Quelle Nat.thy Sprache: Isabelle

(*  Title:      ZF/Nat.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1994  University of Cambridge
*)

section\<open>The Natural numbers As a Least Fixed Point\<close>

theory Nat imports OrdQuant Bool begin

definition
  nat :: i  where
    "nat \ lfp(Inf, \X. {0} \ {succ(i). i \ X})"

definition
  quasinat :: "i \ o" where
    "quasinat(n) \ n=0 | (\m. n = succ(m))"

definition
  (*Has an unconditional succ case, which is used in "recursor" below.*)
  nat_case :: "[i, i\i, i]\i" where
    "nat_case(a,b,k) \ THE y. k=0 \ y=a | (\x. k=succ(x) \ y=b(x))"

definition
  nat_rec :: "[i, i, [i,i]\i]\i" where
    "nat_rec(k,a,b) \
          wfrec(Memrel(nat), k, \<lambda>n f. nat_case(a, \<lambda>m. b(m, f`m), n))"

  (*Internalized relations on the naturals*)

definition
  Le :: i  where
    "Le \ {\x,y\:nat*nat. x \ y}"

definition
  Lt :: i  where
    "Lt \ {\x, y\:nat*nat. x < y}"

definition
  Ge :: i  where
    "Ge \ {\x,y\:nat*nat. y \ x}"

definition
  Gt :: i  where
    "Gt \ {\x,y\:nat*nat. y < x}"

definition
  greater_than :: "i\i" where
    "greater_than(n) \ {i \ nat. n < i}"

text\<open>No need for a less-than operator: a natural number is its list of
predecessors!\<close>

lemma nat_bnd_mono: "bnd_mono(Inf, \X. {0} \ {succ(i). i \ X})"
apply (rule bnd_monoI)
apply (cut_tac infinity, blast, blast)
done

(* @{term"nat = {0} \<union> {succ(x). x \<in> nat}"} *)
lemmas nat_unfold = nat_bnd_mono [THEN nat_def [THEN def_lfp_unfold]]

(** Type checking of 0 and successor **)

lemma nat_0I [iff,TC]: "0 \ nat"
apply (subst nat_unfold)
apply (rule singletonI [THEN UnI1])
done

lemma nat_succI [intro!,TC]: "n \ nat \ succ(n) \ nat"
apply (subst nat_unfold)
apply (erule RepFunI [THEN UnI2])
done

lemma nat_1I [iff,TC]: "1 \ nat"
by (rule nat_0I [THEN nat_succI])

lemma nat_2I [iff,TC]: "2 \ nat"
by (rule nat_1I [THEN nat_succI])

lemma bool_subset_nat: "bool \ nat"
by (blast elim!: boolE)

lemmas bool_into_nat = bool_subset_nat [THEN subsetD]

subsection\<open>Injectivity Properties and Induction\<close>

(*Mathematical induction*)
lemma nat_induct [case_names 0 succ, induct set: nat]:
    "\n \ nat; P(0); \x. \x \ nat; P(x)\ \ P(succ(x))\ \ P(n)"
by (erule def_induct [OF nat_def nat_bnd_mono], blast)

lemma natE:
assumes "n \ nat"
obtains ("0") "n=0" | (succ) x where "x \ nat" "n=succ(x)"
using assms
by (rule nat_unfold [THEN equalityD1, THEN subsetD, THEN UnE]) auto

lemma nat_into_Ord [simp]: "n \ nat \ Ord(n)"
by (erule nat_induct, auto)

(* @{term"i \<in> nat \<Longrightarrow> 0 \<le> i"}; same thing as @{term"0<succ(i)"}  *)
lemmas nat_0_le = nat_into_Ord [THEN Ord_0_le]

