--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/thys2/SizeBoundStrong.thy Wed Feb 16 17:20:40 2022 +0000
@@ -0,0 +1,1644 @@
+
+theory SizeBoundStrong
+ imports "Lexer"
+begin
+
+section \<open>Bit-Encodings\<close>
+
+datatype bit = Z | S
+
+fun code :: "val \<Rightarrow> bit list"
+where
+ "code Void = []"
+| "code (Char c) = []"
+| "code (Left v) = Z # (code v)"
+| "code (Right v) = S # (code v)"
+| "code (Seq v1 v2) = (code v1) @ (code v2)"
+| "code (Stars []) = [S]"
+| "code (Stars (v # vs)) = (Z # code v) @ code (Stars vs)"
+
+
+fun
+ Stars_add :: "val \<Rightarrow> val \<Rightarrow> val"
+where
+ "Stars_add v (Stars vs) = Stars (v # vs)"
+
+function
+ decode' :: "bit list \<Rightarrow> rexp \<Rightarrow> (val * bit list)"
+where
+ "decode' ds ZERO = (Void, [])"
+| "decode' ds ONE = (Void, ds)"
+| "decode' ds (CH d) = (Char d, ds)"
+| "decode' [] (ALT r1 r2) = (Void, [])"
+| "decode' (Z # ds) (ALT r1 r2) = (let (v, ds') = decode' ds r1 in (Left v, ds'))"
+| "decode' (S # ds) (ALT r1 r2) = (let (v, ds') = decode' ds r2 in (Right v, ds'))"
+| "decode' ds (SEQ r1 r2) = (let (v1, ds') = decode' ds r1 in
+ let (v2, ds'') = decode' ds' r2 in (Seq v1 v2, ds''))"
+| "decode' [] (STAR r) = (Void, [])"
+| "decode' (S # ds) (STAR r) = (Stars [], ds)"
+| "decode' (Z # ds) (STAR r) = (let (v, ds') = decode' ds r in
+ let (vs, ds'') = decode' ds' (STAR r)
+ in (Stars_add v vs, ds''))"
+by pat_completeness auto
+
+lemma decode'_smaller:
+ assumes "decode'_dom (ds, r)"
+ shows "length (snd (decode' ds r)) \<le> length ds"
+using assms
+apply(induct ds r)
+apply(auto simp add: decode'.psimps split: prod.split)
+using dual_order.trans apply blast
+by (meson dual_order.trans le_SucI)
+
+termination "decode'"
+apply(relation "inv_image (measure(%cs. size cs) <*lex*> measure(%s. size s)) (%(ds,r). (r,ds))")
+apply(auto dest!: decode'_smaller)
+by (metis less_Suc_eq_le snd_conv)
+
+definition
+ decode :: "bit list \<Rightarrow> rexp \<Rightarrow> val option"
+where
+ "decode ds r \<equiv> (let (v, ds') = decode' ds r
+ in (if ds' = [] then Some v else None))"
+
+lemma decode'_code_Stars:
+ assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> (\<forall>x. decode' (code v @ x) r = (v, x)) \<and> flat v \<noteq> []"
+ shows "decode' (code (Stars vs) @ ds) (STAR r) = (Stars vs, ds)"
+ using assms
+ apply(induct vs)
+ apply(auto)
+ done
+
+lemma decode'_code:
+ assumes "\<Turnstile> v : r"
+ shows "decode' ((code v) @ ds) r = (v, ds)"
+using assms
+ apply(induct v r arbitrary: ds)
+ apply(auto)
+ using decode'_code_Stars by blast
+
+lemma decode_code:
+ assumes "\<Turnstile> v : r"
+ shows "decode (code v) r = Some v"
+ using assms unfolding decode_def
+ by (smt append_Nil2 decode'_code old.prod.case)
+
+
+section {* Annotated Regular Expressions *}
+
+datatype arexp =
+ AZERO
+| AONE "bit list"
+| ACHAR "bit list" char
+| ASEQ "bit list" arexp arexp
+| AALTs "bit list" "arexp list"
+| ASTAR "bit list" arexp
+
+abbreviation
+ "AALT bs r1 r2 \<equiv> AALTs bs [r1, r2]"
+
+fun asize :: "arexp \<Rightarrow> nat" where
+ "asize AZERO = 1"
+| "asize (AONE cs) = 1"
+| "asize (ACHAR cs c) = 1"
+| "asize (AALTs cs rs) = Suc (sum_list (map asize rs))"
+| "asize (ASEQ cs r1 r2) = Suc (asize r1 + asize r2)"
+| "asize (ASTAR cs r) = Suc (asize r)"
+
+fun
+ erase :: "arexp \<Rightarrow> rexp"
+where
+ "erase AZERO = ZERO"
+| "erase (AONE _) = ONE"
+| "erase (ACHAR _ c) = CH c"
+| "erase (AALTs _ []) = ZERO"
+| "erase (AALTs _ [r]) = (erase r)"
+| "erase (AALTs bs (r#rs)) = ALT (erase r) (erase (AALTs bs rs))"
+| "erase (ASEQ _ r1 r2) = SEQ (erase r1) (erase r2)"
+| "erase (ASTAR _ r) = STAR (erase r)"
+
+
+fun nonalt :: "arexp \<Rightarrow> bool"
+ where
+ "nonalt (AALTs bs2 rs) = False"
+| "nonalt r = True"
+
+
+fun fuse :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp" where
+ "fuse bs AZERO = AZERO"
+| "fuse bs (AONE cs) = AONE (bs @ cs)"
+| "fuse bs (ACHAR cs c) = ACHAR (bs @ cs) c"
+| "fuse bs (AALTs cs rs) = AALTs (bs @ cs) rs"
+| "fuse bs (ASEQ cs r1 r2) = ASEQ (bs @ cs) r1 r2"
+| "fuse bs (ASTAR cs r) = ASTAR (bs @ cs) r"
+
+lemma fuse_append:
+ shows "fuse (bs1 @ bs2) r = fuse bs1 (fuse bs2 r)"
+ apply(induct r)
+ apply(auto)
+ done
+
+lemma fuse_Nil:
+ shows "fuse [] r = r"
+ by (induct r)(simp_all)
+
+lemma map_fuse_Nil:
+ shows "map (fuse []) rs = rs"
+ by (induct rs)(simp_all add: fuse_Nil)
+
+
+fun intern :: "rexp \<Rightarrow> arexp" where
+ "intern ZERO = AZERO"
+| "intern ONE = AONE []"
+| "intern (CH c) = ACHAR [] c"
+| "intern (ALT r1 r2) = AALT [] (fuse [Z] (intern r1))
+ (fuse [S] (intern r2))"
+| "intern (SEQ r1 r2) = ASEQ [] (intern r1) (intern r2)"
+| "intern (STAR r) = ASTAR [] (intern r)"
+
+
+fun retrieve :: "arexp \<Rightarrow> val \<Rightarrow> bit list" where
+ "retrieve (AONE bs) Void = bs"
+| "retrieve (ACHAR bs c) (Char d) = bs"
+| "retrieve (AALTs bs [r]) v = bs @ retrieve r v"
+| "retrieve (AALTs bs (r#rs)) (Left v) = bs @ retrieve r v"
+| "retrieve (AALTs bs (r#rs)) (Right v) = bs @ retrieve (AALTs [] rs) v"
+| "retrieve (ASEQ bs r1 r2) (Seq v1 v2) = bs @ retrieve r1 v1 @ retrieve r2 v2"
+| "retrieve (ASTAR bs r) (Stars []) = bs @ [S]"
+| "retrieve (ASTAR bs r) (Stars (v#vs)) =
+ bs @ [Z] @ retrieve r v @ retrieve (ASTAR [] r) (Stars vs)"
+
+
+
+fun
+ bnullable :: "arexp \<Rightarrow> bool"
+where
+ "bnullable (AZERO) = False"
+| "bnullable (AONE bs) = True"
+| "bnullable (ACHAR bs c) = False"
+| "bnullable (AALTs bs rs) = (\<exists>r \<in> set rs. bnullable r)"
+| "bnullable (ASEQ bs r1 r2) = (bnullable r1 \<and> bnullable r2)"
+| "bnullable (ASTAR bs r) = True"
+
+fun
+ bmkeps :: "arexp \<Rightarrow> bit list"
+where
+ "bmkeps(AONE bs) = bs"
+| "bmkeps(ASEQ bs r1 r2) = bs @ (bmkeps r1) @ (bmkeps r2)"
+| "bmkeps(AALTs bs [r]) = bs @ (bmkeps r)"
+| "bmkeps(AALTs bs (r#rs)) = (if bnullable(r) then bs @ (bmkeps r) else (bmkeps (AALTs bs rs)))"
+| "bmkeps(ASTAR bs r) = bs @ [S]"
+
+
+fun
+ bder :: "char \<Rightarrow> arexp \<Rightarrow> arexp"
+where
+ "bder c (AZERO) = AZERO"
+| "bder c (AONE bs) = AZERO"
+| "bder c (ACHAR bs d) = (if c = d then AONE bs else AZERO)"
+| "bder c (AALTs bs rs) = AALTs bs (map (bder c) rs)"
+| "bder c (ASEQ bs r1 r2) =
+ (if bnullable r1
+ then AALT bs (ASEQ [] (bder c r1) r2) (fuse (bmkeps r1) (bder c r2))
+ else ASEQ bs (bder c r1) r2)"
