author | wenzelm |
Wed, 17 Jun 2015 11:03:05 +0200 | |
changeset 60500 | 903bb1495239 |
parent 60352 | d46de31a50c4 |
child 62046 | 2c9f68fbf047 |
permissions | -rw-r--r-- |
41959 | 1 |
(* Title: HOL/Library/Nat_Bijection.thy |
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Author: Brian Huffman |
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Author: Florian Haftmann |
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Author: Stefan Richter |
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Author: Tobias Nipkow |
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Author: Alexander Krauss |
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*) |
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section \<open>Bijections between natural numbers and other types\<close> |
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theory Nat_Bijection |
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imports Main |
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begin |
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subsection \<open>Type @{typ "nat \<times> nat"}\<close> |
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text "Triangle numbers: 0, 1, 3, 6, 10, 15, ..." |
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definition |
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triangle :: "nat \<Rightarrow> nat" |
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where |
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"triangle n = n * Suc n div 2" |
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lemma triangle_0 [simp]: "triangle 0 = 0" |
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unfolding triangle_def by simp |
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lemma triangle_Suc [simp]: "triangle (Suc n) = triangle n + Suc n" |
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unfolding triangle_def by simp |
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definition |
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prod_encode :: "nat \<times> nat \<Rightarrow> nat" |
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where |
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"prod_encode = (\<lambda>(m, n). triangle (m + n) + m)" |
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text \<open>In this auxiliary function, @{term "triangle k + m"} is an invariant.\<close> |
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fun |
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prod_decode_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" |
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where |
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"prod_decode_aux k m = |
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(if m \<le> k then (m, k - m) else prod_decode_aux (Suc k) (m - Suc k))" |
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declare prod_decode_aux.simps [simp del] |
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definition |
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prod_decode :: "nat \<Rightarrow> nat \<times> nat" |
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where |
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"prod_decode = prod_decode_aux 0" |
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lemma prod_encode_prod_decode_aux: |
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"prod_encode (prod_decode_aux k m) = triangle k + m" |
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apply (induct k m rule: prod_decode_aux.induct) |
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apply (subst prod_decode_aux.simps) |
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apply (simp add: prod_encode_def) |
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done |
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lemma prod_decode_inverse [simp]: "prod_encode (prod_decode n) = n" |
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unfolding prod_decode_def by (simp add: prod_encode_prod_decode_aux) |
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lemma prod_decode_triangle_add: |
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"prod_decode (triangle k + m) = prod_decode_aux k m" |
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apply (induct k arbitrary: m) |
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apply (simp add: prod_decode_def) |
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apply (simp only: triangle_Suc add.assoc) |
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apply (subst prod_decode_aux.simps, simp) |
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done |
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lemma prod_encode_inverse [simp]: "prod_decode (prod_encode x) = x" |
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unfolding prod_encode_def |
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apply (induct x) |
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apply (simp add: prod_decode_triangle_add) |
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apply (subst prod_decode_aux.simps, simp) |
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done |
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lemma inj_prod_encode: "inj_on prod_encode A" |
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by (rule inj_on_inverseI, rule prod_encode_inverse) |
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lemma inj_prod_decode: "inj_on prod_decode A" |
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by (rule inj_on_inverseI, rule prod_decode_inverse) |
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lemma surj_prod_encode: "surj prod_encode" |
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by (rule surjI, rule prod_decode_inverse) |
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lemma surj_prod_decode: "surj prod_decode" |
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by (rule surjI, rule prod_encode_inverse) |
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lemma bij_prod_encode: "bij prod_encode" |
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by (rule bijI [OF inj_prod_encode surj_prod_encode]) |
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lemma bij_prod_decode: "bij prod_decode" |
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by (rule bijI [OF inj_prod_decode surj_prod_decode]) |
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lemma prod_encode_eq: "prod_encode x = prod_encode y \<longleftrightarrow> x = y" |
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by (rule inj_prod_encode [THEN inj_eq]) |
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lemma prod_decode_eq: "prod_decode x = prod_decode y \<longleftrightarrow> x = y" |
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by (rule inj_prod_decode [THEN inj_eq]) |
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text \<open>Ordering properties\<close> |
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lemma le_prod_encode_1: "a \<le> prod_encode (a, b)" |
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unfolding prod_encode_def by simp |
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lemma le_prod_encode_2: "b \<le> prod_encode (a, b)" |