(* @{term"i \<in> nat \<Longrightarrow> i \<le> i"}; same thing as @{term"i<succ(i)"}  *)
lemmas nat_le_refl = nat_into_Ord [THEN le_refl]

lemma Ord_nat [iff]: "Ord(nat)"
apply (rule OrdI)
apply (erule_tac [2] nat_into_Ord [THEN Ord_is_Transset])
  unfolding Transset_def
apply (rule ballI)
apply (erule nat_induct, auto)
done

lemma Limit_nat [iff]: "Limit(nat)"
  unfolding Limit_def
apply (safe intro!: ltI Ord_nat)
apply (erule ltD)
done

lemma naturals_not_limit: "a \ nat \ \ Limit(a)"
by (induct a rule: nat_induct, auto)

lemma succ_natD: "succ(i): nat \ i \ nat"
by (rule Ord_trans [OF succI1], auto)

lemma nat_succ_iff [iff]: "succ(n): nat \ n \ nat"
by (blast dest!: succ_natD)

lemma nat_le_Limit: "Limit(i) \ nat \ i"
apply (rule subset_imp_le)
apply (simp_all add: Limit_is_Ord)
apply (rule subsetI)
apply (erule nat_induct)
apply (erule Limit_has_0 [THEN ltD])
apply (blast intro: Limit_has_succ [THEN ltD] ltI Limit_is_Ord)
done

(* \<lbrakk>succ(i): k;  k \<in> nat\<rbrakk> \<Longrightarrow> i \<in> k *)
lemmas succ_in_naturalD = Ord_trans [OF succI1 _ nat_into_Ord]

lemma lt_nat_in_nat: "\m nat\ \ m \ nat"
apply (erule ltE)
apply (erule Ord_trans, assumption, simp)
done

lemma le_in_nat: "\m \ n; n \ nat\ \ m \ nat"
by (blast dest!: lt_nat_in_nat)

subsection\<open>Variations on Mathematical Induction\<close>

(*complete induction*)

lemmas complete_induct = Ord_induct [OF _ Ord_nat, case_names less, consumes 1]

lemma complete_induct_rule [case_names less, consumes 1]:
  "i \ nat \ (\x. x \ nat \ (\y. y \ x \ P(y)) \ P(x)) \ P(i)"
  using complete_induct [of i P] by simp

(*Induction starting from m rather than 0*)
lemma nat_induct_from:
  assumes "m \ n" "m \ nat" "n \ nat"
    and "P(m)"
    and "\x. \x \ nat; m \ x; P(x)\ \ P(succ(x))"
  shows "P(n)"
proof -
  from assms(3) have "m \ n \ P(m) \ P(n)"
    by (rule nat_induct) (use assms(5) in \<open>simp_all add: distrib_simps le_succ_iff\<close>)
  with assms(1,2,4) show ?thesis by blast
qed

(*Induction suitable for subtraction and less-than*)
lemma diff_induct [case_names 0 0_succ succ_succ, consumes 2]:
    "\m \ nat; n \ nat;
        \<And>x. x \<in> nat \<Longrightarrow> P(x,0);
        \<And>y. y \<in> nat \<Longrightarrow> P(0,succ(y));
        \<And>x y. \<lbrakk>x \<in> nat;  y \<in> nat;  P(x,y)\<rbrakk> \<Longrightarrow> P(succ(x),succ(y))\<rbrakk>
     \<Longrightarrow> P(m,n)"
apply (erule_tac x = m in rev_bspec)
apply (erule nat_induct, simp)
apply (rule ballI)
apply (rename_tac i j)
apply (erule_tac n=j in nat_induct, auto)
done

(** Induction principle analogous to trancl_induct **)

lemma succ_lt_induct_lemma [rule_format]:
     "m \ nat \ P(m,succ(m)) \ (\x\nat. P(m,x) \ P(m,succ(x))) \
                 (\<forall>n\<in>nat. m<n \<longrightarrow> P(m,n))"
apply (erule nat_induct)
apply (intro impI, rule nat_induct [THEN ballI])
   prefer 4 apply (intro impI, rule nat_induct [THEN ballI])
apply (auto simp add: le_iff)
done

