--- a/src/HOL/Complex/NSComplex.thy Thu Apr 22 10:43:06 2004 +0200
+++ b/src/HOL/Complex/NSComplex.thy Thu Apr 22 10:45:56 2004 +0200
@@ -7,7 +7,7 @@
header{*Nonstandard Complex Numbers*}
-theory NSComplex = NSInduct:
+theory NSComplex = Complex:
constdefs
hcomplexrel :: "((nat=>complex)*(nat=>complex)) set"
--- a/src/HOL/Complex/NSInduct.thy Thu Apr 22 10:43:06 2004 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,127 +0,0 @@
-(* Title: NSInduct.thy
- Author: Jacques D. Fleuriot
- Copyright: 2001 University of Edinburgh
-*)
-
-header{*Nonstandard Characterization of Induction*}
-
-theory NSInduct = Complex:
-
-constdefs
-
- starPNat :: "(nat => bool) => hypnat => bool" ("*pNat* _" [80] 80)
- "*pNat* P == (%x. \<exists>X. (X \<in> Rep_hypnat(x) &
- {n. P(X n)} \<in> FreeUltrafilterNat))"
-
-
- starPNat2 :: "(nat => nat => bool) => hypnat =>hypnat => bool"
- ("*pNat2* _" [80] 80)
- "*pNat2* P == (%x y. \<exists>X. \<exists>Y. (X \<in> Rep_hypnat(x) & Y \<in> Rep_hypnat(y) &
- {n. P(X n) (Y n)} \<in> FreeUltrafilterNat))"
-
- hSuc :: "hypnat => hypnat"
- "hSuc n == n + 1"
-
-
-lemma starPNat:
- "(( *pNat* P) (Abs_hypnat(hypnatrel``{%n. X n}))) =
- ({n. P (X n)} \<in> FreeUltrafilterNat)"
-by (auto simp add: starPNat_def, ultra)
-
-lemma starPNat_hypnat_of_nat [simp]: "( *pNat* P) (hypnat_of_nat n) = P n"
-by (auto simp add: starPNat hypnat_of_nat_eq)
-
-lemma hypnat_induct_obj:
- "(( *pNat* P) 0 &
- (\<forall>n. ( *pNat* P)(n) --> ( *pNat* P)(n + 1)))
- --> ( *pNat* P)(n)"
-apply (cases n)
-apply (auto simp add: hypnat_zero_def hypnat_one_def starPNat, ultra)
-apply (erule nat_induct)
-apply (drule_tac x = "hypnat_of_nat n" in spec)
-apply (rule ccontr)
-apply (auto simp add: starPNat hypnat_of_nat_eq hypnat_add)
-done
-
-lemma hypnat_induct:
- "[| ( *pNat* P) 0;
- !!n. ( *pNat* P)(n) ==> ( *pNat* P)(n + 1)|]
- ==> ( *pNat* P)(n)"
-by (insert hypnat_induct_obj [of P n], auto)
-
-lemma starPNat2:
-"(( *pNat2* P) (Abs_hypnat(hypnatrel``{%n. X n}))
- (Abs_hypnat(hypnatrel``{%n. Y n}))) =
- ({n. P (X n) (Y n)} \<in> FreeUltrafilterNat)"
-by (auto simp add: starPNat2_def, ultra)
-
-lemma starPNat2_eq_iff: "( *pNat2* (op =)) = (op =)"
-apply (simp add: starPNat2_def)
-apply (rule ext)
-apply (rule ext)
-apply (rule_tac z = x in eq_Abs_hypnat)
-apply (rule_tac z = y in eq_Abs_hypnat)
-apply (auto, ultra)
-done
-
-lemma starPNat2_eq_iff2: "( *pNat2* (%x y. x = y)) X Y = (X = Y)"
-by (simp add: starPNat2_eq_iff)
-
-lemma lemma_hyp: "(\<exists>h. P(h::hypnat)) = (\<exists>x. P(Abs_hypnat(hypnatrel `` {x})))"
-apply auto
-apply (rule_tac z = h in eq_Abs_hypnat, auto)
-done
-
-lemma hSuc_not_zero [iff]: "hSuc m \<noteq> 0"
-by (simp add: hSuc_def)
-
-lemmas zero_not_hSuc = hSuc_not_zero [THEN not_sym, standard, iff]
-
-lemma hSuc_hSuc_eq [iff]: "(hSuc m = hSuc n) = (m = n)"
-by (simp add: hSuc_def hypnat_one_def)
-
-lemma nonempty_nat_set_Least_mem: "c \<in> (S :: nat set) ==> (LEAST n. n \<in> S) \<in> S"
-by (erule LeastI)
-
-lemma nonempty_InternalNatSet_has_least:
- "[| S \<in> InternalNatSets; S \<noteq> {} |] ==> \<exists>n \<in> S. \<forall>m \<in> S. n \<le> m"
