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theory Efficient_Nat_examples(* 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