| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/Isabelle/Archive-of-Formal-Proofs/thys/WorkerWrapper/ (Sammlung formaler Beweise Version 2026-5©) Datei vom 29.4.2026 mit Größe 17 kB |
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Quelle IPv6.thySprache: Isabelle |
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(* Title: IPv6.thy Authors: Cornelius Diekmann *) theory IPv6 imports IP_Address NumberWang_IPv6 (* include "HOL-Library.Code_Target_Nat" if you need to work with actual numbers.*) begin section ‹IPv6 Addresses› text‹An IPv6 address is basically a 128 bit unsigned integer. RFC 4291, Section type_synonym ipv6addr = "128 word" text‹Conversion between natural numbers and IPv6 adresses› definition nat_of_ipv6addr :: "ipv6addr ==> nat" where "nat_of_ipv6addr a = unat a" definition ipv6addr_of_nat :: "nat ==> ipv6addr" where "ipv6addr_of_nat n = of_nat n" lemma "ipv6addr_of_nat n = word_of_int (int n)" by(simp add: ipv6addr_of_nat_def) text‹The maximum IPv6 address› definition max_ipv6_addr :: "ipv6addr" where "max_ipv6_addr ≡ ipv6addr_of_nat ((2^128) - 1)" lemma max_ipv6_addr_number: "max_ipv6_addr = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF" unfolding max_ipv6_addr_def ipv6addr_of_nat_def by(simp) lemma "max_ipv6_addr = 340282366920938463463374607431768211455" by(fact max_ipv6_addr_number) lemma max_ipv6_addr_max_word: "max_ipv6_addr = - 1" by(simp add: max_ipv6_addr_number) lemma max_ipv6_addr_max: "∀a. a ≤ max_ipv6_addr" by(simp add: max_ipv6_addr_max_word) lemma UNIV_ipv6addrset: "UNIV = {0 .. max_ipv6_addr}" (*not in the simp set, for a reason* by(simp add: max_ipv6_addr_max_word) fastforce text‹identity functions› lemma nat_of_ipv6addr_ipv6addr_of_nat_mod: "nat_of_ipv6addr (ipv6addr_of_nat n) = by (simp add: ipv6addr_of_nat_def nat_of_ipv6addr_def unat_of_nat take_bit_eq_mod) lemma nat_of_ipv6addr_ipv6addr_of_nat: "n ≤ nat_of_ipv6addr max_ipv6_addr ==> nat_of_ipv6addr (ipv6addr_of_nat n) = n" by (simp add: nat_of_ipv6addr_ipv6addr_of_nat_mod max_ipv6_addr_def) lemma ipv6addr_of_nat_nat_of_ipv6addr: "ipv6addr_of_nat (nat_of_ipv6addr addr) = a by(simp add: ipv6addr_of_nat_def nat_of_ipv6addr_def) subsection‹Syntax of IPv6 Adresses› text‹RFC 4291, Section 2.2.: Text Representation of Addresses› text‹Quoting the RFC (note: errata exists):› text_raw‹ \begin{verbatim} 1. The preferred form is x:x:x:x:x:x:x:x, where the 'x's are one to four hexadecimal digits of the eight 16-bit pieces of the address. Examples: ABCD:EF01:2345:6789:ABCD:EF01:2345:6789 2001:DB8:0:0:8:800:200C:417A \end{verbatim} › datatype ipv6addr_syntax = IPv6AddrPreferred "16 word" "16 word" "16 word" "16 word" "16 word" "16 word" "16 word" text_raw‹ \begin{verbatim} 2. [...] In order to make writing addresses containing zero bits easier, a special syntax is available to compress the zeros. The use of "::" indicates one or more groups of 16 bits of zeros. The "::" can only appear once in an address. The "::" can also be used to compress leading or trailing zeros in an address. For example, the following addresses 2001:DB8:0:0:8:800:200C:417A a unicast address FF01:0:0:0:0:0:0:101 a multicast address 0:0:0:0:0:0:0:1 the loopback address 0:0:0:0:0:0:0:0 the unspecified address may be represented as 2001:DB8::8:800:200C:417A a unicast address FF01::101 a multicast address ::1 the loopback address :: the unspecified address \end{verbatim} › (*datatype may take some minutes to load*) datatype ipv6addr_syntax_compressed = ― ‹using @{typ unit} for the omission @{text "::"}. Naming convention of the datatype: The first number is the position where the omission occurs. The second number is the length of the specified address pieces. I.e. `8 minus the second number' pieces are omitted.› IPv6AddrCompressed1_0 unit | IPv6AddrCompressed1_1 unit "16 word" | IPv6AddrCompressed1_2 unit "16 word" "16 word" | IPv6AddrCompressed1_3 unit "16 word" "16 word" "16 word" | IPv6AddrCompressed1_4 unit "16 word" "16 word" "16 word" "16 word" | IPv6AddrCompressed1_5 unit "16 word" "16 word" "16 word" "16 word" "16 word" | IPv6AddrCompressed1_6 unit "16 word" "16 word" "16 word" "16 word" "16 word" "16 word" | IPv6AddrCompressed1_7 unit "16 word" "16 word" "16 word" "16 word" "16 word" "16 word" | IPv6AddrCompressed2_1 "16 word" unit | IPv6AddrCompressed2_2 "16 word" unit "16 word" | IPv6AddrCompressed2_3 "16 word" unit "16 word" "16 word" | IPv6AddrCompressed2_4 "16 word" unit "16 word" "16 word" "16 word" | IPv6AddrCompressed2_5 "16 word" unit "16 word" "16 word" "16 word" "16 word" | IPv6AddrCompressed2_6 "16 word" unit "16 word" "16 word" "16 word" "16 word" "16 word" | IPv6AddrCompressed2_7 "16 word" unit "16 word" "16 word" "16 word" "16 word" "16 word" | IPv6AddrCompressed3_2 "16 word" "16 word" unit | IPv6AddrCompressed3_3 "16 word" "16 word" unit "16 word" | IPv6AddrCompressed3_4 "16 word" "16 word" unit "16 word" "16 word" | IPv6AddrCompressed3_5 "16 word" "16 word" unit "16 word" "16 word" "16 word" | IPv6AddrCompressed3_6 "16 word" "16 word" unit "16 word" "16 word" "16 word" "16 word" | IPv6AddrCompressed3_7 "16 word" "16 word" unit "16 word" "16 word" "16 word" "16 word" | IPv6AddrCompressed4_3 "16 word" "16 word" "16 word" unit | IPv6AddrCompressed4_4 "16 word" "16 word" "16 word" unit "16 word" | IPv6AddrCompressed4_5 "16 word" "16 word" "16 word" unit "16 word" "16 word" | IPv6AddrCompressed4_6 "16 word" "16 word" "16 word" unit "16 word" "16 