src/HOL/ex/Efficient_Nat_examples.thy

explicit class Counter;

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

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

theory Efficient_Nat_examples
imports Complex_Main "~~/src/HOL/Library/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