src/HOL/Code_Numeral.thy
author haftmann
Thu, 29 Dec 2011 10:47:55 +0100
changeset 46028 9f113cdf3d66
parent 45694 4a8743618257
child 46547 d1dcb91a512e
permissions -rw-r--r--
attribute code_abbrev superseedes code_unfold_post
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(* Author: Florian Haftmann, TU Muenchen *)
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header {* Type of target language numerals *}
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theory Code_Numeral
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imports Nat_Numeral Nat_Transfer Divides
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begin
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text {*
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  Code numerals are isomorphic to HOL @{typ nat} but
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  mapped to target-language builtin numerals.
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*}
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subsection {* Datatype of target language numerals *}
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typedef (open) code_numeral = "UNIV \<Colon> nat set"
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  morphisms nat_of of_nat ..
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lemma of_nat_nat_of [simp]:
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  "of_nat (nat_of k) = k"
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  by (rule nat_of_inverse)
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lemma nat_of_of_nat [simp]:
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  "nat_of (of_nat n) = n"
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  by (rule of_nat_inverse) (rule UNIV_I)
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lemma [measure_function]:
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  "is_measure nat_of" by (rule is_measure_trivial)
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lemma code_numeral:
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  "(\<And>n\<Colon>code_numeral. PROP P n) \<equiv> (\<And>n\<Colon>nat. PROP P (of_nat n))"
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proof
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  fix n :: nat
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  assume "\<And>n\<Colon>code_numeral. PROP P n"
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  then show "PROP P (of_nat n)" .
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next
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  fix n :: code_numeral
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  assume "\<And>n\<Colon>nat. PROP P (of_nat n)"
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  then have "PROP P (of_nat (nat_of n))" .
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  then show "PROP P n" by simp
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qed
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lemma code_numeral_case:
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  assumes "\<And>n. k = of_nat n \<Longrightarrow> P"
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  shows P
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  by (rule assms [of "nat_of k"]) simp
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lemma code_numeral_induct_raw:
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  assumes "\<And>n. P (of_nat n)"
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  shows "P k"
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proof -
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  from assms have "P (of_nat (nat_of k))" .
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  then show ?thesis by simp
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qed
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lemma nat_of_inject [simp]:
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  "nat_of k = nat_of l \<longleftrightarrow> k = l"
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  by (rule nat_of_inject)
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lemma of_nat_inject [simp]:
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  "of_nat n = of_nat m \<longleftrightarrow> n = m"
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  by (rule of_nat_inject) (rule UNIV_I)+
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instantiation code_numeral :: zero
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begin
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definition [simp, code del]:
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  "0 = of_nat 0"
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instance ..
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end
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definition [simp]:
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  "Suc_code_numeral k = of_nat (Suc (nat_of k))"
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rep_datatype "0 \<Colon> code_numeral" Suc_code_numeral
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proof -
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  fix P :: "code_numeral \<Rightarrow> bool"
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  fix k :: code_numeral
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  assume "P 0" then have init: "P (of_nat 0)" by simp
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  assume "\<And>k. P k \<Longrightarrow> P (Suc_code_numeral k)"
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    then have "\<And>n. P (of_nat n) \<Longrightarrow> P (Suc_code_numeral (of_nat n))" .
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    then have step: "\<And>n. P (of_nat n) \<Longrightarrow> P (of_nat (Suc n))" by simp
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  from init step have "P (of_nat (nat_of k))"
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    by (induct ("nat_of k")) simp_all
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  then show "P k" by simp
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qed simp_all
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declare code_numeral_case [case_names nat, cases type: code_numeral]
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declare code_numeral.induct [case_names nat, induct type: code_numeral]
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lemma code_numeral_decr [termination_simp]:
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  "k \<noteq> of_nat 0 \<Longrightarrow> nat_of k - Suc 0 < nat_of k"
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  by (cases k) simp
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lemma [simp, code]:
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  "code_numeral_size = nat_of"
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proof (rule ext)
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  fix k
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  have "code_numeral_size k = nat_size (nat_of k)"
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    by (induct k rule: code_numeral.induct) (simp_all del: zero_code_numeral_def Suc_code_numeral_def, simp_all)
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  also have "nat_size (nat_of k) = nat_of k" by (induct ("nat_of k")) simp_all
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  finally show "code_numeral_size k = nat_of k" .
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qed
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lemma [simp, code]:
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  "size = nat_of"
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proof (rule ext)
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  fix k
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  show "size k = nat_of k"
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  by (induct k) (simp_all del: zero_code_numeral_def Suc_code_numeral_def, simp_all)
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qed
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lemmas [code del] = code_numeral.recs code_numeral.cases
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lemma [code]:
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  "HOL.equal k l \<longleftrightarrow> HOL.equal (nat_of k) (nat_of l)"
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  by (cases k, cases l) (simp add: equal)
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lemma [code nbe]:
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  "HOL.equal (k::code_numeral) k \<longleftrightarrow> True"
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  by (rule equal_refl)
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subsection {* Code numerals as datatype of ints *}
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instantiation code_numeral :: number
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begin
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definition
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  "number_of = of_nat o nat"
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instance ..
