# HG changeset patch
# User Christian Urban <christian dot urban at kcl dot ac dot uk>
# Date 1465904483 -3600
# Node ID 5c063eeda6229298a26b658a91c3a96b055d34e8
# Parent  2585e2a7a7ab8ab373fd082f2a31d41f515b212d
deleted afp submission

diff -r 2585e2a7a7ab -r 5c063eeda622 AFP-Submission/Derivatives.thy
--- a/AFP-Submission/Derivatives.thy	Tue Jun 14 12:37:46 2016 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,370 +0,0 @@
-section "Derivatives of regular expressions"
-
-(* Author: Christian Urban *)
-
-theory Derivatives
-imports Regular_Exp
-begin
-
-text{* This theory is based on work by Brozowski \cite{Brzozowski64} and Antimirov \cite{Antimirov95}. *}
-
-subsection {* Brzozowski's derivatives of regular expressions *}
-
-primrec
-  deriv :: "'a \<Rightarrow> 'a rexp \<Rightarrow> 'a rexp"
-where
-  "deriv c (Zero) = Zero"
-| "deriv c (One) = Zero"
-| "deriv c (Atom c') = (if c = c' then One else Zero)"
-| "deriv c (Plus r1 r2) = Plus (deriv c r1) (deriv c r2)"
-| "deriv c (Times r1 r2) = 
-    (if nullable r1 then Plus (Times (deriv c r1) r2) (deriv c r2) else Times (deriv c r1) r2)"
-| "deriv c (Star r) = Times (deriv c r) (Star r)"
-
-primrec 
-  derivs :: "'a list \<Rightarrow> 'a rexp \<Rightarrow> 'a rexp"
-where
-  "derivs [] r = r"
-| "derivs (c # s) r = derivs s (deriv c r)"
-
-
-lemma atoms_deriv_subset: "atoms (deriv x r) \<subseteq> atoms r"
-by (induction r) (auto)
-
-lemma atoms_derivs_subset: "atoms (derivs w r) \<subseteq> atoms r"
-by (induction w arbitrary: r) (auto dest: atoms_deriv_subset[THEN subsetD])
-
-lemma lang_deriv: "lang (deriv c r) = Deriv c (lang r)"
-by (induct r) (simp_all add: nullable_iff)
-
-lemma lang_derivs: "lang (derivs s r) = Derivs s (lang r)"
-by (induct s arbitrary: r) (simp_all add: lang_deriv)
-
-text {* A regular expression matcher: *}
-
-definition matcher :: "'a rexp \<Rightarrow> 'a list \<Rightarrow> bool" where
-"matcher r s = nullable (derivs s r)"
-
-lemma matcher_correctness: "matcher r s \<longleftrightarrow> s \<in> lang r"
-by (induct s arbitrary: r)
-   (simp_all add: nullable_iff lang_deriv matcher_def Deriv_def)
-
-
-subsection {* Antimirov's partial derivatives *}
-
-abbreviation
-  "Timess rs r \<equiv> (\<Union>r' \<in> rs. {Times r' r})"
-
-primrec
-  pderiv :: "'a \<Rightarrow> 'a rexp \<Rightarrow> 'a rexp set"
-where
-  "pderiv c Zero = {}"
-| "pderiv c One = {}"
-| "pderiv c (Atom c') = (if c = c' then {One} else {})"
-| "pderiv c (Plus r1 r2) = (pderiv c r1) \<union> (pderiv c r2)"
-| "pderiv c (Times r1 r2) = 
-    (if nullable r1 then Timess (pderiv c r1) r2 \<union> pderiv c r2 else Timess (pderiv c r1) r2)"
-| "pderiv c (Star r) = Timess (pderiv c r) (Star r)"
-
-primrec
-  pderivs :: "'a list \<Rightarrow> 'a rexp \<Rightarrow> ('a rexp) set"
-where
-  "pderivs [] r = {r}"
-| "pderivs (c # s) r = \<Union> (pderivs s ` pderiv c r)"
-
-abbreviation
- pderiv_set :: "'a \<Rightarrow> 'a rexp set \<Rightarrow> 'a rexp set"
-where
-  "pderiv_set c rs \<equiv> \<Union> (pderiv c ` rs)"
-
-abbreviation
-  pderivs_set :: "'a list \<Rightarrow> 'a rexp set \<Rightarrow> 'a rexp set"
-where
-  "pderivs_set s rs \<equiv> \<Union> (pderivs s ` rs)"
-
-lemma pderivs_append:
-  "pderivs (s1 @ s2) r = \<Union> (pderivs s2 ` pderivs s1 r)"
-by (induct s1 arbitrary: r) (simp_all)
-
-lemma pderivs_snoc:
-  shows "pderivs (s @ [c]) r = pderiv_set c (pderivs s r)"
-by (simp add: pderivs_append)
-
-lemma pderivs_simps [simp]:
-  shows "pderivs s Zero = (if s = [] then {Zero} else {})"
-  and   "pderivs s One = (if s = [] then {One} else {})"
-  and   "pderivs s (Plus r1 r2) = (if s = [] then {Plus r1 r2} else (pderivs s r1) \<union> (pderivs s r2))"
-by (induct s) (simp_all)
-
-lemma pderivs_Atom:
-  shows "pderivs s (Atom c) \<subseteq> {Atom c, One}"
-by (induct s) (simp_all)
-
-subsection {* Relating left-quotients and partial derivatives *}
-
-lemma Deriv_pderiv:
-  shows "Deriv c (lang r) = \<Union> (lang ` pderiv c r)"
-by (induct r) (auto simp add: nullable_iff conc_UNION_distrib)
-
-lemma Derivs_pderivs:
-  shows "Derivs s (lang r) = \<Union> (lang ` pderivs s r)"
-proof (induct s arbitrary: r)
-  case (Cons c s)
-  have ih: "\<And>r. Derivs s (lang r) = \<Union> (lang ` pderivs s r)" by fact
-  have "Derivs (c # s) (lang r) = Derivs s (Deriv c (lang r))" by simp
-  also have "\<dots> = Derivs s (\<Union> (lang ` pderiv c r))" by (simp add: Deriv_pderiv)
-  also have "\<dots> = Derivss s (lang ` (pderiv c r))"
-    by (auto simp add:  Derivs_def)
-  also have "\<dots> = \<Union> (lang ` (pderivs_set s (pderiv c r)))"
-    using ih by auto
-  also have "\<dots> = \<Union> (lang ` (pderivs (c # s) r))" by simp
-  finally show "Derivs (c # s) (lang r) = \<Union> (lang ` pderivs (c # s) r)" .
-qed (simp add: Derivs_def)
-
-subsection {* Relating derivatives and partial derivatives *}
-
-lemma deriv_pderiv:
-  shows "\<Union> (lang ` (pderiv c r)) = lang (deriv c r)"
-unfolding lang_deriv Deriv_pderiv by simp
-
-lemma derivs_pderivs:
-  shows "\<Union> (lang ` (pderivs s r)) = lang (derivs s r)"
-unfolding lang_derivs Derivs_pderivs by simp
-
-
-subsection {* Finiteness property of partial derivatives *}
-
-definition
-  pderivs_lang :: "'a lang \<Rightarrow> 'a rexp \<Rightarrow> 'a rexp set"
-where
-  "pderivs_lang A r \<equiv> \<Union>x \<in> A. pderivs x r"
-
-lemma pderivs_lang_subsetI:
-  assumes "\<And>s. s \<in> A \<Longrightarrow> pderivs s r \<subseteq> C"
-  shows "pderivs_lang A r \<subseteq> C"
-using assms unfolding pderivs_lang_def by (rule UN_least)
-
-lemma pderivs_lang_union:
-  shows "pderivs_lang (A \<union> B) r = (pderivs_lang A r \<union> pderivs_lang B r)"
-by (simp add: pderivs_lang_def)
-
-lemma pderivs_lang_subset:
-  shows "A \<subseteq> B \<Longrightarrow> pderivs_lang A r \<subseteq> pderivs_lang B r"
-by (auto simp add: pderivs_lang_def)
-
-definition
-  "UNIV1 \<equiv> UNIV - {[]}"
-
-lemma pderivs_lang_Zero [simp]:
-  shows "pderivs_lang UNIV1 Zero = {}"
-unfolding UNIV1_def pderivs_lang_def by auto
-
-lemma pderivs_lang_One [simp]:
-  shows "pderivs_lang UNIV1 One = {}"
-unfolding UNIV1_def pderivs_lang_def by (auto split: if_splits)
-
-lemma pderivs_lang_Atom [simp]:
-  shows "pderivs_lang UNIV1 (Atom c) = {One}"
-unfolding UNIV1_def pderivs_lang_def 
-apply(auto)
-apply(frule rev_subsetD)
-apply(rule pderivs_Atom)
-apply(simp)
-apply(case_tac xa)
-apply(auto split: if_splits)
-done
-
-lemma pderivs_lang_Plus [simp]:
-  shows "pderivs_lang UNIV1 (Plus r1 r2) = pderivs_lang UNIV1 r1 \<union> pderivs_lang UNIV1 r2"
-unfolding UNIV1_def pderivs_lang_def by auto
-
-
-text {* Non-empty suffixes of a string (needed for the cases of @{const Times} and @{const Star} below) *}
-
-definition
-  "PSuf s \<equiv> {v. v \<noteq> [] \<and> (\<exists>u. u @ v = s)}"
-
-lemma PSuf_snoc:
-  shows "PSuf (s @ [c]) = (PSuf s) @@ {[c]} \<union> {[c]}"
-unfolding PSuf_def conc_def
-by (auto simp add: append_eq_append_conv2 append_eq_Cons_conv)
-
-lemma PSuf_Union:
-  shows "(\<Union>v \<in> PSuf s @@ {[c]}. f v) = (\<Union>v \<in> PSuf s. f (v @ [c]))"
-by (auto simp add: conc_def)
-
-lemma pderivs_lang_snoc:
-  shows "pderivs_lang (PSuf s @@ {[c]}) r = (pderiv_set c (pderivs_lang (PSuf s) r))"
-unfolding pderivs_lang_def
-by (simp add: PSuf_Union pderivs_snoc)
-
-lemma pderivs_Times:
-  shows "pderivs s (Times r1 r2) \<subseteq> Timess (pderivs s r1) r2 \<union> (pderivs_lang (PSuf s) r2)"
-proof (induct s rule: rev_induct)
-  case (snoc c s)
-  have ih: "pderivs s (Times r1 r2) \<subseteq> Timess (pderivs s r1) r2 \<union> (pderivs_lang (PSuf s) r2)" 
-    by fact
-  have "pderivs (s @ [c]) (Times r1 r2) = pderiv_set c (pderivs s (Times r1 r2))" 
-    by (simp add: pderivs_snoc)
-  also have "\<dots> \<subseteq> pderiv_set c (Timess (pderivs s r1) r2 \<union> (pderivs_lang (PSuf s) r2))"
-    using ih by fast
-  also have "\<dots> = pderiv_set c (Timess (pderivs s r1) r2) \<union> pderiv_set c (pderivs_lang (PSuf s) r2)"
-    by (simp)
-  also have "\<dots> = pderiv_set c (Timess (pderivs s r1) r2) \<union> pderivs_lang (PSuf s @@ {[c]}) r2"
-    by (simp add: pderivs_lang_snoc)
-  also 
-  have "\<dots> \<subseteq> pderiv_set c (Timess (pderivs s r1) r2) \<union> pderiv c r2 \<union> pderivs_lang (PSuf s @@ {[c]}) r2"
-    by auto
-  also 
-  have "\<dots> \<subseteq> Timess (pderiv_set c (pderivs s r1)) r2 \<union> pderiv c r2 \<union> pderivs_lang (PSuf s @@ {[c]}) r2"
-    by (auto simp add: if_splits)
-  also have "\<dots> = Timess (pderivs (s @ [c]) r1) r2 \<union> pderiv c r2 \<union> pderivs_lang (PSuf s @@ {[c]}) r2"
-    by (simp add: pderivs_snoc)
-  also have "\<dots> \<subseteq> Timess (pderivs (s @ [c]) r1) r2 \<union> pderivs_lang (PSuf (s @ [c])) r2"
-    unfolding pderivs_lang_def by (auto simp add: PSuf_snoc)  
-  finally show ?case .