lemma succ_lt_induct:
    "\m nat;
        P(m,succ(m));
        \<And>x. \<lbrakk>x \<in> nat;  P(m,x)\<rbrakk> \<Longrightarrow> P(m,succ(x))\<rbrakk>
     \<Longrightarrow> P(m,n)"
by (blast intro: succ_lt_induct_lemma lt_nat_in_nat)

subsection\<open>quasinat: to allow a case-split rule for \<^term>\<open>nat_case\<close>\<close>

text\<open>True if the argument is zero or any successor\<close>
lemma [iff]: "quasinat(0)"
by (simp add: quasinat_def)

lemma [iff]: "quasinat(succ(x))"
by (simp add: quasinat_def)

lemma nat_imp_quasinat: "n \ nat \ quasinat(n)"
by (erule natE, simp_all)

lemma non_nat_case: "\ quasinat(x) \ nat_case(a,b,x) = 0"
by (simp add: quasinat_def nat_case_def)

lemma nat_cases_disj: "k=0 | (\y. k = succ(y)) | \ quasinat(k)"
apply (case_tac "k=0", simp)
apply (case_tac "\m. k = succ(m)")
apply (simp_all add: quasinat_def)
done

lemma nat_cases:
     "\k=0 \ P; \y. k = succ(y) \ P; \ quasinat(k) \ P\ \ P"
by (insert nat_cases_disj [of k], blast)

(** nat_case **)

lemma nat_case_0 [simp]: "nat_case(a,b,0) = a"
by (simp add: nat_case_def)

lemma nat_case_succ [simp]: "nat_case(a,b,succ(n)) = b(n)"
by (simp add: nat_case_def)

lemma nat_case_type [TC]:
    "\n \ nat; a \ C(0); \m. m \ nat \ b(m): C(succ(m))\
     \<Longrightarrow> nat_case(a,b,n) \<in> C(n)"
by (erule nat_induct, auto)

lemma split_nat_case:
  "P(nat_case(a,b,k)) \
   ((k=0 \<longrightarrow> P(a)) \<and> (\<forall>x. k=succ(x) \<longrightarrow> P(b(x))) \<and> (\<not> quasinat(k) \<longrightarrow> P(0)))"
apply (rule nat_cases [of k])
apply (auto simp add: non_nat_case)
done

subsection\<open>Recursion on the Natural Numbers\<close>

(** nat_rec is used to define eclose and transrec, then becomes obsolete.
    The operator rec, from arith.thy, has fewer typing conditions **)

lemma nat_rec_0: "nat_rec(0,a,b) = a"
apply (rule nat_rec_def [THEN def_wfrec, THEN trans])
apply (rule wf_Memrel)
apply (rule nat_case_0)
done

lemma nat_rec_succ: "m \ nat \ nat_rec(succ(m),a,b) = b(m, nat_rec(m,a,b))"
apply (rule nat_rec_def [THEN def_wfrec, THEN trans])
apply (rule wf_Memrel)
apply (simp add: vimage_singleton_iff)
done

(** The union of two natural numbers is a natural number -- their maximum **)

lemma Un_nat_type [TC]: "\i \ nat; j \ nat\ \ i \ j \ nat"
apply (rule Un_least_lt [THEN ltD])
apply (simp_all add: lt_def)
done

lemma Int_nat_type [TC]: "\i \ nat; j \ nat\ \ i \ j \ nat"
apply (rule Int_greatest_lt [THEN ltD])
apply (simp_all add: lt_def)
done

(*needed to simplify unions over nat*)
lemma nat_nonempty [simp]: "nat \ 0"
by blast

text\<open>A natural number is the set of its predecessors\<close>
lemma nat_eq_Collect_lt: "i \ nat \ {j\nat. j
apply (rule equalityI)
apply (blast dest: ltD)
apply (auto simp add: Ord_mem_iff_lt)
apply (blast intro: lt_trans)
done

lemma Le_iff [iff]: "\x,y\ \ Le \ x \ y \ x \ nat \ y \ nat"
by (force simp add: Le_def)

end

quality97%

¤ Dauer der Verarbeitung: 0.13 Sekunden (vorverarbeitet) ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.