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unfolding prod_encode_def by (induct b, simp_all) |
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subsection \<open>Type @{typ "nat + nat"}\<close> |
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definition |
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sum_encode :: "nat + nat \<Rightarrow> nat" |
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where |
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"sum_encode x = (case x of Inl a \<Rightarrow> 2 * a | Inr b \<Rightarrow> Suc (2 * b))" |
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definition |
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sum_decode :: "nat \<Rightarrow> nat + nat" |
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where |
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"sum_decode n = (if even n then Inl (n div 2) else Inr (n div 2))" |
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lemma sum_encode_inverse [simp]: "sum_decode (sum_encode x) = x" |
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unfolding sum_decode_def sum_encode_def |
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by (induct x) simp_all |
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lemma sum_decode_inverse [simp]: "sum_encode (sum_decode n) = n" |
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by (simp add: even_two_times_div_two sum_decode_def sum_encode_def) |
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lemma inj_sum_encode: "inj_on sum_encode A" |
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by (rule inj_on_inverseI, rule sum_encode_inverse) |
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lemma inj_sum_decode: "inj_on sum_decode A" |
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by (rule inj_on_inverseI, rule sum_decode_inverse) |
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lemma surj_sum_encode: "surj sum_encode" |
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by (rule surjI, rule sum_decode_inverse) |
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lemma surj_sum_decode: "surj sum_decode" |
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by (rule surjI, rule sum_encode_inverse) |
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lemma bij_sum_encode: "bij sum_encode" |
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by (rule bijI [OF inj_sum_encode surj_sum_encode]) |
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lemma bij_sum_decode: "bij sum_decode" |
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by (rule bijI [OF inj_sum_decode surj_sum_decode]) |
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lemma sum_encode_eq: "sum_encode x = sum_encode y \<longleftrightarrow> x = y" |
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by (rule inj_sum_encode [THEN inj_eq]) |
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lemma sum_decode_eq: "sum_decode x = sum_decode y \<longleftrightarrow> x = y" |
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by (rule inj_sum_decode [THEN inj_eq]) |
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subsection \<open>Type @{typ "int"}\<close> |
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definition |
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int_encode :: "int \<Rightarrow> nat" |
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where |
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"int_encode i = sum_encode (if 0 \<le> i then Inl (nat i) else Inr (nat (- i - 1)))" |
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definition |
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int_decode :: "nat \<Rightarrow> int" |
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where |
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"int_decode n = (case sum_decode n of Inl a \<Rightarrow> int a | Inr b \<Rightarrow> - int b - 1)" |
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lemma int_encode_inverse [simp]: "int_decode (int_encode x) = x" |
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unfolding int_decode_def int_encode_def by simp |
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lemma int_decode_inverse [simp]: "int_encode (int_decode n) = n" |
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unfolding int_decode_def int_encode_def using sum_decode_inverse [of n] |
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by (cases "sum_decode n", simp_all) |
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lemma inj_int_encode: "inj_on int_encode A" |
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by (rule inj_on_inverseI, rule int_encode_inverse) |
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lemma inj_int_decode: "inj_on int_decode A" |
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by (rule inj_on_inverseI, rule int_decode_inverse) |
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lemma surj_int_encode: "surj int_encode" |
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by (rule surjI, rule int_decode_inverse) |
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lemma surj_int_decode: "surj int_decode" |
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by (rule surjI, rule int_encode_inverse) |
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lemma bij_int_encode: "bij int_encode" |
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by (rule bijI [OF inj_int_encode surj_int_encode]) |
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lemma bij_int_decode: "bij int_decode" |
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by (rule bijI [OF inj_int_decode surj_int_decode]) |
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lemma int_encode_eq: "int_encode x = int_encode y \<longleftrightarrow> x = y" |
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by (rule inj_int_encode [THEN inj_eq]) |
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lemma int_decode_eq: "int_decode x = int_decode y \<longleftrightarrow> x = y" |
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by (rule inj_int_decode [THEN inj_eq]) |
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subsection \<open>Type @{typ "nat list"}\<close> |
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fun |
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list_encode :: "nat list \<Rightarrow> nat" |
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where |
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"list_encode [] = 0" |
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| "list_encode (x # xs) = Suc (prod_encode (x, list_encode xs))" |
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function |
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list_decode :: "nat \<Rightarrow> nat list" |
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where |
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"list_decode 0 = []" |
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| "list_decode (Suc n) = (case prod_decode n of (x, y) \<Rightarrow> x # list_decode y)" |
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by pat_completeness auto |