-qed (simp) 
-
-lemma pderivs_lang_Times_aux1:
-  assumes a: "s \<in> UNIV1"
-  shows "pderivs_lang (PSuf s) r \<subseteq> pderivs_lang UNIV1 r"
-using a unfolding UNIV1_def PSuf_def pderivs_lang_def by auto
-
-lemma pderivs_lang_Times_aux2:
-  assumes a: "s \<in> UNIV1"
-  shows "Timess (pderivs s r1) r2 \<subseteq> Timess (pderivs_lang UNIV1 r1) r2"
-using a unfolding pderivs_lang_def by auto
-
-lemma pderivs_lang_Times:
-  shows "pderivs_lang UNIV1 (Times r1 r2) \<subseteq> Timess (pderivs_lang UNIV1 r1) r2 \<union> pderivs_lang UNIV1 r2"
-apply(rule pderivs_lang_subsetI)
-apply(rule subset_trans)
-apply(rule pderivs_Times)
-using pderivs_lang_Times_aux1 pderivs_lang_Times_aux2
-apply(blast)
-done
-
-lemma pderivs_Star:
-  assumes a: "s \<noteq> []"
-  shows "pderivs s (Star r) \<subseteq> Timess (pderivs_lang (PSuf s) r) (Star r)"
-using a
-proof (induct s rule: rev_induct)
-  case (snoc c s)
-  have ih: "s \<noteq> [] \<Longrightarrow> pderivs s (Star r) \<subseteq> Timess (pderivs_lang (PSuf s) r) (Star r)" by fact
-  { assume asm: "s \<noteq> []"
-    have "pderivs (s @ [c]) (Star r) = pderiv_set c (pderivs s (Star r))" by (simp add: pderivs_snoc)
-    also have "\<dots> \<subseteq> pderiv_set c (Timess (pderivs_lang (PSuf s) r) (Star r))"
-      using ih[OF asm] by fast
-    also have "\<dots> \<subseteq> Timess (pderiv_set c (pderivs_lang (PSuf s) r)) (Star r) \<union> pderiv c (Star r)"
-      by (auto split: if_splits)
-    also have "\<dots> \<subseteq> Timess (pderivs_lang (PSuf (s @ [c])) r) (Star r) \<union> (Timess (pderiv c r) (Star r))"
-      by (simp only: PSuf_snoc pderivs_lang_snoc pderivs_lang_union)
-         (auto simp add: pderivs_lang_def)
-    also have "\<dots> = Timess (pderivs_lang (PSuf (s @ [c])) r) (Star r)"
-      by (auto simp add: PSuf_snoc PSuf_Union pderivs_snoc pderivs_lang_def)
-    finally have ?case .
-  }
-  moreover
-  { assume asm: "s = []"
-    then have ?case by (auto simp add: pderivs_lang_def pderivs_snoc PSuf_def)
-  }
-  ultimately show ?case by blast
-qed (simp)
-
-lemma pderivs_lang_Star:
-  shows "pderivs_lang UNIV1 (Star r) \<subseteq> Timess (pderivs_lang UNIV1 r) (Star r)"
-apply(rule pderivs_lang_subsetI)
-apply(rule subset_trans)
-apply(rule pderivs_Star)
-apply(simp add: UNIV1_def)
-apply(simp add: UNIV1_def PSuf_def)
-apply(auto simp add: pderivs_lang_def)
-done
-
-lemma finite_Timess [simp]:
-  assumes a: "finite A"
-  shows "finite (Timess A r)"
-using a by auto
-
-lemma finite_pderivs_lang_UNIV1:
-  shows "finite (pderivs_lang UNIV1 r)"
-apply(induct r)
-apply(simp_all add: 
-  finite_subset[OF pderivs_lang_Times]
-  finite_subset[OF pderivs_lang_Star])
-done
-    
-lemma pderivs_lang_UNIV:
-  shows "pderivs_lang UNIV r = pderivs [] r \<union> pderivs_lang UNIV1 r"
-unfolding UNIV1_def pderivs_lang_def
-by blast
-
-lemma finite_pderivs_lang_UNIV:
-  shows "finite (pderivs_lang UNIV r)"
-unfolding pderivs_lang_UNIV
-by (simp add: finite_pderivs_lang_UNIV1)
-
-lemma finite_pderivs_lang:
-  shows "finite (pderivs_lang A r)"
-by (metis finite_pderivs_lang_UNIV pderivs_lang_subset rev_finite_subset subset_UNIV)
-
-
-text{* The following relationship between the alphabetic width of regular expressions
-(called @{text awidth} below) and the number of partial derivatives was proved
-by Antimirov~\cite{Antimirov95} and formalized by Max Haslbeck. *}
-
-fun awidth :: "'a rexp \<Rightarrow> nat" where
-"awidth Zero = 0" |
-"awidth One = 0" |
-"awidth (Atom a) = 1" |
-"awidth (Plus r1 r2) = awidth r1 + awidth r2" |
-"awidth (Times r1 r2) = awidth r1 + awidth r2" |
-"awidth (Star r1) = awidth r1"
-
-lemma card_Timess_pderivs_lang_le:
-  "card (Timess (pderivs_lang A r) s) \<le> card (pderivs_lang A r)"
-by (metis card_image_le finite_pderivs_lang image_eq_UN)
-
-lemma card_pderivs_lang_UNIV1_le_awidth: "card (pderivs_lang UNIV1 r) \<le> awidth r"
-proof (induction r)
-  case (Plus r1 r2)
-  have "card (pderivs_lang UNIV1 (Plus r1 r2)) = card (pderivs_lang UNIV1 r1 \<union> pderivs_lang UNIV1 r2)" by simp
-  also have "\<dots> \<le> card (pderivs_lang UNIV1 r1) + card (pderivs_lang UNIV1 r2)"
-    by(simp add: card_Un_le)
-  also have "\<dots> \<le> awidth (Plus r1 r2)" using Plus.IH by simp
-  finally show ?case .
-next
-  case (Times r1 r2)
-  have "card (pderivs_lang UNIV1 (Times r1 r2)) \<le> card (Timess (pderivs_lang UNIV1 r1) r2 \<union> pderivs_lang UNIV1 r2)"
-    by (simp add: card_mono finite_pderivs_lang pderivs_lang_Times)
-  also have "\<dots> \<le> card (Timess (pderivs_lang UNIV1 r1) r2) + card (pderivs_lang UNIV1 r2)"
-    by (simp add: card_Un_le)
-  also have "\<dots> \<le> card (pderivs_lang UNIV1 r1) + card (pderivs_lang UNIV1 r2)"
-    by (simp add: card_Timess_pderivs_lang_le)
-  also have "\<dots> \<le> awidth (Times r1 r2)" using Times.IH by simp
-  finally show ?case .
-next
-  case (Star r)
-  have "card (pderivs_lang UNIV1 (Star r)) \<le> card (Timess (pderivs_lang UNIV1 r) (Star r))"
-    by (simp add: card_mono finite_pderivs_lang pderivs_lang_Star)
-  also have "\<dots> \<le> card (pderivs_lang UNIV1 r)" by (rule card_Timess_pderivs_lang_le)
-  also have "\<dots> \<le> awidth (Star r)" by (simp add: Star.IH)
-  finally show ?case .
-qed (auto)
-
-text{* Antimirov's Theorem 3.4: *}
-theorem card_pderivs_lang_UNIV_le_awidth: "card (pderivs_lang UNIV r) \<le> awidth r + 1"
-proof -
-  have "card (insert r (pderivs_lang UNIV1 r)) \<le> Suc (card (pderivs_lang UNIV1 r))"
-    by(auto simp: card_insert_if[OF finite_pderivs_lang_UNIV1])
-  also have "\<dots> \<le> Suc (awidth r)" by(simp add: card_pderivs_lang_UNIV1_le_awidth)
-  finally show ?thesis by(simp add: pderivs_lang_UNIV)
-qed 
-
-text{* Antimirov's Corollary 3.5: *}
-corollary card_pderivs_lang_le_awidth: "card (pderivs_lang A r) \<le> awidth r + 1"
-by(rule order_trans[OF
-  card_mono[OF finite_pderivs_lang_UNIV pderivs_lang_subset[OF subset_UNIV]]
-  card_pderivs_lang_UNIV_le_awidth])
-
-end
\ No newline at end of file
diff -r 2585e2a7a7ab -r 5c063eeda622 AFP-Submission/Lexer.thy
--- a/AFP-Submission/Lexer.thy	Tue Jun 14 12:37:46 2016 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,493 +0,0 @@
-(*  Title:       POSIX Lexing with Derivatives of Regular Expressions
-    Authors:     Fahad Ausaf <fahad.ausaf at icloud.com>, 2016
-                 Roy Dyckhoff <roy.dyckhoff at st-andrews.ac.uk>, 2016
-                 Christian Urban <christian.urban at kcl.ac.uk>, 2016
-    Maintainer:  Christian Urban <christian.urban at kcl.ac.uk>
-*) 
-
-theory Lexer
-  imports Derivatives
-begin
-
-section {* Values *}
-
-datatype 'a val = 
-  Void
-| Atm 'a
-| Seq "'a val" "'a val"
-| Right "'a val"
-| Left "'a val"
-| Stars "('a val) list"
-
-
-section {* The string behind a value *}
-
-fun 
-  flat :: "'a val \<Rightarrow> 'a list"
-where
-  "flat (Void) = []"
-| "flat (Atm c) = [c]"
-| "flat (Left v) = flat v"
-| "flat (Right v) = flat v"
-| "flat (Seq v1 v2) = (flat v1) @ (flat v2)"
-| "flat (Stars []) = []"
-| "flat (Stars (v#vs)) = (flat v) @ (flat (Stars vs))" 
-
-lemma flat_Stars [simp]:
- "flat (Stars vs) = concat (map flat vs)"
-by (induct vs) (auto)
-
-section {* Relation between values and regular expressions *}
-
-inductive 
-  Prf :: "'a val \<Rightarrow> 'a rexp \<Rightarrow> bool" ("\<turnstile> _ : _" [100, 100] 100)
-where
- "\<lbrakk>\<turnstile> v1 : r1; \<turnstile> v2 : r2\<rbrakk> \<Longrightarrow> \<turnstile> Seq v1 v2 : Times r1 r2"
-| "\<turnstile> v1 : r1 \<Longrightarrow> \<turnstile> Left v1 : Plus r1 r2"
-| "\<turnstile> v2 : r2 \<Longrightarrow> \<turnstile> Right v2 : Plus r1 r2"
-| "\<turnstile> Void : One"
-| "\<turnstile> Atm c : Atom c"
-| "\<turnstile> Stars [] : Star r"
-| "\<lbrakk>\<turnstile> v : r; \<turnstile> Stars vs : Star r\<rbrakk> \<Longrightarrow> \<turnstile> Stars (v # vs) : Star r"
-
-inductive_cases Prf_elims:
-  "\<turnstile> v : Zero"
-  "\<turnstile> v : Times r1 r2"
-  "\<turnstile> v : Plus r1 r2"
-  "\<turnstile> v : One"
-  "\<turnstile> v : Atom c"
-(*  "\<turnstile> vs : Star r"*)
-
-lemma Prf_flat_lang:
-  assumes "\<turnstile> v : r" shows "flat v \<in> lang r"
-using assms
-by(induct v r rule: Prf.induct) (auto)
-
-lemma Prf_Stars:
-  assumes "\<forall>v \<in> set vs. \<turnstile> v : r"
-  shows "\<turnstile> Stars vs : Star r"
-using assms
-by(induct vs) (auto intro: Prf.intros)
-
-lemma Star_string:
-  assumes "s \<in> star A"
-  shows "\<exists>ss. concat ss = s \<and> (\<forall>s \<in> set ss. s \<in> A)"
-using assms
-by (metis in_star_iff_concat set_mp)
-
-lemma Star_val:
-  assumes "\<forall>s\<in>set ss. \<exists>v. s = flat v \<and> \<turnstile> v : r"
-  shows "\<exists>vs. concat (map flat vs) = concat ss \<and> (\<forall>v\<in>set vs. \<turnstile> v : r)"
-using assms
-apply(induct ss)
-apply(auto)
-apply (metis empty_iff list.set(1))
-by (metis concat.simps(2) list.simps(9) set_ConsD)
-
-lemma L_flat_Prf1:
-  assumes "\<turnstile> v : r" shows "flat v \<in> lang r"
-using assms
-by (induct)(auto)
-
-lemma L_flat_Prf2:
-  assumes "s \<in> lang r" shows "\<exists>v. \<turnstile> v : r \<and> flat v = s"
-using assms
-apply(induct r arbitrary: s)
-apply(auto intro: Prf.intros)
-using Prf.intros(2) flat.simps(3) apply blast
-using Prf.intros(3) flat.simps(4) apply blast
-apply (metis Prf.intros(1) concE flat.simps(5))
-apply(subgoal_tac "\<exists>vs::('a val) list. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. \<turnstile> v : r)")
-apply(auto)[1]
-apply(rule_tac x="Stars vs" in exI)
-apply(simp)
-apply (simp add: Prf_Stars)
-apply(drule Star_string)
-apply(auto)
-apply(rule Star_val)
-apply(auto)
-done
-
-lemma L_flat_Prf:
-  "lang r = {flat v | v. \<turnstile> v : r}"
-using L_flat_Prf1 L_flat_Prf2 by blast
-
-
-section {* Sulzmann and Lu functions *}
-
-fun 
-  mkeps :: "'a rexp \<Rightarrow> 'a val"
-where
-  "mkeps(One) = Void"
-| "mkeps(Times r1 r2) = Seq (mkeps r1) (mkeps r2)"
-| "mkeps(Plus r1 r2) = (if nullable(r1) then Left (mkeps r1) else Right (mkeps r2))"
-| "mkeps(Star r) = Stars []"
-
-fun injval :: "'a rexp \<Rightarrow> 'a \<Rightarrow> 'a val \<Rightarrow> 'a val"
-where
-  "injval (Atom d) c Void = Atm d"
-| "injval (Plus r1 r2) c (Left v1) = Left(injval r1 c v1)"
-| "injval (Plus r1 r2) c (Right v2) = Right(injval r2 c v2)"