-apply (auto simp add: InternalNatSets_def starsetNat_n_def lemma_hyp)
-apply (rule_tac z = x in eq_Abs_hypnat)
-apply (auto dest!: bspec simp add: hypnat_le)
-apply (drule_tac x = "%m. LEAST n. n \<in> As m" in spec, auto)
-apply (ultra, force dest: nonempty_nat_set_Least_mem)
-apply (drule_tac x = x in bspec, auto)
-apply (ultra, auto intro: Least_le)
-done
-
-text{* Goldblatt page 129 Thm 11.3.2*}
-lemma internal_induct:
- "[| X \<in> InternalNatSets; 0 \<in> X; \<forall>n. n \<in> X --> n + 1 \<in> X |]
- ==> X = (UNIV:: hypnat set)"
-apply (rule ccontr)
-apply (frule InternalNatSets_Compl)
-apply (drule_tac S = "- X" in nonempty_InternalNatSet_has_least, auto)
-apply (subgoal_tac "1 \<le> n")
-apply (drule_tac x = "n - 1" in bspec, safe)
-apply (drule_tac x = "n - 1" in spec)
-apply (drule_tac [2] c = 1 and a = n in add_right_mono)
-apply (auto simp add: linorder_not_less [symmetric] iff del: hypnat_neq0_conv)
-done
-
-ML
-{*
-val starPNat = thm "starPNat";
-val starPNat_hypnat_of_nat = thm "starPNat_hypnat_of_nat";
-val hypnat_induct = thm "hypnat_induct";
-val starPNat2 = thm "starPNat2";
-val starPNat2_eq_iff = thm "starPNat2_eq_iff";
-val starPNat2_eq_iff2 = thm "starPNat2_eq_iff2";
-val hSuc_not_zero = thm "hSuc_not_zero";
-val zero_not_hSuc = thms "zero_not_hSuc";
-val hSuc_hSuc_eq = thm "hSuc_hSuc_eq";
-val nonempty_nat_set_Least_mem = thm "nonempty_nat_set_Least_mem";
-val nonempty_InternalNatSet_has_least = thm "nonempty_InternalNatSet_has_least";
-val internal_induct = thm "internal_induct";
-*}
-
-end
--- a/src/HOL/Complex/README.html Thu Apr 22 10:43:06 2004 +0200
+++ b/src/HOL/Complex/README.html Thu Apr 22 10:45:56 2004 +0200
@@ -7,52 +7,52 @@
with numeric constants and some complex analysis. The development includes
nonstandard analysis for the complex numbers. Note that the image
<KBD>HOL-Complex</KBD> includes theories from the directories
-<KBD><a href="#Anchor-Real">HOL/Real</a></KBD> and <KBD><a href="#Anchor-Hyperreal">HOL/Hyperreal</a></KBD>. They define other types including <kbd>real</kbd> (the real numbers) and <kbd>hypreal</kbd> (the hyperreal or non-standard reals).<ul>
- <li><a href="CLim.html">CLim</a> Limits, continuous functions, and derivatives for the complex numbers
- <li><a href="CSeries.html">CSeries</a> Finite summation and infinite series for the complex numbers
- <li><a href="CStar.html">CStar</a> Star-transforms for the complex numbers, to form non-standard extensions of sets and functions
- <li><a href="Complex.html">Complex</a> The complex numbers
- <li><a href="NSCA.html">NSCA</a> Nonstandard complex analysis
- <li><a href="NSComplex.html">NSComplex</a> Ultrapower construction of the nonstandard complex numbers
-
- <li><a href="NSInduct.html">NSInduct</a> Nonstandard induction for the hypernatural numbers
-
-
- </ul>
- <h2><a name="Anchor-Real" id="Anchor-Real"></a>Real: Dedekind Cut Construction of the Real Line</h2>
- <ul>
- <li><a href="Lubs.html">Lubs</a> Definition of upper bounds, lubs and so on, to support completeness proofs.