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termination list_decode |
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apply (relation "measure id", simp_all) |
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apply (drule arg_cong [where f="prod_encode"]) |
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apply (drule sym) |
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apply (simp add: le_imp_less_Suc le_prod_encode_2) |
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done |
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lemma list_encode_inverse [simp]: "list_decode (list_encode x) = x" |
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by (induct x rule: list_encode.induct) simp_all |
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lemma list_decode_inverse [simp]: "list_encode (list_decode n) = n" |
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apply (induct n rule: list_decode.induct, simp) |
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apply (simp split: prod.split) |
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apply (simp add: prod_decode_eq [symmetric]) |
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done |
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lemma inj_list_encode: "inj_on list_encode A" |
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by (rule inj_on_inverseI, rule list_encode_inverse) |
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lemma inj_list_decode: "inj_on list_decode A" |
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by (rule inj_on_inverseI, rule list_decode_inverse) |
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lemma surj_list_encode: "surj list_encode" |
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by (rule surjI, rule list_decode_inverse) |
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lemma surj_list_decode: "surj list_decode" |
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by (rule surjI, rule list_encode_inverse) |
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lemma bij_list_encode: "bij list_encode" |
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by (rule bijI [OF inj_list_encode surj_list_encode]) |
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lemma bij_list_decode: "bij list_decode" |
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by (rule bijI [OF inj_list_decode surj_list_decode]) |
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lemma list_encode_eq: "list_encode x = list_encode y \<longleftrightarrow> x = y" |
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by (rule inj_list_encode [THEN inj_eq]) |
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lemma list_decode_eq: "list_decode x = list_decode y \<longleftrightarrow> x = y" |
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by (rule inj_list_decode [THEN inj_eq]) |
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subsection \<open>Finite sets of naturals\<close> |
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subsubsection \<open>Preliminaries\<close> |
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lemma finite_vimage_Suc_iff: "finite (Suc -` F) \<longleftrightarrow> finite F" |
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apply (safe intro!: finite_vimageI inj_Suc) |
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apply (rule finite_subset [where B="insert 0 (Suc ` Suc -` F)"]) |
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apply (rule subsetI, case_tac x, simp, simp) |
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apply (rule finite_insert [THEN iffD2]) |
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apply (erule finite_imageI) |
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done |
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lemma vimage_Suc_insert_0: "Suc -` insert 0 A = Suc -` A" |
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by auto |
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lemma vimage_Suc_insert_Suc: |
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"Suc -` insert (Suc n) A = insert n (Suc -` A)" |
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by auto |
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lemma div2_even_ext_nat: |
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fixes x y :: nat |
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assumes "x div 2 = y div 2" |
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and "even x \<longleftrightarrow> even y" |
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shows "x = y" |
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proof - |
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from \<open>even x \<longleftrightarrow> even y\<close> have "x mod 2 = y mod 2" |
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by (simp only: even_iff_mod_2_eq_zero) auto |
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with assms have "x div 2 * 2 + x mod 2 = y div 2 * 2 + y mod 2" |
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by simp |
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then show ?thesis |
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by simp |
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qed |
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subsubsection \<open>From sets to naturals\<close> |
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definition |
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set_encode :: "nat set \<Rightarrow> nat" |
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where |
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"set_encode = setsum (op ^ 2)" |
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lemma set_encode_empty [simp]: "set_encode {} = 0" |
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by (simp add: set_encode_def) |
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lemma set_encode_inf: "~ finite A \<Longrightarrow> set_encode A = 0" |
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by (simp add: set_encode_def) |
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lemma set_encode_insert [simp]: |
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"\<lbrakk>finite A; n \<notin> A\<rbrakk> \<Longrightarrow> set_encode (insert n A) = 2^n + set_encode A" |
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by (simp add: set_encode_def) |
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lemma even_set_encode_iff: "finite A \<Longrightarrow> even (set_encode A) \<longleftrightarrow> 0 \<notin> A" |
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unfolding set_encode_def by (induct set: finite, auto) |
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lemma set_encode_vimage_Suc: "set_encode (Suc -` A) = set_encode A div 2" |
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apply (cases "finite A") |
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apply (erule finite_induct, simp) |
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apply (case_tac x) |
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apply (simp add: even_set_encode_iff vimage_Suc_insert_0) |
