src/HOL/ex/Efficient_Nat_examples.thy

depend on ../pdfsetup.sty;

(*  Title:      HOL/ex/Efficient_Nat_examples.thy
    ID:         $Id$
    Author:     Florian Haftmann, TU Muenchen
*)

header {* Simple examples for Efficient\_Nat theory.  *}

theory Efficient_Nat_examples
imports Main "~~/src/HOL/Real/RealDef" Efficient_Nat
begin

fun
  to_n :: "nat \<Rightarrow> nat list"
where
  "to_n 0 = []"
  | "to_n (Suc 0) = []"
  | "to_n (Suc (Suc 0)) = []"
  | "to_n (Suc n) = n # to_n n"

definition
  naive_prime :: "nat \<Rightarrow> bool"
where
  "naive_prime n \<longleftrightarrow> n \<ge> 2 \<and> filter (\<lambda>m. n mod m = 0) (to_n n) = []"

primrec
  fac :: "nat \<Rightarrow> nat"
where
  "fac 0 = 1"
  | "fac (Suc n) = Suc n * fac n"

primrec
  rat_of_nat :: "nat \<Rightarrow> rat"
where
  "rat_of_nat 0 = 0"
  | "rat_of_nat (Suc n) = rat_of_nat n + 1"

primrec
  harmonic :: "nat \<Rightarrow> rat"
where
  "harmonic 0 = 0"
  | "harmonic (Suc n) = 1 / rat_of_nat (Suc n) + harmonic n"

lemma "harmonic 200 \<ge> 5"
  by eval

lemma "harmonic 200 \<ge> 5"
  by evaluation

lemma "harmonic 20 \<ge> 3"
  by normalization

lemma "naive_prime 89"
  by eval

lemma "naive_prime 89"
  by evaluation

lemma "naive_prime 89"
  by normalization

lemma "\<not> naive_prime 87"
  by eval

lemma "\<not> naive_prime 87"
  by evaluation

lemma "\<not> naive_prime 87"
  by normalization

lemma "fac 10 > 3000000"
  by eval

lemma "fac 10 > 3000000"
  by evaluation

lemma "fac 10 > 3000000"
  by normalization

end