--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Omega_Words_Fun.thy Tue Sep 15 11:18:25 2015 +0200
@@ -0,0 +1,972 @@
+(* Author: Stefan Merz
+ Additions by Salomon Sickert, Julian Brunner, Peter Lammich.
+*)
+section {* $\omega$-words *}
+theory Omega_Words_Fun
+imports "~~/src/HOL/Library/Infinite_Set"
+begin
+text {* Note: This theory is based on Stefan Merz's work. *}
+
+text {*
+ Automata recognize languages, which are sets of words. For the
+ theory of $\omega$-automata, we are mostly interested in
+ $\omega$-words, but it is sometimes useful to reason about
+ finite words, too. We are modeling finite words as lists; this
+ lets us benefit from the existing library. Other formalizations
+ could be investigated, such as representing words as functions
+ whose domains are initial intervals of the natural numbers.
+*}
+
+subsection {* Type declaration and elementary operations *}
+
+text {*
+ We represent $\omega$-words as functions from the natural numbers
+ to the alphabet type. Other possible formalizations include
+ a coinductive definition or a uniform encoding of finite and
+ infinite words, as studied by M\"uller et al.
+*}
+
+type_synonym
+ 'a word = "nat \<Rightarrow> 'a"
+
+text {*
+ We can prefix a finite word to an $\omega$-word, and a way
+ to obtain an $\omega$-word from a finite, non-empty word is by
+ $\omega$-iteration.
+*}
+
+definition
+ conc :: "['a list, 'a word] \<Rightarrow> 'a word" (infixr "conc" 65)
+ where "w conc x == \<lambda>n. if n < length w then w!n else x (n - length w)"
+
+definition
+ iter :: "'a list \<Rightarrow> 'a word"
+ where "iter w == if w = [] then undefined else (\<lambda>n. w!(n mod (length w)))"
+
+notation (xsymbols)
+ conc (infixr "\<frown>" 65) and
+ iter ("(_\<^sup>\<omega>)" [1000])
+
+lemma conc_empty[simp]: "[] \<frown> w = w"
+ unfolding conc_def by auto
+
+lemma conc_fst[simp]:
+ "n < length w \<Longrightarrow> (w \<frown> x) n = w!n"
+by (simp add: conc_def)
+
+lemma conc_snd[simp]:
+ "\<not>(n < length w) \<Longrightarrow> (w \<frown> x) n = x (n - length w)"
+by (simp add: conc_def)
+
+lemma iter_nth [simp]:
+ "0 < length w \<Longrightarrow> w\<^sup>\<omega> n = w!(n mod (length w))"
+by (simp add: iter_def)
+
+lemma conc_conc[simp]:
+ "u \<frown> v \<frown> w = (u @ v) \<frown> w" (is "?lhs = ?rhs")
+proof
+ fix n
+ have u: "n < length u \<Longrightarrow> ?lhs n = ?rhs n"
+ by (simp add: conc_def nth_append)
+ have v: "\<lbrakk> \<not>(n < length u); n < length u + length v \<rbrakk> \<Longrightarrow> ?lhs n = ?rhs n"
+ by (simp add: conc_def nth_append, arith)
+ have w: "\<not>(n < length u + length v) \<Longrightarrow> ?lhs n = ?rhs n"
+ by (simp add: conc_def nth_append, arith)
+ from u v w show "?lhs n = ?rhs n" by blast
+qed
+
+lemma range_conc[simp]: "range (w\<^sub>1 \<frown> w\<^sub>2) = set w\<^sub>1 \<union> range w\<^sub>2"
+proof (intro equalityI subsetI)
+ case (goal1 a)
+ obtain i where 1: "a = (w\<^sub>1 \<frown> w\<^sub>2) i" using goal1 by auto
+ show ?case unfolding 1 by (cases "i < length w\<^sub>1", simp+)
+next
+ case (goal2 a)
+ show ?case
+ using goal2
+ proof
+ case (goal1)
+ obtain i where 1: "i < length w\<^sub>1" "a = w\<^sub>1 ! i" using goal1 in_set_conv_nth by metis
+ show ?thesis
+ proof
+ show "a = (w\<^sub>1 \<frown> w\<^sub>2) i" using 1 by auto
+ show "i \<in> UNIV" by rule
+ qed
+ next
+ case (goal2)
+ obtain i where 1: "a = w\<^sub>2 i" using goal2 by auto
+ show ?thesis
+ proof
+ show "a = (w\<^sub>1 \<frown> w\<^sub>2) (length w\<^sub>1 + i)" using 1 by simp
+ show "length w\<^sub>1 + i \<in> UNIV" by rule
+ qed
+ qed
+qed
+
+
+lemma iter_unroll:
+ "0 < length w \<Longrightarrow> w\<^sup>\<omega> = w \<frown> w\<^sup>\<omega>"
+by (rule ext, simp add: conc_def mod_geq)
+
+subsection \<open>Subsequence, Prefix, and Suffix\<close>
+definition
+ suffix :: "[nat, 'a word] \<Rightarrow> 'a word"
+ where "suffix k x \<equiv> \<lambda>n. x (k+n)"
+
+definition subsequence :: "'a word \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a list"
+ ("_ [_ \<rightarrow> _]" 900)
+where
+ "subsequence w i j \<equiv> map w [i..<j]"
+
+abbreviation prefix :: "nat \<Rightarrow> 'a word \<Rightarrow> 'a list"
+where
+ "prefix n w \<equiv> subsequence w 0 n"
+
+lemma suffix_nth [simp]:
+ "(suffix k x) n = x (k+n)"
+by (simp add: suffix_def)
+
+lemma suffix_0 [simp]:
+ "suffix 0 x = x"
+by (simp add: suffix_def)
+
+lemma suffix_suffix [simp]:
+ "suffix m (suffix k x) = suffix (k+m) x"
+by (rule ext, simp add: suffix_def add.assoc)
+
+lemma subsequence_append:
+ "prefix (i + j) w = prefix i w @ (w [i \<rightarrow> i + j])"
+ unfolding map_append[symmetric] upt_add_eq_append[OF le0] subsequence_def ..