-| "injval (Times r1 r2) c (Seq v1 v2) = Seq (injval r1 c v1) v2"
-| "injval (Times r1 r2) c (Left (Seq v1 v2)) = Seq (injval r1 c v1) v2"
-| "injval (Times r1 r2) c (Right v2) = Seq (mkeps r1) (injval r2 c v2)"
-| "injval (Star r) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)" 
-
-
-section {* Mkeps, injval *}
-
-lemma mkeps_nullable:
-  assumes "nullable r" 
-  shows "\<turnstile> mkeps r : r"
-using assms
-by (induct r) 
-   (auto intro: Prf.intros)
-
-lemma mkeps_flat:
-  assumes "nullable r" 
-  shows "flat (mkeps r) = []"
-using assms
-by (induct r) (auto)
-
-
-lemma Prf_injval:
-  assumes "\<turnstile> v : deriv c r" 
-  shows "\<turnstile> (injval r c v) : r"
-using assms
-apply(induct r arbitrary: c v rule: rexp.induct)
-apply(auto intro!: Prf.intros mkeps_nullable elim!: Prf_elims split: if_splits)
-(* Star *)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)[7]
-apply(auto)
-apply (metis Prf.intros(6) Prf.intros(7))
-by (metis Prf.intros(7))
-
-lemma Prf_injval_flat:
-  assumes "\<turnstile> v : deriv c r" 
-  shows "flat (injval r c v) = c # (flat v)"
-using assms
-apply(induct r arbitrary: v c)
-apply(auto elim!: Prf_elims split: if_splits)
-apply(metis mkeps_flat)
-apply(rotate_tac 2)
-apply(erule Prf.cases)
-apply(simp_all)[7]
-done
-
-(* HERE *)
-
-section {* Our Alternative Posix definition *}
-
-inductive 
-  Posix :: "'a list \<Rightarrow> 'a rexp \<Rightarrow> 'a val \<Rightarrow> bool" ("_ \<in> _ \<rightarrow> _" [100, 100, 100] 100)
-where
-  Posix_One: "[] \<in> One \<rightarrow> Void"
-| Posix_Atom: "[c] \<in> (Atom c) \<rightarrow> (Atm c)"
-| Posix_Plus1: "s \<in> r1 \<rightarrow> v \<Longrightarrow> s \<in> (Plus r1 r2) \<rightarrow> (Left v)"
-| Posix_Plus2: "\<lbrakk>s \<in> r2 \<rightarrow> v; s \<notin> lang r1\<rbrakk> \<Longrightarrow> s \<in> (Plus r1 r2) \<rightarrow> (Right v)"
-| Posix_Times: "\<lbrakk>s1 \<in> r1 \<rightarrow> v1; s2 \<in> r2 \<rightarrow> v2;
-    \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> lang r1 \<and> s\<^sub>4 \<in> lang r2)\<rbrakk> \<Longrightarrow> 
-    (s1 @ s2) \<in> (Times r1 r2) \<rightarrow> (Seq v1 v2)"
-| Posix_Star1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> Star r \<rightarrow> Stars vs; flat v \<noteq> [];
-    \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> lang r \<and> s\<^sub>4 \<in> lang (Star r))\<rbrakk>
-    \<Longrightarrow> (s1 @ s2) \<in> Star r \<rightarrow> Stars (v # vs)"
-| Posix_Star2: "[] \<in> Star r \<rightarrow> Stars []"
-
-inductive_cases Posix_elims:
-  "s \<in> Zero \<rightarrow> v"
-  "s \<in> One \<rightarrow> v"
-  "s \<in> Atom c \<rightarrow> v"
-  "s \<in> Plus r1 r2 \<rightarrow> v"
-  "s \<in> Times r1 r2 \<rightarrow> v"
-  "s \<in> Star r \<rightarrow> v"
-
-lemma Posix1:
-  assumes "s \<in> r \<rightarrow> v"
-  shows "s \<in> lang r" "flat v = s"
-using assms
-by (induct s r v rule: Posix.induct) (auto)
-
-
-lemma Posix1a:
-  assumes "s \<in> r \<rightarrow> v"
-  shows "\<turnstile> v : r"
-using assms
-by (induct s r v rule: Posix.induct)(auto intro: Prf.intros)
-
-
-lemma Posix_mkeps:
-  assumes "nullable r"
-  shows "[] \<in> r \<rightarrow> mkeps r"
-using assms
-apply(induct r)
-apply(auto intro: Posix.intros simp add: nullable_iff)
-apply(subst append.simps(1)[symmetric])
-apply(rule Posix.intros)
-apply(auto)
-done
-
-
-lemma Posix_determ:
-  assumes "s \<in> r \<rightarrow> v1" "s \<in> r \<rightarrow> v2"
-  shows "v1 = v2"
-using assms
-proof (induct s r v1 arbitrary: v2 rule: Posix.induct)
-  case (Posix_One v2)
-  have "[] \<in> One \<rightarrow> v2" by fact
-  then show "Void = v2" by cases auto
-next 
-  case (Posix_Atom c v2)
-  have "[c] \<in> Atom c \<rightarrow> v2" by fact
-  then show "Atm c = v2" by cases auto
-next 
-  case (Posix_Plus1 s r1 v r2 v2)
-  have "s \<in> Plus r1 r2 \<rightarrow> v2" by fact
-  moreover
-  have "s \<in> r1 \<rightarrow> v" by fact
-  then have "s \<in> lang r1" by (simp add: Posix1)
-  ultimately obtain v' where eq: "v2 = Left v'" "s \<in> r1 \<rightarrow> v'" by cases auto 
-  moreover
-  have IH: "\<And>v2. s \<in> r1 \<rightarrow> v2 \<Longrightarrow> v = v2" by fact
-  ultimately have "v = v'" by simp
-  then show "Left v = v2" using eq by simp
-next 
-  case (Posix_Plus2 s r2 v r1 v2)
-  have "s \<in> Plus r1 r2 \<rightarrow> v2" by fact
-  moreover
-  have "s \<notin> lang r1" by fact
-  ultimately obtain v' where eq: "v2 = Right v'" "s \<in> r2 \<rightarrow> v'" 
-    by cases (auto simp add: Posix1) 
-  moreover
-  have IH: "\<And>v2. s \<in> r2 \<rightarrow> v2 \<Longrightarrow> v = v2" by fact
-  ultimately have "v = v'" by simp
-  then show "Right v = v2" using eq by simp
-next
-  case (Posix_Times s1 r1 v1 s2 r2 v2 v')
-  have "(s1 @ s2) \<in> Times r1 r2 \<rightarrow> v'" 
-       "s1 \<in> r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2"
-       "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> lang r1 \<and> s\<^sub>4 \<in> lang r2)" by fact+
-  then obtain v1' v2' where "v' = Seq v1' v2'" "s1 \<in> r1 \<rightarrow> v1'" "s2 \<in> r2 \<rightarrow> v2'"
-  apply(cases) apply (auto simp add: append_eq_append_conv2)
-  using Posix1(1) by fastforce+
-  moreover
-  have IHs: "\<And>v1'. s1 \<in> r1 \<rightarrow> v1' \<Longrightarrow> v1 = v1'"
-            "\<And>v2'. s2 \<in> r2 \<rightarrow> v2' \<Longrightarrow> v2 = v2'" by fact+
-  ultimately show "Seq v1 v2 = v'" by simp
-next
-  case (Posix_Star1 s1 r v s2 vs v2)
-  have "(s1 @ s2) \<in> Star r \<rightarrow> v2" 
-       "s1 \<in> r \<rightarrow> v" "s2 \<in> Star r \<rightarrow> Stars vs" "flat v \<noteq> []"
-       "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> lang r \<and> s\<^sub>4 \<in> lang (Star r))" by fact+
-  then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (Star r) \<rightarrow> (Stars vs')"
-  apply(cases) apply (auto simp add: append_eq_append_conv2)
-  using Posix1(1) apply fastforce
-  apply (metis Posix1(1) Posix_Star1.hyps(6) append_Nil append_Nil2)
-  using Posix1(2) by blast
-  moreover
-  have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"
-            "\<And>v2. s2 \<in> Star r \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact+
-  ultimately show "Stars (v # vs) = v2" by auto
-next
-  case (Posix_Star2 r v2)
-  have "[] \<in> Star r \<rightarrow> v2" by fact
-  then show "Stars [] = v2" by cases (auto simp add: Posix1)
-qed
-
-
-lemma Posix_injval:
-  assumes "s \<in> (deriv c r) \<rightarrow> v"
-  shows "(c # s) \<in> r \<rightarrow> (injval r c v)"
-using assms
-proof(induct r arbitrary: s v rule: rexp.induct)
-  case Zero
-  have "s \<in> deriv c Zero \<rightarrow> v" by fact
-  then have "s \<in> Zero \<rightarrow> v" by simp
-  then have "False" by cases
-  then show "(c # s) \<in> Zero \<rightarrow> (injval Zero c v)" by simp
-next
-  case One
-  have "s \<in> deriv c One \<rightarrow> v" by fact
-  then have "s \<in> Zero \<rightarrow> v" by simp
-  then have "False" by cases
-  then show "(c # s) \<in> One \<rightarrow> (injval One c v)" by simp
-next 
-  case (Atom d)
-  consider (eq) "c = d" | (ineq) "c \<noteq> d" by blast
-  then show "(c # s) \<in> (Atom d) \<rightarrow> (injval (Atom d) c v)"
-  proof (cases)
-    case eq
-    have "s \<in> deriv c (Atom d) \<rightarrow> v" by fact
-    then have "s \<in> One \<rightarrow> v" using eq by simp
-    then have eqs: "s = [] \<and> v = Void" by cases simp
-    show "(c # s) \<in> Atom d \<rightarrow> injval (Atom d) c v" using eq eqs 
-    by (auto intro: Posix.intros)
-  next
-    case ineq
-    have "s \<in> deriv c (Atom d) \<rightarrow> v" by fact
-    then have "s \<in> Zero \<rightarrow> v" using ineq by simp
-    then have "False" by cases
-    then show "(c # s) \<in> Atom d \<rightarrow> injval (Atom d) c v" by simp
-  qed
-next
-  case (Plus r1 r2)
-  have IH1: "\<And>s v. s \<in> deriv c r1 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r1 \<rightarrow> injval r1 c v" by fact
-  have IH2: "\<And>s v. s \<in> deriv c r2 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r2 \<rightarrow> injval r2 c v" by fact
-  have "s \<in> deriv c (Plus r1 r2) \<rightarrow> v" by fact
-  then have "s \<in> Plus (deriv c r1) (deriv c r2) \<rightarrow> v" by simp
-  then consider (left) v' where "v = Left v'" "s \<in> deriv c r1 \<rightarrow> v'" 
-              | (right) v' where "v = Right v'" "s \<notin> lang (deriv c r1)" "s \<in> deriv c r2 \<rightarrow> v'" 
-              by cases auto
-  then show "(c # s) \<in> Plus r1 r2 \<rightarrow> injval (Plus r1 r2) c v"
-  proof (cases)
-    case left
-    have "s \<in> deriv c r1 \<rightarrow> v'" by fact
-    then have "(c # s) \<in> r1 \<rightarrow> injval r1 c v'" using IH1 by simp
-    then have "(c # s) \<in> Plus r1 r2 \<rightarrow> injval (Plus r1 r2) c (Left v')" by (auto intro: Posix.intros)
-    then show "(c # s) \<in> Plus r1 r2 \<rightarrow> injval (Plus r1 r2) c v" using left by simp
-  next 
-    case right
-    have "s \<notin> lang (deriv c r1)" by fact
-    then have "c # s \<notin> lang r1" by (simp add: lang_deriv Deriv_def)
-    moreover 
-    have "s \<in> deriv c r2 \<rightarrow> v'" by fact
-    then have "(c # s) \<in> r2 \<rightarrow> injval r2 c v'" using IH2 by simp
-    ultimately have "(c # s) \<in> Plus r1 r2 \<rightarrow> injval (Plus r1 r2) c (Right v')" 
-      by (auto intro: Posix.intros)
-    then show "(c # s) \<in> Plus r1 r2 \<rightarrow> injval (Plus r1 r2) c v" using right by simp
-  qed
-next
-  case (Times r1 r2)
-  have IH1: "\<And>s v. s \<in> deriv c r1 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r1 \<rightarrow> injval r1 c v" by fact
-  have IH2: "\<And>s v. s \<in> deriv c r2 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r2 \<rightarrow> injval r2 c v" by fact
-  have "s \<in> deriv c (Times r1 r2) \<rightarrow> v" by fact
-  then consider 
-        (left_nullable) v1 v2 s1 s2 where 
-        "v = Left (Seq v1 v2)"  "s = s1 @ s2" 
-        "s1 \<in> deriv c r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" "nullable r1" 
-        "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> lang (deriv c r1) \<and> s\<^sub>4 \<in> lang r2)"
-      | (right_nullable) v1 s1 s2 where 
