lexing: comparison thys3/RegLangs.thy

equal deleted inserted replaced

-:56057198e4f5
+:70c10dc41606
 theory RegLangs
 imports Main "HOL-Library.Sublist"
 begin
 section \<open>Sequential Composition of Languages\<close>
 lemma Sequ_empty [simp]:
 shows "A ;; {} = {}"
 and   "{} ;; A = {}"
 by (simp_all add: Sequ_def)
+lemma concI[simp,intro]: "u : A \<Longrightarrow> v : B \<Longrightarrow> u@v : A ;; B"
+by (auto simp add: Sequ_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: Sequ_def)
+lemma concI_if_Nil2: "[] \<in> B \<Longrightarrow> xs : A \<Longrightarrow> xs \<in> A ;; B"
+by (metis append_Nil2 concI)
+lemma conc_assoc: "(A ;; B) ;; C = A ;; (B ;; C)"
+by (auto elim!: concE) (simp only: append_assoc[symmetric] concI)
+text \<open>Language power operations\<close>
+overloading lang_pow == "compow :: nat \<Rightarrow> string set \<Rightarrow> string set"
+begin
+primrec lang_pow :: "nat \<Rightarrow> string set \<Rightarrow> string set" where
+"lang_pow 0 A = {[]}" |
+"lang_pow (Suc n) A = A ;; (lang_pow n A)"
+end
+lemma conc_pow_comm:
+shows "A ;; (A ^^ n) = (A ^^ n) ;; A"
+by (induct n) (simp_all add: conc_assoc[symmetric])
+lemma lang_pow_add: "A ^^ (n + m) = (A ^^ n) ;; (A ^^ m)"
+by (induct n) (auto simp: conc_assoc)
+lemma lang_empty:
+fixes A::"string set"
+shows "A ^^ 0 = {[]}"
+by simp
 section \<open>Semantic Derivative (Left Quotient) of Languages\<close>
 definition
 Der :: "char \<Rightarrow> string set \<Rightarrow> string set"
 lemma Star_Der_Sequ:
 shows "Der c (A\<star>) \<subseteq> (Der c A) ;; A\<star>"
 unfolding Der_def Sequ_def
 by(auto simp add: Star_decomp)
+lemma Der_inter[simp]:   "Der a (A \<inter> B) = Der a A \<inter> Der a B"
+and Der_compl[simp]:   "Der a (-A) = - Der a A"
+and Der_Union[simp]:   "Der a (Union M) = Union(Der a ` M)"
+and Der_UN[simp]:      "Der a (UN x:I. S x) = (UN x:I. Der a (S x))"
+by (auto simp: Der_def)
 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>)"
 also have "... =  (Der c A) ;; A\<star>"
 using Star_Der_Sequ by auto
 finally show "Der c (A\<star>) = (Der c A) ;; A\<star>" .
 qed
+lemma Der_pow[simp]:
+shows "Der c (A ^^ n) = (if n = 0 then {} else (Der c A) ;; (A ^^ (n - 1)))"
+apply(induct n arbitrary: A)
+apply(auto simp add: Cons_eq_append_conv)
+by (metis Suc_pred concI_if_Nil2 conc_assoc conc_pow_comm lang_pow.simps(2))
 lemma Star_concat:
 assumes "\<forall>s \<in> set ss. s \<in> A"
 shows "concat ss \<in> A\<star>"
 using assms by (induct ss) (auto)
 | ONE
 | CH char
 | SEQ rexp rexp
 | ALT rexp rexp
 | STAR rexp
+| NTIMES rexp nat
 section \<open>Semantics of Regular Expressions\<close>
 fun
 L :: "rexp \<Rightarrow> string set"
 | "L (ONE) = {[]}"
 | "L (CH 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 (NTIMES r n) = (L r) ^^ n"