+
+lemma subsequence_drop[simp]:
+ "drop i (w [j \<rightarrow> k]) = w [j + i \<rightarrow> k]"
+ by (simp add: subsequence_def drop_map)
+
+lemma subsequence_empty[simp]:
+ "w [i \<rightarrow> j] = [] \<longleftrightarrow> j \<le> i"
+ by (auto simp add: subsequence_def)
+
+lemma subsequence_length[simp]:
+ "length (subsequence w i j) = j - i"
+ by (simp add: subsequence_def)
+
+lemma subsequence_nth[simp]:
+ "k < j - i \<Longrightarrow> (w [i \<rightarrow> j]) ! k = w (i + k)"
+ unfolding subsequence_def
+ by auto
+
+lemma subseq_to_zero[simp]: "w[i\<rightarrow>0] = []" by simp
+lemma subseq_to_smaller[simp]: "i\<ge>j \<Longrightarrow> w[i\<rightarrow>j] = []" by simp
+lemma subseq_to_Suc[simp]: "i\<le>j \<Longrightarrow> w [i \<rightarrow> Suc j] = w [ i \<rightarrow> j ] @ [w j]"
+ by (auto simp: subsequence_def)
+
+lemma subsequence_singleton[simp]: "w [i \<rightarrow> Suc i] = [w i]"
+ by (auto simp: subsequence_def)
+
+
+lemma subsequence_prefix_suffix:
+ "prefix (j - i) (suffix i w) = w [i \<rightarrow> j]"
+proof (cases "i \<le> j")
+ case True
+ have "w [i \<rightarrow> j] = map w (map (\<lambda>n. n + i) [0..<j - i])"
+ unfolding map_add_upt subsequence_def
+ using le_add_diff_inverse2[OF True] by force
+ also
+ have "\<dots> = map (\<lambda>n. w (n + i)) [0..<j - i]"
+ unfolding map_map comp_def by blast
+ finally
+ show ?thesis
+ unfolding subsequence_def suffix_def add.commute[of i] by simp
+qed (simp add: subsequence_def)
+
+lemma prefix_suffix:
+ "x = prefix n x \<frown> (suffix n x)"
+ by (rule ext, simp add: subsequence_def conc_def)
+
+declare prefix_suffix[symmetric, simp]
+
+
+lemma word_split:
+ obtains v\<^sub>1 v\<^sub>2
+ where "v = v\<^sub>1 \<frown> v\<^sub>2" "length v\<^sub>1 = k"
+proof
+ show "v = prefix k v \<frown> suffix k v" using prefix_suffix by this
+ show "length (prefix k v) = k" by simp
+qed
+
+
+lemma set_subsequence[simp]: "set (w[i\<rightarrow>j]) = w`{i..<j}"
+ unfolding subsequence_def by auto
+
+lemma subsequence_take[simp]:
+ "take i (w [j \<rightarrow> k]) = w [j \<rightarrow> min (j + i) k]"
+ by (simp add: subsequence_def take_map min_def)
+
+lemma subsequence_shift[simp]:
+ "(suffix i w) [j \<rightarrow> k] = w [i + j \<rightarrow> i + k]"
+ by (metis add_diff_cancel_left subsequence_prefix_suffix suffix_suffix)
+
+lemma suffix_subseq_join[simp]: "i \<le> j \<Longrightarrow> v [i \<rightarrow> j] \<frown> suffix j v = suffix i v"
+ by (metis (no_types, lifting) Nat.add_0_right le_add_diff_inverse prefix_suffix
+ subsequence_shift suffix_suffix)
+
+lemma prefix_conc_fst[simp]:
+ assumes "j \<le> length w"
+ shows "prefix j (w \<frown> w') = take j w"
+proof -
+ have "\<forall>i < j. (prefix j (w \<frown> w')) ! i = (take j w) ! i"
+ using assms by (simp add: conc_fst subsequence_def)
+ thus ?thesis
+ by (simp add: assms list_eq_iff_nth_eq min.absorb2)
+qed
+
+lemma prefix_conc_snd[simp]:
+ assumes "n \<ge> length u"
+ shows "prefix n (u \<frown> v) = u @ prefix (n - length u) v"
+proof (intro nth_equalityI allI impI)
+ case (goal1)
+ show ?case using assms by simp
+next
+ case (goal2 i)
+ show ?case using goal2
+ by (cases "i < length u")
+ (auto simp: nth_append)
+
+qed
+
+lemma prefix_conc_length[simp]:
+ "prefix (length w) (w \<frown> w') = w"
+ by simp
+
+lemma suffix_conc_fst[simp]:
+ assumes "n \<le> length u"
+ shows "suffix n (u \<frown> v) = drop n u \<frown> v"
+proof
+ case (goal1 i)
+ show ?case using assms by (cases "n + i < length u", auto simp: algebra_simps)
+qed
+
+lemma suffix_conc_snd[simp]:
+ assumes "n \<ge> length u"
+ shows "suffix n (u \<frown> v) = suffix (n - length u) v"
+proof
+ case (goal1 i)
+ show ?case using assms by simp
+qed
+
+lemma suffix_conc_length[simp]:
+ "suffix (length w) (w \<frown> w') = w'"
+ unfolding conc_def by force
+
+lemma concat_eq[iff]:
+ assumes "length v\<^sub>1 = length v\<^sub>2"