-        "v = Right v1" "s = s1 @ s2"  
-        "s \<in> deriv c r2 \<rightarrow> v1" "nullable r1" "s1 @ s2 \<notin> lang (Times (deriv c r1) r2)"
-      | (not_nullable) v1 v2 s1 s2 where
-        "v = Seq v1 v2" "s = s1 @ s2" 
-        "s1 \<in> deriv c r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" "\<not>nullable r1" 
-        "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> lang (deriv c r1) \<and> s\<^sub>4 \<in> lang r2)"
-        by (force split: if_splits elim!: Posix_elims simp add: lang_deriv Deriv_def)   
-  then show "(c # s) \<in> Times r1 r2 \<rightarrow> injval (Times r1 r2) c v" 
-    proof (cases)
-      case left_nullable
-      have "s1 \<in> deriv c r1 \<rightarrow> v1" by fact
-      then have "(c # s1) \<in> r1 \<rightarrow> injval r1 c v1" using IH1 by simp
-      moreover
-      have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> lang (deriv c r1) \<and> s\<^sub>4 \<in> lang r2)" by fact
-      then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> lang r1 \<and> s\<^sub>4 \<in> lang r2)" 
-         by (simp add: lang_deriv Deriv_def)
-      ultimately have "((c # s1) @ s2) \<in> Times r1 r2 \<rightarrow> Seq (injval r1 c v1) v2" using left_nullable by (rule_tac Posix.intros)
-      then show "(c # s) \<in> Times r1 r2 \<rightarrow> injval (Times r1 r2) c v" using left_nullable by simp
-    next
-      case right_nullable
-      have "nullable r1" by fact
-      then have "[] \<in> r1 \<rightarrow> (mkeps r1)" by (rule Posix_mkeps)
-      moreover
-      have "s \<in> deriv c r2 \<rightarrow> v1" by fact
-      then have "(c # s) \<in> r2 \<rightarrow> (injval r2 c v1)" using IH2 by simp
-      moreover
-      have "s1 @ s2 \<notin> lang (Times (deriv c r1) r2)" by fact
-      then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = c # s \<and> [] @ s\<^sub>3 \<in> lang r1 \<and> s\<^sub>4 \<in> lang r2)" 
-        using right_nullable 
-        apply (auto simp add: lang_deriv Deriv_def append_eq_Cons_conv)
-        by (metis concI mem_Collect_eq)
-      ultimately have "([] @ (c # s)) \<in> Times r1 r2 \<rightarrow> Seq (mkeps r1) (injval r2 c v1)"
-      by(rule Posix.intros)
-      then show "(c # s) \<in> Times r1 r2 \<rightarrow> injval (Times r1 r2) c v" using right_nullable by simp
-    next
-      case not_nullable
-      have "s1 \<in> deriv c r1 \<rightarrow> v1" by fact
-      then have "(c # s1) \<in> r1 \<rightarrow> injval r1 c v1" using IH1 by simp
-      moreover
-      have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> lang (deriv c r1) \<and> s\<^sub>4 \<in> lang r2)" by fact
-      then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> lang r1 \<and> s\<^sub>4 \<in> lang r2)" by (simp add: lang_deriv Deriv_def)
-      ultimately have "((c # s1) @ s2) \<in> Times r1 r2 \<rightarrow> Seq (injval r1 c v1) v2" using not_nullable 
-        by (rule_tac Posix.intros) (simp_all) 
-      then show "(c # s) \<in> Times r1 r2 \<rightarrow> injval (Times r1 r2) c v" using not_nullable by simp
-    qed
-next
-  case (Star r)
-  have IH: "\<And>s v. s \<in> deriv c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact
-  have "s \<in> deriv c (Star r) \<rightarrow> v" by fact
-  then consider
-      (cons) v1 vs s1 s2 where 
-        "v = Seq v1 (Stars vs)" "s = s1 @ s2" 
-        "s1 \<in> deriv c r \<rightarrow> v1" "s2 \<in> (Star r) \<rightarrow> (Stars vs)"
-        "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> lang (deriv c r) \<and> s\<^sub>4 \<in> lang (Star r))" 
-        apply(auto elim!: Posix_elims(1-5) simp add: lang_deriv Deriv_def intro: Posix.intros)
-        apply(rotate_tac 3)
-        apply(erule_tac Posix_elims(6))
-        apply (simp add: Posix.intros(6))
-        using Posix.intros(7) by blast
-    then show "(c # s) \<in> Star r \<rightarrow> injval (Star r) c v" 
-    proof (cases)
-      case cons
-          have "s1 \<in> deriv c r \<rightarrow> v1" by fact
-          then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp
-        moreover
-          have "s2 \<in> Star r \<rightarrow> Stars vs" by fact
-        moreover 
-          have "(c # s1) \<in> r \<rightarrow> injval r c v1" by fact 
-          then have "flat (injval r c v1) = (c # s1)" by (rule Posix1)
-          then have "flat (injval r c v1) \<noteq> []" by simp
-        moreover 
-          have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> lang (deriv c r) \<and> s\<^sub>4 \<in> lang (Star r))" by fact
-          then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> lang r \<and> s\<^sub>4 \<in> lang (Star r))" 
-            by (simp add: lang_deriv Deriv_def)
-        ultimately 
-        have "((c # s1) @ s2) \<in> Star r \<rightarrow> Stars (injval r c v1 # vs)" by (rule Posix.intros)
-        then show "(c # s) \<in> Star r \<rightarrow> injval (Star r) c v" using cons by(simp)
-    qed
-qed
-
-
-section {* The Lexer by Sulzmann and Lu  *}
-
-fun 
-  lexer :: "'a rexp \<Rightarrow> 'a list \<Rightarrow> ('a val) option"
-where
-  "lexer r [] = (if nullable r then Some(mkeps r) else None)"
-| "lexer r (c#s) = (case (lexer (deriv c r) s) of  
-                    None \<Rightarrow> None
-                  | Some(v) \<Rightarrow> Some(injval r c v))"
-
-
-lemma lexer_correct_None:
-  shows "s \<notin> lang r \<longleftrightarrow> lexer r s = None"
-using assms
-apply(induct s arbitrary: r)
-apply(simp add: nullable_iff)
-apply(drule_tac x="deriv a r" in meta_spec)
-apply(auto simp add: lang_deriv Deriv_def)
-done
-
-lemma lexer_correct_Some:
-  shows "s \<in> lang r \<longleftrightarrow> (\<exists>v. lexer r s = Some(v) \<and> s \<in> r \<rightarrow> v)"
-using assms
-apply(induct s arbitrary: r)
-apply(auto simp add: Posix_mkeps nullable_iff)[1]
-apply(drule_tac x="deriv a r" in meta_spec)
-apply(simp add: lang_deriv Deriv_def)
-apply(rule iffI)
-apply(auto intro: Posix_injval simp add: Posix1(1))
-done 
-
-lemma lexer_correctness:
-  shows "(lexer r s = Some v) \<longleftrightarrow> s \<in> r \<rightarrow> v"
-  and   "(lexer r s = None) \<longleftrightarrow> \<not>(\<exists>v. s \<in> r \<rightarrow> v)"
-apply(auto)
-using lexer_correct_None lexer_correct_Some apply fastforce
-using Posix1(1) Posix_determ lexer_correct_Some apply blast
-using Posix1(1) lexer_correct_None apply blast
-using lexer_correct_None lexer_correct_Some by blast
-
-
-end
\ No newline at end of file
diff -r 2585e2a7a7ab -r 5c063eeda622 AFP-Submission/README
--- a/AFP-Submission/README	Tue Jun 14 12:37:46 2016 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,53 +0,0 @@
-Title:
-======
-POSIX Lexing with Derivatives of Regular Expressions
-
-
-Authors:
-========
-Fahad Ausaf <fahad.ausaf at icloud.com>, 2016
-Roy Dyckhoff <roy.dyckhoff at st-andrews.ac.uk>, 2016
-Christian Urban <christian.urban at kcl.ac.uk>, 2016
-
-
-Abstract:
-=========
-
-Brzozowski introduced the notion of derivatives for regular
-expressions. They can be used for a very simple regular expression
-matching algorithm. Sulzmann and Lu cleverly extended this algorithm
-in order to deal with POSIX matching, which is the underlying
-disambiguation strategy for regular expressions needed in
-lexers. Sulzmann and Lu have made available on-line what they call a
-``rigorous proof'' of the correctness of their algorithm w.r.t. their
-specification; regrettably, it appears to us to have unfillable
-gaps. In the first part of this paper we give our inductive definition
-of what a POSIX value is and show (i) that such a value is unique (for
-given regular expression and string being matched) and (ii) that
-Sulzmann and Lu's algorithm always generates such a value (provided
-that the regular expression matches the string). We also prove the
-correctness of an optimised version of the POSIX matching
-algorithm. Our definitions and proof are much simpler than those by
-Sulzmann and Lu and can be easily formalised in Isabelle/HOL. In the
-second part we analyse the correctness argument by Sulzmann and Lu and
-explain why the gaps in this argument cannot be filled easily.
-
-
-New Theories:
-=============
-
- Lexer.thy
- Simplifying.thy
-
-The repository can be checked using Isabelle 2016.
-
- isabelle build -c -v -d . Posix-Lexing
-
-
-  
-
-
-
-
-
-
diff -r 2585e2a7a7ab -r 5c063eeda622 AFP-Submission/ROOT
--- a/AFP-Submission/ROOT	Tue Jun 14 12:37:46 2016 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,23 +0,0 @@
-chapter AFP
-
-(* Session name, add to AFP group, list base session: *)
-session "Posix-Lexing" (AFP) = HOL +
-
-(* Timeout (in sec) in case of non-termination problems *)
-  options [timeout = 600]
-
-(* The top-level theories of the submission: *)
-  theories [document = false]
-     "Regular_Set" 
-     "Regular_Exp"
-     "Derivatives" 
-
-  theories
-    "Lexer" 
-    "Simplifying"
-    
-(* Dependencies on document source files: *)
-  document_files
-    "root.bib"
-    "root.tex"
-
diff -r 2585e2a7a7ab -r 5c063eeda622 AFP-Submission/Regular_Exp.thy
--- a/AFP-Submission/Regular_Exp.thy	Tue Jun 14 12:37:46 2016 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,160 +0,0 @@
-(*  Author: Tobias Nipkow *)
-
-section "Regular expressions"
-
-theory Regular_Exp
-imports Regular_Set
-begin
-
-datatype (atoms: 'a) rexp =
-  is_Zero: Zero |
-  is_One: One |
-  Atom 'a |
-  Plus "('a rexp)" "('a rexp)" |
-  Times "('a rexp)" "('a rexp)" |
-  Star "('a rexp)"
-
-primrec lang :: "'a rexp => 'a lang" where
-"lang Zero = {}" |
-"lang One = {[]}" |
-"lang (Atom a) = {[a]}" |
-"lang (Plus r s) = (lang r) Un (lang s)" |
-"lang (Times r s) = conc (lang r) (lang s)" |
-"lang (Star r) = star(lang r)"
-
-primrec nullable :: "'a rexp \<Rightarrow> bool" where
-"nullable Zero = False" |
-"nullable One = True" |
-"nullable (Atom c) = False" |
-"nullable (Plus r1 r2) = (nullable r1 \<or> nullable r2)" |
-"nullable (Times r1 r2) = (nullable r1 \<and> nullable r2)" |
-"nullable (Star r) = True"
-
-lemma nullable_iff: "nullable r \<longleftrightarrow> [] \<in> lang r"
-by (induct r) (auto simp add: conc_def split: if_splits)
-
-text{* Composition on rhs usually complicates matters: *}
-lemma map_map_rexp:
-  "map_rexp f (map_rexp g r) = map_rexp (\<lambda>r. f (g r)) r"
-  unfolding rexp.map_comp o_def ..
