haftmann@29752
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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 by rule
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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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"eq_class.eq k l \<longleftrightarrow> eq_class.eq (nat_of k) (nat_of l)"
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by (cases k, cases l) (simp add: eq)
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lemma [code nbe]:
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"eq_class.eq (k::code_numeral) k \<longleftrightarrow> True"
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by (rule HOL.eq_refl)
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subsection {* Indices 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, ordered_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 div_mult_self1)
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end
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lemma zero_code_numeral_code [code, code_unfold]:
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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_post]: "Numeral0 = (0\<Colon>code_numeral)"
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using zero_code_numeral_code ..
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lemma one_code_numeral_code [code, code_unfold]:
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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_post]: "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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haftmann@28708
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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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haftmann@31259
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by simp
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haftmann@31259
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lemma Suc_code_numeral_minus_one:
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"Suc_code_numeral n - 1 = n"
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haftmann@31259
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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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haftmann@25918
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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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haftmann@25918
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by (rule sym)
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haftmann@25918
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qed
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haftmann@25918
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haftmann@31205
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lemma code_numeral_not_eq_zero: "i \<noteq> of_nat 0 \<longleftrightarrow> i \<ge> 1"
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haftmann@25928
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by (cases i) auto
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haftmann@25928
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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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haftmann@25928
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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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haftmann@31205
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by (auto simp add: nat_of_aux_def code_numeral_not_eq_zero)
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haftmann@25928
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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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haftmann@29752
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by (simp add: nat_of_aux_def)
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haftmann@25918
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haftmann@31205
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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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haftmann@31205
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[code del]: "div_mod_code_numeral n m = (n div m, n mod m)"
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haftmann@26009
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lemma [code]:
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haftmann@31205
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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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haftmann@31205
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unfolding div_mod_code_numeral_def by auto
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haftmann@26009
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haftmann@28562
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lemma [code]:
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haftmann@31205
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"n div m = fst (div_mod_code_numeral n m)"
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haftmann@31205
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unfolding div_mod_code_numeral_def by simp
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haftmann@26009
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haftmann@28562
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lemma [code]:
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haftmann@31205
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"n mod m = snd (div_mod_code_numeral n m)"
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haftmann@31205
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unfolding div_mod_code_numeral_def by simp
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haftmann@26009
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haftmann@31205
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definition int_of :: "code_numeral \<Rightarrow> int" where
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haftmann@31192
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"int_of = Nat.of_nat o nat_of"
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haftmann@26009
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haftmann@31192
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lemma int_of_code [code]:
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haftmann@31192
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"int_of k = (if k = 0 then 0
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haftmann@31192
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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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haftmann@31192
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by (auto simp add: int_of_def mod_div_equality')
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haftmann@28708
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haftmann@31192
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hide (open) const of_nat nat_of int_of
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haftmann@28708
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haftmann@28708
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haftmann@28228
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subsection {* Code generator setup *}
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haftmann@24999
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haftmann@25767
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text {* Implementation of indices by bounded integers *}
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haftmann@25767
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haftmann@31205
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code_type code_numeral
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(SML "int")
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haftmann@31377
|
282 |
(OCaml "Big'_int.big'_int")
|
haftmann@25967
|
283 |
(Haskell "Int")
|
haftmann@24999
|
284 |
|
haftmann@31205
|
285 |
code_instance code_numeral :: eq
|
haftmann@24999
|
286 |
(Haskell -)
|
haftmann@24999
|
287 |
|
haftmann@24999
|
288 |
setup {*
|
haftmann@31205
|
289 |
fold (Numeral.add_code @{const_name number_code_numeral_inst.number_of_code_numeral}
|
haftmann@31377
|
290 |
false false) ["SML", "Haskell"]
|
haftmann@31377
|
291 |
#> Numeral.add_code @{const_name number_code_numeral_inst.number_of_code_numeral} false true "OCaml"
|
haftmann@24999
|
292 |
*}
|
haftmann@24999
|
293 |
|
haftmann@25918
|
294 |
code_reserved SML Int int
|
haftmann@31377
|
295 |
code_reserved OCaml Big_int
|
haftmann@24999
|
296 |
|
haftmann@31205
|
297 |
code_const "op + \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
|
haftmann@25928
|
298 |
(SML "Int.+/ ((_),/ (_))")
|
haftmann@31377
|
299 |
(OCaml "Big'_int.add'_big'_int")
|
haftmann@24999
|
300 |
(Haskell infixl 6 "+")
|
haftmann@24999
|
301 |
|
haftmann@31205
|
302 |
code_const "subtract_code_numeral \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
|
haftmann@25918
|
303 |
(SML "Int.max/ (_/ -/ _,/ 0 : int)")
|
haftmann@31377
|
304 |
(OCaml "Big'_int.max'_big'_int/ (Big'_int.sub'_big'_int/ _/ _)/ Big'_int.zero'_big'_int")
|
haftmann@25918
|
305 |
(Haskell "max/ (_/ -/ _)/ (0 :: Int)")
|
haftmann@24999
|
306 |
|
haftmann@31205
|
307 |
code_const "op * \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
|
haftmann@25928
|
308 |
(SML "Int.*/ ((_),/ (_))")
|
haftmann@31377
|
309 |
(OCaml "Big'_int.mult'_big'_int")
|
haftmann@24999
|
310 |
(Haskell infixl 7 "*")
|
haftmann@24999
|
311 |
|
haftmann@31205
|
312 |
code_const div_mod_code_numeral
|
haftmann@29760
|
313 |
(SML "(fn n => fn m =>/ if m = 0/ then (0, n) else/ (n div m, n mod m))")
|
haftmann@31377
|
314 |
(OCaml "(fun k -> fun l ->/ Big'_int.quomod'_big'_int/ (Big'_int.abs'_big'_int k)/ (Big'_int.abs'_big'_int l))")
|
haftmann@26009
|
315 |
(Haskell "divMod")
|
haftmann@25928
|
316 |
|
haftmann@31205
|
317 |
code_const "eq_class.eq \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
|
haftmann@24999
|
318 |
(SML "!((_ : Int.int) = _)")
|
haftmann@31377
|
319 |
(OCaml "Big'_int.eq'_big'_int")
|
haftmann@24999
|
320 |
(Haskell infixl 4 "==")
|
haftmann@24999
|
321 |
|
haftmann@31205
|
322 |
code_const "op \<le> \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
|
haftmann@25928
|
323 |
(SML "Int.<=/ ((_),/ (_))")
|
haftmann@31377
|
324 |
(OCaml "Big'_int.le'_big'_int")
|
haftmann@24999
|
325 |
(Haskell infix 4 "<=")
|
haftmann@24999
|
326 |
|
haftmann@31205
|
327 |
code_const "op < \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
|
haftmann@25928
|
328 |
(SML "Int.</ ((_),/ (_))")
|
haftmann@31377
|
329 |
(OCaml "Big'_int.lt'_big'_int")
|
haftmann@24999
|
330 |
(Haskell infix 4 "<")
|
haftmann@24999
|
331 |
|
haftmann@24999
|
332 |
end
|