author | haftmann |
Fri, 10 Oct 2008 06:45:53 +0200 | |
changeset 28562 | 4e74209f113e |
parent 28351 | abfc66969d1f |
child 28708 | a1a436f09ec6 |
permissions | -rw-r--r-- |
24999 | 1 |
(* ID: $Id$ |
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Author: Florian Haftmann, TU Muenchen |
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*) |
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header {* Type of indices *} |
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theory Code_Index |
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imports Plain "~~/src/HOL/Code_Eval" "~~/src/HOL/Presburger" |
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begin |
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text {* |
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Indices are isomorphic to HOL @{typ nat} but |
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mapped to target-language builtin integers. |
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*} |
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subsection {* Datatype of indices *} |
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typedef index = "UNIV \<Colon> nat set" |
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morphisms nat_of_index index_of_nat by rule |
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lemma index_of_nat_nat_of_index [simp]: |
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"index_of_nat (nat_of_index k) = k" |
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by (rule nat_of_index_inverse) |
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lemma nat_of_index_index_of_nat [simp]: |
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"nat_of_index (index_of_nat n) = n" |
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by (rule index_of_nat_inverse) |
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(unfold index_def, rule UNIV_I) |
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lemma index: |
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"(\<And>n\<Colon>index. PROP P n) \<equiv> (\<And>n\<Colon>nat. PROP P (index_of_nat n))" |
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proof |
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fix n :: nat |
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assume "\<And>n\<Colon>index. PROP P n" |
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then show "PROP P (index_of_nat n)" . |
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next |
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fix n :: index |
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assume "\<And>n\<Colon>nat. PROP P (index_of_nat n)" |
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then have "PROP P (index_of_nat (nat_of_index n))" . |
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then show "PROP P n" by simp |
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qed |
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lemma index_case: |
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assumes "\<And>n. k = index_of_nat n \<Longrightarrow> P" |
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shows P |
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by (rule assms [of "nat_of_index k"]) simp |
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lemma index_induct_raw: |
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assumes "\<And>n. P (index_of_nat n)" |
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shows "P k" |
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proof - |
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from assms have "P (index_of_nat (nat_of_index k))" . |
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then show ?thesis by simp |
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qed |
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lemma nat_of_index_inject [simp]: |
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"nat_of_index k = nat_of_index l \<longleftrightarrow> k = l" |
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by (rule nat_of_index_inject) |
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lemma index_of_nat_inject [simp]: |
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"index_of_nat n = index_of_nat m \<longleftrightarrow> n = m" |
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by (auto intro!: index_of_nat_inject simp add: index_def) |
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instantiation index :: zero |
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begin |
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definition [simp, code del]: |
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"0 = index_of_nat 0" |
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instance .. |
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end |
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definition [simp]: |
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"Suc_index k = index_of_nat (Suc (nat_of_index k))" |
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||
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rep_datatype "0 \<Colon> index" Suc_index |
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proof - |
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fix P :: "index \<Rightarrow> bool" |
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fix k :: index |
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assume "P 0" then have init: "P (index_of_nat 0)" by simp |
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assume "\<And>k. P k \<Longrightarrow> P (Suc_index k)" |
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then have "\<And>n. P (index_of_nat n) \<Longrightarrow> P (Suc_index (index_of_nat n))" . |
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then have step: "\<And>n. P (index_of_nat n) \<Longrightarrow> P (index_of_nat (Suc n))" by simp |
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from init step have "P (index_of_nat (nat_of_index k))" |
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by (induct "nat_of_index k") simp_all |
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then show "P k" by simp |
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qed simp_all |
26140 | 89 |
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lemmas [code del] = index.recs index.cases |
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declare index_case [case_names nat, cases type: index] |
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declare index.induct [case_names nat, induct type: index] |
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lemma [code]: |
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"index_size = nat_of_index" |
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proof (rule ext) |
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fix k |
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have "index_size k = nat_size (nat_of_index k)" |
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by (induct k rule: index.induct) (simp_all del: zero_index_def Suc_index_def, simp_all) |
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also have "nat_size (nat_of_index k) = nat_of_index k" by (induct "nat_of_index k") simp_all |
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finally show "index_size k = nat_of_index k" . |
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qed |
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lemma [code]: |