word" "16 word" | IPv6AddrCompressed4_7 "16 word" "16 word" "16 word" unit "16 word" "16 word" "16 word" | IPv6AddrCompressed5_4 "16 word" "16 word" "16 word" "16 word" unit | IPv6AddrCompressed5_5 "16 word" "16 word" "16 word" "16 word" unit "16 word" | IPv6AddrCompressed5_6 "16 word" "16 word" "16 word" "16 word" unit "16 word" "16 word" | IPv6AddrCompressed5_7 "16 word" "16 word" "16 word" "16 word" unit "16 word" "16 word" | IPv6AddrCompressed6_5 "16 word" "16 word" "16 word" "16 word" "16 word" unit | IPv6AddrCompressed6_6 "16 word" "16 word" "16 word" "16 word" "16 word" unit "16 word" | IPv6AddrCompressed6_7 "16 word" "16 word" "16 word" "16 word" "16 word" unit "16 word" | IPv6AddrCompressed7_6 "16 word" "16 word" "16 word" "16 word" "16 word" "16 word" unit | IPv6AddrCompressed7_7 "16 word" "16 word" "16 word" "16 word" "16 word" "16 word" unit | IPv6AddrCompressed8_7 "16 word" "16 word" "16 word" "16 word" "16 word" "16 word" "16 (*RFC 5952: """ 4. A Recommendation for IPv6 Text Representation 4.2.2. Handling One 16-Bit 0 Field The symbol "::" MUST NOT be used to shorten just one 16-bit 0 field. For example, the representation 2001:db8:0:1:1:1:1:1 is correct, but 2001:db8::1:1:1:1:1 is not correct. """ So we could remove all IPv6AddrCompressed*_7 constructors. But these are `recommendations', we might still see these non-recommended definitions. "[...] all implementations must accept and be able to handle any legitimate RFC 4291 format." *) (*More convenient parser helper function for compressed IPv6 addresses: Input list (from parser): Some 16word \<longrightarrow> address piece None \<longrightarrow> omission '::' Basically, the parser must only do the following (python syntax): split the string which is an ipv6 address at ':' map empty string to None map everything else to Some (string_to_16word str) sanitize empty strings at the start and the end (see toString and parser theories) Example: "1:2:3".split(":") = ['1', '2', '3'] ":2:3:4".split(":") = ['', '2', '3', '4'] ":2::3".split(":") = ['', '2', '', '3'] "1:2:3:".split(":") = ['1', '2', '3', ''] *) definition parse_ipv6_address_compressed :: "((16 word) option) list ==> ipv6addr_s "parse_ipv6_address_compressed as = (case as of [None] ==> Some (IPv6AddrCompressed1_0 ()) | [None, Some a] ==> Some (IPv6AddrCompressed1_1 () a) | [None, Some a, Some b] ==> Some (IPv6AddrCompressed1_2 () a b) | [None, Some a, Some b, Some c] ==> Some (IPv6AddrCompressed1_3 () a b c) | [None, Some a, Some b, Some c, Some d] ==> Some (IPv6AddrCompressed1_4 () a b | [None, Some a, Some b, Some c, Some d, Some e] ==> Some (IPv6AddrCompressed1_5 | [None, Some a, Some b, Some c, Some d, Some e, Some f] ==> Some (IPv6AddrCompr | [None, Some a, Some b, Some c, Some d, Some e, Some f, Some g] ==> Some (IPv6A | [Some a, None] ==> Some (IPv6AddrCompressed2_1 a ()) | [Some a, None, Some b] ==> Some (IPv6AddrCompressed2_2 a () b) | [Some a, None, Some b, Some c] ==> Some (IPv6AddrCompressed2_3 a () b c) | [Some a, None, Some b, Some c, Some d] ==> Some (IPv6AddrCompressed2_4 a () b | [Some a, None, Some b, Some c, Some d, Some e] ==> Some (IPv6AddrCompressed2_5 | [Some a, None, Some b, Some c, Some d, Some e, Some f] ==> Some (IPv6AddrCompr | [Some a, None, Some b, Some c, Some d, Some e, Some f, Some g] ==> Some (IPv6A | [Some a, Some b, None] ==> Some (IPv6AddrCompressed3_2 a b ()) | [Some a, Some b, None, Some c] ==> Some (IPv6AddrCompressed3_3 a b () c) | [Some a, Some b, None, Some c, Some d] ==> Some (IPv6AddrCompressed3_4 a b () | [Some a, Some b, None, Some c, Some d, Some e] ==> Some (IPv6AddrCompressed3_5 | [Some a, Some b, None, Some c, Some d, Some e, Some f] ==> Some (IPv6AddrCompr | [Some a, Some b, None, Some c, Some d, Some e, Some f, Some g] ==> Some (IPv6A | [Some a, Some b, Some c, None] ==> Some (IPv6AddrCompressed4_3 a b c ()) | [Some a, Some b, Some c, None, Some d] ==> Some (IPv6AddrCompressed4_4 a b c ( | [Some a, Some b, Some c, None, Some d, Some e] ==> Some (IPv6AddrCompressed4_5 | [Some a, Some b, Some c, None, Some d, Some e, Some f] ==> Some (IPv6AddrCompr | [Some a, Some b, Some c, None, Some d, Some e, Some f, Some g] ==> Some (IPv6A | [Some a, Some b, Some c, Some d, None] ==> Some (IPv6AddrCompressed5_4 a b c d | [Some a, Some b, Some c, Some d, None, Some e] ==> Some (IPv6AddrCompressed5_5 | [Some a, Some b, Some c, Some d, None, Some e, Some f] ==> Some (IPv6AddrCompr | [Some a, Some b, Some c, Some d, None, Some e, Some f, Some g] ==> Some (IPv6A | [Some a, Some b, Some c, Some d, Some e, None] ==> Some (IPv6AddrCompressed6_5 | [Some a, Some b, Some c, Some d, Some e, None, Some f] ==> Some (IPv6AddrCompr | [Some a, Some b, Some c, Some d, Some e, None, Some f, Some g] ==> Some (IPv6A | [Some a, Some b, Some c, Some d, Some e, Some f, None] ==> Some (IPv6AddrCompr | [Some a, Some b, Some c, Some d, Some e, Some f, None, Some g] ==> Some (IPv6A | [Some a, Some b, Some c, Some d, Some e, Some f, Some g, None] ==> Some (IPv6A | _ ==> None ― ‹invalid ipv6 copressed address.