# Theory Code_Abstract_Char

```(*  Title:      HOL/Library/Code_Abstract_Char.thy
Author:     Florian Haftmann, TU Muenchen
Author:     René Thiemann, UIBK
*)

theory Code_Abstract_Char
imports
Main
"HOL-Library.Char_ord"
begin

definition Chr :: ‹integer ⇒ char›
where [simp]: ‹Chr = char_of›

lemma char_of_integer_of_char [code abstype]:
‹Chr (integer_of_char c) = c›

lemma char_of_integer_code [code]:
‹integer_of_char (char_of_integer k) = (if 0 ≤ k ∧ k < 256 then k else k mod 256)›
by (simp add: integer_of_char_def char_of_integer_def integer_eq_iff integer_less_eq_iff integer_less_iff)

lemma of_char_code [code]:
‹of_char c = of_nat (nat_of_integer (integer_of_char c))›
proof -
have ‹int_of_integer (of_char c) = of_char c›
by (cases c) simp
then show ?thesis
by (simp add: integer_of_char_def nat_of_integer_def of_nat_of_char)
qed

definition byte :: ‹bool ⇒ bool ⇒ bool ⇒ bool ⇒ bool ⇒ bool ⇒ bool ⇒ bool ⇒ integer›
where [simp]: ‹byte b0 b1 b2 b3 b4 b5 b6 b7 = horner_sum of_bool 2 [b0, b1, b2, b3, b4, b5, b6, b7]›

lemma byte_code [code]:
‹byte b0 b1 b2 b3 b4 b5 b6 b7 = (
let
s0 = if b0 then 1 else 0;
s1 = if b1 then s0 + 2 else s0;
s2 = if b2 then s1 + 4 else s1;
s3 = if b3 then s2 + 8 else s2;
s4 = if b4 then s3 + 16 else s3;
s5 = if b5 then s4 + 32 else s4;
s6 = if b6 then s5 + 64 else s5;
s7 = if b7 then s6 + 128 else s6
in s7)›
by simp

lemma Char_code [code]:
‹integer_of_char (Char b0 b1 b2 b3 b4 b5 b6 b7) = byte b0 b1 b2 b3 b4 b5 b6 b7›

lemma digit_0_code [code]:
‹digit0 c ⟷ bit (integer_of_char c) 0›
by (cases c) (simp add: integer_of_char_def)

lemma digit_1_code [code]:
‹digit1 c ⟷ bit (integer_of_char c) 1›
by (cases c) (simp add: integer_of_char_def)

lemma digit_2_code [code]:
‹digit2 c ⟷ bit (integer_of_char c) 2›
by (cases c) (simp add: integer_of_char_def)

lemma digit_3_code [code]:
‹digit3 c ⟷ bit (integer_of_char c) 3›
by (cases c) (simp add: integer_of_char_def)

lemma digit_4_code [code]:
‹digit4 c ⟷ bit (integer_of_char c) 4›
by (cases c) (simp add: integer_of_char_def)

lemma digit_5_code [code]:
‹digit5 c ⟷ bit (integer_of_char c) 5›
by (cases c) (simp add: integer_of_char_def)

lemma digit_6_code [code]:
‹digit6 c ⟷ bit (integer_of_char c) 6›
by (cases c) (simp add: integer_of_char_def)

lemma digit_7_code [code]:
‹digit7 c ⟷ bit (integer_of_char c) 7›
by (cases c) (simp add: integer_of_char_def)

lemma case_char_code [code]:
‹case_char f c = f (digit0 c) (digit1 c) (digit2 c) (digit3 c) (digit4 c) (digit5 c) (digit6 c) (digit7 c)›
by (fact char.case_eq_if)

lemma rec_char_code [code]:
‹rec_char f c = f (digit0 c) (digit1 c) (digit2 c) (digit3 c) (digit4 c) (digit5 c) (digit6 c) (digit7 c)›
by (cases c) simp

lemma char_of_code [code]:
‹integer_of_char (char_of a) =
byte (bit a 0) (bit a 1) (bit a 2) (bit a 3) (bit a 4) (bit a 5) (bit a 6) (bit a 7)›

lemma ascii_of_code [code]:
‹integer_of_char (String.ascii_of c) = (let k = integer_of_char c in if k < 128 then k else k - 128)›
proof (cases ‹of_char c < (128 :: integer)›)
