imports "Main" 

begin

lemma Suc_Union:

  "(\<Union> x\<le>Suc m. B x) = (B (Suc m) \<union> (\<Union> x\<le>m. B x))"

by (metis UN_insert atMost_Suc)

lemma Suc_reduce_Union:

  "(\<Union>x\<in>{Suc n..Suc m}. B x) = (\<Union>x\<in>{n..m}. B (Suc x))"

by (metis UN_extend_simps(10) image_Suc_atLeastAtMost)

section {* Regular Expressions *}

datatype rexp =

  NULL

| EMPTY

| CHAR char

| SEQ rexp rexp

| ALT rexp rexp

| STAR rexp

| NOT rexp

| PLUS rexp

| OPT rexp

| NTIMES rexp nat

| NMTIMES rexp nat nat

fun M :: "rexp \<Rightarrow> nat"

where

  "M (NULL) = 0"

| "M (EMPTY) = 0"

| "M (CHAR char) = 0"

| "M (SEQ r1 r2) = Suc ((M r1) + (M r2))"

| "M (ALT r1 r2) = Suc ((M r1) + (M r2))"

| "M (STAR r) = Suc (M r)"

| "M (NOT r) = Suc (M r)"

| "M (PLUS r) = Suc (M r)"

| "M (OPT r) = Suc (M r)"

| "M (NTIMES r n) = Suc (M r) * 2 * (Suc n)"

| "M (NMTIMES r n m) = Suc (M r) * 2 * (Suc m - Suc n)"

section {* Sequential Composition of Sets *}

definition

  Seq :: "string set \<Rightarrow> string set \<Rightarrow> string set" ("_ ;; _" [100,100] 100)

where 

  "A ;; B = {s1 @ s2 | s1 s2. s1 \<in> A \<and> s2 \<in> B}"

text {* Two Simple Properties about Sequential Composition *}

lemma seq_empty [simp]:

  shows "A ;; {[]} = A"

  and   "{[]} ;; A = A"

by (simp_all add: Seq_def)

lemma seq_null [simp]:

  shows "A ;; {} = {}"

  and   "{} ;; A = {}"

by (simp_all add: Seq_def)

lemma seq_union:

  shows "A ;; (B \<union> C) = A ;; B \<union> A ;; C"

  and   "(B \<union> C) ;; A = B ;; A \<union> C ;; A"

by (auto simp add: Seq_def)

lemma seq_Union:

  shows "A ;; (\<Union>x\<in>B. C x) = (\<Union>x\<in>B. A ;; C x)"

by (auto simp add: Seq_def)

lemma seq_empty_in [simp]:

  "[] \<in> A ;; B \<longleftrightarrow> ([] \<in> A \<and> [] \<in> B)"

by (simp add: Seq_def)

lemma seq_assoc: 

  shows "A ;; (B ;; C) = (A ;; B) ;; C" 

apply(auto simp add: Seq_def)

apply(metis append_assoc)

apply(metis)

done

section {* Power for Sets *}

fun 

  pow :: "string set \<Rightarrow> nat \<Rightarrow> string set" ("_ \<up> _" [101, 102] 101)

where

   "A \<up> 0 = {[]}"

|  "A \<up> (Suc n) = A ;; (A \<up> n)"

lemma pow_empty [simp]:

  shows "[] \<in> A \<up> n \<longleftrightarrow> (n = 0 \<or> [] \<in> A)"

by (induct n) (auto)

lemma pow_plus:

  "A \<up> (n + m) = A \<up> n ;; A \<up> m"

by (induct n) (simp_all add: seq_assoc)

section {* Kleene Star for Sets *}

inductive_set

  Star :: "string set \<Rightarrow> string set" ("_\<star>" [101] 102)

  for A :: "string set"

where

  start[intro]: "[] \<in> A\<star>"

| step[intro]:  "\<lbrakk>s1 \<in> A; s2 \<in> A\<star>\<rbrakk> \<Longrightarrow> s1 @ s2 \<in> A\<star>"

text {* A Standard Property of Star *}

lemma star_decomp: 

  assumes a: "c # x \<in> A\<star>" 

  shows "\<exists>a b. x = a @ b \<and> c # a \<in> A \<and> b \<in> A\<star>"

using a 

using a

by (induct x\<equiv>"c # x" rule: Star.induct) 

