wenzelm@41959: (* Title: HOL/Library/Nat_Bijection.thy huffman@35700: Author: Brian Huffman huffman@35700: Author: Florian Haftmann huffman@35700: Author: Stefan Richter huffman@35700: Author: Tobias Nipkow huffman@35700: Author: Alexander Krauss huffman@35700: *) huffman@35700: huffman@35700: header {* Bijections between natural numbers and other types *} huffman@35700: huffman@35700: theory Nat_Bijection huffman@35700: imports Main Parity huffman@35700: begin huffman@35700: huffman@35700: subsection {* Type @{typ "nat \ nat"} *} huffman@35700: huffman@35700: text "Triangle numbers: 0, 1, 3, 6, 10, 15, ..." huffman@35700: huffman@35700: definition huffman@35700: triangle :: "nat \ nat" huffman@35700: where huffman@35700: "triangle n = n * Suc n div 2" huffman@35700: huffman@35700: lemma triangle_0 [simp]: "triangle 0 = 0" huffman@35700: unfolding triangle_def by simp huffman@35700: huffman@35700: lemma triangle_Suc [simp]: "triangle (Suc n) = triangle n + Suc n" huffman@35700: unfolding triangle_def by simp huffman@35700: huffman@35700: definition huffman@35700: prod_encode :: "nat \ nat \ nat" huffman@35700: where huffman@35700: "prod_encode = (\(m, n). triangle (m + n) + m)" huffman@35700: huffman@35700: text {* In this auxiliary function, @{term "triangle k + m"} is an invariant. *} huffman@35700: huffman@35700: fun huffman@35700: prod_decode_aux :: "nat \ nat \ nat \ nat" huffman@35700: where huffman@35700: "prod_decode_aux k m = huffman@35700: (if m \ k then (m, k - m) else prod_decode_aux (Suc k) (m - Suc k))" huffman@35700: huffman@35700: declare prod_decode_aux.simps [simp del] huffman@35700: huffman@35700: definition huffman@35700: prod_decode :: "nat \ nat \ nat" huffman@35700: where huffman@35700: "prod_decode = prod_decode_aux 0" huffman@35700: huffman@35700: lemma prod_encode_prod_decode_aux: huffman@35700: "prod_encode (prod_decode_aux k m) = triangle k + m" huffman@35700: apply (induct k m rule: prod_decode_aux.induct) huffman@35700: apply (subst prod_decode_aux.simps) huffman@35700: apply (simp add: prod_encode_def) huffman@35700: done huffman@35700: huffman@35700: lemma prod_decode_inverse [simp]: "prod_encode (prod_decode n) = n" huffman@35700: unfolding prod_decode_def by (simp add: prod_encode_prod_decode_aux) huffman@35700: huffman@35700: lemma prod_decode_triangle_add: huffman@35700: "prod_decode (triangle k + m) = prod_decode_aux k m" huffman@35700: apply (induct k arbitrary: m) huffman@35700: apply (simp add: prod_decode_def) huffman@35700: apply (simp only: triangle_Suc add_assoc) huffman@35700: apply (subst prod_decode_aux.simps, simp) huffman@35700: done huffman@35700: huffman@35700: lemma prod_encode_inverse [simp]: "prod_decode (prod_encode x) = x" huffman@35700: unfolding prod_encode_def huffman@35700: apply (induct x) huffman@35700: apply (simp add: prod_decode_triangle_add) huffman@35700: apply (subst prod_decode_aux.simps, simp) huffman@35700: done huffman@35700: huffman@35700: lemma inj_prod_encode: "inj_on prod_encode A" huffman@35700: by (rule inj_on_inverseI, rule prod_encode_inverse) huffman@35700: huffman@35700: lemma inj_prod_decode: "inj_on prod_decode A" huffman@35700: by (rule inj_on_inverseI, rule prod_decode_inverse) huffman@35700: huffman@35700: lemma surj_prod_encode: "surj prod_encode" huffman@35700: by (rule surjI, rule prod_decode_inverse) huffman@35700: huffman@35700: lemma surj_prod_decode: "surj prod_decode" huffman@35700: by (rule surjI, rule prod_encode_inverse) huffman@35700: huffman@35700: lemma bij_prod_encode: "bij prod_encode" huffman@35700: by (rule bijI [OF inj_prod_encode surj_prod_encode]) huffman@35700: huffman@35700: lemma bij_prod_decode: "bij prod_decode" huffman@35700: by (rule bijI [OF inj_prod_decode surj_prod_decode]) huffman@35700: huffman@35700: lemma prod_encode_eq: "prod_encode x = prod_encode y \ x = y" huffman@35700: by (rule inj_prod_encode [THEN inj_eq]) huffman@35700: huffman@35700: lemma prod_decode_eq: "prod_decode x = prod_decode y \ x = y" huffman@35700: by (rule inj_prod_decode [THEN inj_eq]) huffman@35700: huffman@35700: text {* Ordering properties *} huffman@35700: huffman@35700: lemma le_prod_encode_1: "a \ prod_encode (a, b)" huffman@35700: unfolding prod_encode_def by simp huffman@35700: huffman@35700: lemma le_prod_encode_2: "b \ prod_encode (a, b)" huffman@35700: unfolding prod_encode_def by (induct b, simp_all) huffman@35700: huffman@35700: huffman@35700: subsection {* Type @{typ "nat + nat"} *} huffman@35700: huffman@35700: definition huffman@35700: sum_encode :: "nat + nat \ nat" huffman@35700: where huffman@35700: "sum_encode x = (case x of Inl a \ 2 * a | Inr b \ Suc (2 * b))" huffman@35700: huffman@35700: definition huffman@35700: sum_decode :: "nat \ nat + nat" huffman@35700: where huffman@35700: "sum_decode n = (if even n then Inl (n div 2) else Inr (n div 2))" huffman@35700: huffman@35700: lemma sum_encode_inverse [simp]: "sum_decode (sum_encode x) = x" huffman@35700: unfolding sum_decode_def sum_encode_def huffman@35700: by (induct x) simp_all huffman@35700: huffman@35700: lemma sum_decode_inverse [simp]: "sum_encode (sum_decode n) = n" huffman@35700: unfolding sum_decode_def sum_encode_def numeral_2_eq_2 huffman@35700: by (simp add: even_nat_div_two_times_two odd_nat_div_two_times_two_plus_one huffman@35700: del: mult_Suc) huffman@35700: huffman@35700: lemma inj_sum_encode: "inj_on sum_encode A" huffman@35700: by (rule inj_on_inverseI, rule sum_encode_inverse) huffman@35700: huffman@35700: lemma inj_sum_decode: "inj_on sum_decode A" huffman@35700: by (rule inj_on_inverseI, rule sum_decode_inverse) huffman@35700: huffman@35700: lemma surj_sum_encode: "surj sum_encode" huffman@35700: by (rule surjI, rule sum_decode_inverse) huffman@35700: huffman@35700: lemma surj_sum_decode: "surj sum_decode" huffman@35700: by (rule surjI, rule sum_encode_inverse) huffman@35700: huffman@35700: lemma bij_sum_encode: "bij sum_encode" huffman@35700: by (rule bijI [OF inj_sum_encode surj_sum_encode]) huffman@35700: huffman@35700: lemma bij_sum_decode: "bij sum_decode" huffman@35700: by (rule bijI [OF inj_sum_decode surj_sum_decode]) huffman@35700: huffman@35700: lemma sum_encode_eq: "sum_encode x = sum_encode y \ x = y" huffman@35700: by (rule inj_sum_encode [THEN inj_eq]) huffman@35700: huffman@35700: lemma sum_decode_eq: "sum_decode x = sum_decode y \ x = y" huffman@35700: by (rule inj_sum_decode [THEN inj_eq]) huffman@35700: huffman@35700: huffman@35700: subsection {* Type @{typ "int"} *} huffman@35700: huffman@35700: definition huffman@35700: int_encode :: "int \ nat" huffman@35700: where huffman@35700: "int_encode i = sum_encode (if 0 \ i then Inl (nat i) else Inr (nat (- i - 1)))" huffman@35700: huffman@35700: definition huffman@35700: int_decode :: "nat \ int" huffman@35700: where huffman@35700: "int_decode n = (case sum_decode n of Inl a \ int a | Inr b \ - int b - 1)" huffman@35700: huffman@35700: lemma int_encode_inverse [simp]: "int_decode (int_encode x) = x" huffman@35700: unfolding int_decode_def int_encode_def by simp huffman@35700: huffman@35700: lemma int_decode_inverse [simp]: "int_encode (int_decode n) = n" huffman@35700: unfolding int_decode_def int_encode_def using sum_decode_inverse [of n] huffman@35700: by (cases "sum_decode n", simp_all) huffman@35700: huffman@35700: lemma inj_int_encode: "inj_on int_encode A" huffman@35700: by (rule inj_on_inverseI, rule int_encode_inverse) huffman@35700: huffman@35700: lemma inj_int_decode: "inj_on int_decode A" huffman@35700: by (rule inj_on_inverseI, rule int_decode_inverse) huffman@35700: huffman@35700: lemma surj_int_encode: "surj int_encode" huffman@35700: by (rule surjI, rule int_decode_inverse) huffman@35700: huffman@35700: lemma surj_int_decode: "surj int_decode" huffman@35700: by (rule surjI, rule int_encode_inverse) huffman@35700: huffman@35700: lemma bij_int_encode: "bij int_encode" huffman@35700: by (rule bijI [OF inj_int_encode surj_int_encode]) huffman@35700: huffman@35700: lemma bij_int_decode: "bij int_decode" huffman@35700: by (rule bijI [OF inj_int_decode surj_int_decode]) huffman@35700: huffman@35700: lemma int_encode_eq: "int_encode x = int_encode y \ x = y" huffman@35700: by (rule inj_int_encode [THEN inj_eq]) huffman@35700: huffman@35700: lemma int_decode_eq: "int_decode x = int_decode y \ x = y" huffman@35700: by (rule inj_int_decode [THEN inj_eq]) huffman@35700: huffman@35700: huffman@35700: subsection {* Type @{typ "nat list"} *} huffman@35700: huffman@35700: fun huffman@35700: list_encode :: "nat list \ nat" huffman@35700: where huffman@35700: "list_encode [] = 0" huffman@35700: | "list_encode (x # xs) = Suc (prod_encode (x, list_encode xs))" huffman@35700: huffman@35700: function huffman@35700: list_decode :: "nat \ nat list" huffman@35700: where huffman@35700: "list_decode 0 = []" huffman@35700: | "list_decode (Suc n) = (case prod_decode n of (x, y) \ x # list_decode y)" huffman@35700: by pat_completeness auto huffman@35700: huffman@35700: termination list_decode huffman@35700: apply (relation "measure id", simp_all) huffman@35700: apply (drule arg_cong [where f="prod_encode"]) haftmann@37591: apply (drule sym) huffman@35700: apply (simp add: le_imp_less_Suc le_prod_encode_2) huffman@35700: done huffman@35700: huffman@35700: lemma list_encode_inverse [simp]: "list_decode (list_encode x) = x" huffman@35700: by (induct x rule: list_encode.induct) simp_all huffman@35700: huffman@35700: lemma list_decode_inverse [simp]: "list_encode (list_decode n) = n" huffman@35700: apply (induct n rule: list_decode.induct, simp) huffman@35700: apply (simp split: prod.split) huffman@35700: apply (simp add: prod_decode_eq [symmetric]) huffman@35700: done huffman@35700: huffman@35700: lemma