src/HOL/Library/Nat_Infinity.thy

tuned

+
−(*  Title:      HOL/Library/Nat_Infinity.thy
+
−    ID:         $Id$
+
−    Author:     David von Oheimb, TU Muenchen;  Florian Haftmann, TU Muenchen
+
−*)
+
−
+
−header {* Natural numbers with infinity *}
+
−
+
−theory Nat_Infinity
+
−imports Plain "~~/src/HOL/Presburger"
+
−begin
+
−
+
−subsection {* Type definition *}
+
−
+
−text {*
+
−  We extend the standard natural numbers by a special value indicating
+
−  infinity.
+
−*}
+
−
+
−datatype inat = Fin nat | Infty
+
−
+
−notation (xsymbols)
+
−  Infty  ("\<infinity>")
+
−
+
−notation (HTML output)
+
−  Infty  ("\<infinity>")
+
−
+
−
+
−subsection {* Constructors and numbers *}
+
−
+
−instantiation inat :: "{zero, one, number}"
+
−begin
+
−
+
−definition
+
−  "0 = Fin 0"
+
−
+
−definition
+
−  [code inline]: "1 = Fin 1"
+
−
+
−definition
+
−  [code inline, code del]: "number_of k = Fin (number_of k)"
+
−
+
−instance ..
+
−
+
−end
+
−
+
−definition iSuc :: "inat \<Rightarrow> inat" where
+
−  "iSuc i = (case i of Fin n \<Rightarrow> Fin (Suc n) | \<infinity> \<Rightarrow> \<infinity>)"
+
−
+
−lemma Fin_0: "Fin 0 = 0"
+
−  by (simp add: zero_inat_def)
+
−
+
−lemma Fin_1: "Fin 1 = 1"
+
−  by (simp add: one_inat_def)
+
−
+
−lemma Fin_number: "Fin (number_of k) = number_of k"
+
−  by (simp add: number_of_inat_def)
+
−
+
−lemma one_iSuc: "1 = iSuc 0"
+
−  by (simp add: zero_inat_def one_inat_def iSuc_def)
+
−
+
−lemma Infty_ne_i0 [simp]: "\<infinity> \<noteq> 0"
+
−  by (simp add: zero_inat_def)
+
−
+
−lemma i0_ne_Infty [simp]: "0 \<noteq> \<infinity>"
+
−  by (simp add: zero_inat_def)
+
−
+
−lemma zero_inat_eq [simp]:
+
−  "number_of k = (0\<Colon>inat) \<longleftrightarrow> number_of k = (0\<Colon>nat)"
+
−  "(0\<Colon>inat) = number_of k \<longleftrightarrow> number_of k = (0\<Colon>nat)"
+
−  unfolding zero_inat_def number_of_inat_def by simp_all
+
−
+
−lemma one_inat_eq [simp]:
+
−  "number_of k = (1\<Colon>inat) \<longleftrightarrow> number_of k = (1\<Colon>nat)"
+
−  "(1\<Colon>inat) = number_of k \<longleftrightarrow> number_of k = (1\<Colon>nat)"
+
−  unfolding one_inat_def number_of_inat_def by simp_all
+
−
+
−lemma zero_one_inat_neq [simp]:
+
−  "\<not> 0 = (1\<Colon>inat)"
+
−  "\<not> 1 = (0\<Colon>inat)"
+
−  unfolding zero_inat_def one_inat_def by simp_all
+
−
+
−lemma Infty_ne_i1 [simp]: "\<infinity> \<noteq> 1"
+
−  by (simp add: one_inat_def)
+
−
+
−lemma i1_ne_Infty [simp]: "1 \<noteq> \<infinity>"
+
−  by (simp add: one_inat_def)