+| "bder c (ASTAR bs r) = ASEQ bs (fuse [Z] (bder c r)) (ASTAR [] r)"
+
+
+fun
+ bders :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
+where
+ "bders r [] = r"
+| "bders r (c#s) = bders (bder c r) s"
+
+lemma bders_append:
+ "bders r (s1 @ s2) = bders (bders r s1) s2"
+ apply(induct s1 arbitrary: r s2)
+ apply(simp_all)
+ done
+
+lemma bnullable_correctness:
+ shows "nullable (erase r) = bnullable r"
+ apply(induct r rule: erase.induct)
+ apply(simp_all)
+ done
+
+lemma erase_fuse:
+ shows "erase (fuse bs r) = erase r"
+ apply(induct r rule: erase.induct)
+ apply(simp_all)
+ done
+
+lemma erase_intern [simp]:
+ shows "erase (intern r) = r"
+ apply(induct r)
+ apply(simp_all add: erase_fuse)
+ done
+
+lemma erase_bder [simp]:
+ shows "erase (bder a r) = der a (erase r)"
+ apply(induct r rule: erase.induct)
+ apply(simp_all add: erase_fuse bnullable_correctness)
+ done
+
+lemma erase_bders [simp]:
+ shows "erase (bders r s) = ders s (erase r)"
+ apply(induct s arbitrary: r )
+ apply(simp_all)
+ done
+
+lemma bnullable_fuse:
+ shows "bnullable (fuse bs r) = bnullable r"
+ apply(induct r arbitrary: bs)
+ apply(auto)
+ done
+
+lemma retrieve_encode_STARS:
+ assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> code v = retrieve (intern r) v"
+ shows "code (Stars vs) = retrieve (ASTAR [] (intern r)) (Stars vs)"
+ using assms
+ apply(induct vs)
+ apply(simp_all)
+ done
+
+
+lemma retrieve_fuse2:
+ assumes "\<Turnstile> v : (erase r)"
+ shows "retrieve (fuse bs r) v = bs @ retrieve r v"
+ using assms
+ apply(induct r arbitrary: v bs)
+ apply(auto elim: Prf_elims)[4]
+ defer
+ using retrieve_encode_STARS
+ apply(auto elim!: Prf_elims)[1]
+ apply(case_tac vs)
+ apply(simp)
+ apply(simp)
+ (* AALTs case *)
+ apply(simp)
+ apply(case_tac x2a)
+ apply(simp)
+ apply(auto elim!: Prf_elims)[1]
+ apply(simp)
+ apply(case_tac list)
+ apply(simp)
+ apply(auto)
+ apply(auto elim!: Prf_elims)[1]
+ done
+
+lemma retrieve_fuse:
+ assumes "\<Turnstile> v : r"
+ shows "retrieve (fuse bs (intern r)) v = bs @ retrieve (intern r) v"
+ using assms
+ by (simp_all add: retrieve_fuse2)
+
+
+lemma retrieve_code:
+ assumes "\<Turnstile> v : r"
+ shows "code v = retrieve (intern r) v"
+ using assms
+ apply(induct v r )
+ apply(simp_all add: retrieve_fuse retrieve_encode_STARS)
+ done
+
+
+lemma bnullable_Hdbmkeps_Hd:
+ assumes "bnullable a"
+ shows "bmkeps (AALTs bs (a # rs)) = bs @ (bmkeps a)"
+ using assms
+ by (metis bmkeps.simps(3) bmkeps.simps(4) list.exhaust)
+
+lemma r1:
+ assumes "\<not> bnullable a" "bnullable (AALTs bs rs)"
+ shows "bmkeps (AALTs bs (a # rs)) = bmkeps (AALTs bs rs)"
+ using assms
+ apply(induct rs)
+ apply(auto)
+ done
+
+lemma r2:
+ assumes "x \<in> set rs" "bnullable x"
+ shows "bnullable (AALTs bs rs)"
+ using assms
+ apply(induct rs)
+ apply(auto)
+ done
+
+lemma r3:
+ assumes "\<not> bnullable r"
+ " \<exists> x \<in> set rs. bnullable x"
+ shows "retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs))) =
+ retrieve (AALTs bs (r # rs)) (mkeps (erase (AALTs bs (r # rs))))"
+ using assms
+ apply(induct rs arbitrary: r bs)
+ apply(auto)[1]
+ apply(auto)
+ using bnullable_correctness apply blast
+ apply(auto simp add: bnullable_correctness mkeps_nullable retrieve_fuse2)
+ apply(subst retrieve_fuse2[symmetric])
+ apply (smt bnullable.simps(4) bnullable_correctness erase.simps(5) erase.simps(6) insert_iff list.exhaust list.set(2) mkeps.simps(3) mkeps_nullable)
+ apply(simp)
+ apply(case_tac "bnullable a")
+ apply (smt append_Nil2 bnullable.simps(4) bnullable_correctness erase.simps(5) erase.simps(6) fuse.simps(4) insert_iff list.exhaust list.set(2) mkeps.simps(3) mkeps_nullable retrieve_fuse2)
+ apply(drule_tac x="a" in meta_spec)
+ apply(drule_tac x="bs" in meta_spec)
+ apply(drule meta_mp)
+ apply(simp)
+ apply(drule meta_mp)
+ apply(auto)
+ apply(subst retrieve_fuse2[symmetric])
+ apply(case_tac rs)
+ apply(simp)
+ apply(auto)[1]
+ apply (simp add: bnullable_correctness)
+ apply (metis append_Nil2 bnullable_correctness erase_fuse fuse.simps(4) list.set_intros(1) mkeps.simps(3) mkeps_nullable nullable.simps(4) r2)
+ apply (simp add: bnullable_correctness)
+ apply (metis append_Nil2 bnullable_correctness erase.simps(6) erase_fuse fuse.simps(4) list.set_intros(2) mkeps.simps(3) mkeps_nullable r2)
+ apply(simp)
+ done
+
+
+lemma t:
+ assumes "\<forall>r \<in> set rs. nullable (erase r) \<longrightarrow> bmkeps r = retrieve r (mkeps (erase r))"
+ "nullable (erase (AALTs bs rs))"
+ shows " bmkeps (AALTs bs rs) = retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs)))"
+ using assms
+ apply(induct rs arbitrary: bs)
+ apply(simp)
+ apply(auto simp add: bnullable_correctness)
+ apply(case_tac rs)
+ apply(auto simp add: bnullable_correctness)[2]
+ apply(subst r1)
+ apply(simp)
+ apply(rule r2)
+ apply(assumption)
+ apply(simp)
+ apply(drule_tac x="bs" in meta_spec)
+ apply(drule meta_mp)
+ apply(auto)[1]
+ prefer 2
+ apply(case_tac "bnullable a")
+ apply(subst bnullable_Hdbmkeps_Hd)
+ apply blast
+ apply(subgoal_tac "nullable (erase a)")
+ prefer 2
+ using bnullable_correctness apply blast
+ apply (metis (no_types, lifting) erase.simps(5) erase.simps(6) list.exhaust mkeps.simps(3) retrieve.simps(3) retrieve.simps(4))
+ apply(subst r1)
+ apply(simp)
+ using r2 apply blast
+ apply(drule_tac x="bs" in meta_spec)
+ apply(drule meta_mp)
+ apply(auto)[1]
+ apply(simp)
+ using r3 apply blast
+ apply(auto)
+ using r3 by blast
+
+lemma bmkeps_retrieve:
+ assumes "nullable (erase r)"
+ shows "bmkeps r = retrieve r (mkeps (erase r))"
+ using assms
+ apply(induct r)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ defer
+ apply(simp)
+ apply(rule t)
+ apply(auto)
+ done
+
+lemma bder_retrieve:
+ assumes "\<Turnstile> v : der c (erase r)"
+ shows "retrieve (bder c r) v = retrieve r (injval (erase r) c v)"
+ using assms
+ apply(induct r arbitrary: v rule: erase.induct)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(case_tac "c = ca")
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(simp)
+ apply(rename_tac "r\<^sub>1" "r\<^sub>2" rs v)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(simp)
+ apply(case_tac rs)
+ apply(simp)
+ apply(simp)
+ apply (smt Prf_elims(3) injval.simps(2) injval.simps(3) retrieve.simps(4) retrieve.simps(5) same_append_eq)
+ apply(simp)