inj_list_encode: "inj_on list_encode A" huffman@35700: by (rule inj_on_inverseI, rule list_encode_inverse) huffman@35700: huffman@35700: lemma inj_list_decode: "inj_on list_decode A" huffman@35700: by (rule inj_on_inverseI, rule list_decode_inverse) huffman@35700: huffman@35700: lemma surj_list_encode: "surj list_encode" huffman@35700: by (rule surjI, rule list_decode_inverse) huffman@35700: huffman@35700: lemma surj_list_decode: "surj list_decode" huffman@35700: by (rule surjI, rule list_encode_inverse) huffman@35700: huffman@35700: lemma bij_list_encode: "bij list_encode" huffman@35700: by (rule bijI [OF inj_list_encode surj_list_encode]) huffman@35700: huffman@35700: lemma bij_list_decode: "bij list_decode" huffman@35700: by (rule bijI [OF inj_list_decode surj_list_decode]) huffman@35700: huffman@35700: lemma list_encode_eq: "list_encode x = list_encode y \ x = y" huffman@35700: by (rule inj_list_encode [THEN inj_eq]) huffman@35700: huffman@35700: lemma list_decode_eq: "list_decode x = list_decode y \ x = y" huffman@35700: by (rule inj_list_decode [THEN inj_eq]) huffman@35700: huffman@35700: huffman@35700: subsection {* Finite sets of naturals *} huffman@35700: huffman@35700: subsubsection {* Preliminaries *} huffman@35700: huffman@35700: lemma finite_vimage_Suc_iff: "finite (Suc -` F) \ finite F" huffman@35700: apply (safe intro!: finite_vimageI inj_Suc) huffman@35700: apply (rule finite_subset [where B="insert 0 (Suc ` Suc -` F)"]) huffman@35700: apply (rule subsetI, case_tac x, simp, simp) huffman@35700: apply (rule finite_insert [THEN iffD2]) huffman@35700: apply (erule finite_imageI) huffman@35700: done huffman@35700: huffman@35700: lemma vimage_Suc_insert_0: "Suc -` insert 0 A = Suc -` A" huffman@35700: by auto huffman@35700: huffman@35700: lemma vimage_Suc_insert_Suc: huffman@35700: "Suc -` insert (Suc n) A = insert n (Suc -` A)" huffman@35700: by auto huffman@35700: huffman@35700: lemma even_nat_Suc_div_2: "even x \ Suc x div 2 = x div 2" huffman@35700: by (simp only: numeral_2_eq_2 even_nat_plus_one_div_two) huffman@35700: huffman@35700: lemma div2_even_ext_nat: huffman@35700: "\x div 2 = y div 2; even x = even y\ \ (x::nat) = y" huffman@35700: apply (rule mod_div_equality [of x 2, THEN subst]) huffman@35700: apply (rule mod_div_equality [of y 2, THEN subst]) huffman@35700: apply (case_tac "even x") huffman@35700: apply (simp add: numeral_2_eq_2 even_nat_equiv_def) huffman@35700: apply (simp add: numeral_2_eq_2 odd_nat_equiv_def) huffman@35700: done huffman@35700: huffman@35700: subsubsection {* From sets to naturals *} huffman@35700: huffman@35700: definition huffman@35700: set_encode :: "nat set \ nat" huffman@35700: where huffman@35700: "set_encode = setsum (op ^ 2)" huffman@35700: huffman@35700: lemma set_encode_empty [simp]: "set_encode {} = 0" huffman@35700: by (simp add: set_encode_def) huffman@35700: huffman@35700: lemma set_encode_insert [simp]: huffman@35700: "\finite A; n \ A\ \ set_encode (insert n A) = 2^n + set_encode A" huffman@35700: by (simp add: set_encode_def) huffman@35700: huffman@35700: lemma even_set_encode_iff: "finite A \ even (set_encode A) \ 0 \ A" huffman@35700: unfolding set_encode_def by (induct set: finite, auto) huffman@35700: huffman@35700: lemma