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apply (simp add: finite_vimageI add.commute vimage_Suc_insert_Suc) |
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apply (simp add: set_encode_def finite_vimage_Suc_iff) |
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done |
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lemmas set_encode_div_2 = set_encode_vimage_Suc [symmetric] |
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subsubsection \<open>From naturals to sets\<close> |
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definition |
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set_decode :: "nat \<Rightarrow> nat set" |
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where |
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"set_decode x = {n. odd (x div 2 ^ n)}" |
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lemma set_decode_0 [simp]: "0 \<in> set_decode x \<longleftrightarrow> odd x" |
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by (simp add: set_decode_def) |
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lemma set_decode_Suc [simp]: |
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"Suc n \<in> set_decode x \<longleftrightarrow> n \<in> set_decode (x div 2)" |
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by (simp add: set_decode_def div_mult2_eq) |
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lemma set_decode_zero [simp]: "set_decode 0 = {}" |
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by (simp add: set_decode_def) |
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lemma set_decode_div_2: "set_decode (x div 2) = Suc -` set_decode x" |
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by auto |
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lemma set_decode_plus_power_2: |
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"n \<notin> set_decode z \<Longrightarrow> set_decode (2 ^ n + z) = insert n (set_decode z)" |
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339 |
proof (induct n arbitrary: z) |
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340 |
case 0 show ?case |
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separate class for division operator, with particular syntax added in more specific classes
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341 |
proof (rule set_eqI) |
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parents:
59506
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342 |
fix q show "q \<in> set_decode (2 ^ 0 + z) \<longleftrightarrow> q \<in> insert 0 (set_decode z)" |
d46de31a50c4
separate class for division operator, with particular syntax added in more specific classes
haftmann
parents:
59506
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343 |
by (induct q) (insert 0, simp_all) |
d46de31a50c4
separate class for division operator, with particular syntax added in more specific classes
haftmann
parents:
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diff
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344 |
qed |
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separate class for division operator, with particular syntax added in more specific classes
haftmann
parents:
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345 |
next |
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separate class for division operator, with particular syntax added in more specific classes
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346 |
case (Suc n) show ?case |
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separate class for division operator, with particular syntax added in more specific classes
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parents:
59506
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|
347 |
proof (rule set_eqI) |
d46de31a50c4
separate class for division operator, with particular syntax added in more specific classes
haftmann
parents:
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348 |
fix q show "q \<in> set_decode (2 ^ Suc n + z) \<longleftrightarrow> q \<in> insert (Suc n) (set_decode z)" |
d46de31a50c4
separate class for division operator, with particular syntax added in more specific classes
haftmann
parents:
59506
diff
changeset
|
349 |
by (induct q) (insert Suc, simp_all) |
d46de31a50c4
separate class for division operator, with particular syntax added in more specific classes
haftmann
parents:
59506
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|
350 |
qed |
d46de31a50c4
separate class for division operator, with particular syntax added in more specific classes
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|
351 |
qed |
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lemma finite_set_decode [simp]: "finite (set_decode n)" |
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354 |
apply (induct n rule: nat_less_induct) |
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355 |
apply (case_tac "n = 0", simp) |
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356 |
apply (drule_tac x="n div 2" in spec, simp) |
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357 |
apply (simp add: set_decode_div_2) |
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apply (simp add: finite_vimage_Suc_iff) |
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359 |
done |
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||
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subsubsection \<open>Proof of isomorphism\<close> |
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363 |
lemma set_decode_inverse [simp]: "set_encode (set_decode n) = n" |
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364 |
apply (induct n rule: nat_less_induct) |
|
365 |
apply (case_tac "n = 0", simp) |
|
366 |
apply (drule_tac x="n div 2" in spec, simp) |
|
367 |
apply (simp add: set_decode_div_2 set_encode_vimage_Suc) |
|
368 |
apply (erule div2_even_ext_nat) |
|
369 |
apply (simp add: even_set_encode_iff) |
|
370 |
done |
|
371 |
||
372 |
lemma set_encode_inverse [simp]: "finite A \<Longrightarrow> set_decode (set_encode A) = A" |
|
373 |
apply (erule finite_induct, simp_all) |
|
374 |
apply (simp add: set_decode_plus_power_2) |
|
375 |
done |
|
376 |
||
377 |
lemma inj_on_set_encode: "inj_on set_encode (Collect finite)" |
|
378 |
by (rule inj_on_inverseI [where g="set_decode"], simp) |
|
379 |
||
380 |
lemma set_encode_eq: |
|
381 |
"\<lbrakk>finite A; finite B\<rbrakk> \<Longrightarrow> set_encode A = set_encode B \<longleftrightarrow> A = B" |
|
382 |
by (rule iffI, simp add: inj_onD [OF inj_on_set_encode], simp) |
|
383 |
||
51414 | 384 |
lemma subset_decode_imp_le: assumes "set_decode m \<subseteq> set_decode n" shows "m \<le> n" |
385 |
proof - |
|
386 |
have "n = m + set_encode (set_decode n - set_decode m)" |
|
387 |
proof - |
|
388 |
obtain A B where "m = set_encode A" "finite A" |
|
389 |
"n = set_encode B" "finite B" |
|
390 |
by (metis finite_set_decode set_decode_inverse) |
|
391 |
thus ?thesis using assms |
|
392 |
apply auto |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
51489
diff
changeset
|
393 |
apply (simp add: set_encode_def add.commute setsum.subset_diff) |
51414 | 394 |
done |
395 |
qed |
|
396 |
thus ?thesis |
|
397 |
by (metis le_add1) |
|
398 |
qed |
|
399 |
||
35700 | 400 |
end |