-
-lemma map_rexp_ident[simp]: "map_rexp (\<lambda>x. x) = (\<lambda>r. r)"
-  unfolding id_def[symmetric] fun_eq_iff rexp.map_id id_apply by (intro allI refl)
-
-lemma atoms_lang: "w : lang r \<Longrightarrow> set w \<subseteq> atoms r"
-proof(induction r arbitrary: w)
-  case Times thus ?case by fastforce
-next
-  case Star thus ?case by (fastforce simp add: star_conv_concat)
-qed auto
-
-lemma lang_eq_ext: "(lang r = lang s) =
-  (\<forall>w \<in> lists(atoms r \<union> atoms s). w \<in> lang r \<longleftrightarrow> w \<in> lang s)"
-  by (auto simp: atoms_lang[unfolded subset_iff])
-
-lemma lang_eq_ext_Nil_fold_Deriv:
-  fixes r s
-  defines "\<BB> \<equiv> {(fold Deriv w (lang r), fold Deriv w (lang s))| w. w\<in>lists (atoms r \<union> atoms s)}"
-  shows "lang r = lang s \<longleftrightarrow> (\<forall>(K, L) \<in> \<BB>. [] \<in> K \<longleftrightarrow> [] \<in> L)"
-  unfolding lang_eq_ext \<BB>_def by (subst (1 2) in_fold_Deriv[of "[]", simplified, symmetric]) auto
-
-
-subsection {* Term ordering *}
-
-instantiation rexp :: (order) "{order}"
-begin
-
-fun le_rexp :: "('a::order) rexp \<Rightarrow> ('a::order) rexp \<Rightarrow> bool"
-where
-  "le_rexp Zero _ = True"
-| "le_rexp _ Zero = False"
-| "le_rexp One _ = True"
-| "le_rexp _ One = False"
-| "le_rexp (Atom a) (Atom b) = (a <= b)"
-| "le_rexp (Atom _) _ = True"
-| "le_rexp _ (Atom _) = False"
-| "le_rexp (Star r) (Star s) = le_rexp r s"
-| "le_rexp (Star _) _ = True"
-| "le_rexp _ (Star _) = False"
-| "le_rexp (Plus r r') (Plus s s') =
-    (if r = s then le_rexp r' s' else le_rexp r s)"
-| "le_rexp (Plus _ _) _ = True"
-| "le_rexp _ (Plus _ _) = False"
-| "le_rexp (Times r r') (Times s s') =
-    (if r = s then le_rexp r' s' else le_rexp r s)"
-
-(* The class instance stuff is by Dmitriy Traytel *)
-
-definition less_eq_rexp where "r \<le> s \<equiv> le_rexp r s"
-definition less_rexp where "r < s \<equiv> le_rexp r s \<and> r \<noteq> s"
-
-lemma le_rexp_Zero: "le_rexp r Zero \<Longrightarrow> r = Zero"
-by (induction r) auto
-
-lemma le_rexp_refl: "le_rexp r r"
-by (induction r) auto
-
-lemma le_rexp_antisym: "\<lbrakk>le_rexp r s; le_rexp s r\<rbrakk> \<Longrightarrow> r = s"
-by (induction r s rule: le_rexp.induct) (auto dest: le_rexp_Zero)
-
-lemma le_rexp_trans: "\<lbrakk>le_rexp r s; le_rexp s t\<rbrakk> \<Longrightarrow> le_rexp r t"
-proof (induction r s arbitrary: t rule: le_rexp.induct)
-  fix v t assume "le_rexp (Atom v) t" thus "le_rexp One t" by (cases t) auto
-next
-  fix s1 s2 t assume "le_rexp (Plus s1 s2) t" thus "le_rexp One t" by (cases t) auto
-next
-  fix s1 s2 t assume "le_rexp (Times s1 s2) t" thus "le_rexp One t" by (cases t) auto
-next
-  fix s t assume "le_rexp (Star s) t" thus "le_rexp One t" by (cases t) auto
-next
-  fix v u t assume "le_rexp (Atom v) (Atom u)" "le_rexp (Atom u) t"
-  thus "le_rexp (Atom v) t" by (cases t) auto
-next
-  fix v s1 s2 t assume "le_rexp (Plus s1 s2) t" thus "le_rexp (Atom v) t" by (cases t) auto
-next
-  fix v s1 s2 t assume "le_rexp (Times s1 s2) t" thus "le_rexp (Atom v) t" by (cases t) auto
-next
-  fix v s t assume "le_rexp (Star s) t" thus "le_rexp (Atom v) t" by (cases t) auto
-next
-  fix r s t
-  assume IH: "\<And>t. le_rexp r s \<Longrightarrow> le_rexp s t \<Longrightarrow> le_rexp r t"
-    and "le_rexp (Star r) (Star s)" "le_rexp (Star s) t"
-  thus "le_rexp (Star r) t" by (cases t) auto
-next
-  fix r s1 s2 t assume "le_rexp (Plus s1 s2) t" thus "le_rexp (Star r) t" by (cases t) auto
-next
-  fix r s1 s2 t assume "le_rexp (Times s1 s2) t" thus "le_rexp (Star r) t" by (cases t) auto
-next
-  fix r1 r2 s1 s2 t
-  assume "\<And>t. r1 = s1 \<Longrightarrow> le_rexp r2 s2 \<Longrightarrow> le_rexp s2 t \<Longrightarrow> le_rexp r2 t"
-         "\<And>t. r1 \<noteq> s1 \<Longrightarrow> le_rexp r1 s1 \<Longrightarrow> le_rexp s1 t \<Longrightarrow> le_rexp r1 t"
-         "le_rexp (Plus r1 r2) (Plus s1 s2)" "le_rexp (Plus s1 s2) t"
-  thus "le_rexp (Plus r1 r2) t" by (cases t) (auto split: split_if_asm intro: le_rexp_antisym)
-next
-  fix r1 r2 s1 s2 t assume "le_rexp (Times s1 s2) t" thus "le_rexp (Plus r1 r2) t" by (cases t) auto
-next
-  fix r1 r2 s1 s2 t
-  assume "\<And>t. r1 = s1 \<Longrightarrow> le_rexp r2 s2 \<Longrightarrow> le_rexp s2 t \<Longrightarrow> le_rexp r2 t"
-         "\<And>t. r1 \<noteq> s1 \<Longrightarrow> le_rexp r1 s1 \<Longrightarrow> le_rexp s1 t \<Longrightarrow> le_rexp r1 t"
-         "le_rexp (Times r1 r2) (Times s1 s2)" "le_rexp (Times s1 s2) t"
-  thus "le_rexp (Times r1 r2) t" by (cases t) (auto split: split_if_asm intro: le_rexp_antisym)
-qed auto
-
-instance proof
-qed (auto simp add: less_eq_rexp_def less_rexp_def
-       intro: le_rexp_refl le_rexp_antisym le_rexp_trans)
-
-end
-
-instantiation rexp :: (linorder) "{linorder}"
-begin
-
-lemma le_rexp_total: "le_rexp (r :: 'a :: linorder rexp) s \<or> le_rexp s r"
-by (induction r s rule: le_rexp.induct) auto
-
-instance proof
-qed (unfold less_eq_rexp_def less_rexp_def, rule le_rexp_total)
-
-end
-
-end
diff -r 2585e2a7a7ab -r 5c063eeda622 AFP-Submission/Regular_Set.thy
--- a/AFP-Submission/Regular_Set.thy	Tue Jun 14 12:37:46 2016 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,481 +0,0 @@
-(*  Author: Tobias Nipkow, Alex Krauss, Christian Urban  *)
-
-section "Regular sets"
-
-theory Regular_Set
-imports Main
-begin
-
-type_synonym 'a lang = "'a list set"
-
-definition conc :: "'a lang \<Rightarrow> 'a lang \<Rightarrow> 'a lang" (infixr "@@" 75) where
-"A @@ B = {xs@ys | xs ys. xs:A & ys:B}"
-
-text {* checks the code preprocessor for set comprehensions *}
-export_code conc checking SML
-
-overloading lang_pow == "compow :: nat \<Rightarrow> 'a lang \<Rightarrow> 'a lang"
-begin
-  primrec lang_pow :: "nat \<Rightarrow> 'a lang \<Rightarrow> 'a lang" where
-  "lang_pow 0 A = {[]}" |
-  "lang_pow (Suc n) A = A @@ (lang_pow n A)"
-end
-
-text {* for code generation *}
-
-definition lang_pow :: "nat \<Rightarrow> 'a lang \<Rightarrow> 'a lang" where
-  lang_pow_code_def [code_abbrev]: "lang_pow = compow"
-
-lemma [code]:
-  "lang_pow (Suc n) A = A @@ (lang_pow n A)"
-  "lang_pow 0 A = {[]}"
-  by (simp_all add: lang_pow_code_def)
-
-hide_const (open) lang_pow
-
-definition star :: "'a lang \<Rightarrow> 'a lang" where
-"star A = (\<Union>n. A ^^ n)"
-
-
-subsection{* @{term "op @@"} *}
-
-lemma concI[simp,intro]: "u : A \<Longrightarrow> v : B \<Longrightarrow> u@v : A @@ B"
-by (auto simp add: conc_def)
-
-lemma concE[elim]: 
-assumes "w \<in> A @@ B"
-obtains u v where "u \<in> A" "v \<in> B" "w = u@v"
-using assms by (auto simp: conc_def)
-
-lemma conc_mono: "A \<subseteq> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> A @@ B \<subseteq> C @@ D"
-by (auto simp: conc_def) 
-
-lemma conc_empty[simp]: shows "{} @@ A = {}" and "A @@ {} = {}"
-by auto
-
-lemma conc_epsilon[simp]: shows "{[]} @@ A = A" and "A @@ {[]} = A"
-by (simp_all add:conc_def)
-
-lemma conc_assoc: "(A @@ B) @@ C = A @@ (B @@ C)"
-by (auto elim!: concE) (simp only: append_assoc[symmetric] concI)
-
-lemma conc_Un_distrib:
-shows "A @@ (B \<union> C) = A @@ B \<union> A @@ C"
-and   "(A \<union> B) @@ C = A @@ C \<union> B @@ C"
-by auto
-
-lemma conc_UNION_distrib:
-shows "A @@ UNION I M = UNION I (%i. A @@ M i)"
-and   "UNION I M @@ A = UNION I (%i. M i @@ A)"
-by auto
-
-lemma conc_subset_lists: "A \<subseteq> lists S \<Longrightarrow> B \<subseteq> lists S \<Longrightarrow> A @@ B \<subseteq> lists S"
-by(fastforce simp: conc_def in_lists_conv_set)
-
-lemma Nil_in_conc[simp]: "[] \<in> A @@ B \<longleftrightarrow> [] \<in> A \<and> [] \<in> B"
-by (metis append_is_Nil_conv concE concI)
-
-lemma concI_if_Nil1: "[] \<in> A \<Longrightarrow> xs : B \<Longrightarrow> xs \<in> A @@ B"
-by (metis append_Nil concI)
-
-lemma conc_Diff_if_Nil1: "[] \<in> A \<Longrightarrow> A @@ B = (A - {[]}) @@ B \<union> B"
-by (fastforce elim: concI_if_Nil1)
-
-lemma concI_if_Nil2: "[] \<in> B \<Longrightarrow> xs : A \<Longrightarrow> xs \<in> A @@ B"
-by (metis append_Nil2 concI)
-
-lemma conc_Diff_if_Nil2: "[] \<in> B \<Longrightarrow> A @@ B = A @@ (B - {[]}) \<union> A"
-by (fastforce elim: concI_if_Nil2)
-
-lemma singleton_in_conc:
-  "[x] : A @@ B \<longleftrightarrow> [x] : A \<and> [] : B \<or> [] : A \<and> [x] : B"
-by (fastforce simp: Cons_eq_append_conv append_eq_Cons_conv
-       conc_Diff_if_Nil1 conc_Diff_if_Nil2)
-
-
-subsection{* @{term "A ^^ n"} *}
-
-lemma lang_pow_add: "A ^^ (n + m) = A ^^ n @@ A ^^ m"
-by (induct n) (auto simp: conc_assoc)
-
-lemma lang_pow_empty: "{} ^^ n = (if n = 0 then {[]} else {})"
-by (induct n) auto
-
-lemma lang_pow_empty_Suc[simp]: "({}::'a lang) ^^ Suc n = {}"
-by (simp add: lang_pow_empty)
-
-lemma conc_pow_comm:
-  shows "A @@ (A ^^ n) = (A ^^ n) @@ A"
-by (induct n) (simp_all add: conc_assoc[symmetric])
-
-lemma length_lang_pow_ub:
-  "ALL w : A. length w \<le> k \<Longrightarrow> w : A^^n \<Longrightarrow> length w \<le> k*n"
-by(induct n arbitrary: w) (fastforce simp: conc_def)+
-
-lemma length_lang_pow_lb:
-  "ALL w : A. length w \<ge> k \<Longrightarrow> w : A^^n \<Longrightarrow> length w \<ge> k*n"
-by(induct n arbitrary: w) (fastforce simp: conc_def)+
-
-lemma lang_pow_subset_lists: "A \<subseteq> lists S \<Longrightarrow> A ^^ n \<subseteq> lists S"
-by(induction n)(auto simp: conc_subset_lists[OF assms])
-
-
-subsection{* @{const star} *}
-
-lemma star_subset_lists: "A \<subseteq> lists S \<Longrightarrow> star A \<subseteq> lists S"
-unfolding star_def by(blast dest: lang_pow_subset_lists)
-
-lemma star_if_lang_pow[simp]: "w : A ^^ n \<Longrightarrow> w : star A"
-by (auto simp: star_def)
-
-lemma Nil_in_star[iff]: "[] : star A"
-proof (rule star_if_lang_pow)
-  show "[] : A ^^ 0" by simp
-qed
-
-lemma star_if_lang[simp]: assumes "w : A" shows "w : star A"
-proof (rule star_if_lang_pow)
-  show "w : A ^^ 1" using `w : A` by simp
-qed
-
-lemma append_in_starI[simp]:
-assumes "u : star A" and "v : star A" shows "u@v : star A"
-proof -
-  from `u : star A` obtain m where "u : A ^^ m" by (auto simp: star_def)
-  moreover
-  from `v : star A` obtain n where "v : A ^^ n" by (auto simp: star_def)
-  ultimately have "u@v : A ^^ (m+n)" by (simp add: lang_pow_add)
-  thus ?thesis by simp
-qed
-
-lemma conc_star_star: "star A @@ star A = star A"
-by (auto simp: conc_def)
-
-lemma conc_star_comm:
-  shows "A @@ star A = star A @@ A"
-unfolding star_def conc_pow_comm conc_UNION_distrib
-by simp
-
-lemma star_induct[consumes 1, case_names Nil append, induct set: star]:
-assumes "w : star A"
-  and "P []"
-  and step: "!!u v. u : A \<Longrightarrow> v : star A \<Longrightarrow> P v \<Longrightarrow> P (u@v)"
-shows "P w"
-proof -
-  { fix n have "w : A ^^ n \<Longrightarrow> P w"
-    by (induct n arbitrary: w) (auto intro: `P []` step star_if_lang_pow) }
-  with `w : star A` show "P w" by (auto simp: star_def)
-qed
-
-lemma star_empty[simp]: "star {} = {[]}"
-by (auto elim: star_induct)
-
-lemma star_epsilon[simp]: "star {[]} = {[]}"
-by (auto elim: star_induct)
-
-lemma star_idemp[simp]: "star (star A) = star A"
-by (auto elim: star_induct)
-
-lemma star_unfold_left: "star A = A @@ star A \<union> {[]}" (is "?L = ?R")
-proof
-  show "?L \<subseteq> ?R" by (rule, erule star_induct) auto
-qed auto
-
-lemma concat_in_star: "set ws \<subseteq> A \<Longrightarrow> concat ws : star A"
-by (induct ws) simp_all
-
-lemma in_star_iff_concat:
-  "w : star A = (EX ws. set ws \<subseteq> A & w = concat ws)"
-  (is "_ = (EX ws. ?R w ws)")
-proof
-  assume "w : star A" thus "EX ws. ?R w ws"
-  proof induct
-    case Nil have "?R [] []" by simp
-    thus ?case ..
