--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/thys2/SizeBound4CT.thy Wed Feb 02 15:03:20 2022 +0000
@@ -0,0 +1,1182 @@
+
+theory SizeBound4CT
+ imports
+ "Lexer"
+ "PDerivs"
+
+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 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
+
+
+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"
+
+abbreviation
+ bnullables :: "arexp list \<Rightarrow> bool"
+where
+ "bnullables rs \<equiv> (\<exists>r \<in> set rs. bnullable r)"
+
+fun
+ bmkeps :: "arexp \<Rightarrow> bit list" and
+ bmkepss :: "arexp list \<Rightarrow> bit list"
+where
+ "bmkeps(AONE bs) = bs"
+| "bmkeps(ASEQ bs r1 r2) = bs @ (bmkeps r1) @ (bmkeps r2)"
+| "bmkeps(AALTs bs rs) = bs @ (bmkepss rs)"
+| "bmkeps(ASTAR bs r) = bs @ [S]"
+| "bmkepss [] = []"
+| "bmkepss (r # rs) = (if bnullable(r) then (bmkeps r) else (bmkepss rs))"
+
+lemma bmkepss1:
+ assumes "\<not> bnullables rs1"
+ shows "bmkepss (rs1 @ rs2) = bmkepss rs2"
+ using assms
+ by (induct rs1) (auto)
+
+lemma bmkepss2:
+ assumes "bnullables rs1"
+ shows "bmkepss (rs1 @ rs2) = bmkepss rs1"
+ using assms
+ by (induct rs1) (auto)
+
+
+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 c (s1 @ s2) = bders (bders c s1) s2"
+ apply(induct s1 arbitrary: c 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]
+ apply(case_tac x2a)
+ apply(simp)
+ using Prf_elims(1) apply blast
+ apply(case_tac x2a)
+ apply(simp)
+ apply(simp)
+ apply(case_tac list)
+ apply(simp)
+ apply(simp)
+ apply (smt (verit, best) Prf_elims(3) append_assoc retrieve.simps(4) retrieve.simps(5))
+ apply(simp)
+ using retrieve_encode_STARS
+ apply(auto elim!: Prf_elims)[1]
+ apply(case_tac vs)
+ apply(simp)
+ apply(simp)
+ 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 retrieve_AALTs_bnullable1:
+ assumes "bnullable r"
+ shows "retrieve (AALTs bs (r # rs)) (mkeps (erase (AALTs bs (r # rs))))
+ = bs @ retrieve r (mkeps (erase r))"
+ using assms
+ apply(case_tac rs)
+ apply(auto simp add: bnullable_correctness)
+ done
+
+lemma retrieve_AALTs_bnullable2:
+ assumes "\<not>bnullable r" "bnullables rs"
+ shows "retrieve (AALTs bs (r # rs)) (mkeps (erase (AALTs bs (r # rs))))
+ = retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs)))"
+ using assms
+ apply(induct rs arbitrary: r bs)
+ apply(auto)
+ using bnullable_correctness apply blast
+ apply(case_tac rs)
+ apply(auto)
+ using bnullable_correctness apply blast
+ apply(case_tac rs)
+ apply(auto)
+ done
+
+lemma bmkeps_retrieve_AALTs:
+ assumes "\<forall>r \<in> set rs. bnullable r \<longrightarrow> bmkeps r = retrieve r (mkeps (erase r))"
+ "bnullables rs"
+ shows "bs @ bmkepss rs = retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs)))"
+ using assms
+ apply(induct rs arbitrary: bs)
+ apply(auto)
+ using retrieve_AALTs_bnullable1 apply presburger
+ apply (metis retrieve_AALTs_bnullable2)
+ apply (simp add: retrieve_AALTs_bnullable1)
+ by (metis retrieve_AALTs_bnullable2)
+
+
+lemma bmkeps_retrieve:
+ assumes "bnullable r"
+ shows "bmkeps r = retrieve r (mkeps (erase r))"
+ using assms
+ apply(induct r)
+ apply(auto)
+ using bmkeps_retrieve_AALTs by auto
+
+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)
+ using Prf_elims(1) apply auto[1]
+ using Prf_elims(1) apply auto[1]
+ apply(auto)[1]
