section ‹Implementation of natural numbers by target-language integers›
theory Code_Target_Nat
imports Code_Abstract_Nat
begin
subsection ‹Implementation for @{typ nat}›
context
includes natural.lifting integer.lifting
begin
lift_definition Nat :: "integer ⇒ 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 (fact divmod_nat_div_mod)
lemma [code]:
"divmod m n = map_prod nat_of_integer nat_of_integer (divmod m n)"
by (simp only: prod_eq_iff divmod_def map_prod_def case_prod_beta fst_conv snd_conv)
(transfer, simp_all only: nat_div_distrib nat_mod_distrib
zero_le_numeral nat_numeral)
lemma [code]:
"HOL.equal m n = HOL.equal (of_nat m :: integer) (of_nat n)"
by transfer (simp add: equal)
lemma [code]:
"m ≤ n ⟷ (of_nat m :: integer) ≤ of_nat n"
by simp
lemma [code]:
"m < n ⟷ (of_nat m :: integer) < of_nat n"
by simp
lemma num_of_nat_code [code]:
"num_of_nat = num_of_integer ∘ of_nat"
by transfer (simp add: fun_eq_iff)
end
lemma (in semiring_1) of_nat_code_if:
"of_nat n = (if n = 0 then 0
else let
(m, q) = Divides.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 of_nat_add [symmetric])
(simp add: * mult.commute of_nat_mult add.commute)
qed
declare of_nat_code_if [code]
definition int_of_nat :: "nat ⇒ 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)"
including integer.lifting by transfer auto
lemma term_of_nat_code [code]:
― ‹Use @{term Code_Numeral.nat_of_integer} in term reconstruction
instead of @{term Code_Target_Nat.Nat} such that reconstructed
terms can be fed back to the code generator›
"term_of_class.term_of n =
Code_Evaluation.App
(Code_Evaluation.Const (STR ''Code_Numeral.nat_of_integer'')
(typerep.Typerep (STR ''fun'')
[typerep.Typerep (STR ''Code_Numeral.integer'') [],
typerep.Typerep (STR ''Nat.nat'') []]))
(term_of_class.term_of (integer_of_nat n))"
by (simp add: term_of_anything)
lemma nat_of_integer_code_post [code_post]:
"nat_of_integer 0 = 0"
"nat_of_integer 1 = 1"
"nat_of_integer (numeral k) = numeral k"
including integer.lifting by (transfer, simp)+
code_identifier
code_module Code_Target_Nat ⇀
(SML) Arith and (OCaml) Arith and (Haskell) Arith
end