+ shows "v\<^sub>1 \<frown> u\<^sub>1 = v\<^sub>2 \<frown> u\<^sub>2 \<longleftrightarrow> v\<^sub>1 = v\<^sub>2 \<and> u\<^sub>1 = u\<^sub>2"
+proof
+ case (goal1)
+ have 1: "\<And> i. (v\<^sub>1 \<frown> u\<^sub>1) i = (v\<^sub>2 \<frown> u\<^sub>2) i" using goal1 by auto
+ show ?case
+ proof (intro conjI ext nth_equalityI allI impI)
+ show "length v\<^sub>1 = length v\<^sub>2" using assms(1) by this
+ next
+ fix i
+ assume 2: "i < length v\<^sub>1"
+ have 3: "i < length v\<^sub>2" using assms(1) 2 by simp
+ show "v\<^sub>1 ! i = v\<^sub>2 ! i" using 1[of i] 2 3 by simp
+ next
+ fix i
+ show "u\<^sub>1 i = u\<^sub>2 i" using 1[of "length v\<^sub>1 + i"] assms(1) by simp
+ qed
+next
+ case (goal2)
+ show ?case using goal2 by simp
+qed
+lemma same_concat_eq[iff]: "u \<frown> v = u \<frown> w \<longleftrightarrow> v = w" by simp
+
+lemma comp_concat[simp]: "f \<circ> u \<frown> v = map f u \<frown> (f \<circ> v)"
+proof
+ fix i
+ show "(f \<circ> u \<frown> v) i = (map f u \<frown> (f \<circ> v)) i" by (cases "i < length u", simp+)
+qed
+
+subsection \<open>Prepending\<close>
+
+primrec build :: "'a \<Rightarrow> 'a word \<Rightarrow> 'a word" (infixr "##" 65)
+ where "(a ## w) 0 = a" | "(a ## w) (Suc i) = w i"
+
+lemma build_eq[iff]: "a\<^sub>1 ## w\<^sub>1 = a\<^sub>2 ## w\<^sub>2 \<longleftrightarrow> a\<^sub>1 = a\<^sub>2 \<and> w\<^sub>1 = w\<^sub>2"
+proof
+ assume 1: "a\<^sub>1 ## w\<^sub>1 = a\<^sub>2 ## w\<^sub>2"
+ have 2: "\<And> i. (a\<^sub>1 ## w\<^sub>1) i = (a\<^sub>2 ## w\<^sub>2) i" using 1 by auto
+ show "a\<^sub>1 = a\<^sub>2 \<and> w\<^sub>1 = w\<^sub>2"
+ proof (intro conjI ext)
+ show "a\<^sub>1 = a\<^sub>2" using 2[of "0"] by simp
+ next
+ fix i
+ show "w\<^sub>1 i = w\<^sub>2 i" using 2[of "Suc i"] by simp
+ qed
+next
+ assume 1: "a\<^sub>1 = a\<^sub>2 \<and> w\<^sub>1 = w\<^sub>2"
+ show "a\<^sub>1 ## w\<^sub>1 = a\<^sub>2 ## w\<^sub>2" using 1 by simp
+qed
+lemma build_cons[simp]: "(a # u) \<frown> v = a ## u \<frown> v"
+proof
+ fix i
+ show "((a # u) \<frown> v) i = (a ## u \<frown> v) i"
+ proof (cases i)
+ case 0
+ show ?thesis unfolding 0 by simp
+ next
+ case (Suc j)
+ show ?thesis unfolding Suc by (cases "j < length u", simp+)
+ qed
+qed
+lemma build_append[simp]: "(w @ a # u) \<frown> v = w \<frown> a ## u \<frown> v"
+ unfolding conc_conc[symmetric] by simp
+lemma build_first[simp]: "w 0 ## suffix (Suc 0) w = w"
+proof
+ fix i
+ show "(w 0 ## suffix (Suc 0) w) i = w i" by (cases i, simp+)
+qed
+lemma build_split[intro]: "w = w 0 ## suffix 1 w" by simp
+lemma build_range[simp]: "range (a ## w) = insert a (range w)"
+proof safe
+ fix i
+ show "(a ## w) i \<notin> range w \<Longrightarrow> (a ## w) i = a" by (cases i, auto)
+next
+ show "a \<in> range (a ## w)"
+ proof (rule range_eqI)
+ show "a = (a ## w) 0" by simp
+ qed
+next
+ fix i
+ show "w i \<in> range (a ## w)"
+ proof (rule range_eqI)
+ show "w i = (a ## w) (Suc i)" by simp
+ qed
+qed
+
+lemma suffix_singleton_suffix[simp]: "w i ## suffix (Suc i) w = suffix i w"
+ using suffix_subseq_join[of i "Suc i" w]
+ by simp
+
+text \<open>Find the first occurrence of a letter from a given set\<close>
+lemma word_first_split_set:
+ assumes "A \<inter> range w \<noteq> {}"
+ obtains u a v
+ where "w = u \<frown> [a] \<frown> v" "A \<inter> set u = {}" "a \<in> A"
+proof -
+ def i \<equiv> "LEAST i. w i \<in> A"
+ show ?thesis
+ proof
+ show "w = prefix i w \<frown> [w i] \<frown> suffix (Suc i) w" by simp
+ show "A \<inter> set (prefix i w) = {}"
+ proof safe
+ case (goal1 a)
+ obtain k where 3: "k < i" "w k = a" using goal1(2) by auto
+ have 4: "w k \<notin> A" using not_less_Least 3(1) unfolding i_def by this
+ show ?case using goal1(1) 3(2) 4 by auto
+ qed
+ show "w i \<in> A" using LeastI assms(1) unfolding i_def by fast
+ qed
+qed
+
+
+
+subsection {* The limit set of an $\omega$-word *}
+
+text {*
+ The limit set (also called infinity set) of an $\omega$-word
+ is the set of letters that appear infinitely often in the word.
