haftmann@29752
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(* Author: Florian Haftmann, TU Muenchen *)
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haftmann@24999
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haftmann@31205
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header {* Type of target language numerals *}
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haftmann@24999
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haftmann@31205
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theory Code_Numeral
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huffman@48116
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imports Nat_Transfer Divides
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haftmann@24999
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begin
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haftmann@24999
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haftmann@24999
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text {*
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haftmann@31205
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Code numerals are isomorphic to HOL @{typ nat} but
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haftmann@31205
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mapped to target-language builtin numerals.
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haftmann@24999
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*}
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haftmann@24999
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haftmann@31205
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subsection {* Datatype of target language numerals *}
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haftmann@24999
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haftmann@31205
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typedef (open) code_numeral = "UNIV \<Colon> nat set"
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wenzelm@46567
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morphisms nat_of of_nat ..
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haftmann@24999
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haftmann@29752
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lemma of_nat_nat_of [simp]:
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haftmann@29752
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"of_nat (nat_of k) = k"
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haftmann@29752
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by (rule nat_of_inverse)
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haftmann@25967
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haftmann@29752
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lemma nat_of_of_nat [simp]:
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haftmann@29752
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"nat_of (of_nat n) = n"
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haftmann@29752
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by (rule of_nat_inverse) (rule UNIV_I)
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haftmann@24999
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haftmann@28708
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lemma [measure_function]:
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haftmann@29752
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"is_measure nat_of" by (rule is_measure_trivial)
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haftmann@28708
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haftmann@31205
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lemma code_numeral:
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haftmann@31205
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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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haftmann@24999
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proof
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haftmann@25767
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fix n :: nat
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haftmann@31205
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assume "\<And>n\<Colon>code_numeral. PROP P n"
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haftmann@29752
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then show "PROP P (of_nat n)" .
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haftmann@24999
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next
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haftmann@31205
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fix n :: code_numeral
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haftmann@29752
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assume "\<And>n\<Colon>nat. PROP P (of_nat n)"
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haftmann@29752
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then have "PROP P (of_nat (nat_of n))" .
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haftmann@25767
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then show "PROP P n" by simp
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haftmann@24999
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qed
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haftmann@24999
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haftmann@31205
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lemma code_numeral_case:
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haftmann@29752
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assumes "\<And>n. k = of_nat n \<Longrightarrow> P"
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haftmann@26140
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shows P
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haftmann@29752
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by (rule assms [of "nat_of k"]) simp
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haftmann@26140
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haftmann@31205
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lemma code_numeral_induct_raw:
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haftmann@29752
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assumes "\<And>n. P (of_nat n)"
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haftmann@26140
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shows "P k"
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haftmann@26140
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proof -
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haftmann@29752
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from assms have "P (of_nat (nat_of k))" .
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haftmann@26140
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then show ?thesis by simp
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haftmann@26140
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qed
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haftmann@26140
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haftmann@29752
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lemma nat_of_inject [simp]:
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haftmann@29752
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"nat_of k = nat_of l \<longleftrightarrow> k = l"
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haftmann@29752
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by (rule nat_of_inject)
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haftmann@26140
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haftmann@29752
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lemma of_nat_inject [simp]:
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haftmann@29752
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"of_nat n = of_nat m \<longleftrightarrow> n = m"
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haftmann@29752
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by (rule of_nat_inject) (rule UNIV_I)+
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haftmann@26140
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haftmann@31205
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instantiation code_numeral :: zero
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haftmann@26140
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begin
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haftmann@26140
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haftmann@28562
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definition [simp, code del]:
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haftmann@29752
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"0 = of_nat 0"
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haftmann@26140
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haftmann@26140
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instance ..