- <li><a href="PReal.html">PReal</a> The positive reals constructed using Dedekind cuts
- <li><a href="Rational.html">Rational</a> The rational numbers constructed as equivalence classes of integers
- <li><a href="RComplete.html">RComplete</a> The reals are complete: they satisfy the supremum property. They also have the Archimedean property.
- <li><a href="RealDef.html">RealDef</a> The real numbers, their ordering properties, and embedding of the integers and the natural numbers
- <li><a href="RealPow.html">RealPow</a> Real numbers raised to natural number powers
- </ul>
- <h2><a name="Anchor-Hyperreal" id="Anchor-Hyperreal"></a>Hyperreal: Ultrafilter Construction of the Non-Standard Reals</h2>
- See J. D. Fleuriot and L. C. Paulson. Mechanizing Nonstandard Real Analysis. LMS J. Computation and Mathematics 3 (2000), 140-190.
- <ul>
- <li><a href="Filter.html">Filter</a> Theory of Filters and Ultrafilters. Main result is a version of the Ultrafilter Theorem proved using Zorn's Lemma.
- <li><a href="HLog.html">HLog</a> Non-standard logarithms
- <li><a href="HSeries.html">HSeries</a> Non-standard theory of finite summation and infinite series
- <li><a href="HTranscendental.html">HTranscendental</a> Non-standard extensions of transcendental functions
- <li><a href="HyperDef.html">HyperDef</a> Ultrapower construction of the hyperreals
- <li><a href="HyperNat.html">HyperNat</a> Ultrapower construction of the hypernaturals
- <li><a href="HyperPow.html">HyperPow</a> Powers theory for the hyperreals
- <li><a href="IntFloor.html">IntFloor</a> Floor and Ceiling functions relating the reals and integers
- <li><a href="Integration.html">Integration</a> Gage integrals
- <li><a href="Lim.html">Lim</a> Theory of limits, continuous functions, and derivatives
- <li><a href="Log.html">Log</a> Logarithms for the reals
- <li><a href="MacLaurin.html">MacLaurin</a> MacLaurin series
- <li><a href="NatStar.html">NatStar</a> Star-transforms for the hypernaturals, to form non-standard extensions of sets and functions involving the naturals or reals
- <li><a href="NthRoot.html">NthRoot</a> Existence of n-th roots of real numbers
- <li><a href="NSA.html">NSA</a> Theory defining sets of infinite numbers, infinitesimals, the infinitely close relation, and their various algebraic properties.
- <li><a href="Poly.html">Poly</a> Univariate real polynomials
- <li><a href="SEQ.html">SEQ</a> Convergence of sequences and series using standard and nonstandard analysis
- <li><a href="Series.html">Series</a> Finite summation and infinite series for the reals
- <li><a href="Star.html">Star</a> Nonstandard extensions of real sets and real functions
- <li><a href="Transcendental.html">Transcendental</a> Power series and transcendental functions
- </ul>
- <HR>
- <P>Last modified $Date$
+<KBD><a href="#Anchor-Real">HOL/Real</a></KBD> and <KBD><a href="#Anchor-Hyperreal">HOL/Hyperreal</a></KBD>. They define other types including <kbd>real</kbd> (the real numbers) and <kbd>hypreal</kbd> (the hyperreal or non-standard reals).