+ This set plays an important role in defining acceptance conditions
+ of $\omega$-automata.
+*}
+
+definition
+ limit :: "'a word \<Rightarrow> 'a set"
+ where "limit x \<equiv> { a . \<exists>\<^sub>\<infinity>n . x n = a }"
+
+lemma limit_iff_frequent:
+ "(a \<in> limit x) = (\<exists>\<^sub>\<infinity>n . x n = a)"
+by (simp add: limit_def)
+
+text {*
+ The following is a different way to define the limit,
+ using the reverse image, making the laws about reverse
+ image applicable to the limit set.
+ (Might want to change the definition above?)
+*}
+
+lemma limit_vimage:
+ "(a \<in> limit x) = infinite (x -` {a})"
+by (simp add: limit_def Inf_many_def vimage_def)
+
+lemma two_in_limit_iff:
+ "({a,b} \<subseteq> limit x) =
+ ((\<exists>n. x n =a ) \<and> (\<forall>n. x n = a \<longrightarrow> (\<exists>m>n. x m = b)) \<and> (\<forall>m. x m = b \<longrightarrow> (\<exists>n>m. x n = a)))"
+ (is "?lhs = (?r1 \<and> ?r2 \<and> ?r3)")
+proof
+ assume lhs: "?lhs"
+ hence 1: "?r1" by (auto simp: limit_def elim: INFM_EX)
+ from lhs have "\<forall>n. \<exists>m>n. x m = b" by (auto simp: limit_def INFM_nat)
+ hence 2: "?r2" by simp
+ from lhs have "\<forall>m. \<exists>n>m. x n = a" by (auto simp: limit_def INFM_nat)
+ hence 3: "?r3" by simp
+ from 1 2 3 show "?r1 \<and> ?r2 \<and> ?r3" by simp
+next
+ assume "?r1 \<and> ?r2 \<and> ?r3"
+ hence 1: "?r1" and 2: "?r2" and 3: "?r3" by simp+
+ have infa: "\<forall>m. \<exists>n\<ge>m. x n = a"
+ proof
+ fix m
+ show "\<exists>n\<ge>m. x n = a" (is "?A m")
+ proof (induct m)
+ from 1 show "?A 0" by simp
+ next
+ fix m
+ assume ih: "?A m"
+ then obtain n where n: "n \<ge> m" "x n = a" by auto
+ with 2 obtain k where k: "k>n" "x k = b" by auto
+ with 3 obtain l where l: "l>k" "x l = a" by auto
+ from n k l have "l \<ge> Suc m" by auto
+ with l show "?A (Suc m)" by auto
+ qed
+ qed
+ hence infa': "\<exists>\<^sub>\<infinity>n. x n = a" by (simp add: INFM_nat_le)
+ have "\<forall>n. \<exists>m>n. x m = b"
+ proof
+ fix n
+ from infa obtain k where k1: "k\<ge>n" and k2: "x k = a" by auto
+ from 2 k2 obtain l where l1: "l>k" and l2: "x l = b" by auto
+ from k1 l1 have "l > n" by auto
+ with l2 show "\<exists>m>n. x m = b" by auto
+ qed
+ hence "\<exists>\<^sub>\<infinity>m. x m = b" by (simp add: INFM_nat)
+ with infa' show "?lhs" by (auto simp: limit_def)
+qed
+
+text {*
+ For $\omega$-words over a finite alphabet, the limit set is
+ non-empty. Moreover, from some position onward, any such word
+ contains only letters from its limit set.
+*}
+
+lemma limit_nonempty:
+ assumes fin: "finite (range x)"
+ shows "\<exists>a. a \<in> limit x"
+proof -
+ from fin obtain a where "a \<in> range x \<and> infinite (x -` {a})"
+ by (rule inf_img_fin_domE, auto)
+ hence "a \<in> limit x"
+ by (auto simp add: limit_vimage)
+ thus ?thesis ..