+ apply (metis Prf_elims(4) injval.simps(1) retrieve.simps(1) retrieve.simps(2))
+ using Prf_elims(1) apply blast
+ (* AALTs case *)
+ 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)
+ using Prf_elims(3) apply fastforce
+ (* ASEQ case *)
+ apply(simp)
+ apply(case_tac "nullable (erase r1)")
+ apply(simp)
+ apply(erule Prf_elims)
+ using Prf_elims(2) bnullable_correctness apply force
+ apply (simp add: bmkeps_retrieve bnullable_correctness retrieve_fuse2)
+ apply (simp add: bmkeps_retrieve bnullable_correctness retrieve_fuse2)
+ using Prf_elims(2) apply force
+ (* ASTAR case *)
+ apply(rename_tac bs r v)
+ apply(simp)
+ apply(erule Prf_elims)
+ apply(clarify)
+ apply(erule Prf_elims)
+ apply(clarify)
+ by (simp add: retrieve_fuse2)
+
+
+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 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
+ using bnullable_correctness by force
+ 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
+ by (auto simp add: bnullable_correctness[symmetric])
+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)))"
+
+
+
+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"
+
+lemma bsimp_ASEQ0[simp]:
+ shows "bsimp_ASEQ bs r1 AZERO = AZERO"
+ by (case_tac r1)(simp_all)
+
+lemma bsimp_ASEQ1:
+ assumes "r1 \<noteq> AZERO" "r2 \<noteq> AZERO" "\<nexists>bs. r1 = 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[simp]:
+ shows "bsimp_ASEQ bs1 (AONE bs2) r2 = fuse (bs1 @ bs2) r2"
+ by (case_tac r2) (simp_all)
+
+
+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 :: "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"
+
+
+
+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 bmkeps_fuse:
+ assumes "bnullable r"
+ shows "bmkeps (fuse bs r) = bs @ bmkeps r"
+ using assms
+ by (metis bmkeps_retrieve bnullable_correctness erase_fuse mkeps_nullable retrieve_fuse2)
+
+lemma bmkepss_fuse:
+ assumes "bnullables rs"
+ shows "bmkepss (map (fuse bs) rs) = bs @ bmkepss rs"
+ using assms
+ apply(induct rs arbitrary: bs)
+ apply(auto simp add: bmkeps_fuse bnullable_fuse)
+ done
+
+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)
+and
+ srewrite:: "arexp list \<Rightarrow> arexp list \<Rightarrow> bool" (" _ s\<leadsto> _" [100, 100] 100)
+where
+ bs1: "ASEQ bs AZERO r2 \<leadsto> AZERO"
+| bs2: "ASEQ bs r1 AZERO \<leadsto> AZERO"
+| bs3: "ASEQ bs1 (AONE bs2) r \<leadsto> fuse (bs1@bs2) r"
+| bs4: "r1 \<leadsto> r2 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto> ASEQ bs r2 r3"
+| bs5: "r3 \<leadsto> r4 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto> ASEQ bs r1 r4"
+| bs6: "AALTs bs [] \<leadsto> AZERO"
+| bs7: "AALTs bs [r] \<leadsto> fuse bs r"
+| bs8: "rs1 s\<leadsto> rs2 \<Longrightarrow> AALTs bs rs1 \<leadsto> AALTs bs rs2"
+| ss1: "[] s\<leadsto> []"
+| ss2: "rs1 s\<leadsto> rs2 \<Longrightarrow> (r # rs1) s\<leadsto> (r # rs2)"
+| ss3: "r1 \<leadsto> r2 \<Longrightarrow> (r1 # rs) s\<leadsto> (r2 # rs)"
+| ss4: "(AZERO # rs) s\<leadsto> rs"
+| ss5: "(AALTs bs1 rs1 # rsb) s\<leadsto> ((map (fuse bs1) rs1) @ rsb)"
+| ss6: "erase a1 = erase a2 \<Longrightarrow> (rsa@[a1]@rsb@[a2]@rsc) s\<leadsto> (rsa@[a1]@rsb@rsc)"
+| ss7: "erase a01 = erase a02 \<and> (distinctBy as2 erase (set (map erase as1)) = as2p) \<Longrightarrow>
