--- a/src/HOL/Library/Nat_Bijection.thy Sat Aug 06 18:14:59 2016 +0200
+++ b/src/HOL/Library/Nat_Bijection.thy Sun Aug 07 12:10:49 2016 +0200
@@ -9,21 +9,21 @@
section \<open>Bijections between natural numbers and other types\<close>
theory Nat_Bijection
-imports Main
+ imports Main
begin
subsection \<open>Type @{typ "nat \<times> nat"}\<close>
-text "Triangle numbers: 0, 1, 3, 6, 10, 15, ..."
+text \<open>Triangle numbers: 0, 1, 3, 6, 10, 15, ...\<close>
definition triangle :: "nat \<Rightarrow> nat"
where "triangle n = (n * Suc n) div 2"
lemma triangle_0 [simp]: "triangle 0 = 0"
-unfolding triangle_def by simp
+ by (simp add: triangle_def)
lemma triangle_Suc [simp]: "triangle (Suc n) = triangle n + Suc n"
-unfolding triangle_def by simp
+ by (simp add: triangle_def)
definition prod_encode :: "nat \<times> nat \<Rightarrow> nat"
where "prod_encode = (\<lambda>(m, n). triangle (m + n) + m)"
@@ -31,8 +31,7 @@
text \<open>In this auxiliary function, @{term "triangle k + m"} is an invariant.\<close>
fun prod_decode_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat"
-where
- "prod_decode_aux k m =
+ where "prod_decode_aux k m =
(if m \<le> k then (m, k - m) else prod_decode_aux (Suc k) (m - Suc k))"
declare prod_decode_aux.simps [simp del]
@@ -40,200 +39,198 @@
definition prod_decode :: "nat \<Rightarrow> nat \<times> nat"
where "prod_decode = prod_decode_aux 0"
-lemma prod_encode_prod_decode_aux:
- "prod_encode (prod_decode_aux k m) = triangle k + m"
-apply (induct k m rule: prod_decode_aux.induct)
-apply (subst prod_decode_aux.simps)
-apply (simp add: prod_encode_def)
-done
+lemma prod_encode_prod_decode_aux: "prod_encode (prod_decode_aux k m) = triangle k + m"
+ apply (induct k m rule: prod_decode_aux.induct)
+ apply (subst prod_decode_aux.simps)
+ apply (simp add: prod_encode_def)
+ done
lemma prod_decode_inverse [simp]: "prod_encode (prod_decode n) = n"
-unfolding prod_decode_def by (simp add: prod_encode_prod_decode_aux)
+ by (simp add: prod_decode_def prod_encode_prod_decode_aux)
lemma prod_decode_triangle_add: "prod_decode (triangle k + m) = prod_decode_aux k m"
-apply (induct k arbitrary: m)
-apply (simp add: prod_decode_def)
-apply (simp only: triangle_Suc add.assoc)
-apply (subst prod_decode_aux.simps, simp)
-done
+ apply (induct k arbitrary: m)
+ apply (simp add: prod_decode_def)
+ apply (simp only: triangle_Suc add.assoc)
+ apply (subst prod_decode_aux.simps)
+ apply simp
+ done
lemma prod_encode_inverse [simp]: "prod_decode (prod_encode x) = x"
-unfolding prod_encode_def
-apply (induct x)
-apply (simp add: prod_decode_triangle_add)
-apply (subst prod_decode_aux.simps, simp)
-done
+ unfolding prod_encode_def
+ apply (induct x)
+ apply (simp add: prod_decode_triangle_add)
+ apply (subst prod_decode_aux.simps)
+ apply simp
+ done
lemma inj_prod_encode: "inj_on prod_encode A"
-by (rule inj_on_inverseI, rule prod_encode_inverse)
+ by (rule inj_on_inverseI) (rule prod_encode_inverse)
lemma inj_prod_decode: "inj_on prod_decode A"
-by (rule inj_on_inverseI, rule prod_decode_inverse)