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haftmann@26140
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end
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haftmann@26140
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definition Suc where [simp]:
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huffman@47418
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"Suc k = of_nat (Nat.Suc (nat_of k))"
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haftmann@26140
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huffman@47418
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rep_datatype "0 \<Colon> code_numeral" Suc
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haftmann@26140
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proof -
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haftmann@31205
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fix P :: "code_numeral \<Rightarrow> bool"
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haftmann@31205
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fix k :: code_numeral
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haftmann@29752
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assume "P 0" then have init: "P (of_nat 0)" by simp
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huffman@47418
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assume "\<And>k. P k \<Longrightarrow> P (Suc k)"
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huffman@47418
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then have "\<And>n. P (of_nat n) \<Longrightarrow> P (Suc (of_nat n))" .
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huffman@47418
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then have step: "\<And>n. P (of_nat n) \<Longrightarrow> P (of_nat (Nat.Suc n))" by simp
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haftmann@29752
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from init step have "P (of_nat (nat_of k))"
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berghofe@34915
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by (induct ("nat_of k")) simp_all
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haftmann@26140
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then show "P k" by simp
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haftmann@27104
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qed simp_all
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haftmann@26140
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haftmann@31205
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declare code_numeral_case [case_names nat, cases type: code_numeral]
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haftmann@31205
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declare code_numeral.induct [case_names nat, induct type: code_numeral]
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haftmann@26140
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haftmann@31205
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lemma code_numeral_decr [termination_simp]:
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huffman@47418
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"k \<noteq> of_nat 0 \<Longrightarrow> nat_of k - Nat.Suc 0 < nat_of k"
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haftmann@30245
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by (cases k) simp
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haftmann@30245
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haftmann@30245
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lemma [simp, code]:
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haftmann@31205
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"code_numeral_size = nat_of"
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haftmann@26140
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proof (rule ext)
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haftmann@26140
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fix k
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haftmann@31205
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have "code_numeral_size k = nat_size (nat_of k)"
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huffman@47418
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by (induct k rule: code_numeral.induct) (simp_all del: zero_code_numeral_def Suc_def, simp_all)
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berghofe@34915
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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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haftmann@31205
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finally show "code_numeral_size k = nat_of k" .
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haftmann@26140
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qed
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haftmann@26140
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haftmann@30245
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lemma [simp, code]:
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"size = nat_of"
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haftmann@26140
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proof (rule ext)
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haftmann@26140
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fix k
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haftmann@29752
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show "size k = nat_of k"
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huffman@47418
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by (induct k) (simp_all del: zero_code_numeral_def Suc_def, simp_all)
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haftmann@26140
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qed
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haftmann@26140
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haftmann@31205
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lemmas [code del] = code_numeral.recs code_numeral.cases
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haftmann@30245
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haftmann@28562
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lemma [code]:
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haftmann@39086
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"HOL.equal k l \<longleftrightarrow> HOL.equal (nat_of k) (nat_of l)"
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haftmann@39086
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by (cases k, cases l) (simp add: equal)
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haftmann@24999
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haftmann@28351
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lemma [code nbe]:
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haftmann@39086
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"HOL.equal (k::code_numeral) k \<longleftrightarrow> True"
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haftmann@39086
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by (rule equal_refl)
