(*  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

 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)

 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 Bit_def)

 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 func]:

   "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 nat_of_index_code [code func]:

   "nat_of_index n = (if n = 0 then 0 else Suc (nat_of_index (n - 1)))"

   by (induct n) simp

 subsection {* ML interface *}

 ML {*

 structure Index =

 struct

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

 end;

 *}

 subsection {* Code serialization *}

 text {* Implementation of indices by bounded integers *}

 code_type index

   (SML "int")

   (OCaml "int")

   (Haskell "Integer")

 code_instance index :: eq

   (Haskell -)

 setup {*

   fold (fn target => CodeTarget.add_pretty_numeral target false

     @{const_name number_index_inst.number_of_index}

     @{const_name Int.B0} @{const_name Int.B1}

     @{const_name Int.Pls} @{const_name Int.Min}

     @{const_name Int.Bit}

   ) ["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 "op - \<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 "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_const "op div \<Colon> index \<Rightarrow> index \<Rightarrow> index"

   (SML "IntInf.div ((_), (_))")

   (OCaml "Big'_int.div'_big'_int")

   (Haskell "div")

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

   (SML "IntInf.mod ((_), (_))")

   (OCaml "Big'_int.mod'_big'_int")

   (Haskell "mod")

 end