(*  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