-  next
-    case (append u v)
-    moreover
-    then obtain ws where "set ws \<subseteq> A \<and> v = concat ws" by blast
-    ultimately have "?R (u@v) (u#ws)" by auto
-    thus ?case ..
-  qed
-next
-  assume "EX us. ?R w us" thus "w : star A"
-  by (auto simp: concat_in_star)
-qed
-
-lemma star_conv_concat: "star A = {concat ws|ws. set ws \<subseteq> A}"
-by (fastforce simp: in_star_iff_concat)
-
-lemma star_insert_eps[simp]: "star (insert [] A) = star(A)"
-proof-
-  { fix us
-    have "set us \<subseteq> insert [] A \<Longrightarrow> EX vs. concat us = concat vs \<and> set vs \<subseteq> A"
-      (is "?P \<Longrightarrow> EX vs. ?Q vs")
-    proof
-      let ?vs = "filter (%u. u \<noteq> []) us"
-      show "?P \<Longrightarrow> ?Q ?vs" by (induct us) auto
-    qed
-  } thus ?thesis by (auto simp: star_conv_concat)
-qed
-
-lemma star_unfold_left_Nil: "star A = (A - {[]}) @@ (star A) \<union> {[]}"
-by (metis insert_Diff_single star_insert_eps star_unfold_left)
-
-lemma star_Diff_Nil_fold: "(A - {[]}) @@ star A = star A - {[]}"
-proof -
-  have "[] \<notin> (A - {[]}) @@ star A" by simp
-  thus ?thesis using star_unfold_left_Nil by blast
-qed
-
-lemma star_decom: 
-  assumes a: "x \<in> star A" "x \<noteq> []"
-  shows "\<exists>a b. x = a @ b \<and> a \<noteq> [] \<and> a \<in> A \<and> b \<in> star A"
-using a by (induct rule: star_induct) (blast)+
-
-
-subsection {* Left-Quotients of languages *}
-
-definition Deriv :: "'a \<Rightarrow> 'a lang \<Rightarrow> 'a lang"
-where "Deriv x A = { xs. x#xs \<in> A }"
-
-definition Derivs :: "'a list \<Rightarrow> 'a lang \<Rightarrow> 'a lang"
-where "Derivs xs A = { ys. xs @ ys \<in> A }"
-
-abbreviation 
-  Derivss :: "'a list \<Rightarrow> 'a lang set \<Rightarrow> 'a lang"
-where
-  "Derivss s As \<equiv> \<Union> (Derivs s ` As)"
-
-
-lemma Deriv_empty[simp]:   "Deriv a {} = {}"
-  and Deriv_epsilon[simp]: "Deriv a {[]} = {}"
-  and Deriv_char[simp]:    "Deriv a {[b]} = (if a = b then {[]} else {})"
-  and Deriv_union[simp]:   "Deriv a (A \<union> B) = Deriv a A \<union> Deriv a B"
-  and Deriv_inter[simp]:   "Deriv a (A \<inter> B) = Deriv a A \<inter> Deriv a B"
-  and Deriv_compl[simp]:   "Deriv a (-A) = - Deriv a A"
-  and Deriv_Union[simp]:   "Deriv a (Union M) = Union(Deriv a ` M)"
-  and Deriv_UN[simp]:      "Deriv a (UN x:I. S x) = (UN x:I. Deriv a (S x))"
-by (auto simp: Deriv_def)
-
-lemma Der_conc [simp]: 
-  shows "Deriv c (A @@ B) = (Deriv c A) @@ B \<union> (if [] \<in> A then Deriv c B else {})"
-unfolding Deriv_def conc_def
-by (auto simp add: Cons_eq_append_conv)
-
-lemma Deriv_star [simp]: 
-  shows "Deriv c (star A) = (Deriv c A) @@ star A"
-proof -
-  have "Deriv c (star A) = Deriv c ({[]} \<union> A @@ star A)"
-    by (metis star_unfold_left sup.commute)
-  also have "... = Deriv c (A @@ star A)"
-    unfolding Deriv_union by (simp)
-  also have "... = (Deriv c A) @@ (star A) \<union> (if [] \<in> A then Deriv c (star A) else {})"
-    by simp
-  also have "... =  (Deriv c A) @@ star A"
-    unfolding conc_def Deriv_def
-    using star_decom by (force simp add: Cons_eq_append_conv)
-  finally show "Deriv c (star A) = (Deriv c A) @@ star A" .
-qed
-
-lemma Deriv_diff[simp]:   
-  shows "Deriv c (A - B) = Deriv c A - Deriv c B"
-by(auto simp add: Deriv_def)
-
-lemma Deriv_lists[simp]: "c : S \<Longrightarrow> Deriv c (lists S) = lists S"
-by(auto simp add: Deriv_def)
-
-lemma Derivs_simps [simp]:
-  shows "Derivs [] A = A"
-  and   "Derivs (c # s) A = Derivs s (Deriv c A)"
-  and   "Derivs (s1 @ s2) A = Derivs s2 (Derivs s1 A)"
-unfolding Derivs_def Deriv_def by auto
-
-lemma in_fold_Deriv: "v \<in> fold Deriv w L \<longleftrightarrow> w @ v \<in> L"
-  by (induct w arbitrary: L) (simp_all add: Deriv_def)
-
-lemma Derivs_alt_def: "Derivs w L = fold Deriv w L"
-  by (induct w arbitrary: L) simp_all
-
-
-subsection {* Shuffle product *}
-
-fun shuffle where
-  "shuffle [] ys = {ys}"
-| "shuffle xs [] = {xs}"
-| "shuffle (x # xs) (y # ys) =
-    {x # w | w . w \<in> shuffle xs (y # ys)} \<union>
-    {y # w | w . w \<in> shuffle (x # xs) ys}"
-
-lemma shuffle_empty2[simp]: "shuffle xs [] = {xs}"
-  by (cases xs) auto
-
-lemma Nil_in_shuffle[simp]: "[] \<in> shuffle xs ys \<longleftrightarrow> xs = [] \<and> ys = []"
-  by (induct xs ys rule: shuffle.induct) auto
-
-definition Shuffle (infixr "\<parallel>" 80) where
-  "Shuffle A B = \<Union>{shuffle xs ys | xs ys. xs \<in> A \<and> ys \<in> B}"
-
-lemma shuffleE:
-  "zs \<in> shuffle xs ys \<Longrightarrow>
-    (zs = xs \<Longrightarrow> ys = [] \<Longrightarrow> P) \<Longrightarrow>
-    (zs = ys \<Longrightarrow> xs = [] \<Longrightarrow> P) \<Longrightarrow>
-    (\<And>x xs' z zs'. xs = x # xs' \<Longrightarrow> zs = z # zs' \<Longrightarrow> x = z \<Longrightarrow> zs' \<in> shuffle xs' ys \<Longrightarrow> P) \<Longrightarrow>
-    (\<And>y ys' z zs'. ys = y # ys' \<Longrightarrow> zs = z # zs' \<Longrightarrow> y = z \<Longrightarrow> zs' \<in> shuffle xs ys' \<Longrightarrow> P) \<Longrightarrow> P"
-  by (induct xs ys rule: shuffle.induct) auto
-
-lemma Cons_in_shuffle_iff:
-  "z # zs \<in> shuffle xs ys \<longleftrightarrow>
-    (xs \<noteq> [] \<and> hd xs = z \<and> zs \<in> shuffle (tl xs) ys \<or>
-     ys \<noteq> [] \<and> hd ys = z \<and> zs \<in> shuffle xs (tl ys))"
-  by (induct xs ys rule: shuffle.induct) auto
-
-lemma Deriv_Shuffle[simp]:
-  "Deriv a (A \<parallel> B) = Deriv a A \<parallel> B \<union> A \<parallel> Deriv a B"
-  unfolding Shuffle_def Deriv_def by (fastforce simp: Cons_in_shuffle_iff neq_Nil_conv)
-
-lemma shuffle_subset_lists:
-  assumes "A \<subseteq> lists S" "B \<subseteq> lists S"
-  shows "A \<parallel> B \<subseteq> lists S"
-unfolding Shuffle_def proof safe
-  fix x and zs xs ys :: "'a list"
-  assume zs: "zs \<in> shuffle xs ys" "x \<in> set zs" and "xs \<in> A" "ys \<in> B"
-  with assms have "xs \<in> lists S" "ys \<in> lists S" by auto
-  with zs show "x \<in> S" by (induct xs ys arbitrary: zs rule: shuffle.induct) auto
-qed
-
-lemma Nil_in_Shuffle[simp]: "[] \<in> A \<parallel> B \<longleftrightarrow> [] \<in> A \<and> [] \<in> B"
-  unfolding Shuffle_def by force
-
-lemma shuffle_Un_distrib:
-shows "A \<parallel> (B \<union> C) = A \<parallel> B \<union> A \<parallel> C"
-and   "A \<parallel> (B \<union> C) = A \<parallel> B \<union> A \<parallel> C"
-unfolding Shuffle_def by fast+
-
-lemma shuffle_UNION_distrib:
-shows "A \<parallel> UNION I M = UNION I (%i. A \<parallel> M i)"
-and   "UNION I M \<parallel> A = UNION I (%i. M i \<parallel> A)"
-unfolding Shuffle_def by fast+
-
-lemma Shuffle_empty[simp]:
-  "A \<parallel> {} = {}"
-  "{} \<parallel> B = {}"
-  unfolding Shuffle_def by auto
-
-lemma Shuffle_eps[simp]:
-  "A \<parallel> {[]} = A"
-  "{[]} \<parallel> B = B"
-  unfolding Shuffle_def by auto
-
-
-subsection {* Arden's Lemma *}
-
-lemma arden_helper:
-  assumes eq: "X = A @@ X \<union> B"
-  shows "X = (A ^^ Suc n) @@ X \<union> (\<Union>m\<le>n. (A ^^ m) @@ B)"
-proof (induct n)
-  case 0 
-  show "X = (A ^^ Suc 0) @@ X \<union> (\<Union>m\<le>0. (A ^^ m) @@ B)"
-    using eq by simp
-next
-  case (Suc n)
-  have ih: "X = (A ^^ Suc n) @@ X \<union> (\<Union>m\<le>n. (A ^^ m) @@ B)" by fact
-  also have "\<dots> = (A ^^ Suc n) @@ (A @@ X \<union> B) \<union> (\<Union>m\<le>n. (A ^^ m) @@ B)" using eq by simp
-  also have "\<dots> = (A ^^ Suc (Suc n)) @@ X \<union> ((A ^^ Suc n) @@ B) \<union> (\<Union>m\<le>n. (A ^^ m) @@ B)"
-    by (simp add: conc_Un_distrib conc_assoc[symmetric] conc_pow_comm)
-  also have "\<dots> = (A ^^ Suc (Suc n)) @@ X \<union> (\<Union>m\<le>Suc n. (A ^^ m) @@ B)"
-    by (auto simp add: le_Suc_eq)
-  finally show "X = (A ^^ Suc (Suc n)) @@ X \<union> (\<Union>m\<le>Suc n. (A ^^ m) @@ B)" .