› )" fun ipv6addr_syntax_compressed_to_list :: "ipv6addr_syntax_compressed ==> ((16 word where "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed1_0 _) = [None]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed1_1 () a) = [None, Some a]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed1_2 () a b) = [None, Some a, Some b]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed1_3 () a b c) = [None, Some a, Some b, Some c]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed1_4 () a b c d) = [None, Some a, Some b, Some c, Some d]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed1_5 () a b c d e) = [None, Some a, Some b, Some c, Some d, Some e]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed1_6 () a b c d e f) = [None, Some a, Some b, Some c, Some d, Some e, Some f]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed1_7 () a b c d e f g) = [None, Some a, Some b, Some c, Some d, Some e, Some f, Some g]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed2_1 a ()) = [Some a, None]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed2_2 a () b) = [Some a, None, Some b]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed2_3 a () b c) = [Some a, None, Some b, Some c]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed2_4 a () b c d) = [Some a, None, Some b, Some c, Some d]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed2_5 a () b c d e) = [Some a, None, Some b, Some c, Some d, Some e]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed2_6 a () b c d e f) = [Some a, None, Some b, Some c, Some d, Some e, Some f]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed2_7 a () b c d e f g) = [Some a, None, Some b, Some c, Some d, Some e, Some f, Some g]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed3_2 a b ()) = [Some a, S | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed3_3 a b () c) = [Some a, Some b, None, Some c]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed3_4 a b () c d) = [Some a, Some b, None, Some c, Some d]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed3_5 a b () c d e) = [Some a, Some b, None, Some c, Some d, Some e]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed3_6 a b () c d e f) = [Some a, Some b, None, Some c, Some d, Some e, Some f]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed3_7 a b () c d e f g) = [Some a, Some b, None, Some c, Some d, Some e, Some f, Some g]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed4_3 a b c ()) = [Some a, Some b, Some c, None]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed4_4 a b c () d) = [Some a, Some b, Some c, None, Some d]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed4_5 a b c () d e) = [Some a, Some b, Some c, None, Some d, Some e]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed4_6 a b c () d e f) = [Some a, Some b, Some c, None, Some d, Some e, Some f]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed4_7 a b c () d e f g) = [Some a, Some b, Some c, None, Some d, Some e, Some f, Some g]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed5_4 a b c d ()) = [Some a, Some b, Some c, Some d, None]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed5_5 a b c d () e) = [Some a, Some b, Some c, Some d, None, Some e]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed5_6 a b c d () e f) = [Some a, Some b, Some c, Some d, None, Some e, Some f]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed5_7 a b c d () e f g) = [Some a, Some b, Some c, Some d, None, Some e, Some f, Some g]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed6_5 a b c d e ()) = [Some a, Some b, Some c, Some d, Some e, None]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed6_6 a b c d e () f) = [Some a, Some b, Some c, Some d, Some e, None, Some f]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed6_7 a b c d e () f g) = [Some a, Some b, Some c, Some d, Some e, None, Some f, Some g]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed7_6 a b c d e f ()) = [Some a, Some b, Some c, Some d, Some e, Some f, None]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed7_7 a b c d e f () g) = [Some a, Some b, Some c, Some d, Some e, Some f, None, Some g]" | "ipv6addr_syntax_compressed_to_list (IPv6AddrCompressed8_7 a b c d e f g ()) = [Some a, Some b, Some c, Some d, Some e, Some f, Some g, None]" (*for all ipv6_syntax, there is a corresponding list representation*) lemma parse_ipv6_address_compressed_exists: obtains ss where "parse_ipv6_address_compressed ss = Some ipv6_syntax" proof define ss where "ss = ipv6addr_syntax_compressed_to_list ipv6_syntax" thus "parse_ipv6_address_compressed ss = Some ipv6_syntax" by (cases ipv6_syntax; simp add: parse_ipv6_address_compressed_def) qed lemma parse_ipv6_address_compressed_identity: "parse_ipv6_address_compressed (ipv6addr_syntax_compressed_to_list (ipv6_syntax) by(cases ipv6_syntax; simp add: parse_ipv6_address_compressed_def) lemma parse_ipv6_address_compressed_someE: assumes "parse_ipv6_address_compressed as = Some ipv6" obtains "as = [None]" "ipv6 = (IPv6AddrCompressed1_0 ())" | a where "as = [None, Some a]" "ipv6 = (IPv6AddrCompressed1_1 () a)" | a b where "as = [None, Some a, Some b]" "ipv6 = (IPv6AddrCompressed1_2 () a b)" | a b c where "as = [None, Some a, Some b, Some c]" "ipv6 = (IPv6AddrCompressed1_3 () a a b c d where "as = [None, Some a, Some b, Some c, Some d]" "ipv6 = (IPv6AddrCompresse a b c d e where "as = [None, Some a, Some b, Some c, Some d, Some e]" "ipv6 = (IPv6Addr a b c d e f where "as = [None, Some a, Some b, Some c, Some d, Some e, Some f]" "ipv6 = a b c d e f g where "as = [None, Some a, Some b, Some c, Some d, Some e, Some f, Some g] a where "as = [Some a, None]" "ipv6 = (IPv6AddrCompressed2_1 a ())" | a b where "as = [Some a, None, Some b]" "ipv6 = (IPv6AddrCompressed2_2 a () b)" | a b c where "as = [Some a, None, Some