src/HOL/Library/Nat_Infinity.thy
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absolute imports of HOL/*.thy theories
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(*  Title:      HOL/Library/Nat_Infinity.thy
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    ID:         $Id$
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    Author:     David von Oheimb, TU Muenchen;  Florian Haftmann, TU Muenchen
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*)
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header {* Natural numbers with infinity *}
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theory Nat_Infinity
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imports Plain "~~/src/HOL/Presburger"
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begin
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subsection {* Type definition *}
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text {*
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  We extend the standard natural numbers by a special value indicating
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  infinity.
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*}
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datatype inat = Fin nat | Infty
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notation (xsymbols)
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  Infty  ("\<infinity>")
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notation (HTML output)
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  Infty  ("\<infinity>")
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subsection {* Constructors and numbers *}
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instantiation inat :: "{zero, one, number}"
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begin
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definition
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  "0 = Fin 0"
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definition
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  [code inline]: "1 = Fin 1"
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definition
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  [code inline, code func del]: "number_of k = Fin (number_of k)"
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instance ..
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end
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definition iSuc :: "inat \<Rightarrow> inat" where
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  "iSuc i = (case i of Fin n \<Rightarrow> Fin (Suc n) | \<infinity> \<Rightarrow> \<infinity>)"
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lemma Fin_0: "Fin 0 = 0"
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  by (simp add: zero_inat_def)
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lemma Fin_1: "Fin 1 = 1"
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  by (simp add: one_inat_def)
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lemma Fin_number: "Fin (number_of k) = number_of k"
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  by (simp add: number_of_inat_def)
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lemma one_iSuc: "1 = iSuc 0"
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  by (simp add: zero_inat_def one_inat_def iSuc_def)
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lemma Infty_ne_i0 [simp]: "\<infinity> \<noteq> 0"
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  by (simp add: zero_inat_def)
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lemma i0_ne_Infty [simp]: "0 \<noteq> \<infinity>"
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  by (simp add: zero_inat_def)
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lemma zero_inat_eq [simp]:
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  "number_of k = (0\<Colon>inat) \<longleftrightarrow> number_of k = (0\<Colon>nat)"
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  "(0\<Colon>inat) = number_of k \<longleftrightarrow> number_of k = (0\<Colon>nat)"
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  unfolding zero_inat_def number_of_inat_def by simp_all
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lemma one_inat_eq [simp]:
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  "number_of k = (1\<Colon>inat) \<longleftrightarrow> number_of k = (1\<Colon>nat)"
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  "(1\<Colon>inat) = number_of k \<longleftrightarrow> number_of k = (1\<Colon>nat)"
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  unfolding one_inat_def number_of_inat_def by simp_all
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lemma zero_one_inat_neq [simp]:
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  "\<not> 0 = (1\<Colon>inat)"
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  "\<not> 1 = (0\<Colon>inat)"
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  unfolding zero_inat_def one_inat_def by simp_all
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lemma Infty_ne_i1 [simp]: "\<infinity> \<noteq> 1"
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  by (simp add: one_inat_def)
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lemma i1_ne_Infty [simp]: "1 \<noteq> \<infinity>"
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  by (simp add: one_inat_def)
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lemma Infty_ne_number [simp]: "\<infinity> \<noteq> number_of k"
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  by (simp add: number_of_inat_def)
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lemma number_ne_Infty [simp]: "number_of k \<noteq> \<infinity>"
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  by (simp add: number_of_inat_def)
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lemma iSuc_Fin: "iSuc (Fin n) = Fin (Suc n)"
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  by (simp add: iSuc_def)
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lemma iSuc_number_of: "iSuc (number_of k) = Fin (Suc (number_of k))"
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  by (simp add: iSuc_Fin number_of_inat_def)
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lemma iSuc_Infty [simp]: "iSuc \<infinity> = \<infinity>"
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  by (simp add: iSuc_def)
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lemma iSuc_ne_0 [simp]: "iSuc n \<noteq> 0"
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  by (simp add: iSuc_def zero_inat_def split: inat.splits)
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lemma zero_ne_iSuc [simp]: "0 \<noteq> iSuc n"
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  by (rule iSuc_ne_0 [symmetric])
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lemma iSuc_inject [simp]: "iSuc m = iSuc n \<longleftrightarrow> m = n"
