diff -r 6f61794f1ff7 -r edaef19665e6 src/HOL/ContNotDenum.thy
--- a/src/HOL/ContNotDenum.thy	Mon Dec 08 08:56:30 2008 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,579 +0,0 @@
-(*  Title       : HOL/ContNonDenum
-    Author      : Benjamin Porter, Monash University, NICTA, 2005
-*)
-
-header {* Non-denumerability of the Continuum. *}
-
-theory ContNotDenum
-imports RComplete Hilbert_Choice
-begin
-
-subsection {* Abstract *}
-
-text {* The following document presents a proof that the Continuum is
-uncountable. It is formalised in the Isabelle/Isar theorem proving
-system.
-
-{\em Theorem:} The Continuum @{text "\<real>"} is not denumerable. In other
-words, there does not exist a function f:@{text "\<nat>\<Rightarrow>\<real>"} such that f is
-surjective.
-
-{\em Outline:} An elegant informal proof of this result uses Cantor's
-Diagonalisation argument. The proof presented here is not this
-one. First we formalise some properties of closed intervals, then we
-prove the Nested Interval Property. This property relies on the
-completeness of the Real numbers and is the foundation for our
-argument. Informally it states that an intersection of countable
-closed intervals (where each successive interval is a subset of the
-last) is non-empty. We then assume a surjective function f:@{text
-"\<nat>\<Rightarrow>\<real>"} exists and find a real x such that x is not in the range of f
-by generating a sequence of closed intervals then using the NIP. *}
-
-subsection {* Closed Intervals *}
-
-text {* This section formalises some properties of closed intervals. *}
-
-subsubsection {* Definition *}
-
-definition
-  closed_int :: "real \<Rightarrow> real \<Rightarrow> real set" where
-  "closed_int x y = {z. x \<le> z \<and> z \<le> y}"
-
-subsubsection {* Properties *}
-
-lemma closed_int_subset:
-  assumes xy: "x1 \<ge> x0" "y1 \<le> y0"
-  shows "closed_int x1 y1 \<subseteq> closed_int x0 y0"
-proof -
-  {
-    fix x::real
-    assume "x \<in> closed_int x1 y1"
-    hence "x \<ge> x1 \<and> x \<le> y1" by (simp add: closed_int_def)
-    with xy have "x \<ge> x0 \<and> x \<le> y0" by auto
-    hence "x \<in> closed_int x0 y0" by (simp add: closed_int_def)
-  }
-  thus ?thesis by auto
-qed
-
-lemma closed_int_least:
-  assumes a: "a \<le> b"
-  shows "a \<in> closed_int a b \<and> (\<forall>x \<in> closed_int a b. a \<le> x)"
-proof
-  from a have "a\<in>{x. a\<le>x \<and> x\<le>b}" by simp
-  thus "a \<in> closed_int a b" by (unfold closed_int_def)
-next
-  have "\<forall>x\<in>{x. a\<le>x \<and> x\<le>b}. a\<le>x" by simp
-  thus "\<forall>x \<in> closed_int a b. a \<le> x" by (unfold closed_int_def)
-qed
-
-lemma closed_int_most:
-  assumes a: "a \<le> b"
-  shows "b \<in> closed_int a b \<and> (\<forall>x \<in> closed_int a b. x \<le> b)"
-proof
-  from a have "b\<in>{x. a\<le>x \<and> x\<le>b}" by simp
-  thus "b \<in> closed_int a b" by (unfold closed_int_def)
-next
-  have "\<forall>x\<in>{x. a\<le>x \<and> x\<le>b}. x\<le>b" by simp
-  thus "\<forall>x \<in> closed_int a b. x\<le>b" by (unfold closed_int_def)
-qed
-
-lemma closed_not_empty:
-  shows "a \<le> b \<Longrightarrow> \<exists>x. x \<in> closed_int a b" 
-  by (auto dest: closed_int_least)
-
-lemma closed_mem:
-  assumes "a \<le> c" and "c \<le> b"
-  shows "c \<in> closed_int a b"
-  using assms unfolding closed_int_def by auto
-
-lemma closed_subset:
-  assumes ac: "a \<le> b"  "c \<le> d" 
-  assumes closed: "closed_int a b \<subseteq> closed_int c d"
-  shows "b \<ge> c"
-proof -
-  from closed have "\<forall>x\<in>closed_int a b. x\<in>closed_int c d" by auto
-  hence "\<forall>x. a\<le>x \<and> x\<le>b \<longrightarrow> c\<le>x \<and> x\<le>d" by (unfold closed_int_def, auto)
-  with ac have "c\<le>b \<and> b\<le>d" by simp
-  thus ?thesis by auto
-qed
-
-
-subsection {* Nested Interval Property *}
-
-theorem NIP:
-  fixes f::"nat \<Rightarrow> real set"
-  assumes subset: "\<forall>n. f (Suc n) \<subseteq> f n"
-  and closed: "\<forall>n. \<exists>a b. f n = closed_int a b \<and> a \<le> b"
-  shows "(\<Inter>n. f n) \<noteq> {}"
-proof -
-  let ?g = "\<lambda>n. (SOME c. c\<in>(f n) \<and> (\<forall>x\<in>(f n). c \<le> x))"
-  have ne: "\<forall>n. \<exists>x. x\<in>(f n)"
-  proof
-    fix n
-    from closed have "\<exists>a b. f n = closed_int a b \<and> a \<le> b" by simp
-    then obtain a and b where fn: "f n = closed_int a b \<and> a \<le> b" by auto
-    hence "a \<le> b" ..
-    with closed_not_empty have "\<exists>x. x\<in>closed_int a b" by simp
-    with fn show "\<exists>x. x\<in>(f n)" by simp
-  qed
-
-  have gdef: "\<forall>n. (?g n)\<in>(f n) \<and> (\<forall>x\<in>(f n). (?g n)\<le>x)"
-  proof
-    fix n
-    from closed have "\<exists>a b. f n = closed_int a b \<and> a \<le> b" ..
-    then obtain a and b where ff: "f n = closed_int a b" and "a \<le> b" by auto
-    hence "a \<le> b" by simp
-    hence "a\<in>closed_int a b \<and> (\<forall>x\<in>closed_int a b. a \<le> x)" by (rule closed_int_least)
-    with ff have "a\<in>(f n) \<and> (\<forall>x\<in>(f n). a \<le> x)" by simp
-    hence "\<exists>c. c\<in>(f n) \<and> (\<forall>x\<in>(f n). c \<le> x)" ..
-    thus "(?g n)\<in>(f n) \<and> (\<forall>x\<in>(f n). (?g n)\<le>x)" by (rule someI_ex)
-  qed
-
-  -- "A denotes the set of all left-most points of all the intervals ..."
-  moreover obtain A where Adef: "A = ?g ` \<nat>" by simp
-  ultimately have "\<exists>x. x\<in>A"
-  proof -
-    have "(0::nat) \<in> \<nat>" by simp
-    moreover have "?g 0 = ?g 0" by simp
-    ultimately have "?g 0 \<in> ?g ` \<nat>" by (rule  rev_image_eqI)
-    with Adef have "?g 0 \<in> A" by simp
-    thus ?thesis ..
-  qed
-
-  -- "Now show that A is bounded above ..."
-  moreover have "\<exists>y. isUb (UNIV::real set) A y"
-  proof -
-    {
-      fix n
-      from ne have ex: "\<exists>x. x\<in>(f n)" ..
-      from gdef have "(?g n)\<in>(f n) \<and> (\<forall>x\<in>(f n). (?g n)\<le>x)" by simp
-      moreover
-      from closed have "\<exists>a b. f n = closed_int a b \<and> a \<le> b" ..
-      then obtain a and b where "f n = closed_int a b \<and> a \<le> b" by auto
-      hence "b\<in>(f n) \<and> (\<forall>x\<in>(f n). x \<le> b)" using closed_int_most by blast
-      ultimately have "\<forall>x\<in>(f n). (?g n) \<le> b" by simp
-      with ex have "(?g n) \<le> b" by auto
-      hence "\<exists>b. (?g n) \<le> b" by auto
-    }
-    hence aux: "\<forall>n. \<exists>b. (?g n) \<le> b" ..