b, Some c]" "ipv6 = (IPv6AddrCompressed2_3 a () a b c d where "as = [Some a, None, Some b, Some c, Some d]" "ipv6 = (IPv6AddrCompresse a b c d e where "as = [Some a, None, Some b, Some c, Some d, Some e]" "ipv6 = (IPv6Addr a b c d e f where "as = [Some a, None, Some b, Some c, Some d, Some e, Some f]" "ipv6 = a b c d e f g where "as = [Some a, None, Some b, Some c, Some d, Some e, Some f, Some g] a b where "as = [Some a, Some b, None]" "ipv6 = (IPv6AddrCompressed3_2 a b ())" | a b c where "as = [Some a, Some b, None, Some c]" "ipv6 = (IPv6AddrCompressed3_3 a b a b c d where "as = [Some a, Some b, None, Some c, Some d]" "ipv6 = (IPv6AddrCompresse a b c d e where "as = [Some a, Some b, None, Some c, Some d, Some e]" "ipv6 = (IPv6Addr a b c d e f where "as = [Some a, Some b, None, Some c, Some d, Some e, Some f]" "ipv6 = a b c d e f g where "as = [Some a, Some b, None, Some c, Some d, Some e, Some f, Some g] a b c where "as = [Some a, Some b, Some c, None]" "ipv6 = (IPv6AddrCompressed4_3 a b a b c d where "as = [Some a, Some b, Some c, None, Some d]" "ipv6 = (IPv6AddrCompresse a b c d e where "as = [Some a, Some b, Some c, None, Some d, Some e]" "ipv6 = (IPv6Addr a b c d e f where "as = [Some a, Some b, Some c, None, Some d, Some e, Some f]" "ipv6 = a b c d e f g where "as = [Some a, Some b, Some c, None, Some d, Some e, Some f, Some g] a b c d where "as = [Some a, Some b, Some c, Some d, None]" "ipv6 = (IPv6AddrCompresse a b c d e where "as = [Some a, Some b, Some c, Some d, None, Some e]" "ipv6 = (IPv6Addr a b c d e f where "as = [Some a, Some b, Some c, Some d, None, Some e, Some f]" "ipv6 = a b c d e f g where "as = [Some a, Some b, Some c, Some d, None, Some e, Some f, Some g] a b c d e where "as = [Some a, Some b, Some c, Some d, Some e, None]" "ipv6 = (IPv6Addr a b c d e f where "as = [Some a, Some b, Some c, Some d, Some e, None, Some f]" "ipv6 = a b c d e f g where "as = [Some a, Some b, Some c, Some d, Some e, None, Some f, Some g] a b c d e f where "as = [Some a, Some b, Some c, Some d, Some e, Some f, None]" "ipv6 = a b c d e f g where "as = [Some a, Some b, Some c, Some d, Some e, Some f, None, Some g] a b c d e f g where "as = [Some a, Some b, Some c, Some d, Some e, Some f, Some g, None] using assms unfolding parse_ipv6_address_compressed_def by (auto split: list.split_asm option.split_asm) (* takes a minute *) lemma parse_ipv6_address_compressed_identity2: "ipv6addr_syntax_compressed_to_list ipv6_syntax = ls ⟷ (parse_ipv6_address_compressed ls) = Some ipv6_syntax" (is "?lhs = ?rhs") proof assume ?rhs thus ?lhs by (auto elim: parse_ipv6_address_compressed_someE) next assume ?lhs thus ?rhs by (cases ipv6_syntax) (auto simp: parse_ipv6_address_compressed_def) qed text‹Valid IPv6 compressed notation: ▪ at most one omission ▪ at most 7 pieces › lemma RFC_4291_format: "parse_ipv6_address_compressed as ≠ None ⟷ length (filter (λp. p = None) as) = 1 ∧ length (filter (λp. p ≠ None) as) ≤ 7" (is "?lhs = ?rhs") proof assume ?lhs then obtain addr where "parse_ipv6_address_compressed as = Some addr" by blast thus ?rhs by (elim parse_ipv6_address_compressed_someE; simp) next assume ?rhs thus ?lhs unfolding parse_ipv6_address_compressed_def by (auto split: option.split list.split if_split_asm) qed text_raw‹ \begin{verbatim} 3. An alternative form that is sometimes more convenient when dealing with a mixed environment of IPv4 and IPv6 nodes is x:x:x:x:x:x:d.d.d.d, where the 'x's are the hexadecimal values of the six high-order 16-bit pieces of the address, and the 'd's are the decimal values of the four low-order 8-bit pieces of the address (standard IPv4 representation). Examples: 0:0:0:0:0:0:13.1.68.3 0:0:0:0:0:FFFF:129.144.52.38 or in compressed form: ::13.1.68.3 ::FFFF:129.144.52.38 \end{verbatim} This is currently not supported by our library! › (*TODO*) (*TODO: oh boy, they can also be compressed*) subsection‹Semantics› context includes bit_operations_syntax begin fun ipv6preferred_to_int :: "ipv6addr_syntax ==> ipv6addr" where "ipv6preferred_to_int (IPv6AddrPreferred a b c d e f g h) = (ucast a << (16 * 7)) (ucast b << (16 * 6)) OR (ucast c << (16 * 5)) OR (ucast d << (16 * 4)) OR (ucast e << (16 * 3)) OR (ucast f << (16 * 2)) OR (ucast g << (16 * 1)) OR (ucast h << (16 * 0))" lemma "ipv6preferred_to_int (IPv6AddrPreferred 0x2001 0xDB8 0x0 0x0 0x8 0x800 0x2 42540766411282592856906245548098208122" by eval lemma "ipv6preferred_to_int (IPv6AddrPreferred 0xFF01 0x0 0x0 0x0 0x0 0x0 0x0 0x1 338958331222012082418099330867817087233" by eval declare ipv6preferred_to_int.simps[simp del] definition int_to_ipv6preferred :: "ipv6addr ==> ipv6addr_syntax" where "int_to_ipv6preferred i = IPv6AddrPreferred (ucast ((i AND 0xFFFF000000000000000 (ucast ((i AND 0xFFFF000000000000000000000000) >> 16*6)) (ucast ((i AND 0xFFFF00000000000000000000) >> 16*5)) (ucast ((i AND 0xFFFF0000000000000000) >> 16*4)) (ucast ((i AND 0xFFFF000000000000) >> 16*3)) (ucast ((i AND 0xFFFF00000000) >> 16*2)) (ucast ((i AND 0xFFFF0000) >> 16*1)) (ucast ((i AND 0xFFFF)))" lemma "int_to_ipv6preferred 42540766411282592856906245548098208122 = IPv6AddrPreferred 0x2001 0xDB8 0x0 0x0 0x8 0x800 0x200C 0x417A" by eval lemma word128_masks_ipv6pieces: "(0xFFFF0000000000000000000000000000::ipv6addr) = (mask 16) << 112" "(0xFFFF000000000000000000000000::ipv6addr) = (mask 16) << 96" "(0xFFFF00000000000000000000::ipv6addr) = (mask 16) << 80" "(0xFFFF0000000000000000::ipv6addr) = (mask 