section \<open>A simple example demonstrating parallelism for code generated towards Isabelle/ML\<close>
theory Parallel_Example
imports Complex_Main "HOL-Library.Parallel" "HOL-Library.Debug"
begin
subsection \<open>Compute-intensive examples.\<close>
subsubsection \<open>Fragments of the harmonic series\<close>
definition harmonic :: "nat \ rat" where
"harmonic n = sum_list (map (\n. 1 / of_nat n) [1..
subsubsection \<open>The sieve of Erathostenes\<close>
text \<open>
The attentive reader may relate this ad-hoc implementation to the
arithmetic notion of prime numbers as a little exercise.
\<close>
primrec mark :: "nat \ nat \ bool list \ bool list" where
"mark _ _ [] = []"
| "mark m n (p # ps) = (case n of 0 \ False # mark m m ps
| Suc n \<Rightarrow> p # mark m n ps)"
lemma length_mark [simp]:
"length (mark m n ps) = length ps"
by (induct ps arbitrary: n) (simp_all split: nat.split)
function sieve :: "nat \ bool list \ bool list" where
"sieve m ps = (case dropWhile Not ps
of [] \<Rightarrow> ps
| p#ps' \ let n = m - length ps' in takeWhile Not ps @ p # sieve m (mark n n ps'))"
by pat_completeness auto
termination \<comment> \<open>tuning of this proof is left as an exercise to the reader\<close>
apply (relation "measure (length \ snd)")
apply rule
apply (auto simp add: length_dropWhile_le)
proof -
fix ps qs q
assume "dropWhile Not ps = q # qs"
then have "length (q # qs) = length (dropWhile Not ps)" by simp
then have "length qs < length (dropWhile Not ps)" by simp
moreover have "length (dropWhile Not ps) \ length ps"
by (simp add: length_dropWhile_le)
ultimately show "length qs < length ps" by auto
qed
primrec natify :: "nat \ bool list \ nat list" where
"natify _ [] = []"
| "natify n (p#ps) = (if p then n # natify (Suc n) ps else natify (Suc n) ps)"
primrec list_primes where
"list_primes (Suc n) = natify 1 (sieve n (False # replicate n True))"
subsubsection \<open>Naive factorisation\<close>
function factorise_from :: "nat \ nat \ nat list" where
"factorise_from k n = (if 1 < k \ k \ n
then
let (q, r) = Divides.divmod_nat n k
in if r = 0 then k # factorise_from k q
else factorise_from (Suc k) n
else [])"
by pat_completeness auto
termination factorise_from \<comment> \<open>tuning of this proof is left as an exercise to the reader\<close>
apply (relation "measure (\(k, n). 2 * n - k)")
apply (auto simp add: prod_eq_iff algebra_simps elim!: dvdE)
apply (case_tac "k \ ka * 2")
apply (auto intro: diff_less_mono)
done
definition factorise :: "nat \ nat list" where
"factorise n = factorise_from 2 n"
subsection \<open>Concurrent computation via futures\<close>
definition computation_harmonic :: "unit \ rat" where
"computation_harmonic _ = Debug.timing (STR ''harmonic example'') harmonic 300"
definition computation_primes :: "unit \ nat list" where
"computation_primes _ = Debug.timing (STR ''primes example'') list_primes 4000"
definition computation_future :: "unit \ nat list \ rat" where
"computation_future = Debug.timing (STR ''overall computation'')
(\<lambda>() \<Rightarrow> let c = Parallel.fork computation_harmonic
in (computation_primes (), Parallel.join c))"
value "computation_future ()"
definition computation_factorise :: "nat \ nat list" where
"computation_factorise = Debug.timing (STR ''factorise'') factorise"
definition computation_parallel :: "unit \ nat list list" where
"computation_parallel _ = Debug.timing (STR ''overall computation'')
(Parallel.map computation_factorise) [20000..<20100]"
value "computation_parallel ()"
end
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