+ by (rule inj_on_inverseI) (rule prod_decode_inverse)
lemma surj_prod_encode: "surj prod_encode"
-by (rule surjI, rule prod_decode_inverse)
+ by (rule surjI) (rule prod_decode_inverse)
lemma surj_prod_decode: "surj prod_decode"
-by (rule surjI, rule prod_encode_inverse)
+ by (rule surjI) (rule prod_encode_inverse)
lemma bij_prod_encode: "bij prod_encode"
-by (rule bijI [OF inj_prod_encode surj_prod_encode])
+ by (rule bijI [OF inj_prod_encode surj_prod_encode])
lemma bij_prod_decode: "bij prod_decode"
-by (rule bijI [OF inj_prod_decode surj_prod_decode])
+ by (rule bijI [OF inj_prod_decode surj_prod_decode])
lemma prod_encode_eq: "prod_encode x = prod_encode y \<longleftrightarrow> x = y"
-by (rule inj_prod_encode [THEN inj_eq])
+ by (rule inj_prod_encode [THEN inj_eq])
lemma prod_decode_eq: "prod_decode x = prod_decode y \<longleftrightarrow> x = y"
-by (rule inj_prod_decode [THEN inj_eq])
+ by (rule inj_prod_decode [THEN inj_eq])
text \<open>Ordering properties\<close>
lemma le_prod_encode_1: "a \<le> prod_encode (a, b)"
-unfolding prod_encode_def by simp
+ by (simp add: prod_encode_def)
lemma le_prod_encode_2: "b \<le> prod_encode (a, b)"
-unfolding prod_encode_def by (induct b, simp_all)
+ by (induct b) (simp_all add: prod_encode_def)
subsection \<open>Type @{typ "nat + nat"}\<close>
definition sum_encode :: "nat + nat \<Rightarrow> nat"
-where
- "sum_encode x = (case x of Inl a \<Rightarrow> 2 * a | Inr b \<Rightarrow> Suc (2 * b))"
+ where "sum_encode x = (case x of Inl a \<Rightarrow> 2 * a | Inr b \<Rightarrow> Suc (2 * b))"
definition sum_decode :: "nat \<Rightarrow> nat + nat"
-where
- "sum_decode n = (if even n then Inl (n div 2) else Inr (n div 2))"
+ where "sum_decode n = (if even n then Inl (n div 2) else Inr (n div 2))"
lemma sum_encode_inverse [simp]: "sum_decode (sum_encode x) = x"
-unfolding sum_decode_def sum_encode_def
-by (induct x) simp_all
+ by (induct x) (simp_all add: sum_decode_def sum_encode_def)
lemma sum_decode_inverse [simp]: "sum_encode (sum_decode n) = n"
by (simp add: even_two_times_div_two sum_decode_def sum_encode_def)
lemma inj_sum_encode: "inj_on sum_encode A"
-by (rule inj_on_inverseI, rule sum_encode_inverse)
+ by (rule inj_on_inverseI) (rule sum_encode_inverse)
lemma inj_sum_decode: "inj_on sum_decode A"
-by (rule inj_on_inverseI, rule sum_decode_inverse)
+ by (rule inj_on_inverseI) (rule sum_decode_inverse)
lemma surj_sum_encode: "surj sum_encode"
-by (rule surjI, rule sum_decode_inverse)
+ by (rule surjI) (rule sum_decode_inverse)
lemma surj_sum_decode: "surj sum_decode"
-by (rule surjI, rule sum_encode_inverse)
+ by (rule surjI) (rule sum_encode_inverse)
lemma bij_sum_encode: "bij sum_encode"
-by (rule bijI [OF inj_sum_encode surj_sum_encode])
+ by (rule bijI [OF inj_sum_encode surj_sum_encode])
lemma bij_sum_decode: "bij sum_decode"
-by (rule bijI [OF inj_sum_decode surj_sum_decode])
+ by (rule bijI [OF inj_sum_decode surj_sum_decode])
lemma sum_encode_eq: "sum_encode x = sum_encode y \<longleftrightarrow> x = y"