+
−
+
−lemma Infty_ne_number [simp]: "\<infinity> \<noteq> number_of k"
+
−  by (simp add: number_of_inat_def)
+
−
+
−lemma number_ne_Infty [simp]: "number_of k \<noteq> \<infinity>"
+
−  by (simp add: number_of_inat_def)
+
−
+
−lemma iSuc_Fin: "iSuc (Fin n) = Fin (Suc n)"
+
−  by (simp add: iSuc_def)
+
−
+
−lemma iSuc_number_of: "iSuc (number_of k) = Fin (Suc (number_of k))"
+
−  by (simp add: iSuc_Fin number_of_inat_def)
+
−
+
−lemma iSuc_Infty [simp]: "iSuc \<infinity> = \<infinity>"
+
−  by (simp add: iSuc_def)
+
−
+
−lemma iSuc_ne_0 [simp]: "iSuc n \<noteq> 0"
+
−  by (simp add: iSuc_def zero_inat_def split: inat.splits)
+
−
+
−lemma zero_ne_iSuc [simp]: "0 \<noteq> iSuc n"
+
−  by (rule iSuc_ne_0 [symmetric])
+
−
+
−lemma iSuc_inject [simp]: "iSuc m = iSuc n \<longleftrightarrow> m = n"
+
−  by (simp add: iSuc_def split: inat.splits)
+
−
+
−lemma number_of_inat_inject [simp]:
+
−  "(number_of k \<Colon> inat) = number_of l \<longleftrightarrow> (number_of k \<Colon> nat) = number_of l"
+
−  by (simp add: number_of_inat_def)
+
−
+
−
+
−subsection {* Addition *}
+
−
+
−instantiation inat :: comm_monoid_add
+
−begin
+
−
+
−definition
+
−  [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)))"
+
−
+
−lemma plus_inat_simps [simp, code]:
+
−  "Fin m + Fin n = Fin (m + n)"
+
−  "\<infinity> + q = \<infinity>"
+
−  "q + \<infinity> = \<infinity>"
+
−  by (simp_all add: plus_inat_def split: inat.splits)
+
−
+
−instance proof
+
−  fix n m q :: inat
+
−  show "n + m + q = n + (m + q)"
+
−    by (cases n, auto, cases m, auto, cases q, auto)
+
−  show "n + m = m + n"
+
−    by (cases n, auto, cases m, auto)
+
−  show "0 + n = n"
+
−    by (cases n) (simp_all add: zero_inat_def)
+
−qed
+
−
+
−end
+
−
+
−lemma plus_inat_0 [simp]:
+
−  "0 + (q\<Colon>inat) = q"
+
−  "(q\<Colon>inat) + 0 = q"
+
−  by (simp_all add: plus_inat_def zero_inat_def split: inat.splits)
+
−
+
−lemma plus_inat_number [simp]:
+
−  "(number_of k \<Colon> inat) + number_of l = (if neg (number_of k \<Colon> int) then number_of l
+
−    else if neg (number_of l \<Colon> int) then number_of k else number_of (k + l))"
+
−  unfolding number_of_inat_def plus_inat_simps nat_arith(1) if_distrib [symmetric, of _ Fin] ..
+
−
+
−lemma iSuc_number [simp]:
+
−  "iSuc (number_of k) = (if neg (number_of k \<Colon> int) then 1 else number_of (Int.succ k))"
+
−  unfolding iSuc_number_of
+
−  unfolding one_inat_def number_of_inat_def Suc_nat_number_of if_distrib [symmetric] ..
