(*  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)

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 :: 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\<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

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