src/HOL/Library/ContNotDenum.thy
 author paulson Wed Apr 01 16:04:21 2015 +0100 (2015-04-01) changeset 59872 db4000b71fdb parent 59720 f893472fff31 child 60308 f7e406aba90d permissions -rw-r--r--
Theorem "arctan" is no longer a default simprule
     1 (*  Title:      HOL/Library/ContNotDenum.thy

     2     Author:     Benjamin Porter, Monash University, NICTA, 2005

     3     Author:     Johannes Hölzl, TU München

     4 *)

     5

     6 section {* Non-denumerability of the Continuum. *}

     7

     8 theory ContNotDenum

     9 imports Complex_Main Countable_Set

    10 begin

    11

    12 subsection {* Abstract *}

    13

    14 text {* The following document presents a proof that the Continuum is

    15 uncountable. It is formalised in the Isabelle/Isar theorem proving

    16 system.

    17

    18 {\em Theorem:} The Continuum @{text "\<real>"} is not denumerable. In other

    19 words, there does not exist a function @{text "f: \<nat> \<Rightarrow> \<real>"} such that f is

    20 surjective.

    21

    22 {\em Outline:} An elegant informal proof of this result uses Cantor's

    23 Diagonalisation argument. The proof presented here is not this

    24 one. First we formalise some properties of closed intervals, then we

    25 prove the Nested Interval Property. This property relies on the

    26 completeness of the Real numbers and is the foundation for our

    27 argument. Informally it states that an intersection of countable

    28 closed intervals (where each successive interval is a subset of the

    29 last) is non-empty. We then assume a surjective function @{text

    30 "f: \<nat> \<Rightarrow> \<real>"} exists and find a real x such that x is not in the range of f

    31 by generating a sequence of closed intervals then using the NIP. *}

    32

    33 theorem real_non_denum: "\<not> (\<exists>f :: nat \<Rightarrow> real. surj f)"

    34 proof

    35   assume "\<exists>f::nat \<Rightarrow> real. surj f"

    36   then obtain f :: "nat \<Rightarrow> real" where "surj f" ..

    37

    38   txt {* First we construct a sequence of nested intervals, ignoring @{term "range f"}. *}

    39

    40   have "\<forall>a b c::real. a < b \<longrightarrow> (\<exists>ka kb. ka < kb \<and> {ka..kb} \<subseteq> {a..b} \<and> c \<notin> {ka..kb})"

    41     using assms

    42     by (auto simp add: not_le cong: conj_cong)

    43        (metis dense le_less_linear less_linear less_trans order_refl)

    44   then obtain i j where ij:

    45     "\<And>a b c::real. a < b \<Longrightarrow> i a b c < j a b c"

    46     "\<And>a b c. a < b \<Longrightarrow> {i a b c .. j a b c} \<subseteq> {a .. b}"

    47     "\<And>a b c. a < b \<Longrightarrow> c \<notin> {i a b c .. j a b c}"

    48     by metis

    49

    50   def ivl \<equiv> "rec_nat (f 0 + 1, f 0 + 2) (\<lambda>n x. (i (fst x) (snd x) (f n), j (fst x) (snd x) (f n)))"

    51   def I \<equiv> "\<lambda>n. {fst (ivl n) .. snd (ivl n)}"

    52

    53   have ivl[simp]:

    54     "ivl 0 = (f 0 + 1, f 0 + 2)"

    55     "\<And>n. ivl (Suc n) = (i (fst (ivl n)) (snd (ivl n)) (f n), j (fst (ivl n)) (snd (ivl n)) (f n))"

    56     unfolding ivl_def by simp_all

    57

    58   txt {* This is a decreasing sequence of non-empty intervals. *}

    59

    60   { fix n have "fst (ivl n) < snd (ivl n)"

    61       by (induct n) (auto intro!: ij) }

    62   note less = this

    63

    64   have "decseq I"

    65     unfolding I_def decseq_Suc_iff ivl fst_conv snd_conv by (intro ij allI less)

    66

    67   txt {* Now we apply the finite intersection property of compact sets. *}

    68

    69   have "I 0 \<inter> (\<Inter>i. I i) \<noteq> {}"

    70   proof (rule compact_imp_fip_image)

    71     fix S :: "nat set" assume fin: "finite S"

    72     have "{} \<subset> I (Max (insert 0 S))"

    73       unfolding I_def using less[of "Max (insert 0 S)"] by auto

    74     also have "I (Max (insert 0 S)) \<subseteq> (\<Inter>i\<in>insert 0 S. I i)"

    75       using fin decseqD[OF decseq I, of _ "Max (insert 0 S)"] by (auto simp: Max_ge_iff)

    76     also have "(\<Inter>i\<in>insert 0 S. I i) = I 0 \<inter> (\<Inter>i\<in>S. I i)"

    77       by auto

    78     finally show "I 0 \<inter> (\<Inter>i\<in>S. I i) \<noteq> {}"

    79       by auto

    80   qed (auto simp: I_def)

    81   then obtain x where "\<And>n. x \<in> I n"

    82     by blast

    83   moreover from surj f obtain j where "x = f j"

    84     by blast

    85   ultimately have "f j \<in> I (Suc j)"

    86     by blast

    87   with ij(3)[OF less] show False

    88     unfolding I_def ivl fst_conv snd_conv by auto

    89 qed

    90

    91 lemma uncountable_UNIV_real: "uncountable (UNIV::real set)"

    92   using real_non_denum unfolding uncountable_def by auto

    93

    94 lemma bij_betw_open_intervals:

    95   fixes a b c d :: real

    96   assumes "a < b" "c < d"

    97   shows "\<exists>f. bij_betw f {a<..<b} {c<..<d}"

    98 proof -

    99   def f \<equiv> "\<lambda>a b c d x::real. (d - c)/(b - a) * (x - a) + c"

   100   { fix a b c d x :: real assume *: "a < b" "c < d" "a < x" "x < b"

   101     moreover from * have "(d - c) * (x - a) < (d - c) * (b - a)"

   102       by (intro mult_strict_left_mono) simp_all

   103     moreover from * have "0 < (d - c) * (x - a) / (b - a)"

   104       by simp

   105     ultimately have "f a b c d x < d" "c < f a b c d x"

   106       by (simp_all add: f_def field_simps) }

   107   with assms have "bij_betw (f a b c d) {a<..<b} {c<..<d}"

   108     by (intro bij_betw_byWitness[where f'="f c d a b"]) (auto simp: f_def)

   109   thus ?thesis by auto

   110 qed

   111

   112 lemma bij_betw_tan: "bij_betw tan {-pi/2<..<pi/2} UNIV"

   113   using arctan_ubound by (intro bij_betw_byWitness[where f'=arctan]) (auto simp: arctan arctan_tan)

   114

   115 lemma uncountable_open_interval:

   116   fixes a b :: real assumes ab: "a < b"

   117   shows "uncountable {a<..<b}"

   118 proof -

   119   obtain f where "bij_betw f {a <..< b} {-pi/2<..<pi/2}"

   120     using bij_betw_open_intervals[OF a < b, of "-pi/2" "pi/2"] by auto

   121   then show ?thesis

   122     by (metis bij_betw_tan uncountable_bij_betw uncountable_UNIV_real)

   123 qed

   124

   125 end

   126