+ apply(case_tac "nullable (erase r1)")
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(subgoal_tac "bnullable r1")
+ prefer 2
+ using bnullable_correctness apply blast
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ apply(subgoal_tac "bnullable r1")
+ prefer 2
+ using bnullable_correctness apply blast
+ apply(simp)
+ apply(simp add: retrieve_fuse2)
+ apply(simp add: bmkeps_retrieve)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(simp)
+ using bnullable_correctness apply blast
+ apply(rename_tac bs r v)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(clarify)
+ apply(erule Prf_elims)
+ apply(clarify)
+ apply(subst injval.simps)
+ apply(simp del: retrieve.simps)
+ apply(subst retrieve.simps)
+ apply(subst retrieve.simps)
+ apply(simp)
+ apply(simp add: retrieve_fuse2)
+ done
+
+
+
+lemma MAIN_decode:
+ assumes "\<Turnstile> v : ders s r"
+ shows "Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r"
+ using assms
+proof (induct s arbitrary: v rule: rev_induct)
+ case Nil
+ have "\<Turnstile> v : ders [] r" by fact
+ then have "\<Turnstile> v : r" by simp
+ then have "Some v = decode (retrieve (intern r) v) r"
+ using decode_code retrieve_code by auto
+ then show "Some (flex r id [] v) = decode (retrieve (bders (intern r) []) v) r"
+ by simp
+next
+ case (snoc c s v)
+ have IH: "\<And>v. \<Turnstile> v : ders s r \<Longrightarrow>
+ Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r" by fact
+ have asm: "\<Turnstile> v : ders (s @ [c]) r" by fact
+ then have asm2: "\<Turnstile> injval (ders s r) c v : ders s r"
+ by (simp add: Prf_injval ders_append)
+ have "Some (flex r id (s @ [c]) v) = Some (flex r id s (injval (ders s r) c v))"
+ by (simp add: flex_append)
+ also have "... = decode (retrieve (bders (intern r) s) (injval (ders s r) c v)) r"
+ using asm2 IH by simp
+ also have "... = decode (retrieve (bder c (bders (intern r) s)) v) r"
+ using asm by (simp_all add: bder_retrieve ders_append)
+ finally show "Some (flex r id (s @ [c]) v) =
+ decode (retrieve (bders (intern r) (s @ [c])) v) r" by (simp add: bders_append)
+qed
+
+
+definition blex where
+ "blex a s \<equiv> if bnullable (bders a s) then Some (bmkeps (bders a s)) else None"
+
+
+
+definition blexer where
+ "blexer r s \<equiv> if bnullable (bders (intern r) s) then
+ decode (bmkeps (bders (intern r) s)) r else None"
+
+lemma blexer_correctness:
+ shows "blexer r s = lexer r s"
+proof -
+ { define bds where "bds \<equiv> bders (intern r) s"
+ define ds where "ds \<equiv> ders s r"
+ assume asm: "nullable ds"
+ have era: "erase bds = ds"
+ unfolding ds_def bds_def by simp
+ have mke: "\<Turnstile> mkeps ds : ds"
+ using asm by (simp add: mkeps_nullable)
+ have "decode (bmkeps bds) r = decode (retrieve bds (mkeps ds)) r"
+ using bmkeps_retrieve
+ using asm era by (simp add: bmkeps_retrieve)
+ also have "... = Some (flex r id s (mkeps ds))"
+ using mke by (simp_all add: MAIN_decode ds_def bds_def)
+ finally have "decode (bmkeps bds) r = Some (flex r id s (mkeps ds))"
+ unfolding bds_def ds_def .
+ }
+ then show "blexer r s = lexer r s"
+ unfolding blexer_def lexer_flex
+ apply(subst bnullable_correctness[symmetric])
+ apply(simp)
+ done
+qed
+
+
+fun distinctBy :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b set \<Rightarrow> 'a list"
+ where
+ "distinctBy [] f acc = []"
+| "distinctBy (x#xs) f acc =
+ (if (f x) \<in> acc then distinctBy xs f acc
+ else x # (distinctBy xs f ({f x} \<union> acc)))"
+
+(*filter (\<lambda>rt. case rt of
+ SEQ r1p r2p \<Rightarrow> r2p = (erase r2)
+ r \<Rightarrow> False ) allowableTerms*)
+
+
+lemma dB_single_step:
+ shows "distinctBy (a#rs) f {} = a # distinctBy rs f {f a}"
+ by simp
+
+fun flts :: "arexp list \<Rightarrow> arexp list"
+ where
+ "flts [] = []"
+| "flts (AZERO # rs) = flts rs"
+| "flts ((AALTs bs rs1) # rs) = (map (fuse bs) rs1) @ flts rs"
+| "flts (r1 # rs) = r1 # flts rs"
+
+
+
+fun bsimp_ASEQ :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp \<Rightarrow> arexp"
+ where
+ "bsimp_ASEQ _ AZERO _ = AZERO"
+| "bsimp_ASEQ _ _ AZERO = AZERO"
+| "bsimp_ASEQ bs1 (AONE bs2) r2 = fuse (bs1 @ bs2) r2"
+| "bsimp_ASEQ bs1 r1 r2 = ASEQ bs1 r1 r2"
+
+
+fun bsimp_AALTs :: "bit list \<Rightarrow> arexp list \<Rightarrow> arexp"
+ where
+ "bsimp_AALTs _ [] = AZERO"
+| "bsimp_AALTs bs1 [r] = fuse bs1 r"
+| "bsimp_AALTs bs1 rs = AALTs bs1 rs"
+
+
+fun bsimp_ASEQ1 :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp \<Rightarrow> arexp"
+ where
+ "bsimp_ASEQ1 _ AZERO _ = AZERO"
+| "bsimp_ASEQ1 bs (AONE bs1) r2 = fuse (bs @ bs1) r2"
+| "bsimp_ASEQ1 bs r1 r2 = ASEQ bs r1 r2"
+
+
+fun collect where
+ \<open>collect _ [] = []\<close>
+| \<open>collect erasedR2 ((SEQ r1 r2) # rs) = (if r2 = erasedR2 then r1 # (collect erasedR2 rs)
+ else collect erasedR2 rs)\<close>
+| \<open>collect erasedR2 (r # rs) = collect erasedR2 rs\<close>
+
+
+fun pruneRexp where
+ \<open>pruneRexp (ASEQ bs r1 r2) allowableTerms =
+( let termsTruncated = (collect (erase r2) allowableTerms) in (let pruned = pruneRexp r1 termsTruncated in (bsimp_ASEQ1 bs pruned r2)) )\<close>
+| \<open>pruneRexp (AALTs bs rs) allowableTerms = (let rsp = (filter (\<lambda>r. r \<noteq> AZERO) (map (\<lambda>r. pruneRexp r allowableTerms) rs) ) in bsimp_AALTs bs rsp )
+\<close>
+| \<open>pruneRexp r allowableTerms = (if (erase r) \<in> (set allowableTerms) then r else AZERO)\<close>
+
+
+fun oneSimp :: \<open>rexp \<Rightarrow> rexp\<close> where
+ \<open> oneSimp (SEQ ONE r) = r \<close>
+| \<open> oneSimp (SEQ r1 r2) = SEQ (oneSimp r1) r2 \<close>
+| \<open> oneSimp r = r \<close>
+
+fun breakIntoTerms where
+ \<open>breakIntoTerms (SEQ r1 r2) = map (\<lambda>r1p. SEQ r1p r2) (breakIntoTerms r1)\<close>
+| \<open>breakIntoTerms (ALT r1 r2) = (breakIntoTerms r1) @ (breakIntoTerms r2)\<close>
+| \<open>breakIntoTerms r = r # [] \<close>
+
+fun addToAcc :: "arexp \<Rightarrow> rexp list \<Rightarrow> rexp list"
+ where
+\<open>addToAcc r acc = filter (\<lambda>r1. oneSimp r1 \<notin> set acc) (breakIntoTerms (erase r)) \<close>
+
+fun dBStrong :: "arexp list \<Rightarrow> rexp list \<Rightarrow> arexp list"
+ where
+"dBStrong [] acc = []"
+| "dBStrong (r # rs) acc = (if (erase r) \<in> (set acc) then dBStrong rs acc
+ else (case (pruneRexp r (addToAcc r acc)) of
+ AZERO \<Rightarrow> dBStrong rs ((addToAcc r acc) @ acc) |