-by (rule inj_sum_encode [THEN inj_eq])
+ by (rule inj_sum_encode [THEN inj_eq])
lemma sum_decode_eq: "sum_decode x = sum_decode y \<longleftrightarrow> x = y"
-by (rule inj_sum_decode [THEN inj_eq])
+ by (rule inj_sum_decode [THEN inj_eq])
subsection \<open>Type @{typ "int"}\<close>
definition int_encode :: "int \<Rightarrow> nat"
-where
- "int_encode i = sum_encode (if 0 \<le> i then Inl (nat i) else Inr (nat (- i - 1)))"
+ where "int_encode i = sum_encode (if 0 \<le> i then Inl (nat i) else Inr (nat (- i - 1)))"
definition int_decode :: "nat \<Rightarrow> int"
-where
- "int_decode n = (case sum_decode n of Inl a \<Rightarrow> int a | Inr b \<Rightarrow> - int b - 1)"
+ where "int_decode n = (case sum_decode n of Inl a \<Rightarrow> int a | Inr b \<Rightarrow> - int b - 1)"
lemma int_encode_inverse [simp]: "int_decode (int_encode x) = x"
-unfolding int_decode_def int_encode_def by simp
+ by (simp add: int_decode_def int_encode_def)
lemma int_decode_inverse [simp]: "int_encode (int_decode n) = n"
-unfolding int_decode_def int_encode_def using sum_decode_inverse [of n]
-by (cases "sum_decode n", simp_all)
+ unfolding int_decode_def int_encode_def
+ using sum_decode_inverse [of n] by (cases "sum_decode n") simp_all
lemma inj_int_encode: "inj_on int_encode A"
-by (rule inj_on_inverseI, rule int_encode_inverse)
+ by (rule inj_on_inverseI) (rule int_encode_inverse)
lemma inj_int_decode: "inj_on int_decode A"
-by (rule inj_on_inverseI, rule int_decode_inverse)
+ by (rule inj_on_inverseI) (rule int_decode_inverse)
lemma surj_int_encode: "surj int_encode"
-by (rule surjI, rule int_decode_inverse)
+ by (rule surjI) (rule int_decode_inverse)
lemma surj_int_decode: "surj int_decode"
-by (rule surjI, rule int_encode_inverse)
+ by (rule surjI) (rule int_encode_inverse)
lemma bij_int_encode: "bij int_encode"
-by (rule bijI [OF inj_int_encode surj_int_encode])
+ by (rule bijI [OF inj_int_encode surj_int_encode])
lemma bij_int_decode: "bij int_decode"
-by (rule bijI [OF inj_int_decode surj_int_decode])
+ by (rule bijI [OF inj_int_decode surj_int_decode])
lemma int_encode_eq: "int_encode x = int_encode y \<longleftrightarrow> x = y"
-by (rule inj_int_encode [THEN inj_eq])
+ by (rule inj_int_encode [THEN inj_eq])
lemma int_decode_eq: "int_decode x = int_decode y \<longleftrightarrow> x = y"
-by (rule inj_int_decode [THEN inj_eq])
+ by (rule inj_int_decode [THEN inj_eq])
subsection \<open>Type @{typ "nat list"}\<close>
fun list_encode :: "nat list \<Rightarrow> nat"
-where
- "list_encode [] = 0"
-| "list_encode (x # xs) = Suc (prod_encode (x, list_encode xs))"
+ where
+ "list_encode [] = 0"
+ | "list_encode (x # xs) = Suc (prod_encode (x, list_encode xs))"
function list_decode :: "nat \<Rightarrow> nat list"
-where
- "list_decode 0 = []"
-| "list_decode (Suc n) = (case prod_decode n of (x, y) \<Rightarrow> x # list_decode y)"
-by pat_completeness auto
+ where
+ "list_decode 0 = []"
+ | "list_decode (Suc n) = (case prod_decode n of (x, y) \<Rightarrow> x # list_decode y)"
+ by pat_completeness auto