16) << 64" "(0xFFFF000000000000::ipv6addr) = (mask 16) << 48" "(0xFFFF00000000::ipv6addr) = (mask 16) << 32" "(0xFFFF0000::ipv6addr) = (mask 16) << 16" "(0xFFFF::ipv6addr) = (mask 16)" by (simp_all add: mask_eq) text‹Correctness: round trip property one› lemma ipv6preferred_to_int_int_to_ipv6preferred: "ipv6preferred_to_int (int_to_ipv6preferred ip) = ip" proof - have and_mask_shift_helper: "w AND (mask m << n) >> n << n = w AND (mask m << n)" for m n::nat and w::ipv6addr by (metis is_aligned_shift is_aligned_shiftr_shiftl shiftr_and_eq_shiftl) have ucast_ipv6_piece_rule: "length (dropWhile Not (to_bl w)) ≤ 16 ==> (ucast::16 word ==> 128 word) ((ucast for w::ipv6addr by(rule ucast_short_ucast_long_ingoreLeadingZero) (simp_all) have ucast_ipv6_piece: "16 ≤ 128 - n ==> (ucast::16 word ==> 128 word) ((ucast::128 word ==> 16 word) (w AND (mask 16 << n) for w::ipv6addr and n::nat apply(subst ucast_ipv6_piece_rule) apply(rule length_drop_mask_inner) apply(simp; fail) apply(subst and_mask_shift_helper) apply simp done have ucast16_ucast128_masks_highest_bits: "(ucast ((ucast::ipv6addr ==> 16 word) (ip AND 0xFFFF000000000000000000000000000 (ip AND 0xFFFF0000000000000000000000000000)" "(ucast ((ucast::ipv6addr ==> 16 word) (ip AND 0xFFFF000000000000000000000000 >> ip AND 0xFFFF000000000000000000000000" "(ucast ((ucast::ipv6addr ==> 16 word) (ip AND 0xFFFF00000000000000000000 >> 80) ip AND 0xFFFF00000000000000000000" "(ucast ((ucast::ipv6addr ==> 16 word) (ip AND 0xFFFF0000000000000000 >> 64)) << 6 ip AND 0xFFFF0000000000000000" "(ucast ((ucast::ipv6addr ==> 16 word) (ip AND 0xFFFF000000000000 >> 48)) << 48) = ip AND 0xFFFF000000000000" "(ucast ((ucast::ipv6addr ==> 16 word) (ip AND 0xFFFF00000000 >> 32)) << 32) = ip AND 0xFFFF00000000" "(ucast ((ucast::ipv6addr ==> 16 word) (ip AND 0xFFFF0000 >> 16)) << 16) = ip AND 0xFFFF0000" apply (simp_all only: word128_masks_ipv6pieces ucast_ipv6_piece and_mask2 word_size bit apply auto done have ucast16_ucast128_masks_highest_bits0: "(ucast ((ucast::ipv6addr ==> 16 word) (ip AND 0xFFFF))) = ip AND 0xFFFF" apply (simp only: word128_masks_ipv6pieces flip: take_bit_eq_mask) apply (simp add: unsigned_ucast_eq) done have mask_len_word:"n = (LENGTH('a)) ==> w AND mask n = w" for n and w::"'a::len word" by (simp add: mask_eq_iff) have ipv6addr_16word_pieces_compose_or: "ip && (mask 16 << 112) || ip && (mask 16 << 96) || ip && (mask 16 << 80) || ip && (mask 16 << 64) || ip && (mask 16 << 48) || ip && (mask 16 << 32) || ip && (mask 16 << 16) || ip && mask 16 = ip" apply(subst word_ao_dist2[symmetric])+ apply(simp add: mask_numeral) apply(subst mask128) apply(rule mask_len_word) apply simp done show ?thesis apply (simp add: ipv6preferred_to_int.simps int_to_ipv6preferred_def shiftl_def shiftr apply (simp only: word128_masks_ipv6pieces flip: take_bit_eq_mask) apply (simp add: unsigned_ucast_eq push_bit_take_bit) using ipv6addr_16word_pieces_compose_or apply (simp add: take_bit_push_bit slice_eq_mask) apply (simp add: take_bit_eq_mask shiftl_def push_bit_mask_eq) done qed text‹Correctness: round trip property two› lemma int_to_ipv6preferred_ipv6preferred_to_int: "int_to_ipv6preferred (ipv6prefer proof - note ucast_shift_simps=helper_masked_ucast_generic helper_masked_ucast_reverse_gen helper_masked_ucast_generic[where n=0, simplified] helper_masked_ucast_equal_generic note ucast_simps=helper_masked_ucast_reverse_generic[where m=0, simplified] helper_masked_ucast_equal_generic[where n=0, simplified] show ?thesis apply (cases ip, rename_tac a b c d e f g h) apply (simp add: ipv6preferred_to_int.simps int_to_ipv6preferred_def) apply (simp add: word128_masks_ipv6pieces) apply (simp add: word_ao_dist ucast_shift_simps ucast_simps) done qed text‹compressed to preferred format› fun ipv6addr_c2p :: "ipv6addr_syntax_compressed ==> ipv6addr_syntax" where "ipv6addr_c2p (IPv6AddrCompressed1_0 ()) = IPv6AddrPreferred 0 0 0 0 0 0 0 0" | "ipv6addr_c2p (IPv6AddrCompressed1_1 () h) = IPv6AddrPreferred 0 0 0 0 0 0 0 h" | "ipv6addr_c2p (IPv6AddrCompressed1_2 () g h) = IPv6AddrPreferred 0 0 0 0 0 0 g | "ipv6addr_c2p (IPv6AddrCompressed1_3 () f g h) = IPv6AddrPreferred 0 0 0 0 0 f | "ipv6addr_c2p (IPv6AddrCompressed1_4 () e f g h) = IPv6AddrPreferred 0 0 0 0 e | "ipv6addr_c2p (IPv6AddrCompressed1_5 () d e f g h) = IPv6AddrPreferred 0 0 0 d | "ipv6addr_c2p (IPv6AddrCompressed1_6 () c d e f g h) = IPv6AddrPreferred 0 0 c | "ipv6addr_c2p (IPv6AddrCompressed1_7 () b c d e f g h) = IPv6AddrPreferred 0 b | "ipv6addr_c2p (IPv6AddrCompressed2_1 a ()) = IPv6AddrPreferred a 0 0 0 0 0 0 0" | "ipv6addr_c2p (IPv6AddrCompressed2_2 a () h) = IPv6AddrPreferred a 0 0 0 0 0 0 | "ipv6addr_c2p (IPv6AddrCompressed2_3 a () g h) = IPv6AddrPreferred a 0 0 0 0 0 | "ipv6addr_c2p (IPv6AddrCompressed2_4 a () f g h) = IPv6AddrPreferred a 0 0 0 0 | "ipv6addr_c2p (IPv6AddrCompressed2_5 a () e f g h) = IPv6AddrPreferred a 0 0 0 | "ipv6addr_c2p (IPv6AddrCompressed2_6 a () d e f g h) = IPv6AddrPreferred a 0 0 | "ipv6addr_c2p (IPv6AddrCompressed2_7 a () c d e f g h) = IPv6AddrPreferred a 0 | "ipv6addr_c2p (IPv6AddrCompressed3_2 a b ()) = IPv6AddrPreferred a b 0 0 0 0 0 | "ipv6addr_c2p (IPv6AddrCompressed3_3 a b () h) = IPv6AddrPreferred a b 0 0 0 0 | "ipv6addr_c2p (IPv6AddrCompressed3_4 a b () g h) = IPv6AddrPreferred a b 0 0 0 | "ipv6addr_c2p (IPv6AddrCompressed3_5 a b () f g h) = IPv6AddrPreferred a b 0 0 | "ipv6addr_c2p (IPv6AddrCompressed3_6 a b () e f g