-qed
-
-lemma Arden:
-  assumes "[] \<notin> A" 
-  shows "X = A @@ X \<union> B \<longleftrightarrow> X = star A @@ B"
-proof
-  assume eq: "X = A @@ X \<union> B"
-  { fix w assume "w : X"
-    let ?n = "size w"
-    from `[] \<notin> A` have "ALL u : A. length u \<ge> 1"
-      by (metis Suc_eq_plus1 add_leD2 le_0_eq length_0_conv not_less_eq_eq)
-    hence "ALL u : A^^(?n+1). length u \<ge> ?n+1"
-      by (metis length_lang_pow_lb nat_mult_1)
-    hence "ALL u : A^^(?n+1)@@X. length u \<ge> ?n+1"
-      by(auto simp only: conc_def length_append)
-    hence "w \<notin> A^^(?n+1)@@X" by auto
-    hence "w : star A @@ B" using `w : X` using arden_helper[OF eq, where n="?n"]
-      by (auto simp add: star_def conc_UNION_distrib)
-  } moreover
-  { fix w assume "w : star A @@ B"
-    hence "EX n. w : A^^n @@ B" by(auto simp: conc_def star_def)
-    hence "w : X" using arden_helper[OF eq] by blast
-  } ultimately show "X = star A @@ B" by blast 
-next
-  assume eq: "X = star A @@ B"
-  have "star A = A @@ star A \<union> {[]}"
-    by (rule star_unfold_left)
-  then have "star A @@ B = (A @@ star A \<union> {[]}) @@ B"
-    by metis
-  also have "\<dots> = (A @@ star A) @@ B \<union> B"
-    unfolding conc_Un_distrib by simp
-  also have "\<dots> = A @@ (star A @@ B) \<union> B" 
-    by (simp only: conc_assoc)
-  finally show "X = A @@ X \<union> B" 
-    using eq by blast 
-qed
-
-
-lemma reversed_arden_helper:
-  assumes eq: "X = X @@ A \<union> B"
-  shows "X = X @@ (A ^^ Suc n) \<union> (\<Union>m\<le>n. B @@ (A ^^ m))"
-proof (induct n)
-  case 0 
-  show "X = X @@ (A ^^ Suc 0) \<union> (\<Union>m\<le>0. B @@ (A ^^ m))"
-    using eq by simp
-next
-  case (Suc n)
-  have ih: "X = X @@ (A ^^ Suc n) \<union> (\<Union>m\<le>n. B @@ (A ^^ m))" by fact
-  also have "\<dots> = (X @@ A \<union> B) @@ (A ^^ Suc n) \<union> (\<Union>m\<le>n. B @@ (A ^^ m))" using eq by simp
-  also have "\<dots> = X @@ (A ^^ Suc (Suc n)) \<union> (B @@ (A ^^ Suc n)) \<union> (\<Union>m\<le>n. B @@ (A ^^ m))"
-    by (simp add: conc_Un_distrib conc_assoc)
-  also have "\<dots> = X @@ (A ^^ Suc (Suc n)) \<union> (\<Union>m\<le>Suc n. B @@ (A ^^ m))"
-    by (auto simp add: le_Suc_eq)
-  finally show "X = X @@ (A ^^ Suc (Suc n)) \<union> (\<Union>m\<le>Suc n. B @@ (A ^^ m))" .
-qed
-
-theorem reversed_Arden:
-  assumes nemp: "[] \<notin> A"
-  shows "X = X @@ A \<union> B \<longleftrightarrow> X = B @@ star A"
-proof
- assume eq: "X = X @@ A \<union> B"
-  { fix w assume "w : X"
-    let ?n = "size w"
-    from `[] \<notin> A` have "ALL u : A. length u \<ge> 1"
-      by (metis Suc_eq_plus1 add_leD2 le_0_eq length_0_conv not_less_eq_eq)
-    hence "ALL u : A^^(?n+1). length u \<ge> ?n+1"
-      by (metis length_lang_pow_lb nat_mult_1)
-    hence "ALL u : X @@ A^^(?n+1). length u \<ge> ?n+1"
-      by(auto simp only: conc_def length_append)
-    hence "w \<notin> X @@ A^^(?n+1)" by auto
-    hence "w : B @@ star A" using `w : X` using reversed_arden_helper[OF eq, where n="?n"]
-      by (auto simp add: star_def conc_UNION_distrib)
-  } moreover
-  { fix w assume "w : B @@ star A"
-    hence "EX n. w : B @@ A^^n" by (auto simp: conc_def star_def)
-    hence "w : X" using reversed_arden_helper[OF eq] by blast
-  } ultimately show "X = B @@ star A" by blast 
-next 
-  assume eq: "X = B @@ star A"
-  have "star A = {[]} \<union> star A @@ A" 
-    unfolding conc_star_comm[symmetric]
-    by(metis Un_commute star_unfold_left)
-  then have "B @@ star A = B @@ ({[]} \<union> star A @@ A)"
-    by metis
-  also have "\<dots> = B \<union> B @@ (star A @@ A)"
-    unfolding conc_Un_distrib by simp
-  also have "\<dots> = B \<union> (B @@ star A) @@ A" 
-    by (simp only: conc_assoc)
-  finally show "X = X @@ A \<union> B" 
-    using eq by blast 
-qed
-
-end
diff -r 2585e2a7a7ab -r 5c063eeda622 AFP-Submission/Simplifying.thy
--- a/AFP-Submission/Simplifying.thy	Tue Jun 14 12:37:46 2016 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,239 +0,0 @@
-(*  Title:       POSIX Lexing with Derivatives of Regular Expressions
-    Authors:     Fahad Ausaf <fahad.ausaf at icloud.com>, 2016
-                 Roy Dyckhoff <roy.dyckhoff at st-andrews.ac.uk>, 2016
-                 Christian Urban <christian.urban at kcl.ac.uk>, 2016
-    Maintainer:  Christian Urban <christian.urban at kcl.ac.uk>
-*) 
-
-theory Simplifying
-  imports "Lexer" 
-begin
-
-section {* Lexer including simplifications *}
-
-
-fun F_RIGHT where
-  "F_RIGHT f v = Right (f v)"
-
-fun F_LEFT where
-  "F_LEFT f v = Left (f v)"
-
-fun F_Plus where
-  "F_Plus f\<^sub>1 f\<^sub>2 (Right v) = Right (f\<^sub>2 v)"
-| "F_Plus f\<^sub>1 f\<^sub>2 (Left v) = Left (f\<^sub>1 v)"  
-| "F_Plus f1 f2 v = v"
-
-
-fun F_Times1 where
-  "F_Times1 f\<^sub>1 f\<^sub>2 v = Seq (f\<^sub>1 Void) (f\<^sub>2 v)"
-
-fun F_Times2 where 
-  "F_Times2 f\<^sub>1 f\<^sub>2 v = Seq (f\<^sub>1 v) (f\<^sub>2 Void)"
-
-fun F_Times where 
-  "F_Times f\<^sub>1 f\<^sub>2 (Seq v\<^sub>1 v\<^sub>2) = Seq (f\<^sub>1 v\<^sub>1) (f\<^sub>2 v\<^sub>2)"
-| "F_Times f1 f2 v = v"
-
-fun simp_Plus where
-  "simp_Plus (Zero, f\<^sub>1) (r\<^sub>2, f\<^sub>2) = (r\<^sub>2, F_RIGHT f\<^sub>2)"
-| "simp_Plus (r\<^sub>1, f\<^sub>1) (Zero, f\<^sub>2) = (r\<^sub>1, F_LEFT f\<^sub>1)"
-| "simp_Plus (r\<^sub>1, f\<^sub>1) (r\<^sub>2, f\<^sub>2) = (Plus r\<^sub>1 r\<^sub>2, F_Plus f\<^sub>1 f\<^sub>2)"
-
-fun simp_Times where
-  "simp_Times (One, f\<^sub>1) (r\<^sub>2, f\<^sub>2) = (r\<^sub>2, F_Times1 f\<^sub>1 f\<^sub>2)"
-| "simp_Times (r\<^sub>1, f\<^sub>1) (One, f\<^sub>2) = (r\<^sub>1, F_Times2 f\<^sub>1 f\<^sub>2)"
-| "simp_Times (r\<^sub>1, f\<^sub>1) (r\<^sub>2, f\<^sub>2) = (Times r\<^sub>1 r\<^sub>2, F_Times f\<^sub>1 f\<^sub>2)"  
- 
-lemma simp_Times_simps[simp]:
-  "simp_Times p1 p2 = (if (fst p1 = One) then (fst p2, F_Times1 (snd p1) (snd p2))
-                    else (if (fst p2 = One) then (fst p1, F_Times2 (snd p1) (snd p2))
-                    else (Times (fst p1) (fst p2), F_Times (snd p1) (snd p2))))"
-by (induct p1 p2 rule: simp_Times.induct) (auto)
-
-lemma simp_Plus_simps[simp]:
-  "simp_Plus p1 p2 = (if (fst p1 = Zero) then (fst p2, F_RIGHT (snd p2))
-                    else (if (fst p2 = Zero) then (fst p1, F_LEFT (snd p1))
-                    else (Plus (fst p1) (fst p2), F_Plus (snd p1) (snd p2))))"
-by (induct p1 p2 rule: simp_Plus.induct) (auto)
-
-fun 
-  simp :: "'a rexp \<Rightarrow> 'a rexp * ('a val \<Rightarrow> 'a val)"
-where
-  "simp (Plus r1 r2) = simp_Plus (simp r1) (simp r2)" 
-| "simp (Times r1 r2) = simp_Times (simp r1) (simp r2)" 
-| "simp r = (r, id)"
-
-fun 
-  slexer :: "'a rexp \<Rightarrow> 'a list \<Rightarrow> ('a val) option"
-where
-  "slexer r [] = (if nullable r then Some(mkeps r) else None)"
-| "slexer r (c#s) = (let (rs, fr) = simp (deriv c r) in
-                         (case (slexer rs s) of  
-                            None \<Rightarrow> None
-                          | Some(v) \<Rightarrow> Some(injval r c (fr v))))"
-
-lemma slexer_better_simp:
-  "slexer r (c#s) = (case (slexer (fst (simp (deriv c r))) s) of  
-                            None \<Rightarrow> None
-                          | Some(v) \<Rightarrow> Some(injval r c ((snd (simp (deriv c r))) v)))"
-by (auto split: prod.split option.split)
-
-
-lemma L_fst_simp:
-  shows "lang r = lang (fst (simp r))"
-using assms
-by (induct r) (auto)
-
-lemma Posix_simp:
-  assumes "s \<in> (fst (simp r)) \<rightarrow> v" 
-  shows "s \<in> r \<rightarrow> ((snd (simp r)) v)"
-using assms
-proof(induct r arbitrary: s v rule: rexp.induct)
-  case (Plus r1 r2 s v)
-  have IH1: "\<And>s v. s \<in> fst (simp r1) \<rightarrow> v \<Longrightarrow> s \<in> r1 \<rightarrow> snd (simp r1) v" by fact
-  have IH2: "\<And>s v. s \<in> fst (simp r2) \<rightarrow> v \<Longrightarrow> s \<in> r2 \<rightarrow> snd (simp r2) v" by fact
-  have as: "s \<in> fst (simp (Plus r1 r2)) \<rightarrow> v" by fact
-  consider (Zero_Zero) "fst (simp r1) = Zero" "fst (simp r2) = Zero"
-         | (Zero_NZero) "fst (simp r1) = Zero" "fst (simp r2) \<noteq> Zero"
-         | (NZero_Zero) "fst (simp r1) \<noteq> Zero" "fst (simp r2) = Zero"
-         | (NZero_NZero) "fst (simp r1) \<noteq> Zero" "fst (simp r2) \<noteq> Zero" by auto
-  then show "s \<in> Plus r1 r2 \<rightarrow> snd (simp (Plus r1 r2)) v" 
-    proof(cases)
-      case (Zero_Zero)
-      with as have "s \<in> Zero \<rightarrow> v" by simp 
-      then show "s \<in> Plus r1 r2 \<rightarrow> snd (simp (Plus r1 r2)) v" by (rule Posix_elims(1))
-    next
-      case (Zero_NZero)
-      with as have "s \<in> fst (simp r2) \<rightarrow> v" by simp
-      with IH2 have "s \<in> r2 \<rightarrow> snd (simp r2) v" by simp
-      moreover
-      from Zero_NZero have "fst (simp r1) = Zero" by simp
-      then have "lang (fst (simp r1)) = {}" by simp
-      then have "lang r1 = {}" using L_fst_simp by auto
-      then have "s \<notin> lang r1" by simp 
-      ultimately have "s \<in> Plus r1 r2 \<rightarrow> Right (snd (simp r2) v)" by (rule Posix_Plus2)
-      then show "s \<in> Plus r1 r2 \<rightarrow> snd (simp (Plus r1 r2)) v"
-      using Zero_NZero by simp
-    next
-      case (NZero_Zero)