-
-    have fs: "\<forall>n::nat. f n \<subseteq> f 0"
-    proof (rule allI, induct_tac n)
-      show "f 0 \<subseteq> f 0" by simp
-    next
-      fix n
-      assume "f n \<subseteq> f 0"
-      moreover from subset have "f (Suc n) \<subseteq> f n" ..
-      ultimately show "f (Suc n) \<subseteq> f 0" by simp
-    qed
-    have "\<forall>n. (?g n)\<in>(f 0)"
-    proof
-      fix n
-      from gdef have "(?g n)\<in>(f n) \<and> (\<forall>x\<in>(f n). (?g n)\<le>x)" by simp
-      hence "?g n \<in> f n" ..
-      with fs show "?g n \<in> f 0" by auto
-    qed
-    moreover from closed
-      obtain a and b where "f 0 = closed_int a b" and alb: "a \<le> b" by blast
-    ultimately have "\<forall>n. ?g n \<in> closed_int a b" by auto
-    with alb have "\<forall>n. ?g n \<le> b" using closed_int_most by blast
-    with Adef have "\<forall>y\<in>A. y\<le>b" by auto
-    hence "A *<= b" by (unfold setle_def)
-    moreover have "b \<in> (UNIV::real set)" by simp
-    ultimately have "A *<= b \<and> b \<in> (UNIV::real set)" by simp
-    hence "isUb (UNIV::real set) A b" by (unfold isUb_def)
-    thus ?thesis by auto
-  qed
-  -- "by the Axiom Of Completeness, A has a least upper bound ..."
-  ultimately have "\<exists>t. isLub UNIV A t" by (rule reals_complete)
-
-  -- "denote this least upper bound as t ..."
-  then obtain t where tdef: "isLub UNIV A t" ..
-
-  -- "and finally show that this least upper bound is in all the intervals..."
-  have "\<forall>n. t \<in> f n"
-  proof
-    fix n::nat
-    from closed obtain a and b where
-      int: "f n = closed_int a b" and alb: "a \<le> b" by blast
-
-    have "t \<ge> a"
-    proof -
-      have "a \<in> A"
-      proof -
-          (* by construction *)
-        from alb int have ain: "a\<in>f n \<and> (\<forall>x\<in>f n. a \<le> x)"
-          using closed_int_least by blast
-        moreover have "\<forall>e. e\<in>f n \<and> (\<forall>x\<in>f n. e \<le> x) \<longrightarrow> e = a"
-        proof clarsimp
-          fix e
-          assume ein: "e \<in> f n" and lt: "\<forall>x\<in>f n. e \<le> x"
-          from lt ain have aux: "\<forall>x\<in>f n. a \<le> x \<and> e \<le> x" by auto
-  
-          from ein aux have "a \<le> e \<and> e \<le> e" by auto
-          moreover from ain aux have "a \<le> a \<and> e \<le> a" by auto
-          ultimately show "e = a" by simp
-        qed
-        hence "\<And>e.  e\<in>f n \<and> (\<forall>x\<in>f n. e \<le> x) \<Longrightarrow> e = a" by simp
-        ultimately have "(?g n) = a" by (rule some_equality)
-        moreover
-        {
-          have "n = of_nat n" by simp
-          moreover have "of_nat n \<in> \<nat>" by simp
-          ultimately have "n \<in> \<nat>"
-            apply -
-            apply (subst(asm) eq_sym_conv)
-            apply (erule subst)
-            .
-        }
-        with Adef have "(?g n) \<in> A" by auto
-        ultimately show ?thesis by simp
-      qed 
-      with tdef show "a \<le> t" by (rule isLubD2)
-    qed
-    moreover have "t \<le> b"
-    proof -
-      have "isUb UNIV A b"
-      proof -
-        {
-          from alb int have
-            ain: "b\<in>f n \<and> (\<forall>x\<in>f n. x \<le> b)" using closed_int_most by blast
-          
-          have subsetd: "\<forall>m. \<forall>n. f (n + m) \<subseteq> f n"
-          proof (rule allI, induct_tac m)
-            show "\<forall>n. f (n + 0) \<subseteq> f n" by simp
-          next
-            fix m n
-            assume pp: "\<forall>p. f (p + n) \<subseteq> f p"
-            {
-              fix p
-              from pp have "f (p + n) \<subseteq> f p" by simp
-              moreover from subset have "f (Suc (p + n)) \<subseteq> f (p + n)" by auto
-              hence "f (p + (Suc n)) \<subseteq> f (p + n)" by simp
-              ultimately have "f (p + (Suc n)) \<subseteq> f p" by simp
-            }
-            thus "\<forall>p. f (p + Suc n) \<subseteq> f p" ..
