(* Author: Florian Haftmann, TU Muenchen *)

header {* Type of target language numerals *}

theory Code_Numeral

imports Nat_Numeral Nat_Transfer Divides

begin

text {*

  Code numerals are isomorphic to HOL @{typ nat} but

  mapped to target-language builtin numerals.

*}

subsection {* Datatype of target language numerals *}

typedef (open) code_numeral = "UNIV \<Colon> nat set"

  morphisms nat_of of_nat ..

lemma of_nat_nat_of [simp]:

  "of_nat (nat_of k) = k"

  by (rule nat_of_inverse)

lemma nat_of_of_nat [simp]:

  "nat_of (of_nat n) = n"

  by (rule of_nat_inverse) (rule UNIV_I)

lemma [measure_function]:

  "is_measure nat_of" by (rule is_measure_trivial)

lemma code_numeral:

  "(\<And>n\<Colon>code_numeral. PROP P n) \<equiv> (\<And>n\<Colon>nat. PROP P (of_nat n))"

proof

  fix n :: nat

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

  then show "PROP P (of_nat n)" .

next

  fix n :: code_numeral

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

  then have "PROP P (of_nat (nat_of n))" .

  then show "PROP P n" by simp

qed

lemma code_numeral_case:

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

  shows P

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

lemma code_numeral_induct_raw:

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

  shows "P k"

proof -

  from assms have "P (of_nat (nat_of k))" .

  then show ?thesis by simp

qed

lemma nat_of_inject [simp]:

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

  by (rule nat_of_inject)

lemma of_nat_inject [simp]:

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

  by (rule of_nat_inject) (rule UNIV_I)+

instantiation code_numeral :: zero

begin

definition [simp, code del]:

  "0 = of_nat 0"

instance ..

end

definition Suc where [simp]:

  "Suc k = of_nat (Nat.Suc (nat_of k))"

rep_datatype "0 \<Colon> code_numeral" Suc

proof -

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

  fix k :: code_numeral

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

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

    then have "\<And>n. P (of_nat n) \<Longrightarrow> P (Suc (of_nat n))" .

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

  from init step have "P (of_nat (nat_of k))"

    by (induct ("nat_of k")) simp_all

  then show "P k" by simp

qed simp_all

declare code_numeral_case [case_names nat, cases type: code_numeral]

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

lemma code_numeral_decr [termination_simp]:

  "k \<noteq> of_nat 0 \<Longrightarrow> nat_of k - Nat.Suc 0 < nat_of k"

  by (cases k) simp

lemma [simp, code]:

  "code_numeral_size = nat_of"

proof (rule ext)

  fix k

  have "code_numeral_size k = nat_size (nat_of k)"

    by (induct k rule: code_numeral.induct) (simp_all del: zero_code_numeral_def Suc_def, simp_all)

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

  finally show "code_numeral_size k = nat_of k" .

qed

lemma [simp, code]:

  "size = nat_of"

proof (rule ext)

  fix k

  show "size k = nat_of k"

  by (induct k) (simp_all del: zero_code_numeral_def Suc_def, simp_all)

qed

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

lemma [code]:

  "HOL.equal k l \<longleftrightarrow> HOL.equal (nat_of k) (nat_of l)"

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

lemma [code nbe]:

  "HOL.equal (k::code_numeral) k \<longleftrightarrow> True"

  by (rule equal_refl)

subsection {* Code numerals as datatype of ints *}

instantiation code_numeral :: number

begin

definition

  "number_of = of_nat o nat"

instance ..

end

lemma nat_of_number [simp]:

  "nat_of (number_of k) = number_of k"

  by (simp add: number_of_code_numeral_def nat_number_of_def number_of_is_id)

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

subsection {* Basic arithmetic *}

instantiation code_numeral :: "{minus, linordered_semidom, semiring_div, linorder}"

begin

definition [simp, code del]:

  "(1\<Colon>code_numeral) = of_nat 1"

definition [simp, code del]:

  "n + m = of_nat (nat_of n + nat_of m)"

definition [simp, code del]:

  "n - m = of_nat (nat_of n - nat_of m)"

definition [simp, code del]:

  "n * m = of_nat (nat_of n * nat_of m)"

definition [simp, code del]:

  "n div m = of_nat (nat_of n div nat_of m)"

definition [simp, code del]:

  "n mod m = of_nat (nat_of n mod nat_of m)"

definition [simp, code del]:

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

definition [simp, code del]:

  "n < m \<longleftrightarrow> nat_of n < nat_of m"

instance proof

qed (auto simp add: code_numeral left_distrib intro: mult_commute)

end

lemma zero_code_numeral_code [code]:

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

  by (simp add: number_of_code_numeral_def Pls_def)

lemma [code_abbrev]: "Numeral0 = (0\<Colon>code_numeral)"

  using zero_code_numeral_code ..

lemma one_code_numeral_code [code]:

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

  by (simp add: number_of_code_numeral_def Pls_def Bit1_def)

lemma [code_abbrev]: "Numeral1 = (1\<Colon>code_numeral)"

  using one_code_numeral_code ..

lemma plus_code_numeral_code [code nbe]:

  "of_nat n + of_nat m = of_nat (n + m)"

  by simp

definition subtract :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral" where

  [simp]: "subtract = minus"

lemma subtract_code [code nbe]:

  "subtract (of_nat n) (of_nat m) = of_nat (n - m)"

  by simp

lemma minus_code_numeral_code [code]:

  "minus = subtract"

  by simp

lemma times_code_numeral_code [code nbe]:

  "of_nat n * of_nat m = of_nat (n * m)"

  by simp

lemma less_eq_code_numeral_code [code nbe]:

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

  by simp

lemma less_code_numeral_code [code nbe]:

  "of_nat n < of_nat m \<longleftrightarrow> n < m"

  by simp

lemma code_numeral_zero_minus_one:

  "(0::code_numeral) - 1 = 0"

  by simp

lemma Suc_code_numeral_minus_one:

  "Suc n - 1 = n"

  by simp

lemma of_nat_code [code]:

  "of_nat = Nat.of_nat"

proof

  fix n :: nat

  have "Nat.of_nat n = of_nat n"

    by (induct n) simp_all

  then show "of_nat n = Nat.of_nat n"

    by (rule sym)

qed

lemma code_numeral_not_eq_zero: "i \<noteq> of_nat 0 \<longleftrightarrow> i \<ge> 1"

  by (cases i) auto

definition nat_of_aux :: "code_numeral \<Rightarrow> nat \<Rightarrow> nat" where

  "nat_of_aux i n = nat_of i + n"

lemma nat_of_aux_code [code]:

  "nat_of_aux i n = (if i = 0 then n else nat_of_aux (i - 1) (Nat.Suc n))"

  by (auto simp add: nat_of_aux_def code_numeral_not_eq_zero)

lemma nat_of_code [code]:

  "nat_of i = nat_of_aux i 0"

  by (simp add: nat_of_aux_def)

definition div_mod :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral \<times> code_numeral" where

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

lemma [code]:

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

  unfolding div_mod_def by auto

lemma [code]:

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

  unfolding div_mod_def by simp

lemma [code]:

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

  unfolding div_mod_def by simp

definition int_of :: "code_numeral \<Rightarrow> int" where

  "int_of = Nat.of_nat o nat_of"

lemma int_of_code [code]:

  "int_of k = (if k = 0 then 0

    else (if k mod 2 = 0 then 2 * int_of (k div 2) else 2 * int_of (k div 2) + 1))"

proof -

  have "(nat_of k div 2) * 2 + nat_of k mod 2 = nat_of k" 

    by (rule mod_div_equality)

  then have "int ((nat_of k div 2) * 2 + nat_of k mod 2) = int (nat_of k)" 

    by simp

  then have "int (nat_of k) = int (nat_of k div 2) * 2 + int (nat_of k mod 2)" 

    unfolding of_nat_mult of_nat_add by simp

  then show ?thesis by (auto simp add: int_of_def mult_ac)

qed

text {* Lazy Evaluation of an indexed function *}

function iterate_upto :: "(code_numeral \<Rightarrow> 'a) \<Rightarrow> code_numeral \<Rightarrow> code_numeral \<Rightarrow> 'a Predicate.pred"

where

  "iterate_upto f n m =

    Predicate.Seq (%u. if n > m then Predicate.Empty

     else Predicate.Insert (f n) (iterate_upto f (n + 1) m))"

by pat_completeness auto

termination by (relation "measure (%(f, n, m). Code_Numeral.nat_of (m + 1 - n))") auto

hide_const (open) of_nat nat_of Suc  subtract int_of iterate_upto

subsection {* Code generator setup *}

text {* Implementation of code numerals by bounded integers *}

code_type code_numeral

  (SML "int")

  (OCaml "Big'_int.big'_int")

  (Haskell "Integer")

  (Scala "BigInt")

code_instance code_numeral :: equal

  (Haskell -)

setup {*

  Numeral.add_code @{const_name number_code_numeral_inst.number_of_code_numeral}

    false Code_Printer.literal_naive_numeral "SML"

  #> fold (Numeral.add_code @{const_name number_code_numeral_inst.number_of_code_numeral}

    false Code_Printer.literal_numeral) ["OCaml", "Haskell", "Scala"]

*}

code_reserved SML Int int

code_reserved Eval Integer

code_const "plus \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"

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

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

  (Haskell infixl 6 "+")

  (Scala infixl 7 "+")

  (Eval infixl 8 "+")

code_const "Code_Numeral.subtract \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"

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

  (OCaml "Big'_int.max'_big'_int/ (Big'_int.sub'_big'_int/ _/ _)/ Big'_int.zero'_big'_int")

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

  (Scala "!(_/ -/ _).max(0)")

  (Eval "Integer.max/ (_/ -/ _)/ 0")

code_const "times \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"

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

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

  (Haskell infixl 7 "*")

  (Scala infixl 8 "*")

  (Eval infixl 8 "*")

code_const Code_Numeral.div_mod

  (SML "!(fn n => fn m =>/ if m = 0/ then (0, n) else/ (Int.div (n, m), Int.mod (n, m)))")

  (OCaml "Big'_int.quomod'_big'_int/ (Big'_int.abs'_big'_int _)/ (Big'_int.abs'_big'_int _)")

  (Haskell "divMod")

  (Scala "!((k: BigInt) => (l: BigInt) =>/ if (l == 0)/ (BigInt(0), k) else/ (k.abs '/% l.abs))")

  (Eval "!(fn n => fn m =>/ if m = 0/ then (0, n) else/ (Integer.div'_mod n m))")

code_const "HOL.equal \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"

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

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

  (Haskell infix 4 "==")

  (Scala infixl 5 "==")

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

code_const "less_eq \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"

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

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

  (Haskell infix 4 "<=")

  (Scala infixl 4 "<=")

  (Eval infixl 6 "<=")

code_const "less \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"

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

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

  (Haskell infix 4 "<")

  (Scala infixl 4 "<")

  (Eval infixl 6 "<")

code_modulename SML

  Code_Numeral Arith

code_modulename OCaml

  Code_Numeral Arith

code_modulename Haskell

  Code_Numeral Arith

end