-      with as have "s \<in> fst (simp r1) \<rightarrow> v" by simp
-      with IH1 have "s \<in> r1 \<rightarrow> snd (simp r1) v" by simp
-      then have "s \<in> Plus r1 r2 \<rightarrow> Left (snd (simp r1) v)" by (rule Posix_Plus1) 
-      then show "s \<in> Plus r1 r2 \<rightarrow> snd (simp (Plus r1 r2)) v" using NZero_Zero by simp
-    next
-      case (NZero_NZero)
-      with as have "s \<in> Plus (fst (simp r1)) (fst (simp r2)) \<rightarrow> v" by simp
-      then consider (Left) v1 where "v = Left v1" "s \<in> (fst (simp r1)) \<rightarrow> v1"
-                  | (Right) v2 where "v = Right v2" "s \<in> (fst (simp r2)) \<rightarrow> v2" "s \<notin> lang (fst (simp r1))"
-                  by (erule_tac Posix_elims(4)) 
-      then show "s \<in> Plus r1 r2 \<rightarrow> snd (simp (Plus r1 r2)) v"
-      proof(cases)
-        case (Left)
-        then have "v = Left v1" "s \<in> r1 \<rightarrow> (snd (simp r1) v1)" using IH1 by simp_all
-        then show "s \<in> Plus r1 r2 \<rightarrow> snd (simp (Plus r1 r2)) v" using NZero_NZero
-          by (simp_all add: Posix_Plus1)
-      next 
-        case (Right)
-        then have "v = Right v2" "s \<in> r2 \<rightarrow> (snd (simp r2) v2)" "s \<notin> lang r1" using IH2 L_fst_simp by auto
-        then show "s \<in> Plus r1 r2 \<rightarrow> snd (simp (Plus r1 r2)) v" using NZero_NZero
-          by (simp_all add: Posix_Plus2)
-      qed
-    qed
-next
-  case (Times r1 r2 s v)
-  have IH1: "\<And>s v. s \<in> fst (simp r1) \<rightarrow> v \<Longrightarrow> s \<in> r1 \<rightarrow> snd (simp r1) v" by fact
-  have IH2: "\<And>s v. s \<in> fst (simp r2) \<rightarrow> v \<Longrightarrow> s \<in> r2 \<rightarrow> snd (simp r2) v" by fact
-  have as: "s \<in> fst (simp (Times r1 r2)) \<rightarrow> v" by fact
-  consider (One_One) "fst (simp r1) = One" "fst (simp r2) = One"
-         | (One_NOne) "fst (simp r1) = One" "fst (simp r2) \<noteq> One"
-         | (NOne_One) "fst (simp r1) \<noteq> One" "fst (simp r2) = One"
-         | (NOne_NOne) "fst (simp r1) \<noteq> One" "fst (simp r2) \<noteq> One" by auto
-  then show "s \<in> Times r1 r2 \<rightarrow> snd (simp (Times r1 r2)) v" 
-  proof(cases)
-      case (One_One)
-      with as have b: "s \<in> One \<rightarrow> v" by simp 
-      from b have "s \<in> r1 \<rightarrow> snd (simp r1) v" using IH1 One_One by simp
-      moreover
-      from b have c: "s = []" "v = Void" using Posix_elims(2) by auto
-      moreover
-      have "[] \<in> One \<rightarrow> Void" by (simp add: Posix_One)
-      then have "[] \<in> fst (simp r2) \<rightarrow> Void" using One_One by simp
-      then have "[] \<in> r2 \<rightarrow> snd (simp r2) Void" using IH2 by simp
-      ultimately have "([] @ []) \<in> Times r1 r2 \<rightarrow> Seq (snd (simp r1) Void) (snd (simp r2) Void)"
-        using Posix_Times by blast 
-      then show "s \<in> Times r1 r2 \<rightarrow> snd (simp (Times r1 r2)) v" using c One_One by simp
-    next
-      case (One_NOne)
-      with as have b: "s \<in> fst (simp r2) \<rightarrow> v" by simp 
-      from b have "s \<in> r2 \<rightarrow> snd (simp r2) v" using IH2 One_NOne by simp
-      moreover
-      have "[] \<in> One \<rightarrow> Void" by (simp add: Posix_One)
-      then have "[] \<in> fst (simp r1) \<rightarrow> Void" using One_NOne by simp
-      then have "[] \<in> r1 \<rightarrow> snd (simp r1) Void" using IH1 by simp
-      moreover
-      from One_NOne(1) have "lang (fst (simp r1)) = {[]}" by simp
-      then have "lang r1 = {[]}" by (simp add: L_fst_simp[symmetric])
-      ultimately have "([] @ s) \<in> Times r1 r2 \<rightarrow> Seq (snd (simp r1) Void) (snd (simp r2) v)"
-        by(rule_tac Posix_Times) auto
-      then show "s \<in> Times r1 r2 \<rightarrow> snd (simp (Times r1 r2)) v" using One_NOne by simp
-    next
-      case (NOne_One)
-        with as have "s \<in> fst (simp r1) \<rightarrow> v" by simp
-        with IH1 have "s \<in> r1 \<rightarrow> snd (simp r1) v" by simp
-      moreover
-        have "[] \<in> One \<rightarrow> Void" by (simp add: Posix_One)
-        then have "[] \<in> fst (simp r2) \<rightarrow> Void" using NOne_One by simp
-        then have "[] \<in> r2 \<rightarrow> snd (simp r2) Void" using IH2 by simp
-      ultimately have "(s @ []) \<in> Times r1 r2 \<rightarrow> Seq (snd (simp r1) v) (snd (simp r2) Void)"
-        by(rule_tac Posix_Times) auto
-      then show "s \<in> Times r1 r2 \<rightarrow> snd (simp (Times r1 r2)) v" using NOne_One by simp
-    next
-      case (NOne_NOne)
-      with as have "s \<in> Times (fst (simp r1)) (fst (simp r2)) \<rightarrow> v" by simp
-      then obtain s1 s2 v1 v2 where eqs: "s = s1 @ s2" "v = Seq v1 v2"
-                     "s1 \<in> (fst (simp r1)) \<rightarrow> v1" "s2 \<in> (fst (simp r2)) \<rightarrow> v2"
-                     "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> lang r1 \<and> s\<^sub>4 \<in> lang r2)"
-                     by (erule_tac Posix_elims(5)) (auto simp add: L_fst_simp[symmetric]) 
-      then have "s1 \<in> r1 \<rightarrow> (snd (simp r1) v1)" "s2 \<in> r2 \<rightarrow> (snd (simp r2) v2)"
-        using IH1 IH2 by auto             
-      then show "s \<in> Times r1 r2 \<rightarrow> snd (simp (Times r1 r2)) v" using eqs NOne_NOne
-        by(auto intro: Posix_Times)
-    qed
-qed (simp_all)
-
-
-lemma slexer_correctness:
-  shows "slexer r s = lexer r s"
-proof(induct s arbitrary: r)
-  case Nil
-  show "slexer r [] = lexer r []" by simp
-next 
-  case (Cons c s r)
-  have IH: "\<And>r. slexer r s = lexer r s" by fact
-  show "slexer r (c # s) = lexer r (c # s)" 
-   proof (cases "s \<in> lang (deriv c r)")
-     case True
-       assume a1: "s \<in> lang (deriv c r)"
-       then obtain v1 where a2: "lexer (deriv c r) s = Some v1" "s \<in> deriv c r \<rightarrow> v1"
-         using lexer_correct_Some by auto
-       from a1 have "s \<in> lang (fst (simp (deriv c r)))" using L_fst_simp[symmetric] by auto
-       then obtain v2 where a3: "lexer (fst (simp (deriv c r))) s = Some v2" "s \<in> (fst (simp (deriv c r))) \<rightarrow> v2"
-          using lexer_correct_Some by auto
-       then have a4: "slexer (fst (simp (deriv c r))) s = Some v2" using IH by simp
-       from a3(2) have "s \<in> deriv c r \<rightarrow> (snd (simp (deriv c r))) v2" using Posix_simp by auto
-       with a2(2) have "v1 = (snd (simp (deriv c r))) v2" using Posix_determ by auto
-       with a2(1) a4 show "slexer r (c # s) = lexer r (c # s)" by (auto split: prod.split)
-     next 
-     case False
-       assume b1: "s \<notin> lang (deriv c r)"
-       then have "lexer (deriv c r) s = None" using lexer_correct_None by auto
-       moreover
-       from b1 have "s \<notin> lang (fst (simp (deriv c r)))" using L_fst_simp[symmetric] by auto
-       then have "lexer (fst (simp (deriv c r))) s = None" using lexer_correct_None by auto
-       then have "slexer (fst (simp (deriv c r))) s = None" using IH by simp
-       ultimately show "slexer r (c # s) = lexer r (c # s)" 
-         by (simp del: slexer.simps add: slexer_better_simp)
-   qed
-qed  
-
-end
\ No newline at end of file
diff -r 2585e2a7a7ab -r 5c063eeda622 AFP-Submission/document/root.bib
--- a/AFP-Submission/document/root.bib	Tue Jun 14 12:37:46 2016 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,10 +0,0 @@
-
-@inproceedings{Sulzmann2014,
-  author    = {M.~Sulzmann and K.~Lu},
-  title     = {{POSIX} {R}egular {E}xpression {P}arsing with {D}erivatives},
-  booktitle = {Proc.~of the 12th International Conference on Functional and Logic Programming (FLOPS)},
-  pages     = {203--220},
-  year      = {2014},
-  volume =    {8475},
-  series =    {LNCS}
-}
\ No newline at end of file
diff -r 2585e2a7a7ab -r 5c063eeda622 AFP-Submission/document/root.tex
--- a/AFP-Submission/document/root.tex	Tue Jun 14 12:37:46 2016 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,41 +0,0 @@
-\documentclass[11pt,a4paper]{article}
-\usepackage{isabelle,isabellesym}
-
-% this should be the last package used
-\usepackage{pdfsetup}
-
-% urls in roman style, theory text in math-similar italics
-\urlstyle{rm}
-\isabellestyle{it}
-
-
-\begin{document}
-
-\title{POSIX Lexing with Derivatives of Regular Expressions}
-\author{Fahad Ausaf \and Roy Dyckhoff \and Christian Urban}
-\maketitle
-
-\begin{abstract}
-  Brzozowski introduced the notion of derivatives for regular
-  expressions. They can be used for a very simple regular expression
-  matching algorithm. Sulzmann and Lu \cite{Sulzmann2014} cleverly extended this algorithm
-  in order to deal with POSIX matching, which is the underlying
-  disambiguation strategy for regular expressions needed in
-  lexers. In this entry we give our inductive definition
-  of what a POSIX value is and show (i) that such a value is unique (for
-  given regular expression and string being matched) and (ii) that
-  Sulzmann and Lu's algorithm always generates such a value (provided
-  that the regular expression matches the string). We also prove the
-  correctness of an optimised version of the POSIX matching
-  algorithm. 
-\end{abstract}
-
-\tableofcontents
-
-% include generated text of all theories
-\input{session}
-
-\bibliographystyle{abbrv}
-\bibliography{root}
-
-\end{document}