h) = IPv6AddrPreferred a b 0 | "ipv6addr_c2p (IPv6AddrCompressed3_7 a b () d e f g h) = IPv6AddrPreferred a b | "ipv6addr_c2p (IPv6AddrCompressed4_3 a b c ()) = IPv6AddrPreferred a b c 0 0 0 | "ipv6addr_c2p (IPv6AddrCompressed4_4 a b c () h) = IPv6AddrPreferred a b c 0 0 | "ipv6addr_c2p (IPv6AddrCompressed4_5 a b c () g h) = IPv6AddrPreferred a b c 0 | "ipv6addr_c2p (IPv6AddrCompressed4_6 a b c () f g h) = IPv6AddrPreferred a b c | "ipv6addr_c2p (IPv6AddrCompressed4_7 a b c () e f g h) = IPv6AddrPreferred a b | "ipv6addr_c2p (IPv6AddrCompressed5_4 a b c d ()) = IPv6AddrPreferred a b c d 0 | "ipv6addr_c2p (IPv6AddrCompressed5_5 a b c d () h) = IPv6AddrPreferred a b c d | "ipv6addr_c2p (IPv6AddrCompressed5_6 a b c d () g h) = IPv6AddrPreferred a b c | "ipv6addr_c2p (IPv6AddrCompressed5_7 a b c d () f g h) = IPv6AddrPreferred a b | "ipv6addr_c2p (IPv6AddrCompressed6_5 a b c d e ()) = IPv6AddrPreferred a b c d | "ipv6addr_c2p (IPv6AddrCompressed6_6 a b c d e () h) = IPv6AddrPreferred a b c | "ipv6addr_c2p (IPv6AddrCompressed6_7 a b c d e () g h) = IPv6AddrPreferred a b | "ipv6addr_c2p (IPv6AddrCompressed7_6 a b c d e f ()) = IPv6AddrPreferred a b c | "ipv6addr_c2p (IPv6AddrCompressed7_7 a b c d e f () h) = IPv6AddrPreferred a b | "ipv6addr_c2p (IPv6AddrCompressed8_7 a b c d e f g ()) = IPv6AddrPreferred a b definition ipv6_unparsed_compressed_to_preferred :: "((16 word) option) list ==> ip "ipv6_unparsed_compressed_to_preferred ls = ( if length (filter (λp. p = None) ls) ≠ 1 ∨ length (filter (λp. p ≠ None) ls) > 7 then None else let before_omission = map the (takeWhile (λx. x ≠ None) ls); after_omission = map the (drop 1 (dropWhile (λx. x ≠ None) ls)); num_omissions = 8 - (length before_omission + length after_omission); expanded = before_omission @ (replicate num_omissions 0) @ after_omission in case expanded of [a,b,c,d,e,f,g,h] ==> Some (IPv6AddrPreferred a b c d e f g h) | _ ==> None )" lemma "ipv6_unparsed_compressed_to_preferred [Some 0x2001, Some 0xDB8, None, Some 0x8, Some 0x800, Some 0x200C, Some 0x417A] = Some (IPv6AddrPreferred 0x2001 0xDB8 0 0 8 0x800 0x200C 0x417A)" by eval lemma "ipv6_unparsed_compressed_to_preferred [None] = Some (IPv6AddrPreferred 0 0 lemma "ipv6_unparsed_compressed_to_preferred [] = None" by eval lemma ipv6_unparsed_compressed_to_preferred_identity1: "ipv6_unparsed_compressed_to_preferred (ipv6addr_syntax_compressed_to_list ipv6c ⟷ ipv6addr_c2p ipv6compressed = ipv6prferred" by (cases ipv6compressed) (simp_all add: ipv6_unparsed_compressed_to_preferred_def num lemma ipv6_unparsed_compressed_to_preferred_identity2: "ipv6_unparsed_compressed_to_preferred ls = Some ipv6prferred ⟷ (∃ipv6compressed. parse_ipv6_address_compressed ls = Some ipv6compressed ∧ ipv6addr_c2p ipv6compressed = ipv6prferred)" apply(rule iffI) apply(subgoal_tac "parse_ipv6_address_compressed ls ≠ None") prefer 2 apply(subst RFC_4291_format) apply(simp add: ipv6_unparsed_compressed_to_preferred_def split: if_split_asm; fail) apply(simp) apply(erule exE, rename_tac ipv6compressed) apply(rule_tac x="ipv6compressed" in exI) apply(simp) apply(subgoal_tac "(ipv6addr_syntax_compressed_to_list ipv6compressed = ls)") prefer 2 using parse_ipv6_address_compressed_identity2 apply presburger using ipv6_unparsed_compressed_to_preferred_identity1 apply blast apply(erule exE, rename_tac ipv6compressed) apply(subgoal_tac "(ipv6addr_syntax_compressed_to_list ipv6compressed = ls)") prefer 2 using parse_ipv6_address_compressed_identity2 apply presburger using ipv6_unparsed_compressed_to_preferred_identity1 apply blast done end subsection‹IPv6 Pretty Printing (converting to compressed format)› text_raw‹ : begin{verbatim} . A Recommendation for IPv6 Text Representation A recommendation for a canonical text representation format of IPv6 addresses is presented in this section. The recommendation in this document is one that complies fully with [RFC4291], is implemented by various operating systems, and is human friendly. The recommendation in this section SHOULD be followed by systems when generating an address to be represented as text, but all implementations MUST accept and be able to handle any legitimate [RFC4291] format. It is advised that humans also follow these recommendations when spelling an address. .1. Handling Leading Zeros in a 16-Bit Field Leading zeros MUST be suppressed. For example, 2001:0db8::0001 is not acceptable and must be represented as 2001:db8::1. A single 16- bit 0000 field MUST be represented as 0. .2. "::" Usage .2.1. Shorten as Much as Possible The use of the symbol "::" MUST be used to its maximum capability. For example, 2001:db8:0:0:0:0:2:1 must be shortened to 2001:db8::2:1. Likewise, 2001:db8::0:1 is not acceptable, because the symbol "::" could have been used to produce a shorter representation 2001:db8::1. .2.2. Handling One 16-Bit 0 Field The symbol "::" MUST NOT be used to shorten just one 16-bit 0 field. For example, the representation 2001:db8:0:1:1:1:1:1 is correct, but 2001:db8::1:1:1:1:1 is not correct. .2.3. Choice in Placement of "::" When there is an alternative choice in the placement of a "::", the longest run of consecutive 16-bit 0 fields MUST be shortened (i.e., the sequence with three consecutive zero fields is shortened in 2001: 0:0:1:0:0:0:1). When the length of the consecutive 16-bit 0 fields are equal (i.e., 2001:db8:0:0:1:0:0:1), the first sequence of zero bits MUST be shortened. For example, 2001:db8::1:0:0:1 is correct representation. .3. Lowercase The characters "a", "b", "c", "d", "e", and "f" in an IPv6 address MUST be represented in lowercase. end{verbatim} › text‹See @{file ‹IP_Address_toString.thy›} for examples and test cases.