(* Author: Florian Haftmann, TU Muenchen *)
header {* Type of target language numerals *}
theory Code_Numeral
imports Nat_Numeral
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]:
"eq_class.eq k l \<longleftrightarrow> eq_class.eq (nat_of k) (nat_of l)"
by (cases k, cases l) (simp add: eq)
lemma [code nbe]:
"eq_class.eq (k::code_numeral) k \<longleftrightarrow> True"
by (rule HOL.eq_refl)
subsection {* Indices 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, ordered_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 div_mult_self1)
end
lemma zero_code_numeral_code [code inline, code]:
"(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 inline, code]:
"(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))"
by (auto simp add: int_of_def mod_div_equality')
hide (open) const of_nat nat_of int_of
subsection {* Code generator setup *}
text {* Implementation of indices by bounded integers *}
code_type code_numeral
(SML "int")
(OCaml "int")
(Haskell "Int")
code_instance code_numeral :: eq
(Haskell -)
setup {*
fold (Numeral.add_code @{const_name number_code_numeral_inst.number_of_code_numeral}
false false) ["SML", "OCaml", "Haskell"]
*}
code_reserved SML Int int
code_reserved OCaml Pervasives int
code_const "op + \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
(SML "Int.+/ ((_),/ (_))")
(OCaml "Pervasives.( + )")
(Haskell infixl 6 "+")
code_const "subtract_code_numeral \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
(SML "Int.max/ (_/ -/ _,/ 0 : int)")
(OCaml "Pervasives.max/ (_/ -/ _)/ (0 : int) ")
(Haskell "max/ (_/ -/ _)/ (0 :: Int)")
code_const "op * \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
(SML "Int.*/ ((_),/ (_))")
(OCaml "Pervasives.( * )")
(Haskell infixl 7 "*")
code_const div_mod_code_numeral
(SML "(fn n => fn m =>/ if m = 0/ then (0, n) else/ (n div m, n mod m))")
(OCaml "(fun n -> fun m ->/ if m = 0/ then (0, n) else/ (n '/ m, n mod m))")
(Haskell "divMod")
code_const "eq_class.eq \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
(SML "!((_ : Int.int) = _)")
(OCaml "!((_ : int) = _)")
(Haskell infixl 4 "==")
code_const "op \<le> \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
(SML "Int.<=/ ((_),/ (_))")
(OCaml "!((_ : int) <= _)")
(Haskell infix 4 "<=")
code_const "op < \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
(SML "Int.</ ((_),/ (_))")
(OCaml "!((_ : int) < _)")
(Haskell infix 4 "<")
end