+ r1 \<Rightarrow> r1 # (dBStrong rs ((addToAcc r acc) @ acc))
+ )
+ )
+ "
+fun bsimpStrong :: "arexp \<Rightarrow> arexp "
+ where
+ "bsimpStrong (ASEQ bs1 r1 r2) = bsimp_ASEQ bs1 (bsimpStrong r1) (bsimpStrong r2)"
+| "bsimpStrong (AALTs bs1 rs) = bsimp_AALTs bs1 (dBStrong (flts (map bsimpStrong rs)) []) "
+| "bsimpStrong r = r"
+
+
+fun bdersStrong :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
+where
+ "bdersStrong r [] = r"
+| "bdersStrong r (c # s) = bdersStrong (bsimpStrong (bder c r)) s"
+
+
+definition blexerStrong where
+ "blexerStrong r s \<equiv> if bnullable (bdersStrong (intern r) s) then
+ decode (bmkeps (bdersStrong (intern r) s)) r else None"
+
+
+
+fun bsimp :: "arexp \<Rightarrow> arexp"
+ where
+ "bsimp (ASEQ bs1 r1 r2) = bsimp_ASEQ bs1 (bsimp r1) (bsimp r2)"
+| "bsimp (AALTs bs1 rs) = bsimp_AALTs bs1 (distinctBy (flts (map bsimp rs)) erase {}) "
+| "bsimp r = r"
+
+
+fun
+ bders_simp :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
+where
+ "bders_simp r [] = r"
+| "bders_simp r (c # s) = bders_simp (bsimp (bder c r)) s"
+
+definition blexer_simp where
+ "blexer_simp r s \<equiv> if bnullable (bders_simp (intern r) s) then
+ decode (bmkeps (bders_simp (intern r) s)) r else None"
+
+export_code bders_simp in Scala module_name Example
+
+lemma bders_simp_append:
+ shows "bders_simp r (s1 @ s2) = bders_simp (bders_simp r s1) s2"
+ apply(induct s1 arbitrary: r s2)
+ apply(simp_all)
+ done
+
+lemma L_bsimp_ASEQ:
+ "L (SEQ (erase r1) (erase r2)) = L (erase (bsimp_ASEQ bs r1 r2))"
+ apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
+ apply(simp_all)
+ by (metis erase_fuse fuse.simps(4))
+
+lemma L_bsimp_AALTs:
+ "L (erase (AALTs bs rs)) = L (erase (bsimp_AALTs bs rs))"
+ apply(induct bs rs rule: bsimp_AALTs.induct)
+ apply(simp_all add: erase_fuse)
+ done
+
+lemma L_erase_AALTs:
+ shows "L (erase (AALTs bs rs)) = \<Union> (L ` erase ` (set rs))"
+ apply(induct rs)
+ apply(simp)
+ apply(simp)
+ apply(case_tac rs)
+ apply(simp)
+ apply(simp)
+ done
+
+lemma L_erase_flts:
+ shows "\<Union> (L ` erase ` (set (flts rs))) = \<Union> (L ` erase ` (set rs))"
+ apply(induct rs rule: flts.induct)
+ apply(simp_all)
+ apply(auto)
+ using L_erase_AALTs erase_fuse apply auto[1]
+ by (simp add: L_erase_AALTs erase_fuse)
+
+lemma L_erase_dB_acc:
+ shows "( \<Union>(L ` acc) \<union> ( \<Union> (L ` erase ` (set (distinctBy rs erase acc) ) ) )) = \<Union>(L ` acc) \<union> \<Union> (L ` erase ` (set rs))"
+ apply(induction rs arbitrary: acc)
+ apply simp
+ apply simp
+ by (smt (z3) SUP_absorb UN_insert sup_assoc sup_commute)
+
+lemma L_erase_dB:
+ shows " ( \<Union> (L ` erase ` (set (distinctBy rs erase {}) ) ) ) = \<Union> (L ` erase ` (set rs))"
+ by (metis L_erase_dB_acc Un_commute Union_image_empty)
+
+lemma L_bsimp_erase:
+ shows "L (erase r) = L (erase (bsimp r))"
+ apply(induct r)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ apply(auto simp add: Sequ_def)[1]
+ apply(subst L_bsimp_ASEQ[symmetric])
+ apply(auto simp add: Sequ_def)[1]
+ apply(subst (asm) L_bsimp_ASEQ[symmetric])
+ apply(auto simp add: Sequ_def)[1]
+ apply(simp)
+ apply(subst L_bsimp_AALTs[symmetric])
+ defer
+ apply(simp)
+ apply(subst (2)L_erase_AALTs)
+ apply(subst L_erase_dB)
+ apply(subst L_erase_flts)
+ apply(auto)
+ apply (simp add: L_erase_AALTs)
+ using L_erase_AALTs by blast
+
+
+
+lemma bsimp_ASEQ0:
+ shows "bsimp_ASEQ bs r1 AZERO = AZERO"
+ apply(induct r1)
+ apply(auto)
+ done
+
+lemma bsimp_ASEQ1:
+ assumes "r1 \<noteq> AZERO" "r2 \<noteq> AZERO" "\<forall>bs. r1 \<noteq> AONE bs"
+ shows "bsimp_ASEQ bs r1 r2 = ASEQ bs r1 r2"
+ using assms
+ apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
+ apply(auto)
+ done
+
+lemma bsimp_ASEQ2:
+ shows "bsimp_ASEQ bs (AONE bs1) r2 = fuse (bs @ bs1) r2"
+ apply(induct r2)
+ apply(auto)
+ done
+
+
+lemma L_bders_simp:
+ shows "L (erase (bders_simp r s)) = L (erase (bders r s))"
+ apply(induct s arbitrary: r rule: rev_induct)
+ apply(simp)
+ apply(simp)
+ apply(simp add: ders_append)
+ apply(simp add: bders_simp_append)
+ apply(simp add: L_bsimp_erase[symmetric])
+ by (simp add: der_correctness)
+
+
+lemma b2:
+ assumes "bnullable r"
+ shows "bmkeps (fuse bs r) = bs @ bmkeps r"
+ by (simp add: assms bmkeps_retrieve bnullable_correctness erase_fuse mkeps_nullable retrieve_fuse2)
+
+
+lemma b4:
+ shows "bnullable (bders_simp r s) = bnullable (bders r s)"
+ by (metis L_bders_simp bnullable_correctness lexer.simps(1) lexer_correct_None option.distinct(1))
+
+lemma qq1:
+ assumes "\<exists>r \<in> set rs. bnullable r"
+ shows "bmkeps (AALTs bs (rs @ rs1)) = bmkeps (AALTs bs rs)"
+ using assms
+ apply(induct rs arbitrary: rs1 bs)
+ apply(simp)
+ apply(simp)
+ by (metis Nil_is_append_conv bmkeps.simps(4) neq_Nil_conv bnullable_Hdbmkeps_Hd split_list_last)
+
+lemma qq2:
+ assumes "\<forall>r \<in> set rs. \<not> bnullable r" "\<exists>r \<in> set rs1. bnullable r"
+ shows "bmkeps (AALTs bs (rs @ rs1)) = bmkeps (AALTs bs rs1)"
+ using assms
+ apply(induct rs arbitrary: rs1 bs)
+ apply(simp)
+ apply(simp)
+ by (metis append_assoc in_set_conv_decomp r1 r2)
+
+lemma qq3:
+ assumes "bnullable (AALTs bs (rs @ rs1))"
+ "bnullable (AALTs bs (rs @ rs2))"
+ "\<lbrakk>bnullable (AALTs bs rs1); bnullable (AALTs bs rs2); \<forall>r\<in>set rs. \<not>bnullable r\<rbrakk> \<Longrightarrow>
+ bmkeps (AALTs bs rs1) = bmkeps (AALTs bs rs2)"
+ shows "bmkeps (AALTs bs (rs @ rs1)) = bmkeps (AALTs bs (rs @ rs2))"
+ using assms
+ apply(case_tac "\<exists>r \<in> set rs. bnullable r")
+ using qq1 apply auto[1]
+ by (metis UnE bnullable.simps(4) qq2 set_append)
+
+
+lemma flts_append:
+ shows "flts (xs1 @ xs2) = flts xs1 @ flts xs2"
+by (induct xs1 arbitrary: xs2 rule: flts.induct)(auto)
+
+lemma k0a:
+ shows "flts [AALTs bs rs] = map (fuse bs) rs"
+ apply(simp)
+ done
+
+
+lemma bbbbs1:
+ shows "nonalt r \<or> (\<exists>bs rs. r = AALTs bs rs)"
+ using nonalt.elims(3) by auto
+
+
+
+fun nonazero :: "arexp \<Rightarrow> bool"
+ where
+ "nonazero AZERO = False"
+| "nonazero r = True"
+
+
+lemma flts_single1:
+ assumes "nonalt r" "nonazero r"
+ shows "flts [r] = [r]"
+ using assms
+ apply(induct r)
+ apply(auto)
+ done
+
+
+
+lemma q3a:
+ assumes "\<exists>r \<in> set rs. bnullable r"
+ shows "bmkeps (AALTs bs (map (fuse bs1) rs)) = bmkeps (AALTs (bs@bs1) rs)"
+ using assms