-          qed 
-          have subsetm: "\<forall>\<alpha> \<beta>. \<alpha> \<ge> \<beta> \<longrightarrow> (f \<alpha>) \<subseteq> (f \<beta>)"
-          proof ((rule allI)+, rule impI)
-            fix \<alpha>::nat and \<beta>::nat
-            assume "\<beta> \<le> \<alpha>"
-            hence "\<exists>k. \<alpha> = \<beta> + k" by (simp only: le_iff_add)
-            then obtain k where "\<alpha> = \<beta> + k" ..
-            moreover
-            from subsetd have "f (\<beta> + k) \<subseteq> f \<beta>" by simp
-            ultimately show "f \<alpha> \<subseteq> f \<beta>" by auto
-          qed 
-          
-          fix m   
-          {
-            assume "m \<ge> n"
-            with subsetm have "f m \<subseteq> f n" by simp
-            with ain have "\<forall>x\<in>f m. x \<le> b" by auto
-            moreover
-            from gdef have "?g m \<in> f m \<and> (\<forall>x\<in>f m. ?g m \<le> x)" by simp
-            ultimately have "?g m \<le> b" by auto
-          }
-          moreover
-          {
-            assume "\<not>(m \<ge> n)"
-            hence "m < n" by simp
-            with subsetm have sub: "(f n) \<subseteq> (f m)" by simp
-            from closed obtain ma and mb where
-              "f m = closed_int ma mb \<and> ma \<le> mb" by blast
-            hence one: "ma \<le> mb" and fm: "f m = closed_int ma mb" by auto 
-            from one alb sub fm int have "ma \<le> b" using closed_subset by blast
-            moreover have "(?g m) = ma"
-            proof -
-              from gdef have "?g m \<in> f m \<and> (\<forall>x\<in>f m. ?g m \<le> x)" ..
-              moreover from one have
-                "ma \<in> closed_int ma mb \<and> (\<forall>x\<in>closed_int ma mb. ma \<le> x)"
-                by (rule closed_int_least)
-              with fm have "ma\<in>f m \<and> (\<forall>x\<in>f m. ma \<le> x)" by simp
-              ultimately have "ma \<le> ?g m \<and> ?g m \<le> ma" by auto
-              thus "?g m = ma" by auto
-            qed
-            ultimately have "?g m \<le> b" by simp
-          } 
-          ultimately have "?g m \<le> b" by (rule case_split)
-        }
-        with Adef have "\<forall>y\<in>A. y\<le>b" by auto
-        hence "A *<= b" by (unfold setle_def)
-        moreover have "b \<in> (UNIV::real set)" by simp
-        ultimately have "A *<= b \<and> b \<in> (UNIV::real set)" by simp
-        thus "isUb (UNIV::real set) A b" by (unfold isUb_def)
-      qed
-      with tdef show "t \<le> b" by (rule isLub_le_isUb)
-    qed
-    ultimately have "t \<in> closed_int a b" by (rule closed_mem)
-    with int show "t \<in> f n" by simp
-  qed
-  hence "t \<in> (\<Inter>n. f n)" by auto
-  thus ?thesis by auto
-qed
-
-subsection {* Generating the intervals *}
-
-subsubsection {* Existence of non-singleton closed intervals *}
-
-text {* This lemma asserts that given any non-singleton closed
-interval (a,b) and any element c, there exists a closed interval that
-is a subset of (a,b) and that does not contain c and is a
-non-singleton itself. *}
-
-lemma closed_subset_ex:
-  fixes c::real
-  assumes alb: "a < b"
-  shows
-    "\<exists>ka kb. ka < kb \<and> closed_int ka kb \<subseteq> closed_int a b \<and> c \<notin> (closed_int ka kb)"
-proof -
-  {
-    assume clb: "c < b"
-    {
-      assume cla: "c < a"