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"size = nat_of_index" |
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proof (rule ext) |
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fix k |
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show "size k = nat_of_index k" |
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by (induct k) (simp_all del: zero_index_def Suc_index_def, simp_all) |
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qed |
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lemma [code]: |
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"eq_class.eq k l \<longleftrightarrow> eq_class.eq (nat_of_index k) (nat_of_index 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::index) 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 index :: number |
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begin |
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definition |
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"number_of = index_of_nat o nat" |
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instance .. |
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end |
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lemma nat_of_index_number [simp]: |
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"nat_of_index (number_of k) = number_of k" |
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by (simp add: number_of_index_def nat_number_of_def number_of_is_id) |
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code_datatype "number_of \<Colon> int \<Rightarrow> index" |
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subsection {* Basic arithmetic *} |
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instantiation index :: "{minus, ordered_semidom, Divides.div, linorder}" |
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begin |
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lemma zero_index_code [code inline, code]: |
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"(0\<Colon>index) = Numeral0" |
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by (simp add: number_of_index_def Pls_def) |
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lemma [code post]: "Numeral0 = (0\<Colon>index)" |
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using zero_index_code .. |
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definition [simp, code del]: |
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"(1\<Colon>index) = index_of_nat 1" |
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lemma one_index_code [code inline, code]: |
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"(1\<Colon>index) = Numeral1" |
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by (simp add: number_of_index_def Pls_def Bit1_def) |
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lemma [code post]: "Numeral1 = (1\<Colon>index)" |
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using one_index_code .. |
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definition [simp, code del]: |
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"n + m = index_of_nat (nat_of_index n + nat_of_index m)" |
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lemma plus_index_code [code]: |
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"index_of_nat n + index_of_nat m = index_of_nat (n + m)" |
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by simp |
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definition [simp, code del]: |
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"n - m = index_of_nat (nat_of_index n - nat_of_index m)" |
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definition [simp, code del]: |
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"n * m = index_of_nat (nat_of_index n * nat_of_index m)" |
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lemma times_index_code [code]: |
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"index_of_nat n * index_of_nat m = index_of_nat (n * m)" |
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by simp |
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definition [simp, code del]: |
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"n div m = index_of_nat (nat_of_index n div nat_of_index m)" |
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definition [simp, code del]: |
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"n mod m = index_of_nat (nat_of_index n mod nat_of_index m)" |
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lemma div_index_code [code]: |
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"index_of_nat n div index_of_nat m = index_of_nat (n div m)" |
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by simp |
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lemma mod_index_code [code]: |
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"index_of_nat n mod index_of_nat m = index_of_nat (n mod m)" |
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by simp |
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definition [simp, code del]: |
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"n \<le> m \<longleftrightarrow> nat_of_index n \<le> nat_of_index m" |
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definition [simp, code del]: |
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"n < m \<longleftrightarrow> nat_of_index n < nat_of_index m" |
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lemma less_eq_index_code [code]: |
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"index_of_nat n \<le> index_of_nat m \<longleftrightarrow> n \<le> m" |
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by simp |
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lemma less_index_code [code]: |
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"index_of_nat n < index_of_nat m \<longleftrightarrow> n < m" |
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by simp |
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instance by default (auto simp add: left_distrib index) |
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end |
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lemma Suc_index_minus_one: "Suc_index n - 1 = n" by simp |
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lemma index_of_nat_code [code]: |
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"index_of_nat = of_nat" |
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proof |
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fix n :: nat |
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have "of_nat n = index_of_nat n" |
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by (induct n) simp_all |
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then show "index_of_nat n = of_nat n" |
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by (rule sym) |
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qed |
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lemma index_not_eq_zero: "i \<noteq> index_of_nat 0 \<longleftrightarrow> i \<ge> 1" |
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by (cases i) auto |
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definition |
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nat_of_index_aux :: "index \<Rightarrow> nat \<Rightarrow> nat" |
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where |
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"nat_of_index_aux i n = nat_of_index i + n" |