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haftmann@28351
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haftmann@24999
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haftmann@24999
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subsection {* Basic arithmetic *}
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haftmann@35028
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instantiation code_numeral :: "{minus, linordered_semidom, semiring_div, linorder}"
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haftmann@25767
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begin
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haftmann@24999
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haftmann@28708
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definition [simp, code del]:
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haftmann@31205
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"(1\<Colon>code_numeral) = of_nat 1"
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haftmann@28708
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haftmann@28708
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definition [simp, code del]:
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haftmann@29752
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"n + m = of_nat (nat_of n + nat_of m)"
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haftmann@28708
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haftmann@28708
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definition [simp, code del]:
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haftmann@29752
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"n - m = of_nat (nat_of n - nat_of m)"
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haftmann@28708
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haftmann@28708
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definition [simp, code del]:
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haftmann@29752
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"n * m = of_nat (nat_of n * nat_of m)"
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haftmann@28708
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haftmann@28708
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definition [simp, code del]:
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haftmann@29752
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"n div m = of_nat (nat_of n div nat_of m)"
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haftmann@28708
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haftmann@28708
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definition [simp, code del]:
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haftmann@29752
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"n mod m = of_nat (nat_of n mod nat_of m)"
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haftmann@28708
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haftmann@28708
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definition [simp, code del]:
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haftmann@29752
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"n \<le> m \<longleftrightarrow> nat_of n \<le> nat_of m"
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haftmann@28708
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haftmann@28708
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definition [simp, code del]:
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haftmann@29752
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"n < m \<longleftrightarrow> nat_of n < nat_of m"
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haftmann@28708
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haftmann@29752
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instance proof
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haftmann@33335
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qed (auto simp add: code_numeral left_distrib intro: mult_commute)
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haftmann@28708
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haftmann@28708
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end
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haftmann@28708
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huffman@47978
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lemma nat_of_numeral [simp]: "nat_of (numeral k) = numeral k"
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huffman@47978
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by (induct k rule: num_induct) (simp_all add: numeral_inc)
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haftmann@46899
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huffman@47978
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definition Num :: "num \<Rightarrow> code_numeral"
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huffman@47978
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where [simp, code_abbrev]: "Num = numeral"
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huffman@47978
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huffman@47978
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code_datatype "0::code_numeral" Num
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haftmann@25767
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haftmann@46899
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lemma one_code_numeral_code [code]:
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haftmann@31205
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"(1\<Colon>code_numeral) = Numeral1"
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huffman@47978
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by simp
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haftmann@46899
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haftmann@46899
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lemma [code_abbrev]: "Numeral1 = (1\<Colon>code_numeral)"
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haftmann@31205
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using one_code_numeral_code ..
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haftmann@25767
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haftmann@31205
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lemma plus_code_numeral_code [code nbe]:
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haftmann@29752
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"of_nat n + of_nat m = of_nat (n + m)"
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haftmann@24999
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by simp
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haftmann@24999
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huffman@47978
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lemma minus_code_numeral_code [code nbe]:
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huffman@47978
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"of_nat n - of_nat m = of_nat (n - m)"
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haftmann@28708
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by simp