-      from alb cla clb have "c \<notin> closed_int a b" by (unfold closed_int_def, auto)
-      with alb have
-        "a < b \<and> closed_int a b \<subseteq> closed_int a b \<and> c \<notin> closed_int a b"
-        by auto
-      hence
-        "\<exists>ka kb. ka < kb \<and> closed_int ka kb \<subseteq> closed_int a b \<and> c \<notin> (closed_int ka kb)"
-        by auto
-    }
-    moreover
-    {
-      assume ncla: "\<not>(c < a)"
-      with clb have cdef: "a \<le> c \<and> c < b" by simp
-      obtain ka where kadef: "ka = (c + b)/2" by blast
-
-      from kadef clb have kalb: "ka < b" by auto
-      moreover from kadef cdef have kagc: "ka > c" by simp
-      ultimately have "c\<notin>(closed_int ka b)" by (unfold closed_int_def, auto)
-      moreover from cdef kagc have "ka \<ge> a" by simp
-      hence "closed_int ka b \<subseteq> closed_int a b" by (unfold closed_int_def, auto)
-      ultimately have
-        "ka < b  \<and> closed_int ka b \<subseteq> closed_int a b \<and> c \<notin> closed_int ka b"
-        using kalb by auto
-      hence
-        "\<exists>ka kb. ka < kb \<and> closed_int ka kb \<subseteq> closed_int a b \<and> c \<notin> (closed_int ka kb)"
-        by auto
-
-    }
-    ultimately have
-      "\<exists>ka kb. ka < kb \<and> closed_int ka kb \<subseteq> closed_int a b \<and> c \<notin> (closed_int ka kb)"
-      by (rule case_split)
-  }
-  moreover
-  {
-    assume "\<not> (c < b)"
-    hence cgeb: "c \<ge> b" by simp
-
-    obtain kb where kbdef: "kb = (a + b)/2" by blast
-    with alb have kblb: "kb < b" by auto
-    with kbdef cgeb have "a < kb \<and> kb < c" by auto
-    moreover hence "c \<notin> (closed_int a kb)" by (unfold closed_int_def, auto)
-    moreover from kblb have
-      "closed_int a kb \<subseteq> closed_int a b" by (unfold closed_int_def, auto)
-    ultimately have
-      "a < kb \<and>  closed_int a kb \<subseteq> closed_int a b \<and> c\<notin>closed_int a kb"
-      by simp
-    hence
-      "\<exists>ka kb. ka < kb \<and> closed_int ka kb \<subseteq> closed_int a b \<and> c \<notin> (closed_int ka kb)"
-      by auto
-  }
-  ultimately show ?thesis by (rule case_split)
-qed
-
-subsection {* newInt: Interval generation *}
-
-text {* Given a function f:@{text "\<nat>\<Rightarrow>\<real>"}, newInt (Suc n) f returns a
-closed interval such that @{text "newInt (Suc n) f \<subseteq> newInt n f"} and
-does not contain @{text "f (Suc n)"}. With the base case defined such
-that @{text "(f 0)\<notin>newInt 0 f"}. *}
-
-subsubsection {* Definition *}
-
-primrec newInt :: "nat \<Rightarrow> (nat \<Rightarrow> real) \<Rightarrow> (real set)" where
-  "newInt 0 f = closed_int (f 0 + 1) (f 0 + 2)"
-  | "newInt (Suc n) f =
-      (SOME e. (\<exists>e1 e2.
-       e1 < e2 \<and>
-       e = closed_int e1 e2 \<and>
-       e \<subseteq> (newInt n f) \<and>
-       (f (Suc n)) \<notin> e)
-      )"
-
-declare newInt.simps [code del]
-
-subsubsection {* Properties *}
-
-text {* We now show that every application of newInt returns an
-appropriate interval. *}
-
-lemma newInt_ex:
-  "\<exists>a b. a < b \<and>
-   newInt (Suc n) f = closed_int a b \<and>
-   newInt (Suc n) f \<subseteq> newInt n f \<and>
-   f (Suc n) \<notin> newInt (Suc n) f"
-proof (induct n)
-  case 0
-
-  let ?e = "SOME e. \<exists>e1 e2.
