(* Author: Florian Haftmann, TU Muenchen *)
header {* Type of target language numerals *}
theory Code_Numeral
imports 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 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 {* 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 distrib_right intro: mult_commute)
end
lemma nat_of_numeral [simp]: "nat_of (numeral k) = numeral k"
by (induct k rule: num_induct) (simp_all add: numeral_inc)
definition Num :: "num \<Rightarrow> code_numeral"
where [simp, code_abbrev]: "Num = numeral"
code_datatype "0::code_numeral" Num
lemma one_code_numeral_code [code]:
"(1\<Colon>code_numeral) = Numeral1"
by simp
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
lemma minus_code_numeral_code [code nbe]:
"of_nat n - of_nat m = of_nat (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 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
hide_const (open) of_nat nat_of Suc int_of
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 Num}
false Code_Printer.literal_naive_numeral "SML"
#> fold (Numeral.add_code @{const_name Num}
false Code_Printer.literal_numeral) ["OCaml", "Haskell", "Scala"]
*}
code_reserved SML Int int
code_reserved Eval Integer
code_const "0::code_numeral"
(SML "0")
(OCaml "Big'_int.zero'_big'_int")
(Haskell "0")
(Scala "BigInt(0)")
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 "minus \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
(SML "Int.max/ (0 : int,/ Int.-/ ((_),/ (_)))")
(OCaml "Big'_int.max'_big'_int/ Big'_int.zero'_big'_int/ (Big'_int.sub'_big'_int/ _/ _)")
(Haskell "Prelude.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