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  by (simp add: iSuc_def split: inat.splits)
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lemma number_of_inat_inject [simp]:
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  "(number_of k \<Colon> inat) = number_of l \<longleftrightarrow> (number_of k \<Colon> nat) = number_of l"
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  by (simp add: number_of_inat_def)
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subsection {* Addition *}
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instantiation inat :: comm_monoid_add
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begin
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definition
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  [code del]: "m + n = (case m of \<infinity> \<Rightarrow> \<infinity> | Fin m \<Rightarrow> (case n of \<infinity> \<Rightarrow> \<infinity> | Fin n \<Rightarrow> Fin (m + n)))"
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lemma plus_inat_simps [simp, code]:
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  "Fin m + Fin n = Fin (m + n)"
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  "\<infinity> + q = \<infinity>"
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  "q + \<infinity> = \<infinity>"
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  by (simp_all add: plus_inat_def split: inat.splits)
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instance proof
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  fix n m q :: inat
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  show "n + m + q = n + (m + q)"
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    by (cases n, auto, cases m, auto, cases q, auto)
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  show "n + m = m + n"
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    by (cases n, auto, cases m, auto)
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  show "0 + n = n"
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    by (cases n) (simp_all add: zero_inat_def)
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qed
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end
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lemma plus_inat_0 [simp]:
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  "0 + (q\<Colon>inat) = q"
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  "(q\<Colon>inat) + 0 = q"
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  by (simp_all add: plus_inat_def zero_inat_def split: inat.splits)
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lemma plus_inat_number [simp]:
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  "(number_of k \<Colon> inat) + number_of l = (if neg (number_of k \<Colon> int) then number_of l
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    else if neg (number_of l \<Colon> int) then number_of k else number_of (k + l))"
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  unfolding number_of_inat_def plus_inat_simps nat_arith(1) if_distrib [symmetric, of _ Fin] ..
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lemma iSuc_number [simp]:
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  "iSuc (number_of k) = (if neg (number_of k \<Colon> int) then 1 else number_of (Int.succ k))"
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  unfolding iSuc_number_of
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  unfolding one_inat_def number_of_inat_def Suc_nat_number_of if_distrib [symmetric] ..
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   158
lemma iSuc_plus_1:
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   159
  "iSuc n = n + 1"
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   160
  by (cases n) (simp_all add: iSuc_Fin one_inat_def)
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   161
  
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   162
lemma plus_1_iSuc:
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  "1 + q = iSuc q"
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   164
  "q + 1 = iSuc q"
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   165
  unfolding iSuc_plus_1 by (simp_all add: add_ac)
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c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
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   167
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   168
subsection {* Ordering *}
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instantiation inat :: ordered_ab_semigroup_add
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begin
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definition
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  [code del]: "m \<le> n = (case n of Fin n1 \<Rightarrow> (case m of Fin m1 \<Rightarrow> m1 \<le> n1 | \<infinity> \<Rightarrow> False)
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    | \<infinity> \<Rightarrow> True)"
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   176
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definition
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  [code del]: "m < n = (case m of Fin m1 \<Rightarrow> (case n of Fin n1 \<Rightarrow> m1 < n1 | \<infinity> \<Rightarrow> True)
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    | \<infinity> \<Rightarrow> False)"
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lemma inat_ord_simps [simp]:
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  "Fin m \<le> Fin n \<longleftrightarrow> m \<le> n"
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   183
  "Fin m < Fin n \<longleftrightarrow> m < n"
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   184
  "q \<le> \<infinity>"
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   185
  "q < \<infinity> \<longleftrightarrow> q \<noteq> \<infinity>"
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   186
  "\<infinity> \<le> q \<longleftrightarrow> q = \<infinity>"
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   187
  "\<infinity> < q \<longleftrightarrow> False"
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   188
  by (simp_all add: less_eq_inat_def less_inat_def split: inat.splits)
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   189
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   190
lemma inat_ord_code [code]:
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   191
  "Fin m \<le> Fin n \<longleftrightarrow> m \<le> n"
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   192
  "Fin m < Fin n \<longleftrightarrow> m < n"
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   193
  "q \<le> \<infinity> \<longleftrightarrow> True"
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   194
  "Fin m < \<infinity> \<longleftrightarrow> True"
