(* ID: $Id$
Author: Florian Haftmann, TU Muenchen
*)
header {* Type of indices *}
theory Code_Index
imports PreList
begin
text {*
Indices are isomorphic to HOL @{typ int} but
mapped to target-language builtin integers
*}
subsection {* Datatype of indices *}
datatype index = index_of_int int
lemmas [code func del] = index.recs index.cases
fun
int_of_index :: "index \<Rightarrow> int"
where
"int_of_index (index_of_int k) = k"
lemmas [code func del] = int_of_index.simps
lemma index_id [simp]:
"index_of_int (int_of_index k) = k"
by (cases k) simp_all
lemma index:
"(\<And>k\<Colon>index. PROP P k) \<equiv> (\<And>k\<Colon>int. PROP P (index_of_int k))"
proof
fix k :: int
assume "\<And>k\<Colon>index. PROP P k"
then show "PROP P (index_of_int k)" .
next
fix k :: index
assume "\<And>k\<Colon>int. PROP P (index_of_int k)"
then have "PROP P (index_of_int (int_of_index k))" .
then show "PROP P k" by simp
qed
lemma [code func]: "size (k\<Colon>index) = 0"
by (cases k) simp_all
subsection {* Built-in integers as datatype on numerals *}
instance index :: number
"number_of \<equiv> index_of_int" ..
code_datatype "number_of \<Colon> int \<Rightarrow> index"
lemma number_of_index_id [simp]:
"number_of (int_of_index k) = k"
unfolding number_of_index_def by simp
lemma number_of_index_shift:
"number_of k = index_of_int (number_of k)"
by (simp add: number_of_is_id number_of_index_def)
lemma int_of_index_number_of [simp]:
"int_of_index (number_of k) = number_of k"
unfolding number_of_index_def number_of_is_id by simp
subsection {* Basic arithmetic *}
instance index :: zero
[simp]: "0 \<equiv> index_of_int 0" ..
lemmas [code func del] = zero_index_def
instance index :: one
[simp]: "1 \<equiv> index_of_int 1" ..
lemmas [code func del] = one_index_def
instance index :: plus
[simp]: "k + l \<equiv> index_of_int (int_of_index k + int_of_index l)" ..
lemmas [code func del] = plus_index_def
lemma plus_index_code [code func]:
"index_of_int k + index_of_int l = index_of_int (k + l)"
unfolding plus_index_def by simp
instance index :: minus
[simp]: "- k \<equiv> index_of_int (- int_of_index k)"
[simp]: "k - l \<equiv> index_of_int (int_of_index k - int_of_index l)" ..
lemmas [code func del] = uminus_index_def minus_index_def
lemma uminus_index_code [code func]:
"- index_of_int k \<equiv> index_of_int (- k)"
unfolding uminus_index_def by simp
lemma minus_index_code [code func]:
"index_of_int k - index_of_int l = index_of_int (k - l)"
unfolding minus_index_def by simp
instance index :: times
[simp]: "k * l \<equiv> index_of_int (int_of_index k * int_of_index l)" ..
lemmas [code func del] = times_index_def
lemma times_index_code [code func]:
"index_of_int k * index_of_int l = index_of_int (k * l)"
unfolding times_index_def by simp
instance index :: ord
[simp]: "k \<le> l \<equiv> int_of_index k \<le> int_of_index l"
[simp]: "k < l \<equiv> int_of_index k < int_of_index l" ..
lemmas [code func del] = less_eq_index_def less_index_def
lemma less_eq_index_code [code func]:
"index_of_int k \<le> index_of_int l \<longleftrightarrow> k \<le> l"
unfolding less_eq_index_def by simp
lemma less_index_code [code func]:
"index_of_int k < index_of_int l \<longleftrightarrow> k < l"
unfolding less_index_def by simp
instance index :: "Divides.div"
[simp]: "k div l \<equiv> index_of_int (int_of_index k div int_of_index l)"
[simp]: "k mod l \<equiv> index_of_int (int_of_index k mod int_of_index l)" ..
instance index :: ring_1
by default (auto simp add: left_distrib right_distrib)
lemma of_nat_index: "of_nat n = index_of_int (of_nat n)"
proof (induct n)
case 0 show ?case by simp
next
case (Suc n)
then have "int_of_index (index_of_int (int n))
= int_of_index (of_nat n)" by simp
then have "int n = int_of_index (of_nat n)" by simp
then show ?case by simp
qed
instance index :: number_ring
by default
(simp_all add: left_distrib number_of_index_def of_int_of_nat of_nat_index)
lemma zero_index_code [code inline, code func]:
"(0\<Colon>index) = Numeral0"
by simp
lemma one_index_code [code inline, code func]:
"(1\<Colon>index) = Numeral1"
by simp
instance index :: abs
"\<bar>k\<Colon>index\<bar> \<equiv> if k < 0 then -k else k" ..
