--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/thys2/Exercises.thy	Sun Oct 10 18:35:21 2021 +0100
@@ -0,0 +1,253 @@
+theory Exercises
+  imports Spec "~~/src/HOL/Library/Infinite_Set"
+begin
+
+section {* Some Fun Facts *}
+
+fun
+ zeroable :: "rexp \<Rightarrow> bool"
+where
+  "zeroable (ZERO) \<longleftrightarrow> True"
+| "zeroable (ONE) \<longleftrightarrow> False"
+| "zeroable (CH c) \<longleftrightarrow> False"
+| "zeroable (ALT r1 r2) \<longleftrightarrow> zeroable r1 \<and> zeroable r2"
+| "zeroable (SEQ r1 r2) \<longleftrightarrow> zeroable r1 \<or> zeroable r2"
+| "zeroable (STAR r) \<longleftrightarrow> False"
+
+lemma zeroable_correctness:
+  shows "zeroable r \<longleftrightarrow> L r = {}"
+by(induct r)(auto simp add: Sequ_def)
+
+
+fun
+ atmostempty :: "rexp \<Rightarrow> bool"
+where
+  "atmostempty (ZERO) \<longleftrightarrow> True"
+| "atmostempty (ONE) \<longleftrightarrow> True"
+| "atmostempty (CH c) \<longleftrightarrow> False"
+| "atmostempty (ALT r1 r2) \<longleftrightarrow> atmostempty r1 \<and> atmostempty r2"
+| "atmostempty (SEQ r1 r2) \<longleftrightarrow> 
+     zeroable r1 \<or> zeroable r2 \<or> (atmostempty r1 \<and> atmostempty r2)"
+| "atmostempty (STAR r) = atmostempty r"
+
+
+
+fun
+ somechars :: "rexp \<Rightarrow> bool"
+where
+  "somechars (ZERO) \<longleftrightarrow> False"
+| "somechars (ONE) \<longleftrightarrow> False"
+| "somechars (CH c) \<longleftrightarrow> True"
+| "somechars (ALT r1 r2) \<longleftrightarrow> somechars r1 \<or> somechars r2"
+| "somechars (SEQ r1 r2) \<longleftrightarrow> 
+      (\<not> zeroable r1 \<and> somechars r2) \<or> (\<not> zeroable r2 \<and> somechars r1) \<or> 
+      (somechars r1 \<and> nullable r2) \<or> (somechars r2 \<and> nullable r1)"
+| "somechars (STAR r) \<longleftrightarrow> somechars r"
+
+lemma somechars_correctness:
+  shows "somechars r \<longleftrightarrow> (\<exists>s. s \<noteq> [] \<and> s \<in> L r)"
+apply(induct r)
+apply(simp_all add: zeroable_correctness nullable_correctness Sequ_def)
+using Nil_is_append_conv apply blast
+apply blast
+  apply(auto)
+  by (metis Star_decomp hd_Cons_tl list.distinct(1))
+
+lemma atmostempty_correctness_aux:
+  shows "atmostempty r \<longleftrightarrow> \<not> somechars r"
+apply(induct r)
+apply(simp_all)
+apply(auto simp add: zeroable_correctness nullable_correctness somechars_correctness)
+done
+
+lemma atmostempty_correctness:
+  shows "atmostempty r \<longleftrightarrow> L r \<subseteq> {[]}"
+by(auto simp add: atmostempty_correctness_aux somechars_correctness)
+
+fun
+ leastsinglechar :: "rexp \<Rightarrow> bool"
+where
+  "leastsinglechar (ZERO) \<longleftrightarrow> False"
+| "leastsinglechar (ONE) \<longleftrightarrow> False"
+| "leastsinglechar (CH c) \<longleftrightarrow> True"
+| "leastsinglechar (ALT r1 r2) \<longleftrightarrow> leastsinglechar r1 \<or> leastsinglechar r2"
+| "leastsinglechar (SEQ r1 r2) \<longleftrightarrow> 
+      (if (zeroable r1 \<or> zeroable r2) then False
+       else ((nullable r1 \<and> leastsinglechar r2) \<or> (nullable r2 \<and> leastsinglechar r1)))"
+| "leastsinglechar (STAR r) \<longleftrightarrow> leastsinglechar r"
+
+lemma leastsinglechar_correctness:
+  "leastsinglechar r \<longleftrightarrow> (\<exists>c. [c] \<in> L r)"
+  apply(induct r)
+  apply(simp)
+  apply(simp)
+  apply(simp)
+  prefer 2
+  apply(simp)
+  apply(blast)
+  prefer 2
+  apply(simp)
+  using Star.step Star_decomp apply fastforce
+  apply(simp add: Sequ_def zeroable_correctness nullable_correctness)
+  by (metis append_Nil append_is_Nil_conv butlast_append butlast_snoc)
+
+fun
+ infinitestrings :: "rexp \<Rightarrow> bool"
+where
+  "infinitestrings (ZERO) = False"
+| "infinitestrings (ONE) = False"
+| "infinitestrings (CH c) = False"
+| "infinitestrings (ALT r1 r2) = (infinitestrings r1 \<or> infinitestrings r2)"
+| "infinitestrings (SEQ r1 r2) \<longleftrightarrow> 
+      (\<not> zeroable r1 \<and> infinitestrings r2) \<or> (\<not> zeroable r2 \<and> infinitestrings r1)"
+| "infinitestrings (STAR r) = (\<not> atmostempty r)"
+
+
+
+
+
+lemma Star_atmostempty:
+  assumes "A \<subseteq> {[]}"
+  shows "A\<star> \<subseteq> {[]}"
+  using assms
