# Theory Code_Target_Int

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

section ‹Implementation of integer numbers by target-language integers›

theory Code_Target_Int
imports Main
begin

code_datatype int_of_integer

declare [[code drop: integer_of_int]]

context
includes integer.lifting
begin

lemma [code]:
"integer_of_int (int_of_integer k) = k"
by transfer rule

lemma [code]:
"Int.Pos = int_of_integer ∘ integer_of_num"
by transfer (simp add: fun_eq_iff)

lemma [code]:
"Int.Neg = int_of_integer ∘ uminus ∘ integer_of_num"
by transfer (simp add: fun_eq_iff)

lemma [code_abbrev]:
"int_of_integer (numeral k) = Int.Pos k"
by transfer simp

lemma [code_abbrev]:
"int_of_integer (- numeral k) = Int.Neg k"
by transfer simp

context
begin

qualified definition positive :: "num ⇒ int"
where [simp]: "positive = numeral"

qualified definition negative :: "num ⇒ int"
where [simp]: "negative = uminus ∘ numeral"

lemma [code_computation_unfold]:
"numeral = positive"
"Int.Pos = positive"
"Int.Neg = negative"
by (simp_all add: fun_eq_iff)

end

lemma [code, symmetric, code_post]:
"0 = int_of_integer 0"
by transfer simp

lemma [code, symmetric, code_post]:
"1 = int_of_integer 1"
by transfer simp

lemma [code_post]:
"int_of_integer (- 1) = - 1"
by simp

lemma [code]:
"k + l = int_of_integer (of_int k + of_int l)"
by transfer simp

lemma [code]:
"- k = int_of_integer (- of_int k)"
by transfer simp

lemma [code]:
"k - l = int_of_integer (of_int k - of_int l)"
by transfer simp

lemma [code]:
"Int.dup k = int_of_integer (Code_Numeral.dup (of_int k))"
by transfer simp

declare [[code drop: Int.sub]]

lemma [code]:
"k * l = int_of_integer (of_int k * of_int l)"
by simp

lemma [code]:
"k div l = int_of_integer (of_int k div of_int l)"
by simp

lemma [code]:
"k mod l = int_of_integer (of_int k mod of_int l)"
by simp

lemma [code]:
"divmod m n = map_prod int_of_integer int_of_integer (divmod m n)"
unfolding prod_eq_iff divmod_def map_prod_def case_prod_beta fst_conv snd_conv
by transfer simp

lemma [code]:
"HOL.equal k l = HOL.equal (of_int k :: integer) (of_int l)"
by transfer (simp add: equal)

lemma [code]:
"k ≤ l ⟷ (of_int k :: integer) ≤ of_int l"
by transfer rule

lemma [code]:
"k < l ⟷ (of_int k :: integer) < of_int l"
by transfer rule

declare [[code drop: "gcd :: int ⇒ _" "lcm :: int ⇒ _"]]

lemma gcd_int_of_integer [code]:
"gcd (int_of_integer x) (int_of_integer y) = int_of_integer (gcd x y)"
by transfer rule

lemma lcm_int_of_integer [code]:
"lcm (int_of_integer x) (int_of_integer y) = int_of_integer (lcm x y)"
by transfer rule

end

lemma (in ring_1) of_int_code_if:
"of_int k = (if k = 0 then 0
else if k < 0 then - of_int (- k)
else let
l = 2 * of_int (k div 2);
j = k mod 2
in if j = 0 then l else l + 1)"
proof -
from div_mult_mod_eq have *: "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp
show ?thesis
by (simp add: Let_def of_int_add [symmetric]) (simp add: * mult.commute)
qed

declare of_int_code_if [code]

lemma [code]:
"nat = nat_of_integer ∘ of_int"
including integer.lifting by transfer (simp add: fun_eq_iff)

definition char_of_int :: "int ⇒ char"
where [code_abbrev]: "char_of_int = char_of"

definition int_of_char :: "char ⇒ int"
where [code_abbrev]: "int_of_char = of_char"

lemma [code]:
"char_of_int = char_of_integer ∘ integer_of_int"
including integer.lifting unfolding char_of_integer_def char_of_int_def
by transfer (simp add: fun_eq_iff)

lemma [code]:
"int_of_char = int_of_integer ∘ integer_of_char"
including integer.lifting unfolding integer_of_char_def int_of_char_def
by transfer (simp add: fun_eq_iff)

code_identifier
code_module Code_Target_Int ⇀
(SML) Arith and (OCaml) Arith and (Haskell) Arith

end
```