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end
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lemma nat_of_number [simp]:
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  "nat_of (number_of k) = number_of k"
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  by (simp add: number_of_code_numeral_def nat_number_of_def number_of_is_id)
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code_datatype "number_of \<Colon> int \<Rightarrow> code_numeral"
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subsection {* Basic arithmetic *}
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instantiation code_numeral :: "{minus, linordered_semidom, semiring_div, linorder}"
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begin
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definition [simp, code del]:
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  "(1\<Colon>code_numeral) = of_nat 1"
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definition [simp, code del]:
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  "n + m = of_nat (nat_of n + nat_of m)"
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definition [simp, code del]:
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  "n - m = of_nat (nat_of n - nat_of m)"
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definition [simp, code del]:
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  "n * m = of_nat (nat_of n * nat_of m)"
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definition [simp, code del]:
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  "n div m = of_nat (nat_of n div nat_of m)"
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definition [simp, code del]:
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  "n mod m = of_nat (nat_of n mod nat_of m)"
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definition [simp, code del]:
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  "n \<le> m \<longleftrightarrow> nat_of n \<le> nat_of m"
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definition [simp, code del]:
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  "n < m \<longleftrightarrow> nat_of n < nat_of m"
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instance proof
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qed (auto simp add: code_numeral left_distrib intro: mult_commute)
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end
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lemma zero_code_numeral_code [code]:
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  "(0\<Colon>code_numeral) = Numeral0"
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  by (simp add: number_of_code_numeral_def Pls_def)
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lemma [code_abbrev]: "Numeral0 = (0\<Colon>code_numeral)"
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  using zero_code_numeral_code ..
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lemma one_code_numeral_code [code]:
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  "(1\<Colon>code_numeral) = Numeral1"
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  by (simp add: number_of_code_numeral_def Pls_def Bit1_def)
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lemma [code_abbrev]: "Numeral1 = (1\<Colon>code_numeral)"
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  using one_code_numeral_code ..
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lemma plus_code_numeral_code [code nbe]:
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  "of_nat n + of_nat m = of_nat (n + m)"
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  by simp
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definition subtract_code_numeral :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral" where
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  [simp, code del]: "subtract_code_numeral = op -"
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lemma subtract_code_numeral_code [code nbe]:
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  "subtract_code_numeral (of_nat n) (of_nat m) = of_nat (n - m)"
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  by simp
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lemma minus_code_numeral_code [code]:
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  "n - m = subtract_code_numeral n m"
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  by simp
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lemma times_code_numeral_code [code nbe]:
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  "of_nat n * of_nat m = of_nat (n * m)"
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  by simp
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lemma less_eq_code_numeral_code [code nbe]:
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  "of_nat n \<le> of_nat m \<longleftrightarrow> n \<le> m"
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  by simp
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lemma less_code_numeral_code [code nbe]:
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  "of_nat n < of_nat m \<longleftrightarrow> n < m"
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  by simp
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lemma code_numeral_zero_minus_one:
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  "(0::code_numeral) - 1 = 0"
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  by simp
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lemma Suc_code_numeral_minus_one:
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  "Suc_code_numeral n - 1 = n"
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  by simp
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lemma of_nat_code [code]:
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  "of_nat = Nat.of_nat"
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proof
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  fix n :: nat
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  have "Nat.of_nat n = of_nat n"
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    by (induct n) simp_all
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  then show "of_nat n = Nat.of_nat n"
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    by (rule sym)
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qed
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lemma code_numeral_not_eq_zero: "i \<noteq> of_nat 0 \<longleftrightarrow> i \<ge> 1"
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  by (cases i) auto
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definition nat_of_aux :: "code_numeral \<Rightarrow> nat \<Rightarrow> nat" where
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  "nat_of_aux i n = nat_of i + n"
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lemma nat_of_aux_code [code]:
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  "nat_of_aux i n = (if i = 0 then n else nat_of_aux (i - 1) (Suc n))"
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  by (auto simp add: nat_of_aux_def code_numeral_not_eq_zero)
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lemma nat_of_code [code]:
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  "nat_of i = nat_of_aux i 0"
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  by (simp add: nat_of_aux_def)
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definition div_mod_code_numeral :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral \<times> code_numeral" where
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  [code del]: "div_mod_code_numeral n m = (n div m, n mod m)"
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lemma [code]:
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  "div_mod_code_numeral n m = (if m = 0 then (0, n) else (n div m, n mod m))"
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  unfolding div_mod_code_numeral_def by auto
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lemma [code]:
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  "n div m = fst (div_mod_code_numeral n m)"