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   195
  "\<infinity> \<le> Fin n \<longleftrightarrow> False"
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   196
  "\<infinity> < q \<longleftrightarrow> False"
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   197
  by simp_all
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   198
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   199
instance by default
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  (auto simp add: less_eq_inat_def less_inat_def plus_inat_def split: inat.splits)
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   201
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   202
end
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   203
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   204
lemma inat_ord_number [simp]:
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  "(number_of m \<Colon> inat) \<le> number_of n \<longleftrightarrow> (number_of m \<Colon> nat) \<le> number_of n"
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   206
  "(number_of m \<Colon> inat) < number_of n \<longleftrightarrow> (number_of m \<Colon> nat) < number_of n"
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   207
  by (simp_all add: number_of_inat_def)
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   208
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lemma i0_lb [simp]: "(0\<Colon>inat) \<le> n"
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   210
  by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
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   211
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lemma i0_neq [simp]: "n \<le> (0\<Colon>inat) \<longleftrightarrow> n = 0"
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   213
  by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
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   214
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   215
lemma Infty_ileE [elim!]: "\<infinity> \<le> Fin m \<Longrightarrow> R"
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   216
  by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
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   217
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   218
lemma Infty_ilessE [elim!]: "\<infinity> < Fin m \<Longrightarrow> R"
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   219
  by simp
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   220
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   221
lemma not_ilessi0 [simp]: "\<not> n < (0\<Colon>inat)"
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   222
  by (simp add: zero_inat_def less_inat_def split: inat.splits)
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   223
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   224
lemma i0_eq [simp]: "(0\<Colon>inat) < n \<longleftrightarrow> n \<noteq> 0"
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   225
  by (simp add: zero_inat_def less_inat_def split: inat.splits)
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   226
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   227
lemma iSuc_ile_mono [simp]: "iSuc n \<le> iSuc m \<longleftrightarrow> n \<le> m"
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   228
  by (simp add: iSuc_def less_eq_inat_def split: inat.splits)
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   229
 
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   230
lemma iSuc_mono [simp]: "iSuc n < iSuc m \<longleftrightarrow> n < m"
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   231
  by (simp add: iSuc_def less_inat_def split: inat.splits)
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   232
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   233
lemma ile_iSuc [simp]: "n \<le> iSuc n"
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   234
  by (simp add: iSuc_def less_eq_inat_def split: inat.splits)
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   235
11355
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   236
lemma not_iSuc_ilei0 [simp]: "\<not> iSuc n \<le> 0"
27110
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   237
  by (simp add: zero_inat_def iSuc_def less_eq_inat_def split: inat.splits)
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   238
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   239
lemma i0_iless_iSuc [simp]: "0 < iSuc n"
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   240
  by (simp add: zero_inat_def iSuc_def less_inat_def split: inat.splits)
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   241
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   242
lemma ileI1: "m < n \<Longrightarrow> iSuc m \<le> n"
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   243
  by (simp add: iSuc_def less_eq_inat_def less_inat_def split: inat.splits)
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   244
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   245
lemma Suc_ile_eq: "Fin (Suc m) \<le> n \<longleftrightarrow> Fin m < n"
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   246
  by (cases n) auto
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   247
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   248
lemma iless_Suc_eq [simp]: "Fin m < iSuc n \<longleftrightarrow> Fin m \<le> n"
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   249
  by (auto simp add: iSuc_def less_inat_def split: inat.splits)
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   250
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   251
lemma min_inat_simps [simp]:
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   252
  "min (Fin m) (Fin n) = Fin (min m n)"
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   253
  "min q 0 = 0"
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   254
  "min 0 q = 0"
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   255
  "min q \<infinity> = q"
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   256
  "min \<infinity> q = q"
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diff changeset
   257
  by (auto simp add: min_def)
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   258
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   259
lemma max_inat_simps [simp]:
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   260
  "max (Fin m) (Fin n) = Fin (max m n)"
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   261
  "max q 0 = q"
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   262
  "max 0 q = q"
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   263
  "max q \<infinity> = \<infinity>"
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   264
  "max \<infinity> q = \<infinity>"
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   265
  by (simp_all add: max_def)
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   266