termination list_decode
-apply (relation "measure id", simp_all)
-apply (drule arg_cong [where f="prod_encode"])
-apply (drule sym)
-apply (simp add: le_imp_less_Suc le_prod_encode_2)
-done
+ apply (relation "measure id")
+ apply simp_all
+ apply (drule arg_cong [where f="prod_encode"])
+ apply (drule sym)
+ apply (simp add: le_imp_less_Suc le_prod_encode_2)
+ done
lemma list_encode_inverse [simp]: "list_decode (list_encode x) = x"
-by (induct x rule: list_encode.induct) simp_all
+ by (induct x rule: list_encode.induct) simp_all
lemma list_decode_inverse [simp]: "list_encode (list_decode n) = n"
-apply (induct n rule: list_decode.induct, simp)
-apply (simp split: prod.split)
-apply (simp add: prod_decode_eq [symmetric])
-done
+ apply (induct n rule: list_decode.induct)
+ apply simp
+ apply (simp split: prod.split)
+ apply (simp add: prod_decode_eq [symmetric])
+ done
lemma inj_list_encode: "inj_on list_encode A"
-by (rule inj_on_inverseI, rule list_encode_inverse)
+ by (rule inj_on_inverseI) (rule list_encode_inverse)
lemma inj_list_decode: "inj_on list_decode A"
-by (rule inj_on_inverseI, rule list_decode_inverse)
+ by (rule inj_on_inverseI) (rule list_decode_inverse)
lemma surj_list_encode: "surj list_encode"
-by (rule surjI, rule list_decode_inverse)
+ by (rule surjI) (rule list_decode_inverse)
lemma surj_list_decode: "surj list_decode"
-by (rule surjI, rule list_encode_inverse)
+ by (rule surjI) (rule list_encode_inverse)
lemma bij_list_encode: "bij list_encode"
-by (rule bijI [OF inj_list_encode surj_list_encode])
+ by (rule bijI [OF inj_list_encode surj_list_encode])
lemma bij_list_decode: "bij list_decode"
-by (rule bijI [OF inj_list_decode surj_list_decode])
+ by (rule bijI [OF inj_list_decode surj_list_decode])
lemma list_encode_eq: "list_encode x = list_encode y \<longleftrightarrow> x = y"
-by (rule inj_list_encode [THEN inj_eq])
+ by (rule inj_list_encode [THEN inj_eq])
lemma list_decode_eq: "list_decode x = list_decode y \<longleftrightarrow> x = y"
-by (rule inj_list_decode [THEN inj_eq])
+ by (rule inj_list_decode [THEN inj_eq])
subsection \<open>Finite sets of naturals\<close>
@@ -241,24 +238,26 @@
subsubsection \<open>Preliminaries\<close>
lemma finite_vimage_Suc_iff: "finite (Suc -` F) \<longleftrightarrow> finite F"
-apply (safe intro!: finite_vimageI inj_Suc)
-apply (rule finite_subset [where B="insert 0 (Suc ` Suc -` F)"])
-apply (rule subsetI, case_tac x, simp, simp)
-apply (rule finite_insert [THEN iffD2])
-apply (erule finite_imageI)
-done
+ apply (safe intro!: finite_vimageI inj_Suc)
+ apply (rule finite_subset [where B="insert 0 (Suc ` Suc -` F)"])
+ apply (rule subsetI)
+ apply (case_tac x)
+ apply simp
+ apply simp
+ apply (rule finite_insert [THEN iffD2])
+ apply (erule finite_imageI)
+ done
lemma vimage_Suc_insert_0: "Suc -` insert 0 A = Suc -` A"
-by auto
+ by auto
-lemma vimage_Suc_insert_Suc:
- "Suc -` insert (Suc n) A = insert n (Suc -` A)"
-by auto
+lemma vimage_Suc_insert_Suc: "Suc -` insert (Suc n) A = insert n (Suc -` A)"
+ by auto
lemma div2_even_ext_nat:
fixes x y :: nat
assumes "x div 2 = y div 2"
- and "even x \<longleftrightarrow> even y"
+ and "even x \<longleftrightarrow> even y"
shows "x = y"
proof -
from \<open>even x \<longleftrightarrow> even y\<close> have "x mod 2 = y mod 2"
@@ -276,26 +275,26 @@
where "set_encode = setsum (op ^ 2)"
lemma set_encode_empty [simp]: "set_encode {} = 0"
-by (simp add: set_encode_def)
-
-lemma set_encode_inf: "~ finite A \<Longrightarrow> set_encode A = 0"
by (simp add: set_encode_def)
-lemma set_encode_insert [simp]:
- "\<lbrakk>finite A; n \<notin> A\<rbrakk> \<Longrightarrow> set_encode (insert n A) = 2^n + set_encode A"
-by (simp add: set_encode_def)
+lemma set_encode_inf: "\<not> finite A \<Longrightarrow> set_encode A = 0"
+ by (simp add: set_encode_def)
+
+lemma set_encode_insert [simp]: "finite A \<Longrightarrow> n \<notin> A \<Longrightarrow> set_encode (insert n A) = 2^n + set_encode A"
+ by (simp add: set_encode_def)
lemma even_set_encode_iff: "finite A \<Longrightarrow> even (set_encode A) \<longleftrightarrow> 0 \<notin> A"
-unfolding set_encode_def by (induct set: finite, auto)
+ by (induct set: finite) (auto simp: set_encode_def)
lemma set_encode_vimage_Suc: "set_encode (Suc -` A) = set_encode A div 2"
-apply (cases "finite A")
-apply (erule finite_induct, simp)
-apply (case_tac x)
-apply (simp add: even_set_encode_iff vimage_Suc_insert_0)
-apply (simp add: finite_vimageI add.commute vimage_Suc_insert_Suc)
-apply (simp add: set_encode_def finite_vimage_Suc_iff)
-done
+ apply (cases "finite A")
+ apply (erule finite_induct)
+ apply simp
+ apply (case_tac x)
+ apply (simp add: even_set_encode_iff vimage_Suc_insert_0)
+ apply (simp add: finite_vimageI add.commute vimage_Suc_insert_Suc)
+ apply (simp add: set_encode_def finite_vimage_Suc_iff)
+ done
lemmas set_encode_div_2 = set_encode_vimage_Suc [symmetric]
@@ -306,65 +305,70 @@
where "set_decode x = {n. odd (x div 2 ^ n)}"
lemma set_decode_0 [simp]: "0 \<in> set_decode x \<longleftrightarrow> odd x"
-by (simp add: set_decode_def)
+ by (simp add: set_decode_def)
-lemma set_decode_Suc [simp]:
- "Suc n \<in> set_decode x \<longleftrightarrow> n \<in> set_decode (x div 2)"
-by (simp add: set_decode_def div_mult2_eq)
+lemma set_decode_Suc [simp]: "Suc n \<in> set_decode x \<longleftrightarrow> n \<in> set_decode (x div 2)"
+ by (simp add: set_decode_def div_mult2_eq)
lemma set_decode_zero [simp]: "set_decode 0 = {}"
-by (simp add: set_decode_def)
+ by (simp add: set_decode_def)
lemma set_decode_div_2: "set_decode (x div 2) = Suc -` set_decode x"
-by auto
+ by auto
lemma set_decode_plus_power_2:
"n \<notin> set_decode z \<Longrightarrow> set_decode (2 ^ n + z) = insert n (set_decode z)"
proof (induct n arbitrary: z)
- case 0 show ?case
+ case 0
+ show ?case
proof (rule set_eqI)
- fix q show "q \<in> set_decode (2 ^ 0 + z) \<longleftrightarrow> q \<in> insert 0 (set_decode z)"
- by (induct q) (insert 0, simp_all)
+ show "q \<in> set_decode (2 ^ 0 + z) \<longleftrightarrow> q \<in> insert 0 (set_decode z)" for q