   (auto simp add: append_eq_Cons_conv)

lemma star_cases:

  shows "A\<star> = {[]} \<union> A ;; A\<star>"

unfolding Seq_def 

by (auto) (metis Star.simps)

lemma Star_in_Pow:

  assumes a: "s \<in> A\<star>"

  shows "\<exists>n. s \<in> A \<up> n"

using a

apply(induct)

apply(auto)

apply(rule_tac x="Suc n" in exI)

apply(auto simp add: Seq_def)

done

lemma Pow_in_Star:

  assumes a: "s \<in> A \<up> n"

  shows "s \<in> A\<star>"

using a

by (induct n arbitrary: s) (auto simp add: Seq_def)

lemma Star_def2: 

  shows "A\<star> = (\<Union>n. A \<up> n)"

using Star_in_Pow Pow_in_Star

by (auto)

section {* Semantics of Regular Expressions *}

fun

  L :: "rexp \<Rightarrow> string set"

where

  "L (NULL) = {}"

| "L (EMPTY) = {[]}"

| "L (CHAR c) = {[c]}"

| "L (SEQ r1 r2) = (L r1) ;; (L r2)"

| "L (ALT r1 r2) = (L r1) \<union> (L r2)"

| "L (STAR r) = (L r)\<star>"

| "L (NOT r) = UNIV - (L r)"

| "L (PLUS r) = (L r) ;; ((L r)\<star>)"

| "L (OPT r) = (L r) \<union> {[]}"

| "L (NTIMES r n) = (L r) \<up> n"

| "L (NMTIMES r n m) = (\<Union>i\<in> {n..m} . ((L r) \<up> i))" 

lemma "L (NOT NULL) = UNIV"

apply(simp)

done

section {* The Matcher *}

fun

 nullable :: "rexp \<Rightarrow> bool"

where

  "nullable (NULL) = False"

| "nullable (EMPTY) = True"

| "nullable (CHAR c) = False"

| "nullable (ALT r1 r2) = (nullable r1 \<or> nullable r2)"

| "nullable (SEQ r1 r2) = (nullable r1 \<and> nullable r2)"

| "nullable (STAR r) = True"

| "nullable (NOT r) = (\<not>(nullable r))"

| "nullable (PLUS r) = (nullable r)"

| "nullable (OPT r) = True"

| "nullable (NTIMES r n) = (if n = 0 then True else nullable r)"

| "nullable (NMTIMES r n m) = (if m < n then False else (if n = 0 then True else nullable r))"

function der :: "char \<Rightarrow> rexp \<Rightarrow> rexp"

where

  "der c (NULL) = NULL"

| "der c (EMPTY) = NULL"

| "der c (CHAR d) = (if c = d then EMPTY else NULL)"

| "der c (ALT r1 r2) = ALT (der c r1) (der c r2)"

| "der c (SEQ r1 r2) = ALT (SEQ (der c r1) r2) (if nullable r1 then der c r2 else NULL)"

| "der c (STAR r) = SEQ (der c r) (STAR r)"

| "der c (NOT r) = NOT(der c r)"

| "der c (PLUS r) = SEQ (der c r) (STAR r)"

| "der c (OPT r) = der c r"

| "der c (NTIMES r 0) = NULL"

| "der c (NTIMES r (Suc n)) = der c (SEQ r (NTIMES r n))"

| "der c (NMTIMES r n m) = 

     (if m < n then NULL else 

     (if m = 0 then NULL else

     (der c (SEQ r (NMTIMES r (n - 1) (m - 1))))))"

(*

  (if m < n then NULL else 

                              (if n = m then der c (NTIMES r n) else

                                ALT (der c (NTIMES r n)) (der c (NMTIMES r (Suc n) m))))"

*)

by pat_completeness auto

termination der 

apply(relation "measure (\<lambda>(c, r). M r)") 

apply(simp_all)

sorry

fun 

 ders :: "string \<Rightarrow> rexp \<Rightarrow> rexp"

where

  "ders [] r = r"

| "ders (c # s) r = ders s (der c r)"

fun

  matcher :: "rexp \<Rightarrow> string \<Rightarrow> bool"

where

  "matcher r s = nullable (ders s r)"

section {* Correctness Proof of the Matcher *}

lemma nullable_correctness:

  shows "nullable r  \<longleftrightarrow> [] \<in> (L r)"

apply(induct r) 

apply(auto simp add: Seq_def) 

done

section {* Left-Quotient of a Set *}

definition

  Der :: "char \<Rightarrow> string set \<Rightarrow> string set"

where

  "Der c A \<equiv> {s. [c] @ s \<in> A}"

lemma Der_null [simp]:

  shows "Der c {} = {}"

unfolding Der_def

by auto

lemma Der_empty [simp]:

  shows "Der c {[]} = {}"

unfolding Der_def

by auto

lemma Der_char [simp]:

  shows "Der c {[d]} = (if c = d then {[]} else {})"

unfolding Der_def

by auto

lemma Der_union [simp]:

  shows "Der c (A \<union> B) = Der c A \<union> Der c B"

unfolding Der_def

by auto

lemma Der_insert_nil [simp]:

  shows "Der c (insert [] A) = Der c A"

unfolding Der_def 

by auto 

lemma Der_seq [simp]:

  shows "Der c (A ;; B) = (Der c A) ;; B \<union> (if [] \<in> A then Der c B else {})"

unfolding Der_def Seq_def

by (auto simp add: Cons_eq_append_conv)

lemma Der_star [simp]:

  shows "Der c (A\<star>) = (Der c A) ;; A\<star>"

proof -    

  have "Der c (A\<star>) = Der c ({[]} \<union> A ;; A\<star>)"

    by (simp only: star_cases[symmetric])

  also have "... = Der c (A ;; A\<star>)"

    by (simp only: Der_union Der_empty) (simp)

  also have "... = (Der c A) ;; A\<star> \<union> (if [] \<in> A then Der c (A\<star>) else {})"

    by simp

  also have "... =  (Der c A) ;; A\<star>"

    unfolding Seq_def Der_def

    by (auto dest: star_decomp)

  finally show "Der c (A\<star>) = (Der c A) ;; A\<star>" .