+
+<ul>
+<li><a href="CLim.html">CLim</a> Limits, continuous functions, and derivatives for the complex numbers
+<li><a href="CSeries.html">CSeries</a> Finite summation and infinite series for the complex numbers
+<li><a href="CStar.html">CStar</a> Star-transforms for the complex numbers, to form non-standard extensions of sets and functions
+<li><a href="Complex.html">Complex</a> The complex numbers
+<li><a href="NSCA.html">NSCA</a> Nonstandard complex analysis
+<li><a href="NSComplex.html">NSComplex</a> Ultrapower construction of the nonstandard complex numbers
+</ul>
+
+<h2><a name="Anchor-Real" id="Anchor-Real"></a>Real: Dedekind Cut Construction of the Real Line</h2>
+
+<ul>
+<li><a href="Lubs.html">Lubs</a> Definition of upper bounds, lubs and so on, to support completeness proofs.
+<li><a href="PReal.html">PReal</a> The positive reals constructed using Dedekind cuts
+<li><a href="Rational.html">Rational</a> The rational numbers constructed as equivalence classes of integers
+<li><a href="RComplete.html">RComplete</a> The reals are complete: they satisfy the supremum property. They also have the Archimedean property.
+<li><a href="RealDef.html">RealDef</a> The real numbers, their ordering properties, and embedding of the integers and the natural numbers
+<li><a href="RealPow.html">RealPow</a> Real numbers raised to natural number powers
+</ul>
+<h2><a name="Anchor-Hyperreal" id="Anchor-Hyperreal"></a>Hyperreal: Ultrafilter Construction of the Non-Standard Reals</h2>
+See J. D. Fleuriot and L. C. Paulson. Mechanizing Nonstandard Real Analysis. LMS J. Computation and Mathematics 3 (2000), 140-190.
+<ul>
+<li><a href="Filter.html">Filter</a> Theory of Filters and Ultrafilters. Main result is a version of the Ultrafilter Theorem proved using Zorn's Lemma.
+<li><a href="HLog.html">HLog</a> Non-standard logarithms
+<li><a href="HSeries.html">HSeries</a> Non-standard theory of finite summation and infinite series
+<li><a href="HTranscendental.html">HTranscendental</a> Non-standard extensions of transcendental functions
+<li><a href="HyperDef.html">HyperDef</a> Ultrapower construction of the hyperreals
+<li><a href="HyperNat.html">HyperNat</a> Ultrapower construction of the hypernaturals
+<li><a href="HyperPow.html">HyperPow</a> Powers theory for the hyperreals
+<li><a href="IntFloor.html">IntFloor</a> Floor and Ceiling functions relating the reals and integers
+<li><a href="Integration.html">Integration</a> Gage integrals
+<li><a href="Lim.html">Lim</a> Theory of limits, continuous functions, and derivatives
+<li><a href="Log.html">Log</a> Logarithms for the reals
+<li><a href="MacLaurin.html">MacLaurin</a> MacLaurin series
+<li><a href="NatStar.html">NatStar</a> Star-transforms for the hypernaturals, to form non-standard extensions of sets and functions involving the naturals or reals
+<li><a href="NthRoot.html">NthRoot</a> Existence of n-th roots of real numbers
+<li><a href="NSA.html">NSA</a> Theory defining sets of infinite numbers, infinitesimals, the infinitely close relation, and their various algebraic properties.