+
−
+
−lemma iSuc_plus_1:
+
−  "iSuc n = n + 1"
+
−  by (cases n) (simp_all add: iSuc_Fin one_inat_def)
+
−  
+
−lemma plus_1_iSuc:
+
−  "1 + q = iSuc q"
+
−  "q + 1 = iSuc q"
+
−  unfolding iSuc_plus_1 by (simp_all add: add_ac)
+
−
+
−
+
−subsection {* Ordering *}
+
−
+
−instantiation inat :: ordered_ab_semigroup_add
+
−begin
+
−
+
−definition
+
−  [code del]: "m \<le> n = (case n of Fin n1 \<Rightarrow> (case m of Fin m1 \<Rightarrow> m1 \<le> n1 | \<infinity> \<Rightarrow> False)
+
−    | \<infinity> \<Rightarrow> True)"
+
−
+
−definition
+
−  [code del]: "m < n = (case m of Fin m1 \<Rightarrow> (case n of Fin n1 \<Rightarrow> m1 < n1 | \<infinity> \<Rightarrow> True)
+
−    | \<infinity> \<Rightarrow> False)"
+
−
+
−lemma inat_ord_simps [simp]:
+
−  "Fin m \<le> Fin n \<longleftrightarrow> m \<le> n"
+
−  "Fin m < Fin n \<longleftrightarrow> m < n"
+
−  "q \<le> \<infinity>"
+
−  "q < \<infinity> \<longleftrightarrow> q \<noteq> \<infinity>"
+
−  "\<infinity> \<le> q \<longleftrightarrow> q = \<infinity>"
+
−  "\<infinity> < q \<longleftrightarrow> False"
+
−  by (simp_all add: less_eq_inat_def less_inat_def split: inat.splits)
+
−
+
−lemma inat_ord_code [code]:
+
−  "Fin m \<le> Fin n \<longleftrightarrow> m \<le> n"
+
−  "Fin m < Fin n \<longleftrightarrow> m < n"
+
−  "q \<le> \<infinity> \<longleftrightarrow> True"
+
−  "Fin m < \<infinity> \<longleftrightarrow> True"
+
−  "\<infinity> \<le> Fin n \<longleftrightarrow> False"
+
−  "\<infinity> < q \<longleftrightarrow> False"
+
−  by simp_all
+
−
+
−instance by default
+
−  (auto simp add: less_eq_inat_def less_inat_def plus_inat_def split: inat.splits)
+
−
+
−end
+
−
+
−lemma inat_ord_number [simp]:
+
−  "(number_of m \<Colon> inat) \<le> number_of n \<longleftrightarrow> (number_of m \<Colon> nat) \<le> number_of n"
+
−  "(number_of m \<Colon> inat) < number_of n \<longleftrightarrow> (number_of m \<Colon> nat) < number_of n"
+
−  by (simp_all add: number_of_inat_def)
+
−
+
−lemma i0_lb [simp]: "(0\<Colon>inat) \<le> n"
+
−  by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
+
−
+
−lemma i0_neq [simp]: "n \<le> (0\<Colon>inat) \<longleftrightarrow> n = 0"
+
−  by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
+
−
+
−lemma Infty_ileE [elim!]: "\<infinity> \<le> Fin m \<Longrightarrow> R"
+
−  by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
+
−
+
−lemma Infty_ilessE [elim!]: "\<infinity> < Fin m \<Longrightarrow> R"
+
−  by simp
+
−
+
−lemma not_ilessi0 [simp]: "\<not> n < (0\<Colon>inat)"
+
−  by (simp add: zero_inat_def less_inat_def split: inat.splits)
+
−
+
−lemma i0_eq [simp]: "(0\<Colon>inat) < n \<longleftrightarrow> n \<noteq> 0"
+
−  by (simp add: zero_inat_def less_inat_def split: inat.splits)
+
−
+
−lemma iSuc_ile_mono [simp]: "iSuc n \<le> iSuc m \<longleftrightarrow> n \<le> m"
+
−  by (simp add: iSuc_def less_eq_inat_def split: inat.splits)
+
− 
+
−lemma iSuc_mono [simp]: "iSuc n < iSuc m \<longleftrightarrow> n < m"
+
−  by (simp add: iSuc_def less_inat_def split: inat.splits)
+
−
+
