src/HOL/Library/Nat_Infinity.thy
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(*  Title:      HOL/Library/Nat_Infinity.thy
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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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text {* FIXME: move to Nat.thy *}
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instantiation nat :: bot
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begin
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definition bot_nat :: nat where
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  "bot_nat = 0"
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instance proof
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qed (simp add: bot_nat_def)
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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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end
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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 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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c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
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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 k < Int.Pls then number_of l
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    else if l < Int.Pls 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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lemma iSuc_plus_1:
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  "iSuc n = n + 1"
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  by (cases n) (simp_all add: iSuc_Fin one_inat_def)
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lemma plus_1_iSuc:
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  "1 + q = iSuc q"
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  "q + 1 = iSuc q"
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  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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subsection {* Multiplication *}
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instantiation inat :: comm_semiring_1
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begin
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definition
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  times_inat_def [code del]:
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  "m * n = (case m of \<infinity> \<Rightarrow> if n = 0 then 0 else \<infinity> | Fin m \<Rightarrow>
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    (case n of \<infinity> \<Rightarrow> if m = 0 then 0 else \<infinity> | Fin n \<Rightarrow> Fin (m * n)))"
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lemma times_inat_simps [simp, code]:
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  "Fin m * Fin n = Fin (m * n)"
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  "\<infinity> * \<infinity> = \<infinity>"
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  "\<infinity> * Fin n = (if n = 0 then 0 else \<infinity>)"
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  "Fin m * \<infinity> = (if m = 0 then 0 else \<infinity>)"
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  unfolding times_inat_def zero_inat_def
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  by (simp_all split: inat.split)
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instance proof
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  fix a b c :: inat
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  show "(a * b) * c = a * (b * c)"
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    unfolding times_inat_def zero_inat_def
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    by (simp split: inat.split)
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  show "a * b = b * a"
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    unfolding times_inat_def zero_inat_def
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    by (simp split: inat.split)
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  show "1 * a = a"
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    unfolding times_inat_def zero_inat_def one_inat_def
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    by (simp split: inat.split)
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  show "(a + b) * c = a * c + b * c"
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    unfolding times_inat_def zero_inat_def
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    by (simp split: inat.split add: left_distrib)
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  show "0 * a = 0"
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    unfolding times_inat_def zero_inat_def
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    by (simp split: inat.split)
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  show "a * 0 = 0"
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    unfolding times_inat_def zero_inat_def
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    by (simp split: inat.split)
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  show "(0::inat) \<noteq> 1"
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    unfolding zero_inat_def one_inat_def
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    by simp
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qed
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end
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lemma mult_iSuc: "iSuc m * n = n + m * n"
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  unfolding iSuc_plus_1 by (simp add: algebra_simps)
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lemma mult_iSuc_right: "m * iSuc n = m + m * n"
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  unfolding iSuc_plus_1 by (simp add: algebra_simps)
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29023
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lemma of_nat_eq_Fin: "of_nat n = Fin n"
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  apply (induct n)
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  apply (simp add: Fin_0)
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  apply (simp add: plus_1_iSuc iSuc_Fin)
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  done
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instance inat :: semiring_char_0
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  by default (simp add: of_nat_eq_Fin)
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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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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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  "Fin m < Fin n \<longleftrightarrow> m < n"
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  "q \<le> \<infinity>"
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  "q < \<infinity> \<longleftrightarrow> q \<noteq> \<infinity>"
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  "\<infinity> \<le> q \<longleftrightarrow> q = \<infinity>"
