(* ID: $Id$
Author: Florian Haftmann, TU Muenchen
*)
header {* Built-in integers for ML *}
theory ML_Int
imports List
begin
subsection {* Datatype of built-in integers *}
datatype ml_int = ml_int_of_int int
lemmas [code func del] = ml_int.recs ml_int.cases
fun
int_of_ml_int :: "ml_int \<Rightarrow> int"
where
"int_of_ml_int (ml_int_of_int k) = k"
lemmas [code func del] = int_of_ml_int.simps
lemma ml_int_id [simp]:
"ml_int_of_int (int_of_ml_int k) = k"
by (cases k) simp_all
lemma ml_int:
"(\<And>k\<Colon>ml_int. PROP P k) \<equiv> (\<And>k\<Colon>int. PROP P (ml_int_of_int k))"
proof
fix k :: int
assume "\<And>k\<Colon>ml_int. PROP P k"
then show "PROP P (ml_int_of_int k)" .
next
fix k :: ml_int
assume "\<And>k\<Colon>int. PROP P (ml_int_of_int k)"
then have "PROP P (ml_int_of_int (int_of_ml_int k))" .
then show "PROP P k" by simp
qed
lemma [code func]: "size (k\<Colon>ml_int) = 0"
by (cases k) simp_all
subsection {* Built-in integers as datatype on numerals *}
instance ml_int :: number
"number_of \<equiv> ml_int_of_int" ..
lemmas [code inline] = number_of_ml_int_def [symmetric]
code_datatype "number_of \<Colon> int \<Rightarrow> ml_int"
lemma number_of_ml_int_id [simp]:
"number_of (int_of_ml_int k) = k"
unfolding number_of_ml_int_def by simp
subsection {* Basic arithmetic *}
instance ml_int :: zero
[simp]: "0 \<equiv> ml_int_of_int 0" ..
lemmas [code func del] = zero_ml_int_def
instance ml_int :: one
[simp]: "1 \<equiv> ml_int_of_int 1" ..
lemmas [code func del] = one_ml_int_def
instance ml_int :: plus
[simp]: "k + l \<equiv> ml_int_of_int (int_of_ml_int k + int_of_ml_int l)" ..
lemmas [code func del] = plus_ml_int_def
lemma plus_ml_int_code [code func]:
"ml_int_of_int k + ml_int_of_int l = ml_int_of_int (k + l)"
unfolding plus_ml_int_def by simp
instance ml_int :: minus
[simp]: "- k \<equiv> ml_int_of_int (- int_of_ml_int k)"
[simp]: "k - l \<equiv> ml_int_of_int (int_of_ml_int k - int_of_ml_int l)" ..
lemmas [code func del] = uminus_ml_int_def minus_ml_int_def
lemma uminus_ml_int_code [code func]:
"- ml_int_of_int k \<equiv> ml_int_of_int (- k)"
unfolding uminus_ml_int_def by simp
lemma minus_ml_int_code [code func]:
"ml_int_of_int k - ml_int_of_int l = ml_int_of_int (k - l)"
unfolding minus_ml_int_def by simp
instance ml_int :: times
[simp]: "k * l \<equiv> ml_int_of_int (int_of_ml_int k * int_of_ml_int l)" ..
lemmas [code func del] = times_ml_int_def
lemma times_ml_int_code [code func]:
"ml_int_of_int k * ml_int_of_int l = ml_int_of_int (k * l)"
unfolding times_ml_int_def by simp
instance ml_int :: ord
[simp]: "k \<le> l \<equiv> int_of_ml_int k \<le> int_of_ml_int l"
[simp]: "k < l \<equiv> int_of_ml_int k < int_of_ml_int l" ..
lemmas [code func del] = less_eq_ml_int_def less_ml_int_def
lemma less_eq_ml_int_code [code func]:
"ml_int_of_int k \<le> ml_int_of_int l \<longleftrightarrow> k \<le> l"
unfolding less_eq_ml_int_def by simp
lemma less_ml_int_code [code func]:
"ml_int_of_int k < ml_int_of_int l \<longleftrightarrow> k < l"
unfolding less_ml_int_def by simp
instance ml_int :: ring_1
by default (auto simp add: left_distrib right_distrib)
lemma of_nat_ml_int: "of_nat n = ml_int_of_int (of_nat n)"
proof (induct n)
case 0 show ?case by simp
next
case (Suc n)
then have "int_of_ml_int (ml_int_of_int (int n))
= int_of_ml_int (of_nat n)" by simp
then have "int n = int_of_ml_int (of_nat n)" by simp
then show ?case by simp
qed
instance ml_int :: number_ring
by default
(simp_all add: left_distrib number_of_ml_int_def of_int_of_nat of_nat_ml_int)
lemma zero_ml_int_code [code inline, code func]:
"(0\<Colon>ml_int) = Numeral0"
by simp
lemma one_ml_int_code [code inline, code func]:
"(1\<Colon>ml_int) = Numeral1"
by simp
instance ml_int :: abs
"\<bar>k\<bar> \<equiv> if k < 0 then -k else k" ..
subsection {* Conversion to @{typ nat} *}
definition
nat_of_ml_int :: "ml_int \<Rightarrow> nat"
where
"nat_of_ml_int = nat o int_of_ml_int"
definition
nat_of_ml_int_aux :: "ml_int \<Rightarrow> nat \<Rightarrow> nat" where
"nat_of_ml_int_aux i n = nat_of_ml_int i + n"
lemma nat_of_ml_int_aux_code [code]:
"nat_of_ml_int_aux i n = (if i \<le> 0 then n else nat_of_ml_int_aux (i - 1) (Suc n))"
by (auto simp add: nat_of_ml_int_aux_def nat_of_ml_int_def)
lemma nat_of_ml_int_code [code]:
"nat_of_ml_int i = nat_of_ml_int_aux i 0"
by (simp add: nat_of_ml_int_aux_def)
subsection {* ML interface *}
ML {*
structure ML_Int =
struct
fun mk k = @{term ml_int_of_int} $ HOLogic.mk_number @{typ ml_int} k;
end;
*}
subsection {* Code serialization *}
code_type ml_int
(SML "int")
setup {*
CodeTarget.add_pretty_numeral "SML" false
@{const_name ml_int_of_int}
@{const_name Numeral.B0} @{const_name Numeral.B1}
@{const_name Numeral.Pls} @{const_name Numeral.Min}
@{const_name Numeral.Bit}
*}
code_reserved SML int
code_const "op + \<Colon> ml_int \<Rightarrow> ml_int \<Rightarrow> ml_int"
(SML "Int.+ ((_), (_))")
code_const "uminus \<Colon> ml_int \<Rightarrow> ml_int"
(SML "Int.~")
code_const "op - \<Colon> ml_int \<Rightarrow> ml_int \<Rightarrow> ml_int"
(SML "Int.- ((_), (_))")
code_const "op * \<Colon> ml_int \<Rightarrow> ml_int \<Rightarrow> ml_int"
(SML "Int.* ((_), (_))")
code_const "op = \<Colon> ml_int \<Rightarrow> ml_int \<Rightarrow> bool"
(SML "!((_ : Int.int) = _)")
code_const "op \<le> \<Colon> ml_int \<Rightarrow> ml_int \<Rightarrow> bool"
(SML "Int.<= ((_), (_))")
code_const "op < \<Colon> ml_int \<Rightarrow> ml_int \<Rightarrow> bool"
(SML "Int.< ((_), (_))")
end