set_encode_vimage_Suc: "set_encode (Suc -` A) = set_encode A div 2" huffman@35700: apply (cases "finite A") huffman@35700: apply (erule finite_induct, simp) huffman@35700: apply (case_tac x) huffman@35700: apply (simp add: even_nat_Suc_div_2 even_set_encode_iff vimage_Suc_insert_0) huffman@35700: apply (simp add: finite_vimageI add_commute vimage_Suc_insert_Suc) huffman@35700: apply (simp add: set_encode_def finite_vimage_Suc_iff) huffman@35700: done huffman@35700: huffman@35700: lemmas set_encode_div_2 = set_encode_vimage_Suc [symmetric] huffman@35700: huffman@35700: subsubsection {* From naturals to sets *} huffman@35700: huffman@35700: definition huffman@35700: set_decode :: "nat \ nat set" huffman@35700: where huffman@35700: "set_decode x = {n. odd (x div 2 ^ n)}" huffman@35700: huffman@35700: lemma set_decode_0 [simp]: "0 \ set_decode x \ odd x" huffman@35700: by (simp add: set_decode_def) huffman@35700: huffman@35700: lemma set_decode_Suc [simp]: huffman@35700: "Suc n \ set_decode x \ n \ set_decode (x div 2)" huffman@35700: by (simp add: set_decode_def div_mult2_eq) huffman@35700: huffman@35700: lemma set_decode_zero [simp]: "set_decode 0 = {}" huffman@35700: by (simp add: set_decode_def) huffman@35700: huffman@35700: lemma set_decode_div_2: "set_decode (x div 2) = Suc -` set_decode x" huffman@35700: by auto huffman@35700: huffman@35700: lemma set_decode_plus_power_2: huffman@35700: "n \ set_decode z \ set_decode (2 ^ n + z) = insert n (set_decode z)" huffman@35700: apply (induct n arbitrary: z, simp_all) nipkow@39302: apply (rule set_eqI, induct_tac x, simp, simp add: even_nat_Suc_div_2) nipkow@39302: apply (rule set_eqI, induct_tac x, simp, simp add: add_commute) huffman@35700: done huffman@35700: huffman@35700: lemma finite_set_decode [simp]: "finite (set_decode n)" huffman@35700: apply (induct n rule: nat_less_induct) huffman@35700: apply (case_tac "n = 0", simp) huffman@35700: apply (drule_tac x="n div 2" in spec, simp) huffman@35700: apply (simp add: set_decode_div_2) huffman@35700: apply (simp add: finite_vimage_Suc_iff) huffman@35700: done huffman@35700: huffman@35700: subsubsection {* Proof of isomorphism *} huffman@35700: huffman@35700: lemma set_decode_inverse [simp]: "set_encode (set_decode n) = n" huffman@35700: apply (induct n rule: nat_less_induct) huffman@35700: apply (case_tac "n = 0", simp) huffman@35700: apply (drule_tac x="n div 2" in spec, simp) huffman@35700: apply (simp add: set_decode_div_2 set_encode_vimage_Suc) huffman@35700: apply (erule div2_even_ext_nat) huffman@35700: apply (simp add: even_set_encode_iff) huffman@35700: done huffman@35700: huffman@35700: lemma set_encode_inverse [simp]: "finite A \ set_decode (set_encode A) = A" huffman@35700: apply (erule finite_induct, simp_all) huffman@35700: apply (simp add: set_decode_plus_power_2) huffman@35700: done huffman@35700: huffman@35700: lemma inj_on_set_encode: "inj_on set_encode (Collect finite)" huffman@35700: by (rule inj_on_inverseI [where g="set_decode"], simp) huffman@35700: huffman@35700: lemma set_encode_eq: huffman@35700: "\finite A; finite B\ \ set_encode A = set_encode B \ A = B" huffman@35700: by (rule iffI, simp add: inj_onD [OF inj_on_set_encode], simp) huffman@35700: huffman@35700: end