src/HOL/Library/Code_Target_Nat.thy

fundamentals about discrete logarithm and square root

(*  Title:      HOL/Library/Code_Target_Nat.thy
    Author:     Florian Haftmann, TU Muenchen
*)

header {* Implementation of natural numbers by target-language integers *}

theory Code_Target_Nat
imports Code_Abstract_Nat
begin

subsection {* Implementation for @{typ nat} *}

lift_definition Nat :: "integer \<Rightarrow> nat"
  is nat
  .

lemma [code_post]:
  "Nat 0 = 0"
  "Nat 1 = 1"
  "Nat (numeral k) = numeral k"
  by (transfer, simp)+

lemma [code_abbrev]:
  "integer_of_nat = of_nat"
  by transfer rule

lemma [code_unfold]:
  "Int.nat (int_of_integer k) = nat_of_integer k"
  by transfer rule

lemma [code abstype]:
  "Code_Target_Nat.Nat (integer_of_nat n) = n"
  by transfer simp

lemma [code abstract]:
  "integer_of_nat (nat_of_integer k) = max 0 k"
  by transfer auto

lemma [code_abbrev]:
  "nat_of_integer (numeral k) = nat_of_num k"
  by transfer (simp add: nat_of_num_numeral)

lemma [code abstract]:
  "integer_of_nat (nat_of_num n) = integer_of_num n"
  by transfer (simp add: nat_of_num_numeral)

lemma [code abstract]:
  "integer_of_nat 0 = 0"
  by transfer simp

lemma [code abstract]:
  "integer_of_nat 1 = 1"
  by transfer simp

lemma [code]:
  "Suc n = n + 1"
  by simp

lemma [code abstract]:
  "integer_of_nat (m + n) = of_nat m + of_nat n"
  by transfer simp

lemma [code abstract]:
  "integer_of_nat (m - n) = max 0 (of_nat m - of_nat n)"
  by transfer simp

lemma [code abstract]:
  "integer_of_nat (m * n) = of_nat m * of_nat n"
  by transfer (simp add: of_nat_mult)

lemma [code abstract]:
  "integer_of_nat (m div n) = of_nat m div of_nat n"
  by transfer (simp add: zdiv_int)

lemma [code abstract]:
  "integer_of_nat (m mod n) = of_nat m mod of_nat n"
  by transfer (simp add: zmod_int)

lemma [code]:
  "Divides.divmod_nat m n = (m div n, m mod n)"
  by (simp add: prod_eq_iff)

lemma [code]:
  "HOL.equal m n = HOL.equal (of_nat m :: integer) (of_nat n)"
  by transfer (simp add: equal)

lemma [code]:
  "m \<le> n \<longleftrightarrow> (of_nat m :: integer) \<le> of_nat n"
  by simp

lemma [code]:
  "m < n \<longleftrightarrow> (of_nat m :: integer) < of_nat n"
  by simp

lemma num_of_nat_code [code]:
  "num_of_nat = num_of_integer \<circ> of_nat"
  by transfer (simp add: fun_eq_iff)

lemma (in semiring_1) of_nat_code:
  "of_nat n = (if n = 0 then 0
     else let
       (m, q) = divmod_nat n 2;
       m' = 2 * of_nat m
     in if q = 0 then m' else m' + 1)"
proof -
  from mod_div_equality have *: "of_nat n = of_nat (n div 2 * 2 + n mod 2)" by simp
  show ?thesis
    by (simp add: Let_def divmod_nat_div_mod mod_2_not_eq_zero_eq_one_nat
      of_nat_add [symmetric])
      (simp add: * mult_commute of_nat_mult add_commute)
qed

declare of_nat_code [code]

definition int_of_nat :: "nat \<Rightarrow> int" where
  [code_abbrev]: "int_of_nat = of_nat"

lemma [code]:
  "int_of_nat n = int_of_integer (of_nat n)"
  by (simp add: int_of_nat_def)

lemma [code abstract]:
  "integer_of_nat (nat k) = max 0 (integer_of_int k)"
  by transfer auto

code_modulename SML
  Code_Target_Nat Arith

code_modulename OCaml
  Code_Target_Nat Arith

code_modulename Haskell
  Code_Target_Nat Arith

end