case True
moreover have ‹(of_nat 0 :: integer) ≤ of_nat (of_char c)›
by simp
then have ‹(0 :: integer) ≤ of_char c›
by (simp only: of_nat_0 of_nat_of_char)
ultimately show ?thesis
by (simp add: Let_def integer_of_char_def take_bit_eq_mod integer_eq_iff integer_less_eq_iff integer_less_iff)
next
case False
then have ‹(128 :: integer) ≤ of_char c›
by simp
moreover have ‹of_nat (of_char c) < (of_nat 256 :: integer)›
by (simp only: of_nat_less_iff) simp
then have ‹of_char c < (256 :: integer)›
moreover define k :: integer where ‹k = of_char c - 128›
then have ‹of_char c = k + 128›
by simp
ultimately show ?thesis
by (simp add: Let_def integer_of_char_def take_bit_eq_mod integer_eq_iff integer_less_eq_iff integer_less_iff)
qed

lemma equal_char_code [code]:
‹HOL.equal c d ⟷ integer_of_char c = integer_of_char d›

lemma less_eq_char_code [code]:
‹c ≤ d ⟷ integer_of_char c ≤ integer_of_char d› (is ‹?P ⟷ ?Q›)
proof -
have ‹?P ⟷ of_nat (of_char c) ≤ (of_nat (of_char d) :: integer)›
also have ‹… ⟷ ?Q›
finally show ?thesis .
qed

lemma less_char_code [code]:
‹c < d ⟷ integer_of_char c < integer_of_char d› (is ‹?P ⟷ ?Q›)
proof -
have ‹?P ⟷ of_nat (of_char c) < (of_nat (of_char d) :: integer)›
also have ‹… ⟷ ?Q›
finally show ?thesis .
qed

lemma absdef_simps:
‹horner_sum of_bool 2 [] = (0 :: integer)›
‹horner_sum of_bool 2 (False # bs) = (0 :: integer) ⟷ horner_sum of_bool 2 bs = (0 :: integer)›
‹horner_sum of_bool 2 (True # bs) = (1 :: integer) ⟷ horner_sum of_bool 2 bs = (0 :: integer)›
‹horner_sum of_bool 2 (False # bs) = (numeral (Num.Bit0 n) :: integer) ⟷ horner_sum of_bool 2 bs = (numeral n :: integer)›
‹horner_sum of_bool 2 (True # bs) = (numeral (Num.Bit1 n) :: integer) ⟷ horner_sum of_bool 2 bs = (numeral n :: integer)›
by auto (auto simp only: numeral_Bit0 [of n] numeral_Bit1 [of n] mult_2 [symmetric] add.commute [of _ 1] add.left_cancel mult_cancel_left)

local_setup ‹
let
val simps = @{thms absdef_simps integer_of_char_def of_char_Char numeral_One}
fun prove_eqn lthy n lhs def_eqn =
let
val eqn = (HOLogic.mk_Trueprop o HOLogic.mk_eq)
(\<^term>‹integer_of_char› \$ lhs, HOLogic.mk_number \<^typ>‹integer› n)
in
Goal.prove_future lthy [] [] eqn (fn {context = ctxt, ...} =>
unfold_tac ctxt (def_eqn :: simps))
end
fun define n =
let
val s = "Char_" ^ String_Syntax.hex n;
val b = Binding.name s;
val b_def = Thm.def_binding b;
val b_code = Binding.name (s ^ "_code");
in
Local_Theory.define ((b, Mixfix.NoSyn),
((Binding.empty, []), HOLogic.mk_char n))
#-> (fn (lhs, (_, raw_def_eqn)) =>
Local_Theory.note ((b_def, @{attributes [code_abbrev]}), [HOLogic.mk_obj_eq raw_def_eqn])
#-> (fn (_, [def_eqn]) => `(fn lthy => prove_eqn lthy n lhs def_eqn))
#-> (fn raw_code_eqn => Local_Theory.note ((b_code, []), [raw_code_eqn]))
#-> (fn (_, [code_eqn]) => Code.declare_abstract_eqn code_eqn))
end
in
fold define (0 upto 255)
end
›

code_identifier
code_module Code_Abstract_Char ⇀
(SML) Str and (OCaml) Str and (Haskell) Str and (Scala) Str

end
```