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haftmann@28708
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haftmann@31205
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lemma times_code_numeral_code [code nbe]:
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haftmann@29752
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"of_nat n * of_nat m = of_nat (n * m)"
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haftmann@25767
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by simp
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haftmann@25335
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haftmann@31205
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lemma less_eq_code_numeral_code [code nbe]:
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haftmann@29752
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"of_nat n \<le> of_nat m \<longleftrightarrow> n \<le> m"
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haftmann@25767
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by simp
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haftmann@24999
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haftmann@31205
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lemma less_code_numeral_code [code nbe]:
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haftmann@29752
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"of_nat n < of_nat m \<longleftrightarrow> n < m"
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haftmann@25767
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by simp
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haftmann@24999
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haftmann@31259
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lemma code_numeral_zero_minus_one:
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haftmann@31259
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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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haftmann@31259
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lemma Suc_code_numeral_minus_one:
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huffman@47418
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"Suc n - 1 = n"
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haftmann@31259
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by simp
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haftmann@26140
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haftmann@29752
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lemma of_nat_code [code]:
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haftmann@29752
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"of_nat = Nat.of_nat"
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haftmann@25918
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proof
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haftmann@25918
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fix n :: nat
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haftmann@29752
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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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haftmann@29752
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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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haftmann@31205
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definition nat_of_aux :: "code_numeral \<Rightarrow> nat \<Rightarrow> nat" where
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haftmann@29752
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"nat_of_aux i n = nat_of i + n"
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haftmann@25928
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haftmann@29752
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lemma nat_of_aux_code [code]:
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huffman@47418
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"nat_of_aux i n = (if i = 0 then n else nat_of_aux (i - 1) (Nat.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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haftmann@29752
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lemma nat_of_code [code]:
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haftmann@29752
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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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huffman@47418
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definition div_mod :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral \<times> code_numeral" where
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huffman@47418
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[code del]: "div_mod n m = (n div m, n mod m)"
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haftmann@26009
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haftmann@28562
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lemma [code]:
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huffman@47418
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"div_mod n m = (if m = 0 then (0, n) else (n div m, n mod m))"
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huffman@47418
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unfolding div_mod_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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huffman@47418
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"n div m = fst (div_mod n m)"
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huffman@47418
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unfolding div_mod_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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huffman@47418
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"n mod m = snd (div_mod n m)"
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huffman@47418
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unfolding div_mod_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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246 |
"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@33335
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248 |
proof -
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haftmann@33335
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249 |
have "(nat_of k div 2) * 2 + nat_of k mod 2 = nat_of k"
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haftmann@33335
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250 |
by (rule mod_div_equality)
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haftmann@33335
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251 |
then have "int ((nat_of k div 2) * 2 + nat_of k mod 2) = int (nat_of k)"
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haftmann@33335
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252 |
by simp
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haftmann@33335
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253 |
then have "int (nat_of k) = int (nat_of k div 2) * 2 + int (nat_of k mod 2)"