lemma index_of_int [code func]:
"index_of_int k = (if k = 0 then 0
else if k = -1 then -1
else let (l, m) = divAlg (k, 2) in 2 * index_of_int l +
(if m = 0 then 0 else 1))"
by (simp add: number_of_index_shift Let_def split_def divAlg_mod_div) arith
lemma int_of_index [code func]:
"int_of_index k = (if k = 0 then 0
else if k = -1 then -1
else let l = k div 2; m = k mod 2 in 2 * int_of_index l +
(if m = 0 then 0 else 1))"
by (auto simp add: number_of_index_shift Let_def split_def) arith
subsection {* Conversion to and from @{typ nat} *}
definition
nat_of_index :: "index \<Rightarrow> nat"
where
[code func del]: "nat_of_index = nat o int_of_index"
definition
nat_of_index_aux :: "index \<Rightarrow> nat \<Rightarrow> nat" where
[code func del]: "nat_of_index_aux i n = nat_of_index i + n"
lemma nat_of_index_aux_code [code]:
"nat_of_index_aux i n = (if i \<le> 0 then n else nat_of_index_aux (i - 1) (Suc n))"
by (auto simp add: nat_of_index_aux_def nat_of_index_def)
lemma nat_of_index_code [code]:
"nat_of_index i = nat_of_index_aux i 0"
by (simp add: nat_of_index_aux_def)
definition
index_of_nat :: "nat \<Rightarrow> index"
where
[code func del]: "index_of_nat = index_of_int o of_nat"
lemma index_of_nat [code func]:
"index_of_nat 0 = 0"
"index_of_nat (Suc n) = index_of_nat n + 1"
unfolding index_of_nat_def by simp_all
lemma index_nat_id [simp]:
"nat_of_index (index_of_nat n) = n"
"index_of_nat (nat_of_index i) = (if i \<le> 0 then 0 else i)"
unfolding index_of_nat_def nat_of_index_def by simp_all
subsection {* ML interface *}
ML {*
structure Index =
struct
fun mk k = @{term index_of_int} $ HOLogic.mk_number @{typ index} k;
end;
*}
subsection {* Code serialization *}
code_type index
(SML "int")
(OCaml "int")
(Haskell "Integer")
code_instance index :: eq
(Haskell -)
setup {*
fold (fn target => CodeTarget.add_pretty_numeral target true
@{const_name number_index_inst.number_of_index}
@{const_name Numeral.B0} @{const_name Numeral.B1}
@{const_name Numeral.Pls} @{const_name Numeral.Min}
@{const_name Numeral.Bit}
) ["SML", "OCaml", "Haskell"]
*}
code_reserved SML int
code_reserved OCaml int
code_const "op + \<Colon> index \<Rightarrow> index \<Rightarrow> index"
(SML "Int.+ ((_), (_))")
(OCaml "Pervasives.+")
(Haskell infixl 6 "+")
code_const "uminus \<Colon> index \<Rightarrow> index"
(SML "Int.~")
(OCaml "Pervasives.~-")
(Haskell "negate")
code_const "op - \<Colon> index \<Rightarrow> index \<Rightarrow> index"
(SML "Int.- ((_), (_))")
(OCaml "Pervasives.-")
(Haskell infixl 6 "-")
code_const "op * \<Colon> index \<Rightarrow> index \<Rightarrow> index"
(SML "Int.* ((_), (_))")
(OCaml "Pervasives.*")
(Haskell infixl 7 "*")
code_const "op = \<Colon> index \<Rightarrow> index \<Rightarrow> bool"
(SML "!((_ : Int.int) = _)")
(OCaml "!((_ : Pervasives.int) = _)")
(Haskell infixl 4 "==")
code_const "op \<le> \<Colon> index \<Rightarrow> index \<Rightarrow> bool"
(SML "Int.<= ((_), (_))")
(OCaml "!((_ : Pervasives.int) <= _)")
(Haskell infix 4 "<=")
code_const "op < \<Colon> index \<Rightarrow> index \<Rightarrow> bool"
(SML "Int.< ((_), (_))")
(OCaml "!((_ : Pervasives.int) < _)")
(Haskell infix 4 "<")
code_reserved SML Int
code_reserved OCaml Pervasives
end