+<li><a href="Poly.html">Poly</a> Univariate real polynomials
+<li><a href="SEQ.html">SEQ</a> Convergence of sequences and series using standard and nonstandard analysis
+<li><a href="Series.html">Series</a> Finite summation and infinite series for the reals
+<li><a href="Star.html">Star</a> Nonstandard extensions of real sets and real functions
+<li><a href="Transcendental.html">Transcendental</a> Power series and transcendental functions
+</ul>
+<HR>
+<P>Last modified $Date$
</HTML>
--- a/src/HOL/Hyperreal/HTranscendental.thy Thu Apr 22 10:43:06 2004 +0200
+++ b/src/HOL/Hyperreal/HTranscendental.thy Thu Apr 22 10:45:56 2004 +0200
@@ -7,7 +7,7 @@
header{*Nonstandard Extensions of Transcendental Functions*}
-theory HTranscendental = Transcendental + IntFloor:
+theory HTranscendental = Transcendental + Integration:
constdefs
--- a/src/HOL/Hyperreal/IntFloor.thy Thu Apr 22 10:43:06 2004 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,322 +0,0 @@
-(* Title: IntFloor.thy
- Author: Jacques D. Fleuriot
- Copyright: 2001,2002 University of Edinburgh
-Converted to Isar and polished by lcp
-*)
-
-header{*Floor and Ceiling Functions from the Reals to the Integers*}
-
-theory IntFloor = Integration:
-
-constdefs
-
- floor :: "real => int"
- "floor r == (LEAST n. r < real (n + (1::int)))"
-
- 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. real n \<le> r & r < real ((n::int) + 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. real n \<le> r & r < real ((n::int) + 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::int) + 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 na < real (x::int) + 1 ==> na \<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
--- a/src/HOL/Hyperreal/NatStar.thy Thu Apr 22 10:43:06 2004 +0200
+++ b/src/HOL/Hyperreal/NatStar.thy Thu Apr 22 10:45:56 2004 +0200
@@ -533,6 +533,112 @@
val starfunNat_inverse_real_of_nat_Infinitesimal = thm "starfunNat_inverse_real_of_nat_Infinitesimal";
*}
+
+
+subsection{*Nonstandard Characterization of Induction*}
+
+constdefs
+
+ starPNat :: "(nat => bool) => hypnat => bool" ("*pNat* _" [80] 80)
+ "*pNat* P == (%x. \<exists>X. (X \<in> Rep_hypnat(x) &
+ {n. P(X n)} \<in> FreeUltrafilterNat))"
+
+
+ starPNat2 :: "(nat => nat => bool) => hypnat =>hypnat => bool"
+ ("*pNat2* _" [80] 80)
+ "*pNat2* P == (%x y. \<exists>X. \<exists>Y. (X \<in> Rep_hypnat(x) & Y \<in> Rep_hypnat(y) &
+ {n. P(X n) (Y n)} \<in> FreeUltrafilterNat))"
+
+ hSuc :: "hypnat => hypnat"
+ "hSuc n == n + 1"
+
+
+lemma starPNat:
+ "(( *pNat* P) (Abs_hypnat(hypnatrel``{%n. X n}))) =
+ ({n. P (X n)} \<in> FreeUltrafilterNat)"
+by (auto simp add: starPNat_def, ultra)
+
+lemma starPNat_hypnat_of_nat [simp]: "( *pNat* P) (hypnat_of_nat n) = P n"
+by (auto simp add: starPNat hypnat_of_nat_eq)
+
+lemma hypnat_induct_obj:
+ "(( *pNat* P) 0 &
+ (\<forall>n. ( *pNat* P)(n) --> ( *pNat* P)(n + 1)))
+ --> ( *pNat* P)(n)"
+apply (cases n)
+apply (auto simp add: hypnat_zero_def hypnat_one_def starPNat, ultra)
+apply (erule nat_induct)
+apply (drule_tac x = "hypnat_of_nat n" in spec)
+apply (rule ccontr)
+apply (auto simp add: starPNat hypnat_of_nat_eq hypnat_add)
+done
+
+lemma hypnat_induct:
+ "[| ( *pNat* P) 0;
+ !!n. ( *pNat* P)(n) ==> ( *pNat* P)(n + 1)|]
