(*  ID:         $Id$

    Author:     Florian Haftmann, TU Muenchen

*)

header {* Type of indices *}

theory Code_Index

imports ATP_Linkup

begin

text {*

  Indices are isomorphic to HOL @{typ nat} but

  mapped to target-language builtin integers

*}

subsection {* Datatype of indices *}

datatype index = index_of_nat nat

lemma [code func]:

  "index_size k = 0"

  by (cases k) simp

lemmas [code func del] = index.recs index.cases

primrec

  nat_of_index :: "index \<Rightarrow> nat"

where

  "nat_of_index (index_of_nat k) = k"

lemmas [code func del] = nat_of_index.simps

lemma index_id [simp]:

  "index_of_nat (nat_of_index n) = n"

  by (cases n) simp_all

lemma nat_of_index_inject [simp]:

  "nat_of_index n = nat_of_index m \<longleftrightarrow> n = m"

  by (cases n) auto

lemma index:

  "(\<And>n\<Colon>index. PROP P n) \<equiv> (\<And>n\<Colon>nat. PROP P (index_of_nat n))"

proof

  fix n :: nat

  assume "\<And>n\<Colon>index. PROP P n"

  then show "PROP P (index_of_nat n)" .

next

  fix n :: index

  assume "\<And>n\<Colon>nat. PROP P (index_of_nat n)"

  then have "PROP P (index_of_nat (nat_of_index n))" .

  then show "PROP P n" by simp

qed

lemma [code func]: "size (n\<Colon>index) = 0"

  by (cases n) simp_all

subsection {* Indices as datatype of ints *}

instantiation index :: number

begin

definition

  "number_of = index_of_nat o nat"

instance ..

end

code_datatype "number_of \<Colon> int \<Rightarrow> index"

subsection {* Basic arithmetic *}

instantiation index :: "{minus, ordered_semidom, Divides.div, linorder}"

begin

definition [simp, code func del]:

  "(0\<Colon>index) = index_of_nat 0"

lemma zero_index_code [code inline, code func]:

  "(0\<Colon>index) = Numeral0"

  by (simp add: number_of_index_def Pls_def)

lemma [code post]: "Numeral0 = (0\<Colon>index)"

  using zero_index_code ..

definition [simp, code func del]:

  "(1\<Colon>index) = index_of_nat 1"

lemma one_index_code [code inline, code func]:

  "(1\<Colon>index) = Numeral1"

  by (simp add: number_of_index_def Pls_def Bit1_def)

lemma [code post]: "Numeral1 = (1\<Colon>index)"

  using one_index_code ..

definition [simp, code func del]:

  "n + m = index_of_nat (nat_of_index n + nat_of_index m)"

lemma plus_index_code [code func]:

  "index_of_nat n + index_of_nat m = index_of_nat (n + m)"

  by simp

definition [simp, code func del]:

  "n - m = index_of_nat (nat_of_index n - nat_of_index m)"

definition [simp, code func del]:

  "n * m = index_of_nat (nat_of_index n * nat_of_index m)"

lemma times_index_code [code func]:

  "index_of_nat n * index_of_nat m = index_of_nat (n * m)"

  by simp

definition [simp, code func del]:

  "n div m = index_of_nat (nat_of_index n div nat_of_index m)"

definition [simp, code func del]:

  "n mod m = index_of_nat (nat_of_index n mod nat_of_index m)"

lemma div_index_code [code func]:

  "index_of_nat n div index_of_nat m = index_of_nat (n div m)"

  by simp

lemma mod_index_code [code func]:

  "index_of_nat n mod index_of_nat m = index_of_nat (n mod m)"

  by simp

definition [simp, code func del]:

  "n \<le> m \<longleftrightarrow> nat_of_index n \<le> nat_of_index m"

definition [simp, code func del]:

  "n < m \<longleftrightarrow> nat_of_index n < nat_of_index m"

lemma less_eq_index_code [code func]:

  "index_of_nat n \<le> index_of_nat m \<longleftrightarrow> n \<le> m"

  by simp

lemma less_index_code [code func]:

  "index_of_nat n < index_of_nat m \<longleftrightarrow> n < m"

  by simp

instance by default (auto simp add: left_distrib index)

end

lemma index_of_nat_code [code]:

  "index_of_nat = of_nat"

proof

  fix n :: nat

  have "of_nat n = index_of_nat n"

    by (induct n) simp_all

  then show "index_of_nat n = of_nat n"

    by (rule sym)

qed

lemma index_not_eq_zero: "i \<noteq> index_of_nat 0 \<longleftrightarrow> i \<ge> 1"

  by (cases i) auto

definition

  nat_of_index_aux :: "index \<Rightarrow> nat \<Rightarrow> nat"

where

  "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 = 0 then n else nat_of_index_aux (i - 1) (Suc n))"

  by (auto simp add: nat_of_index_aux_def index_not_eq_zero)

lemma nat_of_index_code [code]:

  "nat_of_index i = nat_of_index_aux i 0"

  by (simp add: nat_of_index_aux_def)

subsection {* ML interface *}

ML {*

structure Index =

struct

fun mk k = HOLogic.mk_number @{typ index} k;

end;

*}

subsection {* Specialized @{term "op - \<Colon> index \<Rightarrow> index \<Rightarrow> index"},

  @{term "op div \<Colon> index \<Rightarrow> index \<Rightarrow> index"} and @{term "op mod \<Colon> index \<Rightarrow> index \<Rightarrow> index"}

  operations *}

definition

  minus_index_aux :: "index \<Rightarrow> index \<Rightarrow> index"

where

  [code func del]: "minus_index_aux = op -"

lemma [code func]: "op - = minus_index_aux"

  using minus_index_aux_def ..

definition

  div_mod_index ::  "index \<Rightarrow> index \<Rightarrow> index \<times> index"

where

  [code func del]: "div_mod_index n m = (n div m, n mod m)"

lemma [code func]:

  "div_mod_index n m = (if m = 0 then (0, n) else (n div m, n mod m))"

  unfolding div_mod_index_def by auto

lemma [code func]:

  "n div m = fst (div_mod_index n m)"

  unfolding div_mod_index_def by simp

lemma [code func]:

  "n mod m = snd (div_mod_index n m)"

  unfolding div_mod_index_def by simp

subsection {* Code serialization *}

text {* Implementation of indices by bounded integers *}

code_type index

  (SML "int")

  (OCaml "int")

  (Haskell "Int")

code_instance index :: eq

  (Haskell -)

setup {*

  fold (Numeral.add_code @{const_name number_index_inst.number_of_index}

    false false) ["SML", "OCaml", "Haskell"]

*}

code_reserved SML Int int

code_reserved OCaml Pervasives int

code_const "op + \<Colon> index \<Rightarrow> index \<Rightarrow> index"

  (SML "Int.+/ ((_),/ (_))")

  (OCaml "Pervasives.( + )")

  (Haskell infixl 6 "+")

code_const "minus_index_aux \<Colon> index \<Rightarrow> index \<Rightarrow> index"

  (SML "Int.max/ (_/ -/ _,/ 0 : int)")

  (OCaml "Pervasives.max/ (_/ -/ _)/ (0 : int) ")

  (Haskell "max/ (_/ -/ _)/ (0 :: Int)")

code_const "op * \<Colon> index \<Rightarrow> index \<Rightarrow> index"

  (SML "Int.*/ ((_),/ (_))")

  (OCaml "Pervasives.( * )")

  (Haskell infixl 7 "*")

code_const div_mod_index

  (SML "(fn n => fn m =>/ (n div m, n mod m))")

  (OCaml "(fun n -> fun m ->/ (n '/ m, n mod m))")

  (Haskell "divMod")

code_const "op = \<Colon> index \<Rightarrow> index \<Rightarrow> bool"

  (SML "!((_ : Int.int) = _)")

  (OCaml "!((_ : int) = _)")

  (Haskell infixl 4 "==")

code_const "op \<le> \<Colon> index \<Rightarrow> index \<Rightarrow> bool"

  (SML "Int.<=/ ((_),/ (_))")

  (OCaml "!((_ : int) <= _)")

  (Haskell infix 4 "<=")

code_const "op < \<Colon> index \<Rightarrow> index \<Rightarrow> bool"

  (SML "Int.</ ((_),/ (_))")

  (OCaml "!((_ : int) < _)")

  (Haskell infix 4 "<")

end