+qed
+
+lemmas limit_nonemptyE = limit_nonempty[THEN exE]
+
+lemma limit_inter_INF:
+ assumes hyp: "limit w \<inter> S \<noteq> {}"
+ shows "\<exists>\<^sub>\<infinity> n. w n \<in> S"
+proof -
+ from hyp obtain x where "\<exists>\<^sub>\<infinity> n. w n = x" and "x \<in> S"
+ by (auto simp add: limit_def)
+ thus ?thesis
+ by (auto elim: INFM_mono)
+qed
+
+text {*
+ The reverse implication is true only if $S$ is finite.
+*}
+
+lemma INF_limit_inter:
+ assumes hyp: "\<exists>\<^sub>\<infinity> n. w n \<in> S" and fin: "finite (S \<inter> range w)"
+ shows "\<exists>a. a \<in> limit w \<inter> S"
+proof (rule ccontr)
+ assume contra: "\<not>(\<exists>a. a \<in> limit w \<inter> S)"
+ hence "\<forall>a\<in>S. finite {n. w n = a}"
+ by (auto simp add: limit_def Inf_many_def)
+ with fin have "finite (UN a:S \<inter> range w. {n. w n = a})"
+ by auto
+ moreover
+ have "(UN a:S \<inter> range w. {n. w n = a}) = {n. w n \<in> S}"
+ by auto
+ moreover
+ note hyp
+ ultimately show "False"
+ by (simp add: Inf_many_def)
+qed
+
+lemma fin_ex_inf_eq_limit: "finite A \<Longrightarrow> (\<exists>\<^sub>\<infinity>i. w i \<in> A) \<longleftrightarrow> limit w \<inter> A \<noteq> {}"
+ by (metis INF_limit_inter equals0D finite_Int limit_inter_INF)
+
+lemma limit_in_range_suffix:
+ "limit x \<subseteq> range (suffix k x)"
+proof
+ fix a
+ assume "a \<in> limit x"
+ then obtain l where
+ kl: "k < l" and xl: "x l = a"
+ by (auto simp add: limit_def INFM_nat)
+ from kl obtain m where "l = k+m"
+ by (auto simp add: less_iff_Suc_add)
+ with xl show "a \<in> range (suffix k x)"
+ by auto
+qed
+
+lemma limit_in_range: "limit r \<subseteq> range r"
+ using limit_in_range_suffix[of r 0] by simp
+
+lemmas limit_in_range_suffixD = limit_in_range_suffix[THEN subsetD]
+
+lemma limit_subset:
+ "limit f \<subseteq> f ` {n..}"
+ using limit_in_range_suffix[of f n] unfolding suffix_def by auto
+
+theorem limit_is_suffix:
+ assumes fin: "finite (range x)"
+ shows "\<exists>k. limit x = range (suffix k x)"
+proof -
+ have "\<exists>k. range (suffix k x) \<subseteq> limit x"
+ proof -
+ -- "The set of letters that are not in the limit is certainly finite."
+ from fin have "finite (range x - limit x)"
+ by simp
+ -- "Moreover, any such letter occurs only finitely often"
+ moreover
+ have "\<forall>a \<in> range x - limit x. finite (x -` {a})"
+ by (auto simp add: limit_vimage)
+ -- "Thus, there are only finitely many occurrences of such letters."
+ ultimately have "finite (UN a : range x - limit x. x -` {a})"
+ by (blast intro: finite_UN_I)
+ -- "Therefore these occurrences are within some initial interval."
+ then obtain k where "(UN a : range x - limit x. x -` {a}) \<subseteq> {..<k}"
+ by (blast dest: finite_nat_bounded)
+ -- "This is just the bound we are looking for."
+ hence "\<forall>m. k \<le> m \<longrightarrow> x m \<in> limit x"
+ by (auto simp add: limit_vimage)
+ hence "range (suffix k x) \<subseteq> limit x"
+ by auto
+ thus ?thesis ..
+ qed
+ then obtain k where "range (suffix k x) \<subseteq> limit x" ..
+ with limit_in_range_suffix
+ have "limit x = range (suffix k x)"
+ by (rule subset_antisym)
+ thus ?thesis ..
+qed
+
+theorems limit_is_suffixE = limit_is_suffix[THEN exE]
+
+
+text {*
+ The limit set enjoys some simple algebraic laws with respect
+ to concatenation, suffixes, iteration, and renaming.
+*}
+
+theorem limit_conc [simp]:
+ "limit (w \<frown> x) = limit x"
+proof (auto)
+ fix a assume a: "a \<in> limit (w \<frown> x)"
+ have "\<forall>m. \<exists>n. m<n \<and> x n = a"
+ proof
+ fix m
+ from a obtain n where "m + length w < n \<and> (w \<frown> x) n = a"
+ by (auto simp add: limit_def Inf_many_def infinite_nat_iff_unbounded)
+ hence "m < n - length w \<and> x (n - length w) = a"
+ by (auto simp add: conc_def)
+ thus "\<exists>n. m<n \<and> x n = a" ..