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   267
lemma Fin_ile: "n \<le> Fin m \<Longrightarrow> \<exists>k. n = Fin k"
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   268
  by (cases n) simp_all
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   269
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   270
lemma Fin_iless: "n < Fin m \<Longrightarrow> \<exists>k. n = Fin k"
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   271
  by (cases n) simp_all
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   272
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
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   273
lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j ==> \<exists>j. Fin k < Y j"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
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   274
apply (induct_tac k)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
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   275
 apply (simp (no_asm) only: Fin_0)
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   276
 apply (fast intro: le_less_trans [OF i0_lb])
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3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
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   277
apply (erule exE)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
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   278
apply (drule spec)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
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diff changeset
   279
apply (erule exE)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
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   280
apply (drule ileI1)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
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   281
apply (rule iSuc_Fin [THEN subst])
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
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   282
apply (rule exI)
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   283
apply (erule (1) le_less_trans)
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3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
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   284
done
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   285
26089
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   286
27110
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   287
subsection {* Well-ordering *}
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diff changeset
   288
373221497340 proved linorder and wellorder class instances
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   289
lemma less_FinE:
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   290
  "[| n < Fin m; !!k. n = Fin k ==> k < m ==> P |] ==> P"
373221497340 proved linorder and wellorder class instances
huffman
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diff changeset
   291
by (induct n) auto
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   292
373221497340 proved linorder and wellorder class instances
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diff changeset
   293
lemma less_InftyE:
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diff changeset
   294
  "[| n < Infty; !!k. n = Fin k ==> P |] ==> P"
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   295
by (induct n) auto
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   296
373221497340 proved linorder and wellorder class instances
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diff changeset
   297
lemma inat_less_induct:
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diff changeset
   298
  assumes prem: "!!n. \<forall>m::inat. m < n --> P m ==> P n" shows "P n"
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   299
proof -
373221497340 proved linorder and wellorder class instances
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diff changeset
   300
  have P_Fin: "!!k. P (Fin k)"
373221497340 proved linorder and wellorder class instances
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diff changeset
   301
    apply (rule nat_less_induct)
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   302
    apply (rule prem, clarify)
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   303
    apply (erule less_FinE, simp)
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   304
    done
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   305
  show ?thesis
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   306
  proof (induct n)
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   307
    fix nat
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   308
    show "P (Fin nat)" by (rule P_Fin)
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   309
  next
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
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    show "P Infty"
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      apply (rule prem, clarify)
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      apply (erule less_InftyE)
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      apply (simp add: P_Fin)
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      done
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  qed
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qed
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instance inat :: wellorder
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proof
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  show "wf {(x::inat, y::inat). x < y}"
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  proof (rule wfUNIVI)
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    fix P and x :: inat
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    assume "\<forall>x::inat. (\<forall>y. (y, x) \<in> {(x, y). x < y} \<longrightarrow> P y) \<longrightarrow> P x"
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    hence 1: "!!x::inat. ALL y. y < x --> P y ==> P x" by fast
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    thus "P x" by (rule inat_less_induct)
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  qed
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qed
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27110
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subsection {* Traditional theorem names *}
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lemmas inat_defs = zero_inat_def one_inat_def number_of_inat_def iSuc_def
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  plus_inat_def less_eq_inat_def less_inat_def
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lemmas inat_splits = inat.splits
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11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
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end