# Theory Code_Binary_Nat_examples

theory Code_Binary_Nat_examples
imports Complex_Main Code_Binary_Nat
```(*  Title:      HOL/ex/Code_Binary_Nat_examples.thy
Author:     Florian Haftmann, TU Muenchen
*)

section ‹Simple examples for natural numbers implemented in binary representation.›

theory Code_Binary_Nat_examples
imports Complex_Main "HOL-Library.Code_Binary_Nat"
begin

fun to_n :: "nat ⇒ 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 ⇒ bool"
where
"naive_prime n ⟷ n ≥ 2 ∧ filter (λm. n mod m = 0) (to_n n) = []"

primrec fac :: "nat ⇒ nat"
where
"fac 0 = 1"
| "fac (Suc n) = Suc n * fac n"

primrec harmonic :: "nat ⇒ rat"
where
"harmonic 0 = 0"
| "harmonic (Suc n) = 1 / of_nat (Suc n) + harmonic n"

lemma "harmonic 200 ≥ 5"
by eval

lemma "(let (q, r) = quotient_of (harmonic 8) in q div r) ≥ 2"
by normalization

lemma "naive_prime 89"
by eval

lemma "naive_prime 89"
by normalization

lemma "¬ naive_prime 87"
by eval

lemma "¬ naive_prime 87"
by normalization

lemma "fac 10 > 3000000"
by eval

lemma "fac 10 > 3000000"
by normalization

end

```