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huffman@45692
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unfolding of_nat_mult of_nat_add by simp
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haftmann@33335
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255 |
then show ?thesis by (auto simp add: int_of_def mult_ac)
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haftmann@33335
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256 |
qed
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haftmann@28708
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257 |
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haftmann@28708
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258 |
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huffman@47978
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hide_const (open) of_nat nat_of Suc int_of
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huffman@47418
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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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263 |
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haftmann@38195
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text {* Implementation of code numerals 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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haftmann@24999
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(SML "int")
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haftmann@31377
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(OCaml "Big'_int.big'_int")
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haftmann@38185
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(Haskell "Integer")
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haftmann@38195
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270 |
(Scala "BigInt")
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haftmann@24999
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haftmann@39086
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272 |
code_instance code_numeral :: equal
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haftmann@24999
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273 |
(Haskell -)
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haftmann@24999
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haftmann@24999
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setup {*
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huffman@47978
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276 |
Numeral.add_code @{const_name Num}
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haftmann@38195
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277 |
false Code_Printer.literal_naive_numeral "SML"
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huffman@47978
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#> fold (Numeral.add_code @{const_name Num}
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haftmann@38195
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279 |
false Code_Printer.literal_numeral) ["OCaml", "Haskell", "Scala"]
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haftmann@24999
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*}
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haftmann@24999
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haftmann@25918
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282 |
code_reserved SML Int int
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haftmann@38195
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code_reserved Eval Integer
|
haftmann@24999
|
284 |
|
huffman@47978
|
285 |
code_const "0::code_numeral"
|
huffman@47978
|
286 |
(SML "0")
|
huffman@47978
|
287 |
(OCaml "Big'_int.zero'_big'_int")
|
huffman@47978
|
288 |
(Haskell "0")
|
huffman@47978
|
289 |
(Scala "BigInt(0)")
|
huffman@47978
|
290 |
|
huffman@47418
|
291 |
code_const "plus \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
|
haftmann@25928
|
292 |
(SML "Int.+/ ((_),/ (_))")
|
haftmann@31377
|
293 |
(OCaml "Big'_int.add'_big'_int")
|
haftmann@24999
|
294 |
(Haskell infixl 6 "+")
|
haftmann@34886
|
295 |
(Scala infixl 7 "+")
|
haftmann@38195
|
296 |
(Eval infixl 8 "+")
|
haftmann@24999
|
297 |
|
huffman@47978
|
298 |
code_const "minus \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
|
huffman@47978
|
299 |
(SML "Int.max/ (0 : int,/ Int.-/ ((_),/ (_)))")
|
huffman@47978
|
300 |
(OCaml "Big'_int.max'_big'_int/ Big'_int.zero'_big'_int/ (Big'_int.sub'_big'_int/ _/ _)")
|
haftmann@49446
|
301 |
(Haskell "Prelude.max/ (0 :: Integer)/ (_/ -/ _)")
|
haftmann@34886
|
302 |
(Scala "!(_/ -/ _).max(0)")
|
huffman@47978
|
303 |
(Eval "Integer.max/ 0/ (_/ -/ _)")
|
haftmann@24999
|
304 |
|
huffman@47418
|
305 |
code_const "times \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
|
haftmann@25928
|
306 |
(SML "Int.*/ ((_),/ (_))")
|
haftmann@31377
|
307 |
(OCaml "Big'_int.mult'_big'_int")
|
haftmann@24999
|
308 |
(Haskell infixl 7 "*")
|
haftmann@34886
|
309 |
(Scala infixl 8 "*")
|
haftmann@38195
|
310 |
(Eval infixl 8 "*")
|
haftmann@24999
|
311 |
|
huffman@47418
|
312 |
code_const Code_Numeral.div_mod
|
haftmann@38195
|
313 |
(SML "!(fn n => fn m =>/ if m = 0/ then (0, n) else/ (Int.div (n, m), Int.mod (n, m)))")
|
haftmann@34898
|
314 |
(OCaml "Big'_int.quomod'_big'_int/ (Big'_int.abs'_big'_int _)/ (Big'_int.abs'_big'_int _)")
|
haftmann@26009
|
315 |
(Haskell "divMod")
|
haftmann@38195
|
316 |
(Scala "!((k: BigInt) => (l: BigInt) =>/ if (l == 0)/ (BigInt(0), k) else/ (k.abs '/% l.abs))")
|
haftmann@40060
|
317 |
(Eval "!(fn n => fn m =>/ if m = 0/ then (0, n) else/ (Integer.div'_mod n m))")
|
haftmann@25928
|
318 |
|
haftmann@39086
|
319 |
code_const "HOL.equal \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
|
haftmann@24999
|
320 |
(SML "!((_ : Int.int) = _)")
|
haftmann@31377
|
321 |
(OCaml "Big'_int.eq'_big'_int")
|
haftmann@39499
|
322 |
(Haskell infix 4 "==")
|
haftmann@34886
|
323 |
(Scala infixl 5 "==")
|
haftmann@38195
|
324 |
(Eval "!((_ : int) = _)")
|
haftmann@24999
|
325 |
|
huffman@47418
|
326 |
code_const "less_eq \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
|
haftmann@25928
|
327 |
(SML "Int.<=/ ((_),/ (_))")
|
haftmann@31377
|
328 |
(OCaml "Big'_int.le'_big'_int")
|
haftmann@24999
|
329 |
(Haskell infix 4 "<=")
|
haftmann@34898
|
330 |
(Scala infixl 4 "<=")
|
haftmann@38195
|
331 |
(Eval infixl 6 "<=")
|
haftmann@24999
|
332 |
|
huffman@47418
|
333 |
code_const "less \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
|
haftmann@25928
|
334 |
(SML "Int.</ ((_),/ (_))")
|
haftmann@31377
|
335 |
(OCaml "Big'_int.lt'_big'_int")
|
haftmann@24999
|
336 |
(Haskell infix 4 "<")
|
haftmann@34898
|
337 |
(Scala infixl 4 "<")
|
haftmann@38195
|
338 |
(Eval infixl 6 "<")
|
haftmann@24999
|
339 |
|
huffman@47418
|
340 |
code_modulename SML
|
huffman@47418
|
341 |
Code_Numeral Arith
|
huffman@47418
|
342 |
|
huffman@47418
|
343 |
code_modulename OCaml
|
huffman@47418
|
344 |
Code_Numeral Arith
|
huffman@47418
|
345 |
|
huffman@47418
|
346 |
code_modulename Haskell
|
huffman@47418
|
347 |
Code_Numeral Arith
|
huffman@47418
|
348 |
|
haftmann@24999
|
349 |
end
|
haftmann@47535
|
350 |
|