-   e1 < e2 \<and>
-   e = closed_int e1 e2 \<and>
-   e \<subseteq> closed_int (f 0 + 1) (f 0 + 2) \<and>
-   f (Suc 0) \<notin> e"
-
-  have "newInt (Suc 0) f = ?e" by auto
-  moreover
-  have "f 0 + 1 < f 0 + 2" by simp
-  with closed_subset_ex have
-    "\<exists>ka kb. ka < kb \<and> closed_int ka kb \<subseteq> closed_int (f 0 + 1) (f 0 + 2) \<and>
-     f (Suc 0) \<notin> (closed_int ka kb)" .
-  hence
-    "\<exists>e. \<exists>ka kb. ka < kb \<and> e = closed_int ka kb \<and>
-     e \<subseteq> closed_int (f 0 + 1) (f 0 + 2) \<and> f (Suc 0) \<notin> e" by simp
-  hence
-    "\<exists>ka kb. ka < kb \<and> ?e = closed_int ka kb \<and>
-     ?e \<subseteq> closed_int (f 0 + 1) (f 0 + 2) \<and> f (Suc 0) \<notin> ?e"
-    by (rule someI_ex)
-  ultimately have "\<exists>e1 e2. e1 < e2 \<and>
-   newInt (Suc 0) f = closed_int e1 e2 \<and>
-   newInt (Suc 0) f \<subseteq> closed_int (f 0 + 1) (f 0 + 2) \<and>
-   f (Suc 0) \<notin> newInt (Suc 0) f" by simp
-  thus
-    "\<exists>a b. a < b \<and> newInt (Suc 0) f = closed_int a b \<and>
-     newInt (Suc 0) f \<subseteq> newInt 0 f \<and> f (Suc 0) \<notin> newInt (Suc 0) f"
-    by simp
-next
-  case (Suc n)
-  hence "\<exists>a b.
-   a < b \<and>
-   newInt (Suc n) f = closed_int a b \<and>
-   newInt (Suc n) f \<subseteq> newInt n f \<and>
-   f (Suc n) \<notin> newInt (Suc n) f" by simp
-  then obtain a and b where ab: "a < b \<and>
-   newInt (Suc n) f = closed_int a b \<and>
-   newInt (Suc n) f \<subseteq> newInt n f \<and>
-   f (Suc n) \<notin> newInt (Suc n) f" by auto
-  hence cab: "closed_int a b = newInt (Suc n) f" by simp
-
-  let ?e = "SOME e. \<exists>e1 e2.
-    e1 < e2 \<and>
-    e = closed_int e1 e2 \<and>
-    e \<subseteq> closed_int a b \<and>
-    f (Suc (Suc n)) \<notin> e"
-  from cab have ni: "newInt (Suc (Suc n)) f = ?e" by auto
-
-  from ab have "a < b" by simp
-  with closed_subset_ex have
-    "\<exists>ka kb. ka < kb \<and> closed_int ka kb \<subseteq> closed_int a b \<and>
-     f (Suc (Suc n)) \<notin> closed_int ka kb" .
-  hence
-    "\<exists>e. \<exists>ka kb. ka < kb \<and> e = closed_int ka kb \<and>
-     closed_int ka kb \<subseteq> closed_int a b \<and> f (Suc (Suc n)) \<notin> closed_int ka kb"
-    by simp
-  hence
-    "\<exists>e.  \<exists>ka kb. ka < kb \<and> e = closed_int ka kb \<and>
-     e \<subseteq> closed_int a b \<and> f (Suc (Suc n)) \<notin> e" by simp
-  hence
-    "\<exists>ka kb. ka < kb \<and> ?e = closed_int ka kb \<and>
-     ?e \<subseteq> closed_int a b \<and> f (Suc (Suc n)) \<notin> ?e" by (rule someI_ex)
-  with ab ni show
-    "\<exists>ka kb. ka < kb \<and>
-     newInt (Suc (Suc n)) f = closed_int ka kb \<and>
-     newInt (Suc (Suc n)) f \<subseteq> newInt (Suc n) f \<and>
-     f (Suc (Suc n)) \<notin> newInt (Suc (Suc n)) f" by auto
-qed
-
-lemma newInt_subset:
-  "newInt (Suc n) f \<subseteq> newInt n f"
-  using newInt_ex by auto
-
-
-text {* Another fundamental property is that no element in the range
-of f is in the intersection of all closed intervals generated by
-newInt. *}
-
-lemma newInt_inter:
-  "\<forall>n. f n \<notin> (\<Inter>n. newInt n f)"
-proof
-  fix n::nat
-  {
-    assume n0: "n = 0"
-    moreover have "newInt 0 f = closed_int (f 0 + 1) (f 0 + 2)" by simp
-    ultimately have "f n \<notin> newInt n f" by (unfold closed_int_def, simp)
-  }
-  moreover
-  {
-    assume "\<not> n = 0"
-    hence "n > 0" by simp
-    then obtain m where ndef: "n = Suc m" by (auto simp add: gr0_conv_Suc)
-
-    from newInt_ex have
-      "\<exists>a b. a < b \<and> (newInt (Suc m) f) = closed_int a b \<and>
-       newInt (Suc m) f \<subseteq> newInt m f \<and> f (Suc m) \<notin> newInt (Suc m) f" .