+ ==> ( *pNat* P)(n)"
+by (insert hypnat_induct_obj [of P n], auto)
+
+lemma starPNat2:
+"(( *pNat2* P) (Abs_hypnat(hypnatrel``{%n. X n}))
+ (Abs_hypnat(hypnatrel``{%n. Y n}))) =
+ ({n. P (X n) (Y n)} \<in> FreeUltrafilterNat)"
+by (auto simp add: starPNat2_def, ultra)
+
+lemma starPNat2_eq_iff: "( *pNat2* (op =)) = (op =)"
+apply (simp add: starPNat2_def)
+apply (rule ext)
+apply (rule ext)
+apply (rule_tac z = x in eq_Abs_hypnat)
+apply (rule_tac z = y in eq_Abs_hypnat)
+apply (auto, ultra)
+done
+
+lemma starPNat2_eq_iff2: "( *pNat2* (%x y. x = y)) X Y = (X = Y)"
+by (simp add: starPNat2_eq_iff)
+
+lemma lemma_hyp: "(\<exists>h. P(h::hypnat)) = (\<exists>x. P(Abs_hypnat(hypnatrel `` {x})))"
+apply auto
+apply (rule_tac z = h in eq_Abs_hypnat, auto)
+done
+
+lemma hSuc_not_zero [iff]: "hSuc m \<noteq> 0"
+by (simp add: hSuc_def)
+
+lemmas zero_not_hSuc = hSuc_not_zero [THEN not_sym, standard, iff]
+
+lemma hSuc_hSuc_eq [iff]: "(hSuc m = hSuc n) = (m = n)"
+by (simp add: hSuc_def hypnat_one_def)
+
+lemma nonempty_nat_set_Least_mem: "c \<in> (S :: nat set) ==> (LEAST n. n \<in> S) \<in> S"
+by (erule LeastI)
+
+lemma nonempty_InternalNatSet_has_least:
+ "[| S \<in> InternalNatSets; S \<noteq> {} |] ==> \<exists>n \<in> S. \<forall>m \<in> S. n \<le> m"
+apply (auto simp add: InternalNatSets_def starsetNat_n_def lemma_hyp)
+apply (rule_tac z = x in eq_Abs_hypnat)
+apply (auto dest!: bspec simp add: hypnat_le)
+apply (drule_tac x = "%m. LEAST n. n \<in> As m" in spec, auto)
+apply (ultra, force dest: nonempty_nat_set_Least_mem)
+apply (drule_tac x = x in bspec, auto)
+apply (ultra, auto intro: Least_le)
+done
+
+text{* Goldblatt page 129 Thm 11.3.2*}
+lemma internal_induct:
+ "[| X \<in> InternalNatSets; 0 \<in> X; \<forall>n. n \<in> X --> n + 1 \<in> X |]
+ ==> X = (UNIV:: hypnat set)"
+apply (rule ccontr)
+apply (frule InternalNatSets_Compl)
+apply (drule_tac S = "- X" in nonempty_InternalNatSet_has_least, auto)
+apply (subgoal_tac "1 \<le> n")
+apply (drule_tac x = "n - 1" in bspec, safe)
+apply (drule_tac x = "n - 1" in spec)
+apply (drule_tac [2] c = 1 and a = n in add_right_mono)
+apply (auto simp add: linorder_not_less [symmetric] iff del: hypnat_neq0_conv)
+done
+
+
end
--- a/src/HOL/IsaMakefile Thu Apr 22 10:43:06 2004 +0200
+++ b/src/HOL/IsaMakefile Thu Apr 22 10:45:56 2004 +0200
@@ -150,7 +150,7 @@
Hyperreal/Filter.ML Hyperreal/Filter.thy Hyperreal/HSeries.thy\
Hyperreal/HTranscendental.thy Hyperreal/HyperArith.thy\
Hyperreal/HyperDef.thy Hyperreal/HyperNat.thy\
- Hyperreal/HyperPow.thy Hyperreal/Hyperreal.thy Hyperreal/IntFloor.thy\
+ Hyperreal/HyperPow.thy Hyperreal/Hyperreal.thy\
Hyperreal/Lim.thy Hyperreal/Log.thy\
Hyperreal/MacLaurin.ML Hyperreal/MacLaurin.thy Hyperreal/NatStar.thy\
Hyperreal/NSA.thy Hyperreal/NthRoot.thy Hyperreal/Poly.thy\
@@ -159,7 +159,7 @@
Hyperreal/Transcendental.thy Hyperreal/fuf.ML Hyperreal/hypreal_arith.ML \
Complex/Complex_Main.thy Complex/CLim.thy Complex/CSeries.thy\
Complex/CStar.thy Complex/Complex.thy Complex/ComplexBin.thy\
- Complex/NSCA.thy Complex/NSComplex.thy Complex/NSInduct.thy
+ Complex/NSCA.thy Complex/NSComplex.thy
@cd Complex; $(ISATOOL) usedir -b $(OUT)/HOL HOL-Complex
--- 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
+