+  using Star_decomp concat_eq_Nil_conv empty_iff insert_iff subsetI subset_singletonD 
+  apply(auto)
+proof -
+  fix x :: "char list"
+  assume a1: "x \<in> A\<star>"
+  assume "\<And>c x A. c # x \<in> A\<star> \<Longrightarrow> \<exists>s1 s2. x = s1 @ s2 \<and> c # s1 \<in> A \<and> s2 \<in> A\<star>"
+  then have f2: "\<forall>cs C c. \<exists>csa. c # csa \<in> C \<or> c # cs \<notin> C\<star>"
+    by auto
+  obtain cc :: "char list \<Rightarrow> char" and ccs :: "char list \<Rightarrow> char list" where
+    "\<And>cs. cs = [] \<or> cc cs # ccs cs = cs"
+    by (metis (no_types) list.exhaust)
+  then show "x = []"
+    using f2 a1 by (metis assms empty_iff insert_iff list.distinct(1) subset_singletonD)
+qed
+  
+
+lemma Star_empty_string_finite:
+  shows "finite ({[]}\<star>)"
+using Star_atmostempty infinite_super by auto
+
+lemma Star_empty_finite:
+  shows "finite ({}\<star>)"
+using Star_atmostempty infinite_super by auto
+
+lemma Star_concat_replicate:
+  assumes "s \<in> A"
+  shows "concat (replicate n s) \<in> A\<star>"
+using assms
+by (induct n) (auto)
+
+
+lemma concat_replicate_inj:
+  assumes "concat (replicate n s) = concat (replicate m s)" "s \<noteq> []"
+  shows "n = m"
+using assms
+apply(induct n arbitrary: m)
+apply(auto)[1]
+apply(auto)
+apply(case_tac m)
+apply(clarify)
+apply(simp only: replicate.simps concat.simps)
+apply blast
+by simp
+
+lemma A0:
+  assumes "finite (A ;; B)" "B \<noteq> {}"
+  shows "finite A"
+apply(subgoal_tac "\<exists>s. s \<in> B")
+apply(erule exE)
+apply(subgoal_tac "finite {s1 @ s |s1. s1 \<in> A}")
+apply(rule_tac f="\<lambda>s1. s1 @ s" in finite_imageD)
+apply(simp add: image_def)
+apply(smt Collect_cong)
+apply(simp add: inj_on_def)
+apply(rule_tac B="A ;; B" in finite_subset)
+apply(auto simp add: Sequ_def)[1]
+apply(rule assms(1))
+using assms(2) by auto
+
+lemma A1:
+  assumes "finite (A ;; B)" "A \<noteq> {}"
+  shows "finite B"
+apply(subgoal_tac "\<exists>s. s \<in> A")
+apply(erule exE)
+apply(subgoal_tac "finite {s @ s1 |s1. s1 \<in> B}")
+apply(rule_tac f="\<lambda>s1. s @ s1" in finite_imageD)
+apply(simp add: image_def)
+apply(smt Collect_cong)
+apply(simp add: inj_on_def)
+apply(rule_tac B="A ;; B" in finite_subset)
+apply(auto simp add: Sequ_def)[1]
+apply(rule assms(1))
+using assms(2) by auto
+
+lemma Sequ_Prod_finite:
+  assumes "A \<noteq> {}" "B \<noteq> {}"
+  shows "finite (A ;; B) \<longleftrightarrow> (finite (A \<times> B))"
+apply(rule iffI)
+apply(rule finite_cartesian_product)
+apply(erule A0)
+apply(rule assms(2))
+apply(erule A1)
+apply(rule assms(1))
+apply(simp add: Sequ_def)
+apply(rule finite_image_set2)
+apply(drule finite_cartesian_productD1)
+apply(rule assms(2))
+apply(simp)
+apply(drule finite_cartesian_productD2)
+apply(rule assms(1))
+apply(simp)
+done
+
+
+lemma Star_non_empty_string_infinite:
+  assumes "s \<in> A" " s \<noteq> []"
+  shows "infinite (A\<star>)"
+proof -
+  have "inj (\<lambda>n. concat (replicate n s))" 
+  using assms(2) concat_replicate_inj
+    by(auto simp add: inj_on_def)
+  moreover
+  have "infinite (UNIV::nat set)" by simp
+  ultimately
+  have "infinite ((\<lambda>n. concat (replicate n s)) ` UNIV)"
+   by (simp add: range_inj_infinite)
+  moreover
+  have "((\<lambda>n. concat (replicate n s)) ` UNIV) \<subseteq> (A\<star>)"
+    using Star_concat_replicate assms(1) by auto
+  ultimately show "infinite (A\<star>)" 
+  using infinite_super by auto
+qed
+
+lemma infinitestrings_correctness:
+  shows "infinitestrings r \<longleftrightarrow> infinite (L r)"
+apply(induct r)
+apply(simp_all)
+apply(simp add: zeroable_correctness)
+apply(rule iffI)
+apply(erule disjE)
+apply(subst Sequ_Prod_finite)
+apply(auto)[2]
+using finite_cartesian_productD2 apply blast
+apply(subst Sequ_Prod_finite)
+apply(auto)[2]
+using finite_cartesian_productD1 apply blast
+apply(subgoal_tac "L r1 \<noteq> {} \<and> L r2 \<noteq> {}")
+prefer 2
+apply(auto simp add: Sequ_def)[1]
+apply(subst (asm) Sequ_Prod_finite)
+apply(auto)[2]
+apply(auto)[1]
+apply(simp add: atmostempty_correctness)
+apply(rule iffI)
+apply (metis Star_empty_finite Star_empty_string_finite subset_singletonD)
+using Star_non_empty_string_infinite apply blast
+done
+
+unused_thms
+
+end
\ No newline at end of file