-    then have "f (Suc m) \<notin> newInt (Suc m) f" by auto
-    with ndef have "f n \<notin> newInt n f" by simp
-  }
-  ultimately have "f n \<notin> newInt n f" by (rule case_split)
-  thus "f n \<notin> (\<Inter>n. newInt n f)" by auto
-qed
-
-
-lemma newInt_notempty:
-  "(\<Inter>n. newInt n f) \<noteq> {}"
-proof -
-  let ?g = "\<lambda>n. newInt n f"
-  have "\<forall>n. ?g (Suc n) \<subseteq> ?g n"
-  proof
-    fix n
-    show "?g (Suc n) \<subseteq> ?g n" by (rule newInt_subset)
-  qed
-  moreover have "\<forall>n. \<exists>a b. ?g n = closed_int a b \<and> a \<le> b"
-  proof
-    fix n::nat
-    {
-      assume "n = 0"
-      then have
-        "?g n = closed_int (f 0 + 1) (f 0 + 2) \<and> (f 0 + 1 \<le> f 0 + 2)"
-        by simp
-      hence "\<exists>a b. ?g n = closed_int a b \<and> a \<le> b" by blast
-    }
-    moreover
-    {
-      assume "\<not> n = 0"
-      then have "n > 0" by simp
-      then obtain m where nd: "n = Suc m" by (auto simp add: gr0_conv_Suc)
-
-      have
-        "\<exists>a b. a < b \<and> (newInt (Suc m) f) = closed_int a b \<and>
-        (newInt (Suc m) f) \<subseteq> (newInt m f) \<and> (f (Suc m)) \<notin> (newInt (Suc m) f)"
-        by (rule newInt_ex)
-      then obtain a and b where
-        "a < b \<and> (newInt (Suc m) f) = closed_int a b" by auto
-      with nd have "?g n = closed_int a b \<and> a \<le> b" by auto
-      hence "\<exists>a b. ?g n = closed_int a b \<and> a \<le> b" by blast
-    }
-    ultimately show "\<exists>a b. ?g n = closed_int a b \<and> a \<le> b" by (rule case_split)
-  qed
-  ultimately show ?thesis by (rule NIP)
-qed
-
-
-subsection {* Final Theorem *}
-
-theorem real_non_denum:
-  shows "\<not> (\<exists>f::nat\<Rightarrow>real. surj f)"
-proof -- "by contradiction"
-  assume "\<exists>f::nat\<Rightarrow>real. surj f"
-  then obtain f::"nat\<Rightarrow>real" where "surj f" by auto
-  hence rangeF: "range f = UNIV" by (rule surj_range)
-  -- "We now produce a real number x that is not in the range of f, using the properties of newInt. "
-  have "\<exists>x. x \<in> (\<Inter>n. newInt n f)" using newInt_notempty by blast
-  moreover have "\<forall>n. f n \<notin> (\<Inter>n. newInt n f)" by (rule newInt_inter)
-  ultimately obtain x where "x \<in> (\<Inter>n. newInt n f)" and "\<forall>n. f n \<noteq> x" by blast
-  moreover from rangeF have "x \<in> range f" by simp
-  ultimately show False by blast
-qed
-
-end