qed

lemma Der_UNIV [simp]:

  "Der c (UNIV - A) = UNIV - Der c A"

unfolding Der_def

by (auto)

lemma Der_pow [simp]:

  shows "Der c (A \<up> (Suc n)) = (Der c A) ;; (A \<up> n) \<union> (if [] \<in> A then Der c (A \<up> n) else {})"

unfolding Der_def 

by(auto simp add: Cons_eq_append_conv Seq_def)

lemma Der_UNION [simp]: 

  shows "Der c (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. Der c (B x))"

by (auto simp add: Der_def)

lemma der_correctness:

  shows "L (der c r) = Der c (L r)"

apply(induct rule: der.induct) 

apply(simp_all add: nullable_correctness 

    Suc_Union Suc_reduce_Union seq_Union atLeast0AtMost)

apply(case_tac "[] \<in> L r")

apply(simp)

apply(case_tac n)

apply(auto)[1]

apply(case_tac m)

apply(simp)

apply(auto simp add: )[1]

apply (metis (full_types, hide_lams) Der_pow Der_seq Suc_le_mono UnCI atLeast0AtMost atMost_iff not_less_eq_eq)

apply (metis (poly_guards_query) atLeastAtMost_iff not_le order_refl)

apply (metis Suc_leD atLeastAtMost_iff)

by (metis atLeastAtMost_iff le_antisym not_less_eq_eq)

(*

lemma der_correctness:

  shows "L (der c r) = Der c (L r)"

apply(induct rule: der.induct) 

apply(simp_all add: nullable_correctness 

    Suc_Union Suc_reduce_Union seq_Union atLeast0AtMost)

apply(case_tac m)

apply(simp)

apply(simp)

apply(auto)

apply (metis (poly_guards_query) atLeastAtMost_iff not_le order_refl)

apply (metis Suc_leD atLeastAtMost_iff)

by (metis atLeastAtMost_iff le_antisym not_less_eq_eq)

*)

lemma matcher_correctness:

  shows "matcher r s \<longleftrightarrow> s \<in> L r"

by (induct s arbitrary: r)

   (simp_all add: nullable_correctness der_correctness Der_def)

lemma "L (der c (NMTIMES r n m)) = 

  L ((if m < n then NULL else 

     (if m = 0 then NULL else

     (der c (SEQ r (NMTIMES r (n - 1) (m - 1)))))))"

apply(subst der.simps(12))

apply(auto simp del: der.simps)

apply(simp)

apply(auto)[1]

apply(case_tac m)

apply(simp)

apply(simp)

apply(case_tac m)

apply(simp)

apply(simp)

apply(simp)

apply(subst (asm) der.simps(12))

apply(auto)[1]

apply(case_tac m)

apply(simp)

apply(simp)

apply(case_tac m)

apply(simp)

apply(simp)

apply(auto)[1]

apply(auto simp del: der.simps(12))[1]

apply(subst (asm) der.simps(12))

apply(case_tac n)

apply(simp)

apply(simp)

apply(case_tac m)

apply(simp)

apply(simp)

apply(case_tac "Suc nat = nata")

apply(simp)

apply(auto simp del: der.simps(12))[1]

apply(auto)[1]

apply(case_tac n)

apply(simp)

apply(simp)

apply(case_tac m)

apply(simp)

apply(simp)

apply(simp add: Seq_def)

apply(auto)[1]

apply (metis atLeastAtMost_iff le_eq_less_or_eq linorder_neqE_nat)

apply(simp (no_asm) del: der.simps(12))

apply(auto simp del: der.simps(12))[1]

apply(case_tac "m = Suc n")

apply(simp)

apply(case_tac n)

apply(simp)

apply(simp)

apply(auto simp add: Seq_def del: der.simps(12))[1]

apply(case_tac n)

apply(simp)

apply(case_tac m)

apply(simp)

apply(simp)

apply(simp add: Seq_def)

apply(auto)[1]

apply(case_tac "nat = 0")

apply(simp)

apply(simp)

apply(auto)[1]

apply(case_tac "Suc 0 = nat")

apply(simp)

apply(auto simp del: der.simps(12))[1]

apply(simp)

apply(case_tac "Suc 0 = nat")

apply(simp)

apply(auto simp add: Seq_def del: der.simps(12))[1]

apply(rule_tac x="s1" in exI)

apply(rule_tac x="s2" in exI)

apply(simp)

apply(rule_tac x="1" in bexI)

apply(simp)

apply(simp)

apply(simp)

apply(auto simp add: Seq_def)[1]

apply(case_tac m)

apply(simp)

apply(simp)

end