+ by (induct q) (use 0 in simp_all)
qed
next
- case (Suc n) show ?case
+ case (Suc n)
+ show ?case
proof (rule set_eqI)
- fix q show "q \<in> set_decode (2 ^ Suc n + z) \<longleftrightarrow> q \<in> insert (Suc n) (set_decode z)"
- by (induct q) (insert Suc, simp_all)
+ show "q \<in> set_decode (2 ^ Suc n + z) \<longleftrightarrow> q \<in> insert (Suc n) (set_decode z)" for q
+ by (induct q) (use Suc in simp_all)
qed
qed
lemma finite_set_decode [simp]: "finite (set_decode n)"
-apply (induct n rule: nat_less_induct)
-apply (case_tac "n = 0", simp)
-apply (drule_tac x="n div 2" in spec, simp)
-apply (simp add: set_decode_div_2)
-apply (simp add: finite_vimage_Suc_iff)
-done
+ apply (induct n rule: nat_less_induct)
+ apply (case_tac "n = 0")
+ apply simp
+ apply (drule_tac x="n div 2" in spec)
+ apply simp
+ apply (simp add: set_decode_div_2)
+ apply (simp add: finite_vimage_Suc_iff)
+ done
subsubsection \<open>Proof of isomorphism\<close>
lemma set_decode_inverse [simp]: "set_encode (set_decode n) = n"
-apply (induct n rule: nat_less_induct)
-apply (case_tac "n = 0", simp)
-apply (drule_tac x="n div 2" in spec, simp)
-apply (simp add: set_decode_div_2 set_encode_vimage_Suc)
-apply (erule div2_even_ext_nat)
-apply (simp add: even_set_encode_iff)
-done
+ apply (induct n rule: nat_less_induct)
+ apply (case_tac "n = 0")
+ apply simp
+ apply (drule_tac x="n div 2" in spec)
+ apply simp
+ apply (simp add: set_decode_div_2 set_encode_vimage_Suc)
+ apply (erule div2_even_ext_nat)
+ apply (simp add: even_set_encode_iff)
+ done
lemma set_encode_inverse [simp]: "finite A \<Longrightarrow> set_decode (set_encode A) = A"
-apply (erule finite_induct, simp_all)
-apply (simp add: set_decode_plus_power_2)
-done
+ apply (erule finite_induct)
+ apply simp_all
+ apply (simp add: set_decode_plus_power_2)
+ done
lemma inj_on_set_encode: "inj_on set_encode (Collect finite)"
-by (rule inj_on_inverseI [where g="set_decode"], simp)
+ by (rule inj_on_inverseI [where g = "set_decode"]) simp
-lemma set_encode_eq:
- "\<lbrakk>finite A; finite B\<rbrakk> \<Longrightarrow> set_encode A = set_encode B \<longleftrightarrow> A = B"
-by (rule iffI, simp add: inj_onD [OF inj_on_set_encode], simp)
+lemma set_encode_eq: "finite A \<Longrightarrow> finite B \<Longrightarrow> set_encode A = set_encode B \<longleftrightarrow> A = B"
+ by (rule iffI) (simp_all add: inj_onD [OF inj_on_set_encode])
lemma subset_decode_imp_le:
assumes "set_decode m \<subseteq> set_decode n"
@@ -372,15 +376,14 @@
proof -
have "n = m + set_encode (set_decode n - set_decode m)"
proof -
- obtain A B where "m = set_encode A" "finite A"
- "n = set_encode B" "finite B"
+ obtain A B where
+ "m = set_encode A" "finite A"
+ "n = set_encode B" "finite B"
by (metis finite_set_decode set_decode_inverse)
- thus ?thesis using assms
- apply auto
- apply (simp add: set_encode_def add.commute setsum.subset_diff)
- done
+ with assms show ?thesis
+ by auto (simp add: set_encode_def add.commute setsum.subset_diff)
qed
- thus ?thesis
+ then show ?thesis
by (metis le_add1)
qed