(*  ID:         $Id$

    Author:     Florian Haftmann, TU Muenchen

*)

header {* Type of indices *}

theory Code_Index

imports Plain "~~/src/HOL/Code_Eval" "~~/src/HOL/Presburger"

begin

text {*

  Indices are isomorphic to HOL @{typ nat} but

  mapped to target-language builtin integers.

*}

subsection {* Datatype of indices *}

typedef index = "UNIV \<Colon> nat set"

  morphisms nat_of_index index_of_nat by rule

lemma index_of_nat_nat_of_index [simp]:

  "index_of_nat (nat_of_index k) = k"

  by (rule nat_of_index_inverse)

lemma nat_of_index_index_of_nat [simp]:

  "nat_of_index (index_of_nat n) = n"

  by (rule index_of_nat_inverse) 

    (unfold index_def, rule UNIV_I)

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 index_case:

  assumes "\<And>n. k = index_of_nat n \<Longrightarrow> P"

  shows P

  by (rule assms [of "nat_of_index k"]) simp

lemma index_induct_raw:

  assumes "\<And>n. P (index_of_nat n)"

  shows "P k"

proof -

  from assms have "P (index_of_nat (nat_of_index k))" .

  then show ?thesis by simp

qed

lemma nat_of_index_inject [simp]:

  "nat_of_index k = nat_of_index l \<longleftrightarrow> k = l"

  by (rule nat_of_index_inject)

lemma index_of_nat_inject [simp]:

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

  by (auto intro!: index_of_nat_inject simp add: index_def)

instantiation index :: zero

begin

definition [simp, code del]:

  "0 = index_of_nat 0"

instance ..

end

definition [simp]:

  "Suc_index k = index_of_nat (Suc (nat_of_index k))"

rep_datatype "0 \<Colon> index" Suc_index

proof -

  fix P :: "index \<Rightarrow> bool"

  fix k :: index

  assume "P 0" then have init: "P (index_of_nat 0)" by simp

  assume "\<And>k. P k \<Longrightarrow> P (Suc_index k)"

    then have "\<And>n. P (index_of_nat n) \<Longrightarrow> P (Suc_index (index_of_nat n))" .

    then have step: "\<And>n. P (index_of_nat n) \<Longrightarrow> P (index_of_nat (Suc n))" by simp

  from init step have "P (index_of_nat (nat_of_index k))"

    by (induct "nat_of_index k") simp_all

  then show "P k" by simp

qed simp_all

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

declare index_case [case_names nat, cases type: index]

declare index.induct [case_names nat, induct type: index]

lemma [code]:

  "index_size = nat_of_index"

proof (rule ext)

  fix k

  have "index_size k = nat_size (nat_of_index k)"

    by (induct k rule: index.induct) (simp_all del: zero_index_def Suc_index_def, simp_all)

  also have "nat_size (nat_of_index k) = nat_of_index k" by (induct "nat_of_index k") simp_all

  finally show "index_size k = nat_of_index k" .

qed

lemma [code]:

  "size = nat_of_index"

proof (rule ext)

  fix k

  show "size k = nat_of_index k"

  by (induct k) (simp_all del: zero_index_def Suc_index_def, simp_all)

qed

lemma [code]:

  "eq_class.eq k l \<longleftrightarrow> eq_class.eq (nat_of_index k) (nat_of_index l)"

  by (cases k, cases l) (simp add: eq)

lemma [code nbe]:

  "eq_class.eq (k::index) k \<longleftrightarrow> True"

  by (rule HOL.eq_refl)

subsection {* Indices as datatype of ints *}

instantiation index :: number

begin

definition

  "number_of = index_of_nat o nat"

instance ..

end

lemma nat_of_index_number [simp]:

  "nat_of_index (number_of k) = number_of k"

  by (simp add: number_of_index_def nat_number_of_def number_of_is_id)

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

subsection {* Basic arithmetic *}

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

begin

lemma zero_index_code [code inline, code]:

  "(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 del]:

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

lemma one_index_code [code inline, code]:

  "(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 del]:

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

lemma plus_index_code [code]:

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

  by simp

definition [simp, code del]:

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

definition [simp, code del]:

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

lemma times_index_code [code]:

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

  by simp

definition [simp, code del]:

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

definition [simp, code del]:

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

lemma div_index_code [code]:

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

  by simp

lemma mod_index_code [code]:

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

  by simp

definition [simp, code del]:

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

definition [simp, code del]:

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

lemma less_eq_index_code [code]:

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

  by simp

lemma less_index_code [code]:

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

  by simp

instance by default (auto simp add: left_distrib index)

end

lemma Suc_index_minus_one: "Suc_index n - 1 = n" by simp

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)

text {* Measure function (for termination proofs) *}

lemma [measure_function]:

  "is_measure nat_of_index" by (rule is_measure_trivial)

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 del]: "minus_index_aux = op -"

lemma [code]: "op - = minus_index_aux"

  using minus_index_aux_def ..

definition

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

where

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

lemma [code]:

  "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]:

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

  unfolding div_mod_index_def by simp

lemma [code]:

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

  unfolding div_mod_index_def by simp

subsection {* Code generator setup *}

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 "eq_class.eq \<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 "<")

text {* Evaluation *}

lemma [code, code del]:

  "(Code_Eval.term_of \<Colon> index \<Rightarrow> term) = Code_Eval.term_of" ..

code_const "Code_Eval.term_of \<Colon> index \<Rightarrow> term"

  (SML "HOLogic.mk'_number/ HOLogic.indexT/ (IntInf.fromInt/ _)")

end