+ qed
+ hence "infinite {n . x n = a}"
+ by (simp add: infinite_nat_iff_unbounded)
+ thus "a \<in> limit x"
+ by (simp add: limit_def Inf_many_def)
+next
+ fix a assume a: "a \<in> limit x"
+ have "\<forall>m. length w < m \<longrightarrow> (\<exists>n. m<n \<and> (w \<frown> x) n = a)"
+ proof (clarify)
+ fix m
+ assume m: "length w < m"
+ with a obtain n where "m - length w < n \<and> x n = a"
+ by (auto simp add: limit_def Inf_many_def infinite_nat_iff_unbounded)
+ with m have "m < n + length w \<and> (w \<frown> x) (n + length w) = a"
+ by (simp add: conc_def, arith)
+ thus "\<exists>n. m<n \<and> (w \<frown> x) n = a" ..
+ qed
+ hence "infinite {n . (w \<frown> x) n = a}"
+ by (simp add: unbounded_k_infinite)
+ thus "a \<in> limit (w \<frown> x)"
+ by (simp add: limit_def Inf_many_def)
+qed
+
+theorem limit_suffix [simp]:
+ "limit (suffix n x) = limit x"
+proof -
+ have "x = (prefix n x) \<frown> (suffix n x)"
+ by (simp add: prefix_suffix)
+ hence "limit x = limit (prefix n x \<frown> suffix n x)"
+ by simp
+ also have "\<dots> = limit (suffix n x)"
+ by (rule limit_conc)
+ finally show ?thesis
+ by (rule sym)
+qed
+
+theorem limit_iter [simp]:
+ assumes nempty: "0 < length w"
+ shows "limit w\<^sup>\<omega> = set w"
+proof
+ have "limit w\<^sup>\<omega> \<subseteq> range w\<^sup>\<omega>"
+ by (auto simp add: limit_def dest: INFM_EX)
+ also from nempty have "\<dots> \<subseteq> set w"
+ by auto
+ finally show "limit w\<^sup>\<omega> \<subseteq> set w" .
+next
+ {
+ fix a assume a: "a \<in> set w"
+ then obtain k where k: "k < length w \<and> w!k = a"
+ by (auto simp add: set_conv_nth)
+ -- "the following bound is terrible, but it simplifies the proof"
+ from nempty k
+ have "\<forall>m. w\<^sup>\<omega> ((Suc m)*(length w) + k) = a"
+ by (simp add: mod_add_left_eq)
+ moreover
+ -- "why is the following so hard to prove??"
+ have "\<forall>m. m < (Suc m)*(length w) + k"
+ proof
+ fix m
+ from nempty have "1 \<le> length w" by arith
+ hence "m*1 \<le> m*length w" by simp
+ hence "m \<le> m*length w" by simp
+ with nempty have "m < length w + (m*length w) + k" by arith
+ thus "m < (Suc m)*(length w) + k" by simp
+ qed
+ moreover note nempty
+ ultimately have "a \<in> limit w\<^sup>\<omega>"
+ by (auto simp add: limit_iff_frequent INFM_nat)
+ }
+ then show "set w \<subseteq> limit w\<^sup>\<omega>" by auto
+qed
+
+lemma limit_o [simp]:
+ assumes a: "a \<in> limit w"
+ shows "f a \<in> limit (f \<circ> w)"
+proof -
+ from a
+ have "\<exists>\<^sub>\<infinity>n. w n = a"
+ by (simp add: limit_iff_frequent)
+ hence "\<exists>\<^sub>\<infinity>n. f (w n) = f a"
+ by (rule INFM_mono, simp)
+ thus "f a \<in> limit (f \<circ> w)"
+ by (simp add: limit_iff_frequent)
+qed
+
+text {*
+ The converse relation is not true in general: $f(a)$ can be in the
+ limit of $f \circ w$ even though $a$ is not in the limit of $w$.
+ However, @{text limit} commutes with renaming if the function is
+ injective. More generally, if $f(a)$ is the image of only finitely
+ many elements, some of these must be in the limit of $w$.
+*}
+
+lemma limit_o_inv:
+ assumes fin: "finite (f -` {x})" and x: "x \<in> limit (f \<circ> w)"
+ shows "\<exists>a \<in> (f -` {x}). a \<in> limit w"
+proof (rule ccontr)
+ assume contra: "\<not>(\<exists>a \<in> (f -` {x}). a \<in> limit w)"
+ -- "hence, every element in the pre-image occurs only finitely often"
+ then have "\<forall>a \<in> (f -` {x}). finite {n. w n = a}"
+ by (simp add: limit_def Inf_many_def)
+ -- "so there are only finitely many occurrences of any such element"
+ with fin have "finite (\<Union> a \<in> (f -` {x}). {n. w n = a})"
+ by auto
+ -- {* these are precisely those positions where $x$ occurs in $f \circ w$ *}
+ moreover
+ have "(\<Union> a \<in> (f -` {x}). {n. w n = a}) = {n. f(w n) = x}"
+ by auto
+ ultimately
+ -- "so $x$ can occur only finitely often in the translated word"
+ have "finite {n. f(w n) = x}"
+ by simp
+ -- {* \ldots\ which yields a contradiction *}
+ with x show "False"
+ by (simp add: limit_def Inf_many_def)
+qed
+
+theorem limit_inj [simp]:
+ assumes inj: "inj f"
+ shows "limit (f \<circ> w) = f ` (limit w)"
+proof
+ show "f ` limit w \<subseteq> limit (f \<circ> w)"
+ by auto
+next
+ show "limit (f \<circ> w) \<subseteq> f ` limit w"