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  "\<infinity> < q \<longleftrightarrow> False"
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  by (simp_all add: less_eq_inat_def less_inat_def split: inat.splits)
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lemma inat_ord_code [code]:
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  "Fin m \<le> Fin n \<longleftrightarrow> m \<le> n"
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  "Fin m < Fin n \<longleftrightarrow> m < n"
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  "q \<le> \<infinity> \<longleftrightarrow> True"
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  "Fin m < \<infinity> \<longleftrightarrow> True"
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  "\<infinity> \<le> Fin n \<longleftrightarrow> False"
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  "\<infinity> < q \<longleftrightarrow> False"
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  by simp_all
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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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end
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instance inat :: pordered_comm_semiring
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proof
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  fix a b c :: inat
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  assume "a \<le> b" and "0 \<le> c"
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  thus "c * a \<le> c * b"
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    unfolding times_inat_def less_eq_inat_def zero_inat_def
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    by (simp split: inat.splits)
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qed
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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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  "(number_of m \<Colon> inat) < number_of n \<longleftrightarrow> (number_of m \<Colon> nat) < number_of n"
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  by (simp_all add: number_of_inat_def)
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lemma i0_lb [simp]: "(0\<Colon>inat) \<le> n"
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  by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
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lemma i0_neq [simp]: "n \<le> (0\<Colon>inat) \<longleftrightarrow> n = 0"
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  by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
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lemma Infty_ileE [elim!]: "\<infinity> \<le> Fin m \<Longrightarrow> R"
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  by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
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lemma Infty_ilessE [elim!]: "\<infinity> < Fin m \<Longrightarrow> R"
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  by simp
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lemma not_ilessi0 [simp]: "\<not> n < (0\<Colon>inat)"
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  by (simp add: zero_inat_def less_inat_def split: inat.splits)
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lemma i0_eq [simp]: "(0\<Colon>inat) < n \<longleftrightarrow> n \<noteq> 0"
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  by (simp add: zero_inat_def less_inat_def split: inat.splits)
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lemma iSuc_ile_mono [simp]: "iSuc n \<le> iSuc m \<longleftrightarrow> n \<le> m"
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  by (simp add: iSuc_def less_eq_inat_def split: inat.splits)
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lemma iSuc_mono [simp]: "iSuc n < iSuc m \<longleftrightarrow> n < m"
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  by (simp add: iSuc_def less_inat_def split: inat.splits)
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lemma ile_iSuc [simp]: "n \<le> iSuc n"
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parents: 26089
diff changeset
   316
  by (simp add: iSuc_def less_eq_inat_def split: inat.splits)
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   317
11355
wenzelm
parents: 11351
diff changeset
   318
lemma not_iSuc_ilei0 [simp]: "\<not> iSuc n \<le> 0"
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   319
  by (simp add: zero_inat_def iSuc_def less_eq_inat_def split: inat.splits)
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   320
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   321
lemma i0_iless_iSuc [simp]: "0 < iSuc n"
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   322
  by (simp add: zero_inat_def iSuc_def less_inat_def split: inat.splits)
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   323
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   324
lemma ileI1: "m < n \<Longrightarrow> iSuc m \<le> n"
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   325
  by (simp add: iSuc_def less_eq_inat_def less_inat_def split: inat.splits)
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   326
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   327
lemma Suc_ile_eq: "Fin (Suc m) \<le> n \<longleftrightarrow> Fin m < n"
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   328
  by (cases n) auto
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   329
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   330
lemma iless_Suc_eq [simp]: "Fin m < iSuc n \<longleftrightarrow> Fin m \<le> n"
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   331
  by (auto simp add: iSuc_def less_inat_def split: inat.splits)
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   332
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   333
lemma min_inat_simps [simp]:
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   334
  "min (Fin m) (Fin n) = Fin (min m n)"
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   335
  "min q 0 = 0"
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   336
  "min 0 q = 0"
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   337
  "min q \<infinity> = q"
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   338
  "min \<infinity> q = q"
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   339
  by (auto simp add: min_def)
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   340
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   341
lemma max_inat_simps [simp]:
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   342
  "max (Fin m) (Fin n) = Fin (max m n)"
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   343
  "max q 0 = q"
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   344
  "max 0 q = q"
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   345
  "max q \<infinity> = \<infinity>"
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   346
  "max \<infinity> q = \<infinity>"
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   347
  by (simp_all add: max_def)
