val floor_minus_real_of_int = thm "floor_minus_real_of_int";
+val reals_Archimedean6 = thm "reals_Archimedean6";
+val reals_Archimedean6a = thm "reals_Archimedean6a";
+val reals_Archimedean_6b_int = thm "reals_Archimedean_6b_int";
+val reals_Archimedean_6c_int = thm "reals_Archimedean_6c_int";
+val real_lb_ub_int = thm "real_lb_ub_int";
+val lemma_floor = thm "lemma_floor";
+val real_of_int_floor_le = thm "real_of_int_floor_le";
+val floor_le = thm "floor_le";
+val floor_le2 = thm "floor_le2";
+val lemma_floor2 = thm "lemma_floor2";
+val real_of_int_floor_cancel = thm "real_of_int_floor_cancel";
+val floor_eq = thm "floor_eq";
+val floor_eq2 = thm "floor_eq2";
+val floor_eq3 = thm "floor_eq3";
+val floor_eq4 = thm "floor_eq4";
+val floor_number_of_eq = thm "floor_number_of_eq";
+val real_of_int_floor_ge_diff_one = thm "real_of_int_floor_ge_diff_one";
+val real_of_int_floor_add_one_ge = thm "real_of_int_floor_add_one_ge";
+val ceiling_zero = thm "ceiling_zero";
+val ceiling_real_of_nat = thm "ceiling_real_of_nat";
+val ceiling_real_of_nat_zero = thm "ceiling_real_of_nat_zero";
+val ceiling_floor = thm "ceiling_floor";
+val floor_ceiling = thm "floor_ceiling";
+val real_of_int_ceiling_ge = thm "real_of_int_ceiling_ge";
+val ceiling_le = thm "ceiling_le";
+val ceiling_le2 = thm "ceiling_le2";
+val real_of_int_ceiling_cancel = thm "real_of_int_ceiling_cancel";
+val ceiling_eq = thm "ceiling_eq";
+val ceiling_eq2 = thm "ceiling_eq2";
+val ceiling_eq3 = thm "ceiling_eq3";
+val ceiling_real_of_int = thm "ceiling_real_of_int";
+val ceiling_number_of_eq = thm "ceiling_number_of_eq";
+val real_of_int_ceiling_diff_one_le = thm "real_of_int_ceiling_diff_one_le";
+val real_of_int_ceiling_le_add_one = thm "real_of_int_ceiling_le_add_one";
+*}
+
+
 end
 
 
 
+