+ proof
+ fix x
+ assume x: "x \<in> limit (f \<circ> w)"
+ from inj have "finite (f -` {x})"
+ by (blast intro: finite_vimageI)
+ with x obtain a where a: "a \<in> (f -` {x}) \<and> a \<in> limit w"
+ by (blast dest: limit_o_inv)
+ thus "x \<in> f ` (limit w)"
+ by auto
+ qed
+qed
+
+lemma limit_inter_empty:
+ assumes fin: "finite (range w)"
+ assumes hyp: "limit w \<inter> S = {}"
+ shows "\<forall>\<^sub>\<infinity>n. w n \<notin> S"
+proof -
+ from fin obtain k where k_def: "limit w = range (suffix k w)"
+ using limit_is_suffix by blast
+ have "\<And>k'. w (k + k') \<notin> S"
+ using hyp unfolding k_def suffix_def image_def by blast
+ thus ?thesis
+ unfolding MOST_nat_le using le_Suc_ex by blast
+qed
+
+text \<open>If the limit is the suffix of the sequence's range,
+ we may increase the suffix index arbitrarily\<close>
+lemma limit_range_suffix_incr:
+ assumes "limit r = range (suffix i r)"
+ assumes "j\<ge>i"
+ shows "limit r = range (suffix j r)"
+ (is "?lhs = ?rhs")
+proof -
+ have "?lhs = range (suffix i r)"
+ using assms by simp
+ moreover
+ have "\<dots> \<supseteq> ?rhs" using \<open>j\<ge>i\<close>
+ by (metis (mono_tags, lifting) assms(2) image_subsetI le_Suc_ex range_eqI suffix_def suffix_suffix)
+ moreover
+ have "\<dots> \<supseteq> ?lhs"
+ using limit_in_range_suffix .
+ ultimately
+ show "?lhs = ?rhs"
+ by (metis antisym_conv limit_in_range_suffix)
+qed
+
+text \<open>For two finite sequences, we can find a common suffix index such
+ that the limits can be represented as these suffixes' ranges.\<close>
+lemma common_range_limit:
+ assumes "finite (range x)" and "finite (range y)"
+ obtains i where "limit x = range (suffix i x)"
+ and "limit y = range (suffix i y)"
+proof -
+ obtain i j where
+ 1: "limit x = range (suffix i x)"
+ and 2: "limit y = range (suffix j y)"
+ using assms limit_is_suffix by metis
+ have "limit x = range (suffix (max i j) x)"
+ and "limit y = range (suffix (max i j) y)"
+ using limit_range_suffix_incr[OF 1] limit_range_suffix_incr[OF 2]
+ by auto
+ thus ?thesis
+ using that by metis
+qed
+
+
+subsection {* Index sequences and piecewise definitions *}
+
+text {*
+ A word can be defined piecewise: given a sequence of words $w_0, w_1, \ldots$
+ and a strictly increasing sequence of integers $i_0, i_1, \ldots$ where $i_0=0$,
+ a single word is obtained by concatenating subwords of the $w_n$ as given by
+ the integers: the resulting word is
+ \[
+ (w_0)_{i_0} \ldots (w_0)_{i_1-1} (w_1)_{i_1} \ldots (w_1)_{i_2-1} \ldots
+ \]
+ We prepare the field by proving some trivial facts about such sequences of
+ indexes.
+*}
+
+definition
+ idx_sequence :: "nat word \<Rightarrow> bool"
+ where "idx_sequence idx \<equiv> (idx 0 = 0) \<and> (\<forall>n. idx n < idx (Suc n))"
+
+lemma idx_sequence_less:
+ assumes iseq: "idx_sequence idx"
+ shows "idx n < idx (Suc(n+k))"
+proof (induct k)
+ from iseq show "idx n < idx (Suc (n + 0))"
+ by (simp add: idx_sequence_def)
+next
+ fix k
+ assume ih: "idx n < idx (Suc(n+k))"
+ from iseq have "idx (Suc(n+k)) < idx (Suc(n + Suc k))"
+ by (simp add: idx_sequence_def)
+ with ih show "idx n < idx (Suc(n + Suc k))"
+ by (rule less_trans)
+qed
+
+lemma idx_sequence_inj:
+ assumes iseq: "idx_sequence idx"
+ and eq: "idx m = idx n"
+ shows "m = n"
+proof (rule nat_less_cases)
+ assume "n<m"
+ then obtain k where "m = Suc(n+k)"
+ by (auto simp add: less_iff_Suc_add)
+ with iseq have "idx n < idx m"
+ by (simp add: idx_sequence_less)
+ with eq show ?thesis
+ by simp
+next
+ assume "m<n"
+ then obtain k where "n = Suc(m+k)"
+ by (auto simp add: less_iff_Suc_add)
+ with iseq have "idx m < idx n"
+ by (simp add: idx_sequence_less)
+ with eq show ?thesis
+ by simp
+qed (simp)
+
+lemma idx_sequence_mono:
+ assumes iseq: "idx_sequence idx"
+ and m: "m \<le> n"
+ shows "idx m \<le> idx n"
+proof (cases "m=n")
+ case True
+ thus ?thesis by simp
+next
+ case False
+ with m have "m < n" by simp
+ then obtain k where "n = Suc(m+k)"
+ by (auto simp add: less_iff_Suc_add)
+ with iseq have "idx m < idx n"
+ by (simp add: idx_sequence_less)
+ thus ?thesis by simp
+qed
+
+text {*
+ Given an index sequence, every natural number is contained in the
+ interval defined by two adjacent indexes, and in fact this interval
+ is determined uniquely.
