(* 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 by rule

 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 [simp]:

   "Suc_code_numeral k = of_nat (Suc (nat_of k))"

 rep_datatype "0 \<Colon> code_numeral" Suc_code_numeral

 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_code_numeral k)"

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

     then have step: "\<And>n. P (of_nat n) \<Longrightarrow> P (of_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 - 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_code_numeral_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_code_numeral_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, code_unfold]:

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

   by (simp add: number_of_code_numeral_def Pls_def)

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

   using zero_code_numeral_code ..

 lemma one_code_numeral_code [code, code_unfold]:

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

   by (simp add: number_of_code_numeral_def Pls_def Bit1_def)

 lemma [code_post]: "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 :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral" where

   [simp, code del]: "subtract_code_numeral = op -"

 lemma subtract_code_numeral_code [code nbe]:

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

   by simp

 lemma minus_code_numeral_code [code]:

   "n - m = subtract_code_numeral n m"

   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_code_numeral 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) (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 :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral \<times> code_numeral" where

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

 lemma [code]:

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

   unfolding div_mod_code_numeral_def by auto

 lemma [code]:

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

   unfolding div_mod_code_numeral_def by simp

 lemma [code]:

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

   unfolding div_mod_code_numeral_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 int_mult zadd_int [symmetric] by simp

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

 qed

 hide_const (open) of_nat nat_of int_of

 subsubsection {* Lazy Evaluation of an indexed function *}

 function iterate_upto :: "(code_numeral => 'a) => code_numeral => code_numeral => '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) 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 "op + \<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 "subtract_code_numeral \<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 "op * \<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 div_mod_code_numeral

   (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/ (n div m, n mod m))")

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

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

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

   (Haskell infixl 4 "==")

   (Scala infixl 5 "==")

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

 code_const "op \<le> \<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 "op < \<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 "<")

 end