› context begin private function goup_by_zeros :: "16 word list ==> 16 word list list" where "goup_by_zeros [] = []" | "goup_by_zeros (x#xs) = ( if x = 0 then takeWhile (λx. x = 0) (x#xs) # (goup_by_zeros (dropWhile (λx. x = 0) xs)) else [x]#(goup_by_zeros xs))" by(pat_completeness, auto) termination goup_by_zeros apply(relation "measure (λxs. length xs)") apply(simp_all) by (simp add: le_imp_less_Suc length_dropWhile_le) private lemma "goup_by_zeros [0,1,2,3,0,0,0,0,3,4,0,0,0,2,0,0,2,0,3,0] = [[0], [1], [2], [3], [0, 0, 0, 0], [3], [4], [0, 0, 0], [2], [0, 0], [2] by eval private lemma "concat (goup_by_zeros ls) = ls" by(induction ls rule:goup_by_zeros.induct) simp+ private lemma "[] ∉ set (goup_by_zeros ls)" by(induction ls rule:goup_by_zeros.induct) simp+ private primrec List_replace1 :: "'a ==> 'a ==> 'a list ==> 'a list" where "List_replace1 _ _ [] = []" | "List_replace1 a b (x#xs) = (if a = x then b#xs else x#List_replace1 a b xs)" private lemma "List_replace1 a a ls = ls" by(induction ls) simp_all private lemma "a ∉ set ls ==> List_replace1 a b ls = ls" by(induction ls) simp_all private lemma "a ∈ set ls ==> b ∈ set (List_replace1 a b ls)" apply(induction ls) apply(simp) apply(simp) by blast private fun List_explode :: "'a list list ==> ('a option) list" where "List_explode [] = []" | "List_explode ([]#xs) = None#List_explode xs" | "List_explode (xs1#xs2) = map Some xs1@List_explode xs2" private lemma "List_explode [[0::int], [2,3], [], [3,4]] = [Some 0, Some 2, Some 3 by eval private lemma List_explode_def: "List_explode xss = concat (map (λxs. if xs = [] then [None] else map Some xs) xs by(induction xss rule: List_explode.induct) simp+ private lemma List_explode_no_empty: "[] ∉ set xss ==> List_explode xss = map Some by(induction xss rule: List_explode.induct) simp+ private lemma List_explode_replace1: "[] ∉ set xss ==> foo ∈ set xss ==> List_explode (List_replace1 foo [] xss) = map Some (concat (takeWhile (λxs. xs ≠ foo) xss)) @ [None] @ map Some (concat (tl (dropWhile (λxs. xs ≠ foo) xss)))" apply(induction xss rule: List_explode.induct) apply(simp; fail) apply(simp; fail) apply(simp) apply safe apply(simp_all add: List_explode_no_empty) done fun ipv6_preferred_to_compressed :: "ipv6addr_syntax ==> ((16 word) option) list" wh "ipv6_preferred_to_compressed (IPv6AddrPreferred a b c d e f g h) = ( let lss = goup_by_zeros [a,b,c,d,e,f,g,h]; max_zero_seq = foldr (λxs. max (length xs)) lss 0; shortened = if max_zero_seq > 1 then List_replace1 (replicate max_zero_seq 0) [] in List_explode shortened )" declare ipv6_preferred_to_compressed.simps[simp del] private lemma foldr_max_length: "foldr (λxs. max (length xs)) lss n = fold max (map apply(subst List.foldr_fold) apply fastforce apply(induction lss arbitrary: n) apply(simp; fail) apply(simp) done private lemma List_explode_goup_by_zeros: "List_explode (goup_by_zeros xs) = map So apply(induction xs rule: goup_by_zeros.induct) apply(simp; fail) apply(simp) apply(safe) apply(simp) by (metis map_append takeWhile_dropWhile_id) private definition "max_zero_streak xs ≡ foldr (λxs. max (length xs)) (goup_by_zero private lemma max_zero_streak_def2: "max_zero_streak xs = fold max (map length (gou unfolding max_zero_streak_def by(simp add: foldr_max_length) private lemma ipv6_preferred_to_compressed_pull_out_if: "ipv6_preferred_to_compressed (IPv6AddrPreferred a b c d e f g h) = ( if max_zero_streak [a,b,c,d,e,f,g,h] > 1 then List_explode (List_replace1 (replicate (max_zero_streak [a,b,c,d,e,f,g,h]) 0) [] else map Some [a,b,c,d,e,f,g,h] )" apply (simp only: ipv6_preferred_to_compressed.simps Let_def max_zero_streak_def List_ using List_explode_goup_by_zeros by presburger private lemma "ipv6_preferred_to_compressed (IPv6AddrPreferred 0 0 0 0 0 0 0 0) = private lemma "ipv6_preferred_to_compressed (IPv6AddrPreferred 0x2001 0xDB8 0 0 8 [Some 0x2001, Some 0xDB8, None, Some 8, Some 0x800, Some 0x200C, Some private lemma "ipv6_preferred_to_compressed (IPv6AddrPreferred 0x2001 0xDB8 0 3 8 [Some 0x2001, Some 0xDB8, Some 0, Some 3, Some 8, Some 0x800, Some 0x200C, Some (*the output should even conform to RFC5952, ...*) lemma ipv6_preferred_to_compressed_RFC_4291_format: "ipv6_preferred_to_compressed ip = as ==> length (filter (λp. p = None) as) = 0 ∧ length as = 8 ∨ length (filter (λp. p = None) as) = 1 ∧ length (filter (λp. p ≠ None) as) ≤ 7" apply(cases ip) apply(simp add: ipv6_preferred_to_compressed_pull_out_if) apply(simp only: split: if_split_asm) subgoal for a b c d e f g h apply(rule disjI2) apply(case_tac "a=0",case_tac [!] "b=0",case_tac [!] "c=0",case_tac [!] "d=0", case_tac [!] "e=0",case_tac [!] "f=0",case_tac [!] "g=0",case_tac [!] "h=0") by(auto simp add: max_zero_streak_def) (*1min*) subgoal apply(rule disjI1) apply(simp) by force done ― ‹Idea for the following proof:› private lemma "ipv6_preferred_to_compressed (IPv6AddrPreferred a b c d e f g h) = xs = map Some (dropWhile (λx. x=0) [a,b,c,d,e,f,g,h])" apply(case_tac "a=0",case_tac [!] "b=0",case_tac [!] "c=0",case_tac [!] "d=0", case_tac [!] "e=0",case_tac [!] "f=0",case_tac [!] "g=0",case_tac [!] "h=0") by(simp_all add: ipv6_preferred_to_compressed_pull_out_if max_zero_streak_def) (*20s lemma ipv6_preferred_to_compressed: assumes "ipv6_unparsed_compressed_to_preferred (ipv6_preferred_to_compressed ip) shows "ip = ip'" proof - from assms have 1: "∃ipv6compressed. parse_ipv6_address_compressed (ipv6_preferred_to_compressed ip) = Some ipv6compr ipv6addr_c2p ipv6compressed = ip'" using ipv6_unparsed_compressed_to_preferred_ide obtain a b c d e f g h where ip: "ip = IPv6AddrPreferred a b c d e f g h" by(cases ip) have ipv6_preferred_to_compressed_None1: "ipv6_preferred_to_compressed (IPv6AddrPreferred a b c d e f g h) = None#xs ==> (map Some (dropWhile (λx. x=0) [a,b,c,d,e,f,g,h]) = xs ==> (IPv6AddrPreferred a b (IPv6AddrPreferred a b c d e f g h) = ip'" for xs apply(case_tac "a=0",case_tac [!] "b=0",case_tac [!] "c=0",case_tac [!] "d=0", case_tac [!] "e=0",case_tac [!] "f=0",case_tac [!] "g=0",case_tac [!] "h=0") by(simp_all add: ipv6_preferred_to_compressed_pull_out_if max_zero_streak_def) (*5s* have ipv6_preferred_to_compressed_None2: "ipv6_preferred_to_compressed (IPv6AddrPreferred a b c d e f g h) = (Some a')#No (map Some (dropWhile (λx. x=0) [b,c,d,e,f,g,h]) = xs ==> (IPv6AddrPreferred a' b (IPv6AddrPreferred a b c d e f g h) = ip'" for xs a' apply(case_tac "a=0",case_tac [!] "b=0",case_tac [!] "c=0",case_tac [!] "d=0", case_tac [!] "e=0",case_tac [!] "f=0",case_tac [!] "g=0",case_tac [!] "h=0") by(simp_all add: ipv6_preferred_to_compressed_pull_out_if max_zero_streak_def) (*5s* have ipv6_preferred_to_compressed_None3: "ipv6_preferred_to_compressed (IPv6AddrPreferred a b c d e f g h) = (Some a')#(S (map Some (dropWhile (λx. x=0) [c,d,e,f,g,h]) = xs ==> (IPv6AddrPreferred a' b' c (IPv6AddrPreferred a b c d e f g h) = ip'" for xs a' b' apply(case_tac "a=0",case_tac [!] "b=0",case_tac [!] "c=0",case_tac [!] "d=0", case_tac [!] "e=0",case_tac [!] "f=0",case_tac [!] "g=0",case_tac [!] "h=0") by(simp_all add: ipv6_preferred_to_compressed_pull_out_if max_zero_streak_def) (*5s* have ipv6_preferred_to_compressed_None4: "ipv6_preferred_to_compressed (IPv6AddrPreferred a b c d e f g h) = (Some a')#(S (map Some (dropWhile (λx. x=0) [d,e,f,g,h]) = xs ==> (IPv6AddrPreferred a' b' c' (IPv6AddrPreferred a b c d e f g h) = ip'" for xs a' b' c' apply(case_tac "a=0",case_tac [!] "b=0",case_tac [!] "c=0",case_tac [!] "d=0", case_tac [!] "e=0",case_tac [!] "f=0",case_tac [!] "g=0",case_tac [!] "h=0") by(simp_all add: ipv6_preferred_to_compressed_pull_out_if max_zero_streak_def) (*5s* have ipv6_preferred_to_compressed_None5: "ipv6_preferred_to_compressed (IPv6AddrPreferred a b c d e f g h) = (Some a')#(S (map Some (dropWhile (λx. x=0) [e,f,g,h]) = xs ==> (IPv6AddrPreferred a' b' c' d' (IPv6AddrPreferred a b c d e f g h) = ip'" for xs a' b' c' d' apply(case_tac "a=0",case_tac [!] "b=0",case_tac [!] "c=0",case_tac [!] "d=0", case_tac [!] "e=0",case_tac [!] "f=0",case_tac [!] "g=0",case_tac [!] "h=0") by(simp_all add: ipv6_preferred_to_compressed_pull_out_if max_zero_streak_def) (*5s* have ipv6_preferred_to_compressed_None6: "ipv6_preferred_to_compressed (IPv6AddrPreferred a b c d e f g h) = (Some a')#(S (map Some (dropWhile (λx. x=0) [f,g,h]) = xs ==> (IPv6AddrPreferred a' b' c' d' e (IPv6AddrPreferred a b c d e f g h) = ip'" for xs a' b' c' d' e' apply(case_tac "a=0",case_tac [!] "b=0",case_tac [!] "c=0",case_tac [!] "d=0", case_tac [!] "e=0",case_tac [!] "f=0",case_tac [!] "g=0",case_tac [!] "h=0") by(simp_all add: ipv6_preferred_to_compressed_pull_out_if max_zero_streak_def) (*5s* have ipv6_preferred_to_compressed_None7: "ipv6_preferred_to_compressed (IPv6AddrPreferred a b c d e f g h) = (Some a')#(S (map Some (dropWhile (λx. x=0) [g,h]) = xs ==> (IPv6AddrPreferred a' b' c' d' e' (IPv6AddrPreferred a b c d e f g h) = ip'" for xs a' b' c' d' e' f' apply(case_tac "a=0",case_tac [!] "b=0",case_tac [!] "c=0",case_tac [!] "d=0", case_tac [!] "e=0",case_tac [!] "f=0",case_tac [!] "g=0",case_tac [!] "h=0") by(simp_all add: ipv6_preferred_to_compressed_pull_out_if max_zero_streak_def) (*5s* have ipv6_preferred_to_compressed_None8: "ipv6_preferred_to_compressed (IPv6AddrPreferred a b c d e f g h) = (Some a')#(S (map Some (dropWhile (λx. x=0) [h]) = xs ==> (IPv6AddrPreferred a' b' c' d' e' f' (IPv6AddrPreferred a b c d e f g h) = ip'" for xs a' b' c' d' e' f' g' apply(case_tac "a=0",case_tac [!] "b=0",case_tac [!] "c=0",case_tac [!] "d=0", case_tac [!] "e=0",case_tac [!] "f=0",case_tac [!] "g=0",case_tac [!] "h=0") by(simp_all add: ipv6_preferred_to_compressed_pull_out_if max_zero_streak_def) (*5s* have 2: "parse_ipv6_address_compressed (ipv6_preferred_to_compressed (IPv6AddrPref = Some ipv6compressed ==> ipv6addr_c2p ipv6compressed = ip' ==> IPv6AddrPreferred a b c d e f g h = ip'" for ipv6compressed apply(erule parse_ipv6_address_compressed_someE) apply(simp_all) apply(erule ipv6_preferred_to_compressed_None1, simp split: if_split_asm)+ apply(erule ipv6_preferred_to_compressed_None2, simp split: if_split_asm)+ apply(erule ipv6_preferred_to_compressed_None3, simp split: if_split_asm)+ apply(erule ipv6_preferred_to_compressed_None4, simp split: if_split_asm)+ apply(erule ipv6_preferred_to_compressed_None5, simp split: if_split_asm)+ apply(erule ipv6_preferred_to_compressed_None6, simp split: if_split_asm)+ apply(erule ipv6_preferred_to_compressed_None7, simp split: if_split_asm)+ apply(erule ipv6_preferred_to_compressed_None8, simp split: if_split_asm) done from 1 2 ip show ?thesis by(elim exE conjE, simp) qed end end
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