+ apply(induct rs arbitrary: bs bs1)
+ apply(simp)
+ apply(simp)
+ apply(auto)
+ apply (metis append_assoc b2 bnullable_correctness erase_fuse bnullable_Hdbmkeps_Hd)
+ apply(case_tac "bnullable a")
+ apply (metis append.assoc b2 bnullable_correctness erase_fuse bnullable_Hdbmkeps_Hd)
+ apply(case_tac rs)
+ apply(simp)
+ apply(simp)
+ apply(auto)[1]
+ apply (metis bnullable_correctness erase_fuse)+
+ done
+
+lemma qq4:
+ assumes "\<exists>x\<in>set list. bnullable x"
+ shows "\<exists>x\<in>set (flts list). bnullable x"
+ using assms
+ apply(induct list rule: flts.induct)
+ apply(auto)
+ by (metis UnCI bnullable_correctness erase_fuse imageI)
+
+
+lemma qs3:
+ assumes "\<exists>r \<in> set rs. bnullable r"
+ shows "bmkeps (AALTs bs rs) = bmkeps (AALTs bs (flts rs))"
+ using assms
+ apply(induct rs arbitrary: bs taking: size rule: measure_induct)
+ apply(case_tac x)
+ apply(simp)
+ apply(simp)
+ apply(case_tac a)
+ apply(simp)
+ apply (simp add: r1)
+ apply(simp)
+ apply (simp add: bnullable_Hdbmkeps_Hd)
+ apply(simp)
+ apply(case_tac "flts list")
+ apply(simp)
+ apply (metis L_erase_AALTs L_erase_flts L_flat_Prf1 L_flat_Prf2 Prf_elims(1) bnullable_correctness erase.simps(4) mkeps_nullable r2)
+ apply(simp)
+ apply (simp add: r1)
+ prefer 3
+ apply(simp)
+ apply (simp add: bnullable_Hdbmkeps_Hd)
+ prefer 2
+ apply(simp)
+ apply(case_tac "\<exists>x\<in>set x52. bnullable x")
+ apply(case_tac "list")
+ apply(simp)
+ apply (metis b2 fuse.simps(4) q3a r2)
+ apply(erule disjE)
+ apply(subst qq1)
+ apply(auto)[1]
+ apply (metis bnullable_correctness erase_fuse)
+ apply(simp)
+ apply (metis b2 fuse.simps(4) q3a r2)
+ apply(simp)
+ apply(auto)[1]
+ apply(subst qq1)
+ apply (metis bnullable_correctness erase_fuse image_eqI set_map)
+ apply (metis b2 fuse.simps(4) q3a r2)
+ apply(subst qq1)
+ apply (metis bnullable_correctness erase_fuse image_eqI set_map)
+ apply (metis b2 fuse.simps(4) q3a r2)
+ apply(simp)
+ apply(subst qq2)
+ apply (metis bnullable_correctness erase_fuse imageE set_map)
+ prefer 2
+ apply(case_tac "list")
+ apply(simp)
+ apply(simp)
+ apply (simp add: qq4)
+ apply(simp)
+ apply(auto)
+ apply(case_tac list)
+ apply(simp)
+ apply(simp)
+ apply (simp add: bnullable_Hdbmkeps_Hd)
+ apply(case_tac "bnullable (ASEQ x41 x42 x43)")
+ apply(case_tac list)
+ apply(simp)
+ apply(simp)
+ apply (simp add: bnullable_Hdbmkeps_Hd)
+ apply(simp)
+ using qq4 r1 r2 by auto
+
+lemma bder_fuse:
+ shows "bder c (fuse bs a) = fuse bs (bder c a)"
+ apply(induct a arbitrary: bs c)
+ apply(simp_all)
+ done
+
+
+
+
+inductive
+ rrewrite:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto> _" [99, 99] 99)
+where
+ "ASEQ bs AZERO r2 \<leadsto> AZERO"
+| "ASEQ bs r1 AZERO \<leadsto> AZERO"
+| "ASEQ bs1 (AONE bs2) r \<leadsto> fuse (bs1@bs2) r"
+| "r1 \<leadsto> r2 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto> ASEQ bs r2 r3"
+| "r3 \<leadsto> r4 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto> ASEQ bs r1 r4"
+| "r \<leadsto> r' \<Longrightarrow> (AALTs bs (rs1 @ [r] @ rs2)) \<leadsto> (AALTs bs (rs1 @ [r'] @ rs2))"
+(*context rule for eliminating 0, alts--corresponds to the recursive call flts r::rs = r::(flts rs)*)
+| "AALTs bs (rsa@ [AZERO] @ rsb) \<leadsto> AALTs bs (rsa @ rsb)"
+| "AALTs bs (rsa@ [AALTs bs1 rs1] @ rsb) \<leadsto> AALTs bs (rsa@(map (fuse bs1) rs1)@rsb)"
+| "AALTs bs [] \<leadsto> AZERO"
+| "AALTs bs [r] \<leadsto> fuse bs r"
+| "erase a1 = erase a2 \<Longrightarrow> AALTs bs (rsa@[a1]@rsb@[a2]@rsc) \<leadsto> AALTs bs (rsa@[a1]@rsb@rsc)"
+
+
+inductive
+ rrewrites:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto>* _" [100, 100] 100)
+where
+ rs1[intro, simp]:"r \<leadsto>* r"
+| rs2[intro]: "\<lbrakk>r1 \<leadsto>* r2; r2 \<leadsto> r3\<rbrakk> \<Longrightarrow> r1 \<leadsto>* r3"
+
+
+inductive
+ srewrites:: "arexp list \<Rightarrow> arexp list \<Rightarrow> bool" (" _ s\<leadsto>* _" [100, 100] 100)
+where
+ ss1: "[] s\<leadsto>* []"
+| ss2: "\<lbrakk>r \<leadsto>* r'; rs s\<leadsto>* rs'\<rbrakk> \<Longrightarrow> (r#rs) s\<leadsto>* (r'#rs')"
+
+
+(* rewrites for lists *)
+inductive
+ frewrites:: "arexp list \<Rightarrow> arexp list \<Rightarrow> bool" (" _ f\<leadsto>* _" [100, 100] 100)
+where
+ fs1: "[] f\<leadsto>* []"
+| fs2: "\<lbrakk>rs f\<leadsto>* rs'\<rbrakk> \<Longrightarrow> (AZERO#rs) f\<leadsto>* rs'"
+| fs3: "\<lbrakk>rs f\<leadsto>* rs'\<rbrakk> \<Longrightarrow> ((AALTs bs rs1) # rs) f\<leadsto>* ((map (fuse bs) rs1) @ rs')"
+| fs4: "\<lbrakk>rs f\<leadsto>* rs'; nonalt r; nonazero r\<rbrakk> \<Longrightarrow> (r#rs) f\<leadsto>* (r#rs')"
+
+
+lemma r_in_rstar : "r1 \<leadsto> r2 \<Longrightarrow> r1 \<leadsto>* r2"
+ using rrewrites.intros(1) rrewrites.intros(2) by blast
+
+lemma real_trans[trans]:
+ assumes a1: "r1 \<leadsto>* r2" and a2: "r2 \<leadsto>* r3"
+ shows "r1 \<leadsto>* r3"
+ using a2 a1
+ apply(induct r2 r3 arbitrary: r1 rule: rrewrites.induct)
+ apply(auto)
+ done
+
+
+lemma many_steps_later: "\<lbrakk>r1 \<leadsto> r2; r2 \<leadsto>* r3 \<rbrakk> \<Longrightarrow> r1 \<leadsto>* r3"
+ by (meson r_in_rstar real_trans)
+
+
+lemma contextrewrites1: "r \<leadsto>* r' \<Longrightarrow> (AALTs bs (r#rs)) \<leadsto>* (AALTs bs (r'#rs))"
+ apply(induct r r' rule: rrewrites.induct)
+ apply simp
+ by (metis append_Cons append_Nil rrewrite.intros(6) rs2)
+
+
+lemma contextrewrites2: "r \<leadsto>* r' \<Longrightarrow> (AALTs bs (rs1@[r]@rs)) \<leadsto>* (AALTs bs (rs1@[r']@rs))"
+ apply(induct r r' rule: rrewrites.induct)
+ apply simp
+ using rrewrite.intros(6) by blast
+
+
+
+lemma srewrites_alt: "rs1 s\<leadsto>* rs2 \<Longrightarrow> (AALTs bs (rs@rs1)) \<leadsto>* (AALTs bs (rs@rs2))"
+
+ apply(induct rs1 rs2 arbitrary: bs rs rule: srewrites.induct)
+ apply(rule rs1)
+ apply(drule_tac x = "bs" in meta_spec)
+ apply(drule_tac x = "rsa@[r']" in meta_spec)
+ apply simp
+ apply(rule real_trans)
+ prefer 2
+ apply(assumption)
+ apply(drule contextrewrites2)
+ apply auto
+ done
+
+corollary srewrites_alt1:
+ assumes "rs1 s\<leadsto>* rs2"
+ shows "AALTs bs rs1 \<leadsto>* AALTs bs rs2"
+using assms
+ by (metis append.left_neutral srewrites_alt)
+
+
+lemma star_seq:
+ assumes "r1 \<leadsto>* r2"
+ shows "ASEQ bs r1 r3 \<leadsto>* ASEQ bs r2 r3"
+using assms
+apply(induct r1 r2 arbitrary: r3 rule: rrewrites.induct)
+apply(auto intro: rrewrite.intros)