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  unfolding div_mod_code_numeral_def by simp
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lemma [code]:
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  "n mod m = snd (div_mod_code_numeral n m)"
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  unfolding div_mod_code_numeral_def by simp
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definition int_of :: "code_numeral \<Rightarrow> int" where
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  "int_of = Nat.of_nat o nat_of"
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lemma int_of_code [code]:
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  "int_of k = (if k = 0 then 0
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    else (if k mod 2 = 0 then 2 * int_of (k div 2) else 2 * int_of (k div 2) + 1))"
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proof -
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  have "(nat_of k div 2) * 2 + nat_of k mod 2 = nat_of k" 
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    by (rule mod_div_equality)
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  then have "int ((nat_of k div 2) * 2 + nat_of k mod 2) = int (nat_of k)" 
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    by simp
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  then have "int (nat_of k) = int (nat_of k div 2) * 2 + int (nat_of k mod 2)" 
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    unfolding of_nat_mult of_nat_add by simp
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  then show ?thesis by (auto simp add: int_of_def mult_ac)
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qed
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hide_const (open) of_nat nat_of int_of
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subsubsection {* Lazy Evaluation of an indexed function *}
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function iterate_upto :: "(code_numeral => 'a) => code_numeral => code_numeral => 'a Predicate.pred"
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where
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  "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))"
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by pat_completeness auto
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termination by (relation "measure (%(f, n, m). Code_Numeral.nat_of (m + 1 - n))") auto
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hide_const (open) iterate_upto
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subsection {* Code generator setup *}
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text {* Implementation of code numerals by bounded integers *}
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code_type code_numeral
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  (SML "int")
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  (OCaml "Big'_int.big'_int")
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  (Haskell "Integer")
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  (Scala "BigInt")
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code_instance code_numeral :: equal
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  (Haskell -)
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setup {*
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  Numeral.add_code @{const_name number_code_numeral_inst.number_of_code_numeral}
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    false Code_Printer.literal_naive_numeral "SML"
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  #> fold (Numeral.add_code @{const_name number_code_numeral_inst.number_of_code_numeral}
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    false Code_Printer.literal_numeral) ["OCaml", "Haskell", "Scala"]
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*}
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code_reserved SML Int int
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code_reserved Eval Integer
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code_const "op + \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
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  (SML "Int.+/ ((_),/ (_))")
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  (OCaml "Big'_int.add'_big'_int")
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  (Haskell infixl 6 "+")
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  (Scala infixl 7 "+")
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  (Eval infixl 8 "+")
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code_const "subtract_code_numeral \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
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  (SML "Int.max/ (_/ -/ _,/ 0 : int)")
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  (OCaml "Big'_int.max'_big'_int/ (Big'_int.sub'_big'_int/ _/ _)/ Big'_int.zero'_big'_int")
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  (Haskell "max/ (_/ -/ _)/ (0 :: Integer)")
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  (Scala "!(_/ -/ _).max(0)")
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  (Eval "Integer.max/ (_/ -/ _)/ 0")
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code_const "op * \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
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  (SML "Int.*/ ((_),/ (_))")
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  (OCaml "Big'_int.mult'_big'_int")
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  (Haskell infixl 7 "*")
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  (Scala infixl 8 "*")
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  (Eval infixl 8 "*")
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code_const div_mod_code_numeral
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  (SML "!(fn n => fn m =>/ if m = 0/ then (0, n) else/ (Int.div (n, m), Int.mod (n, m)))")
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  (OCaml "Big'_int.quomod'_big'_int/ (Big'_int.abs'_big'_int _)/ (Big'_int.abs'_big'_int _)")
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  (Haskell "divMod")
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  (Scala "!((k: BigInt) => (l: BigInt) =>/ if (l == 0)/ (BigInt(0), k) else/ (k.abs '/% l.abs))")
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  (Eval "!(fn n => fn m =>/ if m = 0/ then (0, n) else/ (Integer.div'_mod n m))")
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code_const "HOL.equal \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
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  (SML "!((_ : Int.int) = _)")
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  (OCaml "Big'_int.eq'_big'_int")
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  (Haskell infix 4 "==")
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  (Scala infixl 5 "==")
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  (Eval "!((_ : int) = _)")
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code_const "op \<le> \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
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  (SML "Int.<=/ ((_),/ (_))")
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  (OCaml "Big'_int.le'_big'_int")
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  (Haskell infix 4 "<=")
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  (Scala infixl 4 "<=")
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  (Eval infixl 6 "<=")
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code_const "op < \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
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  (SML "Int.</ ((_),/ (_))")
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  (OCaml "Big'_int.lt'_big'_int")
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  (Haskell infix 4 "<")
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  (Scala infixl 4 "<")
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  (Eval infixl 6 "<")
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end