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   348
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   349
lemma Fin_ile: "n \<le> Fin m \<Longrightarrow> \<exists>k. n = Fin k"
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   350
  by (cases n) simp_all
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   351
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   352
lemma Fin_iless: "n < Fin m \<Longrightarrow> \<exists>k. n = Fin k"
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   353
  by (cases n) simp_all
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   354
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   355
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
nipkow
parents: 25112
diff changeset
   356
apply (induct_tac k)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   357
 apply (simp (no_asm) only: Fin_0)
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   358
 apply (fast intro: le_less_trans [OF i0_lb])
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   359
apply (erule exE)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   360
apply (drule spec)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   361
apply (erule exE)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   362
apply (drule ileI1)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   363
apply (rule iSuc_Fin [THEN subst])
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   364
apply (rule exI)
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   365
apply (erule (1) le_less_trans)
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   366
done
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   367
29337
450805a4a91f added instance for bot, top
haftmann
parents: 29023
diff changeset
   368
instantiation inat :: "{bot, top}"
450805a4a91f added instance for bot, top
haftmann
parents: 29023
diff changeset
   369
begin
450805a4a91f added instance for bot, top
haftmann
parents: 29023
diff changeset
   370
450805a4a91f added instance for bot, top
haftmann
parents: 29023
diff changeset
   371
definition bot_inat :: inat where
450805a4a91f added instance for bot, top
haftmann
parents: 29023
diff changeset
   372
  "bot_inat = 0"
450805a4a91f added instance for bot, top
haftmann
parents: 29023
diff changeset
   373
450805a4a91f added instance for bot, top
haftmann
parents: 29023
diff changeset
   374
definition top_inat :: inat where
450805a4a91f added instance for bot, top
haftmann
parents: 29023
diff changeset
   375
  "top_inat = \<infinity>"
450805a4a91f added instance for bot, top
haftmann
parents: 29023
diff changeset
   376
450805a4a91f added instance for bot, top
haftmann
parents: 29023
diff changeset
   377
instance proof
450805a4a91f added instance for bot, top
haftmann
parents: 29023
diff changeset
   378
qed (simp_all add: bot_inat_def top_inat_def)
450805a4a91f added instance for bot, top
haftmann
parents: 29023
diff changeset
   379
450805a4a91f added instance for bot, top
haftmann
parents: 29023
diff changeset
   380
end
450805a4a91f added instance for bot, top
haftmann
parents: 29023
diff changeset
   381
26089
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   382
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   383
subsection {* Well-ordering *}
26089
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   384
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   385
lemma less_FinE:
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   386
  "[| n < Fin m; !!k. n = Fin k ==> k < m ==> P |] ==> P"
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   387
by (induct n) auto
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   388
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   389
lemma less_InftyE:
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   390
  "[| n < Infty; !!k. n = Fin k ==> P |] ==> P"
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   391
by (induct n) auto
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   392
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   393
lemma inat_less_induct:
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   394
  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
   395
proof -
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   396
  have P_Fin: "!!k. P (Fin k)"
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   397
    apply (rule nat_less_induct)
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   398
    apply (rule prem, clarify)
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   399
    apply (erule less_FinE, simp)
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   400
    done
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   401
  show ?thesis
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   402
  proof (induct n)
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   403
    fix nat
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   404
    show "P (Fin nat)" by (rule P_Fin)
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   405
  next
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   406
    show "P Infty"
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   407
      apply (rule prem, clarify)
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   408
      apply (erule less_InftyE)
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   409
      apply (simp add: P_Fin)
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   410
      done
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   411
  qed
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   412
qed
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   413
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   414
instance inat :: wellorder
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   415
proof
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27487
diff changeset
   416
  fix P and n
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27487
diff changeset
   417
  assume hyp: "(\<And>n\<Colon>inat. (\<And>m\<Colon>inat. m < n \<Longrightarrow> P m) \<Longrightarrow> P n)"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27487
diff changeset
   418
  show "P n" by (blast intro: inat_less_induct hyp)
26089
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   419
qed
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   420
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   421
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   422
subsection {* Traditional theorem names *}
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   423
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   424
lemmas inat_defs = zero_inat_def one_inat_def number_of_inat_def iSuc_def
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   425
  plus_inat_def less_eq_inat_def less_inat_def
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   426
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   427
lemmas inat_splits = inat.splits
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   428
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   429
end