+done
+
+lemma star_seq2:
+ assumes "r3 \<leadsto>* r4"
+ shows "ASEQ bs r1 r3 \<leadsto>* ASEQ bs r1 r4"
+using assms
+apply(induct r3 r4 arbitrary: r1 rule: rrewrites.induct)
+apply(auto intro: rrewrite.intros)
+done
+
+lemma continuous_rewrite:
+ assumes "r1 \<leadsto>* AZERO"
+ shows "ASEQ bs1 r1 r2 \<leadsto>* AZERO"
+using assms
+ apply(induction ra\<equiv>"r1" rb\<equiv>"AZERO" arbitrary: bs1 r1 r2 rule: rrewrites.induct)
+ apply(auto intro: rrewrite.intros r_in_rstar star_seq)
+ by (meson rrewrite.intros(1) rs2 star_seq)
+
+
+
+lemma bsimp_aalts_simpcases:
+ shows "AONE bs \<leadsto>* bsimp (AONE bs)"
+ and "AZERO \<leadsto>* bsimp AZERO"
+ and "ACHAR bs c \<leadsto>* bsimp (ACHAR bs c)"
+ by (simp_all)
+
+
+lemma trivialbsimp_srewrites:
+ "\<lbrakk>\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* f x \<rbrakk> \<Longrightarrow> rs s\<leadsto>* (map f rs)"
+
+ apply(induction rs)
+ apply simp
+ apply(rule ss1)
+ by (metis insert_iff list.simps(15) list.simps(9) srewrites.simps)
+
+
+lemma bsimp_AALTs_rewrites:
+ "AALTs bs1 rs \<leadsto>* bsimp_AALTs bs1 rs"
+ apply(induction rs)
+ apply simp
+ apply(rule r_in_rstar)
+ using rrewrite.intros(9) apply blast
+ by (metis bsimp_AALTs.elims list.discI rrewrite.intros(10) rrewrites.simps)
+
+
+
+lemma flts_prepend: "\<lbrakk>nonalt a; nonazero a\<rbrakk> \<Longrightarrow> flts (a#rs) = a # (flts rs)"
+ by (metis append_Cons append_Nil flts_single1 flts_append)
+
+lemma fltsfrewrites: "rs f\<leadsto>* (flts rs)"
+ apply(induction rs)
+ apply simp
+ apply(rule fs1)
+
+ apply(case_tac "a = AZERO")
+
+
+ using fs2 apply auto[1]
+ apply(case_tac "\<exists>bs rs. a = AALTs bs rs")
+ apply(erule exE)+
+
+ apply (simp add: fs3)
+ apply(subst flts_prepend)
+ apply(rule nonalt.elims(2))
+ prefer 2
+ thm nonalt.elims
+
+ apply blast
+
+ using bbbbs1 apply blast
+ apply(simp)+
+
+ apply (meson nonazero.elims(3))
+
+ by (meson fs4 nonalt.elims(3) nonazero.elims(3))
+
+
+lemma rrewrite0away: "AALTs bs (AZERO # rsb) \<leadsto> AALTs bs rsb"
+ by (metis append_Cons append_Nil rrewrite.intros(7))
+
+
+lemma frewritesaalts:"rs f\<leadsto>* rs' \<Longrightarrow> (AALTs bs (rs1@rs)) \<leadsto>* (AALTs bs (rs1@rs'))"
+ apply(induct rs rs' arbitrary: bs rs1 rule:frewrites.induct)
+ apply(rule rs1)
+ apply(drule_tac x = "bs" in meta_spec)
+ apply(drule_tac x = "rs1 @ [AZERO]" in meta_spec)
+ apply(rule real_trans)
+ apply simp
+ using rrewrite.intros(7) apply auto[1]
+ apply(drule_tac x = "bsa" in meta_spec)
+ apply(drule_tac x = "rs1a @ [AALTs bs rs1]" in meta_spec)
+ apply(rule real_trans)
+ apply simp
+ using r_in_rstar rrewrite.intros(8) apply auto[1]
+ apply(drule_tac x = "bs" in meta_spec)
+ apply(drule_tac x = "rs1@[r]" in meta_spec)
+ apply(rule real_trans)
+ apply simp
+ apply auto
+ done
+
+lemma flts_rewrites: " AALTs bs1 rs \<leadsto>* AALTs bs1 (flts rs)"
+ apply(induction rs)
+ apply simp
+ apply(case_tac "a = AZERO")
+ apply (metis flts.simps(2) many_steps_later rrewrite0away)
+
+ apply(case_tac "\<exists>bs2 rs2. a = AALTs bs2 rs2")
+ apply(erule exE)+
+ apply(simp)
+ prefer 2
+
+ apply(subst flts_prepend)
+
+ apply (meson nonalt.elims(3))
+
+ apply (meson nonazero.elims(3))
+ apply(subgoal_tac "(a#rs) f\<leadsto>* (a#flts rs)")
+ apply (metis append_Nil frewritesaalts)
+ apply (meson fltsfrewrites fs4 nonalt.elims(3) nonazero.elims(3))
+ by (metis append_Cons append_Nil fltsfrewrites frewritesaalts flts_append k0a)
+
+(* TEST *)
+lemma r:
+ assumes "AALTs bs rs1 \<leadsto> AALTs bs rs2"
+ shows "AALTs bs (x # rs1) \<leadsto>* AALTs bs (x # rs2)"
+ using assms
+ apply(erule_tac rrewrite.cases)
+ apply(auto)
+ apply (metis append_Cons append_Nil rrewrite.intros(6) r_in_rstar)
+ apply (metis append_Cons append_self_conv2 rrewrite.intros(7) r_in_rstar)
+ apply (metis Cons_eq_appendI append_eq_append_conv2 rrewrite.intros(8) self_append_conv r_in_rstar)
+ apply(case_tac rs2)
+ apply(auto)
+ apply(case_tac r)
+ apply(auto)
+ apply (metis append_Nil2 append_butlast_last_id butlast.simps(2) last.simps list.distinct(1) list.map_disc_iff r_in_rstar rrewrite.intros(8))
+ apply(case_tac r)
+ apply(auto)
+ defer
+ apply(rule r_in_rstar)
+ apply (metis append_Cons append_Nil rrewrite.intros(11))
+ apply(rule real_trans)
+ apply(rule r_in_rstar)
+ using rrewrite.intros(8)[where ?rsb = "[]", of "bs" "[x]" "[]" , simplified]
+ apply(rule_tac rrewrite.intros(8)[where ?rsb = "[]", of "bs" "[x]" "[]" , simplified])
+ apply(simp add: map_fuse_Nil fuse_Nil)
+ done
+
+lemma alts_simpalts:
+ "(\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* bsimp x) \<Longrightarrow>
+ AALTs bs1 rs \<leadsto>* AALTs bs1 (map bsimp rs)"
+ apply(induct rs)
+ apply(auto)[1]
+ using trivialbsimp_srewrites apply auto[1]
+ by (simp add: srewrites_alt1 ss2)
+
+lemma threelistsappend: "rsa@a#rsb = (rsa@[a])@rsb"
+ apply auto
+ done
+
+
+lemma somewhereInside: "r \<in> set rs \<Longrightarrow> \<exists>rs1 rs2. rs = rs1@[r]@rs2"
+ using split_list by fastforce
+
+lemma somewhereMapInside: "f r \<in> f ` set rs \<Longrightarrow> \<exists>rs1 rs2 a. rs = rs1@[a]@rs2 \<and> f a = f r"
+ apply auto
+ by (metis split_list)
+
+lemma alts_dBrewrites_withFront:
+ "AALTs bs (rsa @ rs) \<leadsto>* AALTs bs (rsa @ distinctBy rs erase (erase ` set rsa))"
+ apply(induction rs arbitrary: rsa)
+ apply simp
+ apply(drule_tac x = "rsa@[a]" in meta_spec)
+ apply(subst threelistsappend)
+ apply(rule real_trans)
+ apply simp
+ apply(case_tac "a \<in> set rsa")
+ apply simp
+ apply(drule somewhereInside)
+ apply(erule exE)+
+ apply simp
+ apply(subgoal_tac " AALTs bs
+ (rs1 @
+ a #
+ rs2 @
+ a #
+ distinctBy rs erase
+ (insert (erase a)
+ (erase `
+ (set rs1 \<union> set rs2)))) \<leadsto> AALTs bs (rs1@ a # rs2 @ distinctBy rs erase
+ (insert (erase a)
+ (erase `
+ (set rs1 \<union> set rs2)))) ")
+ prefer 2
+ using rrewrite.intros(11) apply force
+ using r_in_rstar apply force
+ apply(subgoal_tac "erase ` set (rsa @ [a]) = insert (erase a) (erase ` set rsa)")
+ prefer 2
+
+ apply auto[1]
+ apply(case_tac "erase a \<in> erase `set rsa")
+
+ apply simp
+ apply(subgoal_tac "AALTs bs (rsa @ a # distinctBy rs erase (insert (erase a) (erase ` set rsa))) \<leadsto>
+ AALTs bs (rsa @ distinctBy rs erase (insert (erase a) (erase ` set rsa)))")
+ apply force
