Theory Efficient_Nat_examples

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theory Efficient_Nat_examples
imports Complex_Main Efficient_Nat

(*  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 Efficient_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 rat_of_nat :: "nat => rat" where
"rat_of_nat 0 = 0"
| "rat_of_nat (Suc n) = rat_of_nat n + 1"


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


lemma "harmonic 200 ≥ 5"
by eval

lemma "harmonic 200 ≥ 5"
by evaluation

lemma "harmonic 20 ≥ 3"
by normalization

lemma "naive_prime 89"
by eval

lemma "naive_prime 89"
by evaluation

lemma "naive_prime 89"
by normalization

lemma "¬ naive_prime 87"
by eval

lemma "¬ naive_prime 87"
by evaluation

lemma "¬ naive_prime 87"
by normalization

lemma "fac 10 > 3000000"
by eval

lemma "fac 10 > 3000000"
by evaluation

lemma "fac 10 > 3000000"
by normalization

end