# HG changeset patch # User wenzelm # Date 1470564649 -7200 # Node ID 1e7c5bbea36dd2dd56705630f8e456de91b9788a # Parent 994d1a1105ef1679d6352c2a52d17511ee00d81e misc tuning and modernization; diff -r 994d1a1105ef -r 1e7c5bbea36d src/HOL/Library/Nat_Bijection.thy --- 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 \Bijections between natural numbers and other types\ theory Nat_Bijection -imports Main + imports Main begin subsection \Type @{typ "nat \ nat"}\ -text "Triangle numbers: 0, 1, 3, 6, 10, 15, ..." +text \Triangle numbers: 0, 1, 3, 6, 10, 15, ...\ definition triangle :: "nat \ 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 \ nat \ nat" where "prod_encode = (\(m, n). triangle (m + n) + m)" @@ -31,8 +31,7 @@ text \In this auxiliary function, @{term "triangle k + m"} is an invariant.\ fun prod_decode_aux :: "nat \ nat \ nat \ nat" -where - "prod_decode_aux k m = + where "prod_decode_aux k m = (if m \ 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 \ nat \ 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 \ 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 \ x = y" -by (rule inj_prod_decode [THEN inj_eq]) + by (rule inj_prod_decode [THEN inj_eq]) text \Ordering properties\ lemma le_prod_encode_1: "a \ prod_encode (a, b)" -unfolding prod_encode_def by simp + by (simp add: prod_encode_def) lemma le_prod_encode_2: "b \ prod_encode (a, b)" -unfolding prod_encode_def by (induct b, simp_all) + by (induct b) (simp_all add: prod_encode_def) subsection \Type @{typ "nat + nat"}\ definition sum_encode :: "nat + nat \ nat" -where - "sum_encode x = (case x of Inl a \ 2 * a | Inr b \ Suc (2 * b))" + where "sum_encode x = (case x of Inl a \ 2 * a | Inr b \ Suc (2 * b))" definition sum_decode :: "nat \ 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 \ 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 \ x = y" -by (rule inj_sum_decode [THEN inj_eq]) + by (rule inj_sum_decode [THEN inj_eq]) subsection \Type @{typ "int"}\ definition int_encode :: "int \ nat" -where - "int_encode i = sum_encode (if 0 \ i then Inl (nat i) else Inr (nat (- i - 1)))" + where "int_encode i = sum_encode (if 0 \ i then Inl (nat i) else Inr (nat (- i - 1)))" definition int_decode :: "nat \ int" -where - "int_decode n = (case sum_decode n of Inl a \ int a | Inr b \ - int b - 1)" + where "int_decode n = (case sum_decode n of Inl a \ int a | Inr b \ - 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 \ 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 \ x = y" -by (rule inj_int_decode [THEN inj_eq]) + by (rule inj_int_decode [THEN inj_eq]) subsection \Type @{typ "nat list"}\ fun list_encode :: "nat list \ 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 \ nat list" -where - "list_decode 0 = []" -| "list_decode (Suc n) = (case prod_decode n of (x, y) \ x # list_decode y)" -by pat_completeness auto + where + "list_decode 0 = []" + | "list_decode (Suc n) = (case prod_decode n of (x, y) \ 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 \ 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 \ x = y" -by (rule inj_list_decode [THEN inj_eq]) + by (rule inj_list_decode [THEN inj_eq]) subsection \Finite sets of naturals\ @@ -241,24 +238,26 @@ subsubsection \Preliminaries\ lemma finite_vimage_Suc_iff: "finite (Suc -` F) \ 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 \ even y" + and "even x \ even y" shows "x = y" proof - from \even x \ even y\ 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 \ set_encode A = 0" by (simp add: set_encode_def) -lemma set_encode_insert [simp]: - "\finite A; n \ A\ \ set_encode (insert n A) = 2^n + set_encode A" -by (simp add: set_encode_def) +lemma set_encode_inf: "\ finite A \ set_encode A = 0" + by (simp add: set_encode_def) + +lemma set_encode_insert [simp]: "finite A \ n \ A \ set_encode (insert n A) = 2^n + set_encode A" + by (simp add: set_encode_def) lemma even_set_encode_iff: "finite A \ even (set_encode A) \ 0 \ 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 \ set_decode x \ odd x" -by (simp add: set_decode_def) + by (simp add: set_decode_def) -lemma set_decode_Suc [simp]: - "Suc n \ set_decode x \ n \ set_decode (x div 2)" -by (simp add: set_decode_def div_mult2_eq) +lemma set_decode_Suc [simp]: "Suc n \ set_decode x \ n \ 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 \ set_decode z \ 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 \ set_decode (2 ^ 0 + z) \ q \ insert 0 (set_decode z)" - by (induct q) (insert 0, simp_all) + show "q \ set_decode (2 ^ 0 + z) \ q \ 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 \ set_decode (2 ^ Suc n + z) \ q \ insert (Suc n) (set_decode z)" - by (induct q) (insert Suc, simp_all) + show "q \ set_decode (2 ^ Suc n + z) \ q \ 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 \Proof of isomorphism\ 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 \ 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: - "\finite A; finite B\ \ set_encode A = set_encode B \ A = B" -by (rule iffI, simp add: inj_onD [OF inj_on_set_encode], simp) +lemma set_encode_eq: "finite A \ finite B \ set_encode A = set_encode B \ A = B" + by (rule iffI) (simp_all add: inj_onD [OF inj_on_set_encode]) lemma subset_decode_imp_le: assumes "set_decode m \ 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