+ apply (smt (verit, ccfv_threshold) append_Cons append_assoc append_self_conv2 r_in_rstar rrewrite.intros(11) same_append_eq somewhereMapInside)
+ by force
+
+
+
+lemma alts_dBrewrites: "AALTs bs rs \<leadsto>* AALTs bs (distinctBy rs erase {})"
+ apply(induction rs)
+ apply simp
+ apply simp
+ using alts_dBrewrites_withFront
+ by (metis append_Nil dB_single_step empty_set image_empty)
+
+lemma bsimp_rewrite:
+ shows "r \<leadsto>* bsimp r"
+proof (induction r rule: bsimp.induct)
+ case (1 bs1 r1 r2)
+ then show "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)"
+ apply(simp)
+ apply(case_tac "bsimp r1 = AZERO")
+ apply simp
+ using continuous_rewrite apply blast
+ apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
+ apply(erule exE)
+ apply simp
+ apply(subst bsimp_ASEQ2)
+ apply (meson real_trans rrewrite.intros(3) rrewrites.intros(2) star_seq star_seq2)
+ apply (smt (verit, best) bsimp_ASEQ0 bsimp_ASEQ1 real_trans rrewrite.intros(2) rs2 star_seq star_seq2)
+ done
+next
+ case (2 bs1 rs)
+ then show "AALTs bs1 rs \<leadsto>* bsimp (AALTs bs1 rs)"
+ by (metis alts_dBrewrites alts_simpalts bsimp.simps(2) bsimp_AALTs_rewrites flts_rewrites real_trans)
+next
+ case "3_1"
+ then show "AZERO \<leadsto>* bsimp AZERO"
+ by simp
+next
+ case ("3_2" v)
+ then show "AONE v \<leadsto>* bsimp (AONE v)"
+ by simp
+next
+ case ("3_3" v va)
+ then show "ACHAR v va \<leadsto>* bsimp (ACHAR v va)"
+ by simp
+next
+ case ("3_4" v va)
+ then show "ASTAR v va \<leadsto>* bsimp (ASTAR v va)"
+ by simp
+qed
+
+lemma rewrite_non_nullable_strong:
+ assumes "r1 \<leadsto> r2"
+ shows "bnullable r1 = bnullable r2"
+using assms
+apply(induction r1 r2 rule: rrewrite.induct)
+apply(auto)
+apply(metis bnullable_correctness erase_fuse)+
+apply(metis UnCI bnullable_correctness erase_fuse imageI)
+apply(metis bnullable_correctness erase_fuse)+
+done
+
+lemma rewrite_nullable:
+ assumes "r1 \<leadsto> r2" "bnullable r1"
+ shows "bnullable r2"
+using assms rewrite_non_nullable_strong
+by auto
+
+lemma rewritesnullable:
+ assumes "r1 \<leadsto>* r2" "bnullable r1"
+ shows "bnullable r2"
+using assms
+ apply(induction r1 r2 rule: rrewrites.induct)
+ apply simp
+ using rewrite_non_nullable_strong by blast
+
+
+lemma bnullable_segment:
+ "bnullable (AALTs bs (rs1@[r]@rs2)) \<Longrightarrow> bnullable (AALTs bs rs1) \<or> bnullable (AALTs bs rs2) \<or> bnullable r"
+ by auto
+
+lemma bnullablewhichbmkeps: "\<lbrakk>bnullable (AALTs bs (rs1@[r]@rs2)); \<not> bnullable (AALTs bs rs1); bnullable r \<rbrakk>
+ \<Longrightarrow> bmkeps (AALTs bs (rs1@[r]@rs2)) = bs @ (bmkeps r)"
+
+ using qq2 bnullable_Hdbmkeps_Hd by force
+
+lemma spillbmkepslistr: "bnullable (AALTs bs1 rs1)
+ \<Longrightarrow> bmkeps (AALTs bs (AALTs bs1 rs1 # rsb)) = bmkeps (AALTs bs ( map (fuse bs1) rs1 @ rsb))"
+ apply(subst bnullable_Hdbmkeps_Hd)
+
+ apply simp
+ by (metis bmkeps.simps(3) k0a list.set_intros(1) qq1 qq4 qs3)
+
+lemma third_segment_bnullable:
+ "\<lbrakk>bnullable (AALTs bs (rs1@rs2@rs3)); \<not>bnullable (AALTs bs rs1); \<not>bnullable (AALTs bs rs2)\<rbrakk> \<Longrightarrow>
+ bnullable (AALTs bs rs3)"
+ apply(auto)
+ done
+
+lemma third_segment_bmkeps:
+ "\<lbrakk>bnullable (AALTs bs (rs1@rs2@rs3)); \<not>bnullable (AALTs bs rs1); \<not>bnullable (AALTs bs rs2)\<rbrakk> \<Longrightarrow>
+ bmkeps (AALTs bs (rs1@rs2@rs3) ) = bmkeps (AALTs bs rs3)"
+ by (metis bnullable.simps(1) bnullable.simps(4) bsimp_AALTs.simps(1) bsimp_AALTs_rewrites qq2 rewritesnullable self_append_conv third_segment_bnullable)
+
+lemma rewrite_bmkepsalt:
+ "\<lbrakk>bnullable (AALTs bs (rsa @ AALTs bs1 rs1 # rsb)); bnullable (AALTs bs (rsa @ map (fuse bs1) rs1 @ rsb))\<rbrakk>
+ \<Longrightarrow> bmkeps (AALTs bs (rsa @ AALTs bs1 rs1 # rsb)) = bmkeps (AALTs bs (rsa @ map (fuse bs1) rs1 @ rsb))"
+ apply(rule qq3)
+ apply(simp)
+ apply(simp)
+ apply(case_tac "bnullable (AALTs bs1 rs1)")
+ using spillbmkepslistr apply blast
+ apply(subst qq2)
+ apply(auto simp add: bnullable_fuse r1)
+ done
+
+lemma rewrite_bmkeps_aux:
+ assumes "r1 \<leadsto> r2" "bnullable r1" "bnullable r2"
+ shows "bmkeps r1 = bmkeps r2"
+ using assms
+proof (induction r1 r2 rule: rrewrite.induct)
+ case (1 bs r2)
+ then show ?case by simp
+next
+ case (2 bs r1)
+ then show ?case by simp
+next
+ case (3 bs bs1 r)
+ then show ?case by (simp add: b2)
+next
+ case (4 r1 r2 bs r3)
+ then show ?case by simp
+next
+ case (5 r3 r4 bs r1)
+ then show ?case by simp
+next
+ case (6 r r' bs rs1 rs2)
+ then show ?case
+ by (metis append_Cons append_Nil bnullable.simps(4) bnullable_segment bnullablewhichbmkeps qq3 r1 rewrite_non_nullable_strong)
+next
+ case (7 bs rsa rsb)
+ then show ?case
+ by (metis bnullable.simps(1) bnullable.simps(4) bnullable_segment qq1 qq2 rewrite_nullable rrewrite.intros(9) rrewrite0away third_segment_bmkeps)
+next
+ case (8 bs rsa bs1 rs1 rsb)
+ then show ?case
+ by (simp add: rewrite_bmkepsalt)
+next
+ case (9 bs)
+ then show ?case
+ by fastforce
+next
+ case (10 bs r)
+ then show ?case
+ by (simp add: b2)
+next
+ case (11 a1 a2 bs rsa rsb rsc)
+ then show ?case
+ by (smt (verit, ccfv_threshold) append_Cons append_eq_appendI append_self_conv2 bnullable_correctness list.set_intros(1) qq3 r1)
+qed
+
+
+lemma rewrite_bmkeps:
+ assumes "r1 \<leadsto> r2" "bnullable r1"
+ shows "bmkeps r1 = bmkeps r2"
+ using assms(1) assms(2) rewrite_bmkeps_aux rewrite_nullable by blast
+
+
+lemma rewrites_bmkeps:
+ assumes "r1 \<leadsto>* r2" "bnullable r1"
+ shows "bmkeps r1 = bmkeps r2"
+ using assms
+proof(induction r1 r2 rule: rrewrites.induct)
+ case (rs1 r)
+ then show "bmkeps r = bmkeps r" by simp
+next
+ case (rs2 r1 r2 r3)
+ then have IH: "bmkeps r1 = bmkeps r2" by simp
+ have a1: "bnullable r1" by fact
+ have a2: "r1 \<leadsto>* r2" by fact
+ have a3: "r2 \<leadsto> r3" by fact
+ have a4: "bnullable r2" using a1 a2 by (simp add: rewritesnullable)
+ then have "bmkeps r2 = bmkeps r3" using rewrite_bmkeps a3 a4 by simp
+ then show "bmkeps r1 = bmkeps r3" using IH by simp
+qed
+
+lemma alts_rewrite_front: "r \<leadsto> r' \<Longrightarrow> AALTs bs (r # rs) \<leadsto> AALTs bs (r' # rs)"
+ by (metis append_Cons append_Nil rrewrite.intros(6))
+
+lemma to_zero_in_alt: " AALT bs (ASEQ [] AZERO r) r2 \<leadsto> AALT bs AZERO r2"
+ by (simp add: alts_rewrite_front rrewrite.intros(1))
+
+lemma rewrite_fuse:
+ assumes "r2 \<leadsto> r3"