 section \<open>Nullable, Derivatives\<close>
 fun
 nullable :: "rexp \<Rightarrow> bool"
 | "nullable (ONE) = True"
 | "nullable (CH 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 (NTIMES r n) = (if n = 0 then True else nullable r)"
 fun
 der :: "char \<Rightarrow> rexp \<Rightarrow> rexp"
 where
 "der c (ZERO) = ZERO"
 | "der c (SEQ r1 r2) =
 (if nullable r1
 then ALT (SEQ (der c r1) r2) (der c r2)
 else SEQ (der c r1) r2)"
 | "der c (STAR r) = SEQ (der c r) (STAR r)"
+| "der c (NTIMES r n) = (if n = 0 then ZERO else SEQ (der c r) (NTIMES r (n - 1)))"
 fun
 ders :: "string \<Rightarrow> rexp \<Rightarrow> rexp"
 where
 "ders [] r = r"
 | "ders (c # s) r = ders s (der c r)"
+lemma pow_empty_iff:
+shows "[] \<in> (L r) ^^ n \<longleftrightarrow> (if n = 0 then True else [] \<in> (L r))"
+by (induct n) (auto simp add: Sequ_def)
 lemma nullable_correctness:
 shows "nullable r  \<longleftrightarrow> [] \<in> (L r)"
-by (induct r) (auto simp add: Sequ_def)
+by (induct r) (auto simp add: Sequ_def pow_empty_iff)
 lemma der_correctness:
 shows "L (der c r) = Der c (L r)"
-by (induct r) (simp_all add: nullable_correctness)
+apply (induct r)
+apply(auto simp add: nullable_correctness Sequ_def)
+using Der_def apply force
+using Der_def apply auto[1]
+apply (smt (verit, ccfv_SIG) Der_def append_eq_Cons_conv mem_Collect_eq)
+using Der_def apply force
+using Der_Sequ Sequ_def by auto
 lemma ders_correctness:
 shows "L (ders s r) = Ders s (L r)"
 by (induct s arbitrary: r)
 (simp_all add: Ders_def der_correctness Der_def)
 lemma ders_snoc:
 shows "ders (s @ [c]) r = der c (ders s r)"
 by (simp add: ders_append)
-(*
-datatype ctxt =
-SeqC rexp bool
-| AltCL rexp
-| AltCH rexp
-| StarC rexp
-function
-down :: "char \<Rightarrow> rexp \<Rightarrow> ctxt list \<Rightarrow> rexp * ctxt list"
-and  up :: "char \<Rightarrow> rexp \<Rightarrow> ctxt list \<Rightarrow> rexp * ctxt list"
-where
-"down c (SEQ r1 r2) ctxts =
-(if (nullable r1) then down c r1 (SeqC r2 True # ctxts)
-else down c r1 (SeqC r2 False # ctxts))"
-| "down c (CH d) ctxts =
-(if c = d then up c ONE ctxts else up c ZERO ctxts)"
-| "down c ONE ctxts = up c ZERO ctxts"
-| "down c ZERO ctxts = up c ZERO ctxts"
-| "down c (ALT r1 r2) ctxts = down c r1 (AltCH r2 # ctxts)"
-| "down c (STAR r1) ctxts = down c r1 (StarC r1 # ctxts)"
-| "up c r [] = (r, [])"
-| "up c r (SeqC r2 False # ctxts) = up c (SEQ r r2) ctxts"
-| "up c r (SeqC r2 True # ctxts) = down c r2 (AltCL (SEQ r r2) # ctxts)"
-| "up c r (AltCL r1 # ctxts) = up c (ALT r1 r) ctxts"
-| "up c r (AltCH r2 # ctxts) = down c r2 (AltCL r # ctxts)"
-| "up c r (StarC r1 # ctxts) = up c (SEQ r (STAR r1)) ctxts"
-apply(pat_completeness)
-apply(auto)
-done
-termination
-sorry
-*)
 end

changeset 647	70c10dc41606
parent 642	6c13f76c070b