[isabelle-dev] Efficient code for Discrete.log

Manuel Eberl eberlm at in.tum.de
Thu Jun 29 15:29:11 CEST 2017


I'm considering adding efficient code for Discrete.log (the dual
logarithm on natural numbers). PolyML does provide an IntInf.log2
function that seems reasonably efficient so that one can set up code
printing. However, I am struggling with one detail:

Where would the code that does this actually reside? I cannot really put
it into Discrete.thy, because then that would have to import
Code_Target_Numeral. I could put it into Code_Target_Integer.thy, but
then that would have to import Discrete, which does not sound right to
me either.

I attached what I have so far.


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theory Efficient_Log2

includes integer.lifting

qualified lift_definition intlog2 :: "integer \<Rightarrow> integer" is
  "\<lambda>n. int (Discrete.log (nat n))" .

qualified function intlog2_aux :: "integer \<Rightarrow> integer \<Rightarrow> integer" where
  "intlog2_aux n acc = (if n < 2 then acc else intlog2_aux (n div 2) (acc + 1))"
  by auto
termination proof (relation "measure (nat_of_integer \<circ> fst)")
  fix n acc :: integer
  assume "\<not>n < 2"
  hence "nat_of_integer (n div 2) < nat_of_integer n"
    by transfer auto
  thus "((n div 2, acc + 1), (n, acc)) \<in> measure (nat_of_integer \<circ> fst)" 
    by simp
qed simp_all

declare intlog2_aux.simps [simp del]
qualified lemma intlog2_aux_correct: "acc \<ge> 0 \<Longrightarrow> intlog2_aux n acc = intlog2 n + acc"
proof (induction n acc arbitrary: acc rule: intlog2_aux.induct)
  case (1 n acc)
  show ?case
  proof (cases "n < 2")
    case True
    hence "intlog2 n = 0" by transfer (simp add: Discrete.log.simps)
    with True show ?thesis by (subst intlog2_aux.simps) auto
    case False
    hence "intlog2 n = intlog2 (n div 2) + 1"
    proof (transfer, goal_cases)
      case (1 m)
      hence "int (Discrete.log (nat m)) = int (Discrete.log (nat m div 2)) + 1"
        by (subst Discrete.log.simps) auto
      also have "nat m div 2 = nat (m div 2)" by simp
      finally show ?case .
    with False show ?thesis by (subst intlog2_aux.simps) (auto simp: 1)
qualified lemma intlog2_code [code]: "intlog2 n = intlog2_aux n 0"
  by (simp add: intlog2_aux_correct)

lemma Discrete_log_code [code abstract]:
  "integer_of_nat (Discrete.log n) = (if n = 0 then 0 else intlog2 (of_nat n))"
  by transfer simp


  constant "Efficient_Log2.intlog2 :: integer \<Rightarrow> _" \<rightharpoonup> 
    (SML) "IntInf.log2" and
    (Eval) "IntInf.log2"

value [code] "Discrete.log 12345"

export_code Discrete.log checking SML Eval Scala Haskell? OCaml?


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