+ shows "fuse bs r2 \<leadsto>* fuse bs r3"
+ using assms
+proof(induction r2 r3 arbitrary: bs rule: rrewrite.induct)
+ case (1 bs r2)
+ then show ?case
+ by (simp add: continuous_rewrite)
+next
+ case (2 bs r1)
+ then show ?case
+ using rrewrite.intros(2) by force
+next
+ case (3 bs bs1 r)
+ then show ?case
+ by (metis fuse.simps(5) fuse_append r_in_rstar rrewrite.intros(3))
+next
+ case (4 r1 r2 bs r3)
+ then show ?case
+ by (simp add: r_in_rstar star_seq)
+next
+ case (5 r3 r4 bs r1)
+ then show ?case
+ using fuse.simps(5) r_in_rstar star_seq2 by auto
+next
+ case (6 r r' bs rs1 rs2)
+ then show ?case
+ using contextrewrites2 r_in_rstar by force
+next
+ case (7 bs rsa rsb)
+ then show ?case
+ using rrewrite.intros(7) by force
+next
+ case (8 bs rsa bs1 rs1 rsb)
+ then show ?case
+ using rrewrite.intros(8) by force
+next
+ case (9 bs)
+ then show ?case
+ by (simp add: r_in_rstar rrewrite.intros(9))
+next
+ case (10 bs r)
+ then show ?case
+ by (metis fuse.simps(4) fuse_append r_in_rstar rrewrite.intros(10))
+next
+ case (11 a1 a2 bs rsa rsb rsc)
+ then show ?case
+ using fuse.simps(4) r_in_rstar rrewrite.intros(11) by auto
+qed
+
+lemma rewrites_fuse:
+ assumes "r1 \<leadsto>* r2"
+ shows "fuse bs r1 \<leadsto>* fuse bs r2"
+using assms
+apply(induction r1 r2 arbitrary: bs rule: rrewrites.induct)
+apply(auto intro: rewrite_fuse real_trans)
+done
+
+lemma bder_fuse_list:
+ shows "map (bder c \<circ> fuse bs1) rs1 = map (fuse bs1 \<circ> bder c) rs1"
+apply(induction rs1)
+apply(simp_all add: bder_fuse)
+done
+
+
+lemma rewrite_der_altmiddle:
+ "bder c (AALTs bs (rsa @ AALTs bs1 rs1 # rsb)) \<leadsto>* bder c (AALTs bs (rsa @ map (fuse bs1) rs1 @ rsb))"
+ apply simp
+ apply(simp add: bder_fuse_list del: append.simps)
+ by (metis append.assoc map_map r_in_rstar rrewrite.intros(8) threelistsappend)
+
+lemma lock_step_der_removal:
+ shows " erase a1 = erase a2 \<Longrightarrow>
+ bder c (AALTs bs (rsa @ [a1] @ rsb @ [a2] @ rsc)) \<leadsto>*
+ bder c (AALTs bs (rsa @ [a1] @ rsb @ rsc))"
+ apply(simp)
+
+ using rrewrite.intros(11) by auto
+
+lemma rewrite_after_der:
+ assumes "r1 \<leadsto> r2"
+ shows "(bder c r1) \<leadsto>* (bder c r2)"
+ using assms
+proof(induction r1 r2 rule: rrewrite.induct)
+ case (1 bs r2)
+ then show "bder c (ASEQ bs AZERO r2) \<leadsto>* bder c AZERO"
+ by (simp add: continuous_rewrite)
+next
+ case (2 bs r1)
+ then show "bder c (ASEQ bs r1 AZERO) \<leadsto>* bder c AZERO"
+ apply(simp)
+ by (meson contextrewrites1 r_in_rstar real_trans rrewrite.intros(9) rrewrite.intros(2) rrewrite0away)
+next
+ case (3 bs bs1 r)
+ then show "bder c (ASEQ bs (AONE bs1) r) \<leadsto>* bder c (fuse (bs @ bs1) r)"
+ apply(simp)
+ by (metis bder_fuse fuse_append rrewrite.intros(10) rrewrite0away rrewrites.simps to_zero_in_alt)
+next
+ case (4 r1 r2 bs r3)
+ have as: "r1 \<leadsto> r2" by fact
+ have IH: "bder c r1 \<leadsto>* bder c r2" by fact
+ from as IH show "bder c (ASEQ bs r1 r3) \<leadsto>* bder c (ASEQ bs r2 r3)"
+ by (simp add: contextrewrites1 rewrite_bmkeps rewrite_non_nullable_strong star_seq)
+next
+ case (5 r3 r4 bs r1)
+ have as: "r3 \<leadsto> r4" by fact
+ have IH: "bder c r3 \<leadsto>* bder c r4" by fact
+ from as IH show "bder c (ASEQ bs r1 r3) \<leadsto>* bder c (ASEQ bs r1 r4)"
+ using bder.simps(5) r_in_rstar rewrites_fuse srewrites_alt1 ss1 ss2 star_seq2 by presburger
+next
+ case (6 r r' bs rs1 rs2)
+ have as: "r \<leadsto> r'" by fact
+ have IH: "bder c r \<leadsto>* bder c r'" by fact
+ from as IH show "bder c (AALTs bs (rs1 @ [r] @ rs2)) \<leadsto>* bder c (AALTs bs (rs1 @ [r'] @ rs2))"
+ apply(simp)
+ using contextrewrites2 by force
+next
+ case (7 bs rsa rsb)
+ then show "bder c (AALTs bs (rsa @ [AZERO] @ rsb)) \<leadsto>* bder c (AALTs bs (rsa @ rsb))"
+ apply(simp)
+ using rrewrite.intros(7) by auto
+next
+ case (8 bs rsa bs1 rs1 rsb)
+ then show
+ "bder c (AALTs bs (rsa @ [AALTs bs1 rs1] @ rsb)) \<leadsto>* bder c (AALTs bs (rsa @ map (fuse bs1) rs1 @ rsb))"
+ using rewrite_der_altmiddle by auto
+next
+ case (9 bs)
+ then show "bder c (AALTs bs []) \<leadsto>* bder c AZERO"
+ by (simp add: r_in_rstar rrewrite.intros(9))
+next
+ case (10 bs r)
+ then show "bder c (AALTs bs [r]) \<leadsto>* bder c (fuse bs r)"
+ by (simp add: bder_fuse r_in_rstar rrewrite.intros(10))
+next
+ case (11 a1 a2 bs rsa rsb rsc)
+ have as: "erase a1 = erase a2" by fact
+ then show "bder c (AALTs bs (rsa @ [a1] @ rsb @ [a2] @ rsc)) \<leadsto>* bder c (AALTs bs (rsa @ [a1] @ rsb @ rsc))"
+ using lock_step_der_removal by force
+qed
+
+
+lemma rewrites_after_der:
+ assumes "r1 \<leadsto>* r2"
+ shows "bder c r1 \<leadsto>* bder c r2"
+using assms
+apply(induction r1 r2 rule: rrewrites.induct)
+apply(simp_all add: rewrite_after_der real_trans)
+done
+
+
+lemma central:
+ shows "bders r s \<leadsto>* bders_simp r s"
+proof(induct s arbitrary: r rule: rev_induct)
+ case Nil
+ then show "bders r [] \<leadsto>* bders_simp r []" by simp
+next
+ case (snoc x xs)
+ have IH: "\<And>r. bders r xs \<leadsto>* bders_simp r xs" by fact
+ have "bders r (xs @ [x]) = bders (bders r xs) [x]" by (simp add: bders_append)
+ also have "... \<leadsto>* bders (bders_simp r xs) [x]" using IH
+ by (simp add: rewrites_after_der)
+ also have "... \<leadsto>* bders_simp (bders_simp r xs) [x]" using IH
+ by (simp add: bsimp_rewrite)
+ finally show "bders r (xs @ [x]) \<leadsto>* bders_simp r (xs @ [x])"
+ by (simp add: bders_simp_append)
+qed
+
+
+
+
+
+lemma quasi_main:
+ assumes "bnullable (bders r s)"
+ shows "bmkeps (bders r s) = bmkeps (bders_simp r s)"
+proof -
+ have "bders r s \<leadsto>* bders_simp r s" by (rule central)
+ then
+ show "bmkeps (bders r s) = bmkeps (bders_simp r s)" using assms
+ by (rule rewrites_bmkeps)
+qed
+
+
+
+
+theorem main_main:
+ shows "blexer r s = blexer_simp r s"
+ unfolding blexer_def blexer_simp_def
+ using b4 quasi_main by simp
+
+
+theorem blexersimp_correctness:
+ shows "lexer r s = blexer_simp r s"
+ using blexer_correctness main_main by simp
+
+
+
+export_code blexer_simp blexer lexer bders bders_simp in Scala module_name VerifiedLexers
+
+
+unused_thms
+
+
+inductive aggressive:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto>? _" [99, 99] 99)
+ where
+ "ASEQ bs (AALTs bs1 rs) r \<leadsto>? AALTs (bs@bs1) (map (\<lambda>r'. ASEQ [] r' r) rs) "
+
+
+
+end