−lemma ile_iSuc [simp]: "n \<le> iSuc n"
+
−  by (simp add: iSuc_def less_eq_inat_def split: inat.splits)
+
−
+
−lemma not_iSuc_ilei0 [simp]: "\<not> iSuc n \<le> 0"
+
−  by (simp add: zero_inat_def iSuc_def less_eq_inat_def split: inat.splits)
+
−
+
−lemma i0_iless_iSuc [simp]: "0 < iSuc n"
+
−  by (simp add: zero_inat_def iSuc_def less_inat_def split: inat.splits)
+
−
+
−lemma ileI1: "m < n \<Longrightarrow> iSuc m \<le> n"
+
−  by (simp add: iSuc_def less_eq_inat_def less_inat_def split: inat.splits)
+
−
+
−lemma Suc_ile_eq: "Fin (Suc m) \<le> n \<longleftrightarrow> Fin m < n"
+
−  by (cases n) auto
+
−
+
−lemma iless_Suc_eq [simp]: "Fin m < iSuc n \<longleftrightarrow> Fin m \<le> n"
+
−  by (auto simp add: iSuc_def less_inat_def split: inat.splits)
+
−
+
−lemma min_inat_simps [simp]:
+
−  "min (Fin m) (Fin n) = Fin (min m n)"
+
−  "min q 0 = 0"
+
−  "min 0 q = 0"
+
−  "min q \<infinity> = q"
+
−  "min \<infinity> q = q"
+
−  by (auto simp add: min_def)
+
−
+
−lemma max_inat_simps [simp]:
+
−  "max (Fin m) (Fin n) = Fin (max m n)"
+
−  "max q 0 = q"
+
−  "max 0 q = q"
+
−  "max q \<infinity> = \<infinity>"
+
−  "max \<infinity> q = \<infinity>"
+
−  by (simp_all add: max_def)
+
−
+
−lemma Fin_ile: "n \<le> Fin m \<Longrightarrow> \<exists>k. n = Fin k"
+
−  by (cases n) simp_all
+
−
+
−lemma Fin_iless: "n < Fin m \<Longrightarrow> \<exists>k. n = Fin k"
+
−  by (cases n) simp_all
+
−
+
−lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j ==> \<exists>j. Fin k < Y j"
+
−apply (induct_tac k)
+
− apply (simp (no_asm) only: Fin_0)
+
− apply (fast intro: le_less_trans [OF i0_lb])
+
−apply (erule exE)
+
−apply (drule spec)
+
−apply (erule exE)
+
−apply (drule ileI1)
+
−apply (rule iSuc_Fin [THEN subst])
+
−apply (rule exI)
+
−apply (erule (1) le_less_trans)
+
−done
+
−
+
−
+
−subsection {* Well-ordering *}
+
−
+
−lemma less_FinE:
+
−  "[| n < Fin m; !!k. n = Fin k ==> k < m ==> P |] ==> P"
+
−by (induct n) auto
+
−
+
−lemma less_InftyE:
+
−  "[| n < Infty; !!k. n = Fin k ==> P |] ==> P"
+
−by (induct n) auto
+
−
+
−lemma inat_less_induct:
+
−  assumes prem: "!!n. \<forall>m::inat. m < n --> P m ==> P n" shows "P n"
+
−proof -
+
−  have P_Fin: "!!k. P (Fin k)"
+
−    apply (rule nat_less_induct)
+
−    apply (rule prem, clarify)
+
−    apply (erule less_FinE, simp)
+
−    done
+
−  show ?thesis
+
−  proof (induct n)
+
−    fix nat
+
−    show "P (Fin nat)" by (rule P_Fin)
+
−  next
+
−    show "P Infty"
+
−      apply (rule prem, clarify)
+
−      apply (erule less_InftyE)
+
−      apply (simp add: P_Fin)
+
−      done
+
−  qed
+
−qed
+
−
+
−instance inat :: wellorder
+
−proof
+
−  fix P and n
+
−  assume hyp: "(\<And>n\<Colon>inat. (\<And>m\<Colon>inat. m < n \<Longrightarrow> P m) \<Longrightarrow> P n)"
+
−  show "P n" by (blast intro: inat_less_induct hyp)
+
−qed
+
−
+
−
+
−subsection {* Traditional theorem names *}
+
−
+
−lemmas inat_defs = zero_inat_def one_inat_def number_of_inat_def iSuc_def
+
−  plus_inat_def less_eq_inat_def less_inat_def
+
−
+
−lemmas inat_splits = inat.splits
+
−
+
−end