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lemma nat_of_index_aux_code [code]: |
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"nat_of_index_aux i n = (if i = 0 then n else nat_of_index_aux (i - 1) (Suc n))" |
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by (auto simp add: nat_of_index_aux_def index_not_eq_zero) |
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lemma nat_of_index_code [code]: |
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"nat_of_index i = nat_of_index_aux i 0" |
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by (simp add: nat_of_index_aux_def) |
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text {* Measure function (for termination proofs) *} |
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lemma [measure_function]: |
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"is_measure nat_of_index" by (rule is_measure_trivial) |
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subsection {* ML interface *} |
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ML {* |
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structure Index = |
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struct |
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fun mk k = HOLogic.mk_number @{typ index} k; |
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end; |
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*} |
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subsection {* Specialized @{term "op - \<Colon> index \<Rightarrow> index \<Rightarrow> index"}, |
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@{term "op div \<Colon> index \<Rightarrow> index \<Rightarrow> index"} and @{term "op mod \<Colon> index \<Rightarrow> index \<Rightarrow> index"} |
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operations *} |
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definition |
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minus_index_aux :: "index \<Rightarrow> index \<Rightarrow> index" |
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where |
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[code del]: "minus_index_aux = op -" |
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lemma [code]: "op - = minus_index_aux" |
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using minus_index_aux_def .. |
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definition |
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div_mod_index :: "index \<Rightarrow> index \<Rightarrow> index \<times> index" |
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where |
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[code del]: "div_mod_index n m = (n div m, n mod m)" |
26009 | 272 |
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lemma [code]: |
26009 | 274 |
"div_mod_index n m = (if m = 0 then (0, n) else (n div m, n mod m))" |
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unfolding div_mod_index_def by auto |
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lemma [code]: |
26009 | 278 |
"n div m = fst (div_mod_index n m)" |
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unfolding div_mod_index_def by simp |
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28562 | 281 |
lemma [code]: |
26009 | 282 |
"n mod m = snd (div_mod_index n m)" |
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unfolding div_mod_index_def by simp |
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subsection {* Code generator setup *} |
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text {* Implementation of indices by bounded integers *} |
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code_type index |
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(SML "int") |
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(OCaml "int") |
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25967 | 293 |
(Haskell "Int") |
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code_instance index :: eq |
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(Haskell -) |
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setup {* |
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fold (Numeral.add_code @{const_name number_index_inst.number_of_index} |
300 |
false false) ["SML", "OCaml", "Haskell"] |
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*} |
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code_reserved SML Int int |
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code_reserved OCaml Pervasives int |
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code_const "op + \<Colon> index \<Rightarrow> index \<Rightarrow> index" |
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25928 | 307 |
(SML "Int.+/ ((_),/ (_))") |
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(OCaml "Pervasives.( + )") |
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(Haskell infixl 6 "+") |
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code_const "minus_index_aux \<Colon> index \<Rightarrow> index \<Rightarrow> index" |
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(SML "Int.max/ (_/ -/ _,/ 0 : int)") |
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(OCaml "Pervasives.max/ (_/ -/ _)/ (0 : int) ") |
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(Haskell "max/ (_/ -/ _)/ (0 :: Int)") |
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code_const "op * \<Colon> index \<Rightarrow> index \<Rightarrow> index" |
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25928 | 317 |
(SML "Int.*/ ((_),/ (_))") |
25967 | 318 |
(OCaml "Pervasives.( * )") |
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(Haskell infixl 7 "*") |
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26009 | 321 |
code_const div_mod_index |
322 |
(SML "(fn n => fn m =>/ (n div m, n mod m))") |
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(OCaml "(fun n -> fun m ->/ (n '/ m, n mod m))") |
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(Haskell "divMod") |
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25928 | 325 |
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326 |
code_const "eq_class.eq \<Colon> index \<Rightarrow> index \<Rightarrow> bool" |
24999 | 327 |
(SML "!((_ : Int.int) = _)") |
25967 | 328 |
(OCaml "!((_ : int) = _)") |
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(Haskell infixl 4 "==") |
330 |
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331 |
code_const "op \<le> \<Colon> index \<Rightarrow> index \<Rightarrow> bool" |
|
25928 | 332 |
(SML "Int.<=/ ((_),/ (_))") |
25967 | 333 |
(OCaml "!((_ : int) <= _)") |
24999 | 334 |
(Haskell infix 4 "<=") |
335 |
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336 |
code_const "op < \<Colon> index \<Rightarrow> index \<Rightarrow> bool" |
|
25928 | 337 |
(SML "Int.</ ((_),/ (_))") |
25967 | 338 |
(OCaml "!((_ : int) < _)") |
24999 | 339 |
(Haskell infix 4 "<") |
340 |
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28228 | 341 |
text {* Evaluation *} |
342 |
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28562 | 343 |
lemma [code, code del]: |
28228 | 344 |
"(Code_Eval.term_of \<Colon> index \<Rightarrow> term) = Code_Eval.term_of" .. |
345 |
||
346 |
code_const "Code_Eval.term_of \<Colon> index \<Rightarrow> term" |
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347 |
(SML "HOLogic.mk'_number/ HOLogic.indexT/ (IntInf.fromInt/ _)") |
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348 |
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24999 | 349 |
end |