+*}
+
+lemma idx_sequence_idx:
+ assumes "idx_sequence idx"
+ shows "idx k \<in> {idx k ..< idx (Suc k)}"
+using assms by (auto simp add: idx_sequence_def)
+
+lemma idx_sequence_interval:
+ assumes iseq: "idx_sequence idx"
+ shows "\<exists>k. n \<in> {idx k ..< idx (Suc k) }"
+ (is "?P n" is "\<exists>k. ?in n k")
+proof (induct n)
+ from iseq have "0 = idx 0"
+ by (simp add: idx_sequence_def)
+ moreover
+ from iseq have "idx 0 \<in> {idx 0 ..< idx (Suc 0) }"
+ by (rule idx_sequence_idx)
+ ultimately
+ show "?P 0" by auto
+next
+ fix n
+ assume "?P n"
+ then obtain k where k: "?in n k" ..
+ show "?P (Suc n)"
+ proof (cases "Suc n < idx (Suc k)")
+ case True
+ with k have "?in (Suc n) k"
+ by simp
+ thus ?thesis ..
+ next
+ case False
+ with k have "Suc n = idx (Suc k)"
+ by auto
+ with iseq have "?in (Suc n) (Suc k)"
+ by (simp add: idx_sequence_def)
+ thus ?thesis ..
+ qed
+qed
+
+lemma idx_sequence_interval_unique:
+ assumes iseq: "idx_sequence idx"
+ and k: "n \<in> {idx k ..< idx (Suc k) }"
+ and m: "n \<in> {idx m ..< idx (Suc m) }"
+ shows "k = m"
+proof (rule nat_less_cases)
+ assume "k < m"
+ hence "Suc k \<le> m" by simp
+ with iseq have "idx (Suc k) \<le> idx m"
+ by (rule idx_sequence_mono)
+ with m have "idx (Suc k) \<le> n"
+ by auto
+ with k have "False"
+ by simp
+ thus ?thesis ..
+next
+ assume "m < k"
+ hence "Suc m \<le> k" by simp
+ with iseq have "idx (Suc m) \<le> idx k"
+ by (rule idx_sequence_mono)
+ with k have "idx (Suc m) \<le> n"
+ by auto
+ with m have "False"
+ by simp
+ thus ?thesis ..
+qed (simp)
+
+lemma idx_sequence_unique_interval:
+ assumes iseq: "idx_sequence idx"
+ shows "\<exists>! k. n \<in> {idx k ..< idx (Suc k) }"
+proof (rule ex_ex1I)
+ from iseq show "\<exists>k. n \<in> {idx k ..< idx (Suc k)}"
+ by (rule idx_sequence_interval)
+next
+ fix k y
+ assume "n \<in> {idx k..<idx (Suc k)}" and "n \<in> {idx y..<idx (Suc y)}"
+ with iseq show "k = y" by (auto elim: idx_sequence_interval_unique)
+qed
+
+text {*
+ Now we can define the piecewise construction of a word using
+ an index sequence.
+*}
+
+definition
+ merge :: "['a word word, nat word] \<Rightarrow> 'a word"
+ where "merge ws idx \<equiv>
+ \<lambda> n. let i = THE i. n \<in> {idx i ..< idx (Suc i) } in ws i n"
+
+lemma merge:
+ assumes idx: "idx_sequence idx"
+ and n: "n \<in> {idx i ..< idx (Suc i) }"
+ shows "merge ws idx n = ws i n"
+proof -
+ from n have "(THE k. n \<in> {idx k ..< idx (Suc k) }) = i"
+ by (rule the_equality[OF _ sym[OF idx_sequence_interval_unique[OF idx n]]]) simp
+ thus ?thesis
+ by (simp add: merge_def Let_def)
+qed
+
+lemma merge0:
+ assumes idx: "idx_sequence idx"
+ shows "merge ws idx 0 = ws 0 0"
+proof (rule merge[OF idx])
+ from idx have "idx 0 < idx (Suc 0)"
+ by (unfold idx_sequence_def, blast)
+ with idx show "0 \<in> {idx 0 ..< idx (Suc 0)}"
+ by (simp add: idx_sequence_def)
+qed
+
+lemma merge_Suc:
+ assumes idx: "idx_sequence idx"
+ and n: "n \<in> {idx i ..< idx (Suc i) }"
+ shows "merge ws idx (Suc n) =
+ (if Suc n = idx (Suc i) then ws (Suc i) else ws i) (Suc n)"
+proof (auto)
+ assume eq: "Suc n = idx (Suc i)"
+ from idx have "idx (Suc i) < idx (Suc(Suc i))"
+ by (unfold idx_sequence_def, blast)
+ with eq idx show "merge ws idx (idx (Suc i)) = ws (Suc i) (idx (Suc i))"
+ by (simp add: merge)
+next
+ assume neq: "Suc n \<noteq> idx (Suc i)"
+ with n have "Suc n \<in> {idx i ..< idx (Suc i) }"
+ by auto
+ with idx show "merge ws idx (Suc n) = ws i (Suc n)"
+ by (rule merge)
+qed
+
+end