+ (rsa@[ASEQ bs (AALTs bs1 as1) (ASTAR bs01 a01)]@rsb@[ASEQ bs (AALTs bs2 as2) (ASTAR bs02 a02)]@rsc) s\<leadsto>
+ (rsa@[ASEQ bs (AALTs bs1 as1) (ASTAR bs01 a01)]@rsb@[ASEQ bs (AALTs bs2 as2p) (ASTAR bs02 a02)]@rsc)"
+| ss8: "erase a01 = erase a02 \<and> (distinctBy [a2] erase (set (map erase as1)) = []) \<Longrightarrow>
+ (rsa@[ASEQ bs (AALTs bs1 as1) (ASTAR bs01 a01)]@rsb@[ASEQ bs a2 (ASTAR bs02 a02)]@rsc) s\<leadsto>
+ (rsa@[ASEQ bs (AALTs bs1 as1) (ASTAR bs01 a01)]@rsb@rsc)"
+| ss9: "erase a01 = erase a02 \<and> (distinctBy as2 erase {erase a1} = as2p) \<Longrightarrow>
+ (rsa@[ASEQ bs a1 (ASTAR bs01 a01)]@rsb@[ASEQ bs (AALTs bs2 as2) (ASTAR bs02 a02)]@rsc) s\<leadsto>
+ (rsa@[ASEQ bs a1 (ASTAR bs01 a01)]@rsb@[ASEQ bs (AALTs bs2 as2p) (ASTAR bs02 a02)]@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
+ sss1[intro, simp]:"rs s\<leadsto>* rs"
+| sss2[intro]: "\<lbrakk>rs1 s\<leadsto> rs2; rs2 s\<leadsto>* rs3\<rbrakk> \<Longrightarrow> rs1 s\<leadsto>* rs3"
+
+
+
+
+fun fmap :: "arexp \<Rightarrow> rexp set"
+ where
+"fmap AALTs bs rs = flatten (map fmap rs)"
+| "fmap ASEQ bs (AALTs bs1 rs1) r2 = (flatten (map fmap rs1)) ` (ASEQ bs _ r2)"
+| "fmap (ASTAR bs r0) = {ASTAR bs r0}"
+
+(*r1~r2 --\c> r1c~r2 \<longrightarrow> r1\s~r2 + r2\s1 ... \<longrightarrow> (r1\s ~ r2) +r2\s'+r2\s''+........ *)
+(* r* \<longrightarrow>r\c ~ r* ---> r\s1 ~ r* + r\s1' ~ r* \<longrightarrow> s1, s1' \<in> suffix s *)
+(* {r1, r2', r3} {r2,r3,r4} pders\<longrightarrow> {r1-r*, r2-r*, r3-r*, r4-r*} *)
+
+lemma iso_pder:
+"fmap (bders_simp r s) \<subseteq> pderss UNIV r"
+ apply(induction r)
+ prefer 4
+ oops
+
+lemma shape_of_star_after_derssimp:
+ shows "\<forall>bs r s. \<exists>bs1 r1 r2 rs bs2 bsp. (bders_simp (ASTAR bs r) s = (ASEQ bs1 r1 r2)) \<or>
+ (bders_simp (ASTAR bs r) s = (AALTs bs2 rs)) \<or>
+ (bders_simp (ASTAR bs r) s = (ASTAR bsp r))"
+
+ oops
+
+
+
+lemma notManyTerms:
+ shows "card {(bders_simp r (Suffix s))| r s. s \<in> L (erase r) } < (size r)"
+
+ oops
+
+lemma rtozero:
+ shows "\<lbrakk>s \<notin> L (erase r); s \<noteq> Nil\<rbrakk> \<Longrightarrow> bders_simp r s = AZERO"
+ apply(induct r)
+
+ oops
+
+lemma r_in_rstar:
+ shows "r1 \<leadsto> r2 \<Longrightarrow> r1 \<leadsto>* r2"
+ using rrewrites.intros(1) rrewrites.intros(2) by blast
+
+lemma rrewrites_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 srewrites_trans[trans]:
+ assumes a1: "r1 s\<leadsto>* r2" and a2: "r2 s\<leadsto>* r3"
+ shows "r1 s\<leadsto>* r3"
+ using a1 a2
+ apply(induct r1 r2 arbitrary: r3 rule: srewrites.induct)
+ apply(auto)
+ done
+
+
+
+lemma contextrewrites0:
+ "rs1 s\<leadsto>* rs2 \<Longrightarrow> AALTs bs rs1 \<leadsto>* AALTs bs rs2"
+ apply(induct rs1 rs2 rule: srewrites.inducts)
+ apply simp
+ using bs8 r_in_rstar rrewrites_trans by blast
+
+lemma contextrewrites1:
+ "r \<leadsto>* r' \<Longrightarrow> AALTs bs (r # rs) \<leadsto>* AALTs bs (r' # rs)"
+ apply(induct r r' rule: rrewrites.induct)
+ apply simp
+ using bs8 ss3 by blast
+
+lemma srewrite1:
+ shows "rs1 s\<leadsto> rs2 \<Longrightarrow> (rs @ rs1) s\<leadsto> (rs @ rs2)"
+ apply(induct rs)
+ apply(auto)
+ using ss2 by auto
+
+lemma srewrites1:
+ shows "rs1 s\<leadsto>* rs2 \<Longrightarrow> (rs @ rs1) s\<leadsto>* (rs @ rs2)"
+ apply(induct rs1 rs2 rule: srewrites.induct)
+ apply(auto)
+ using srewrite1 by blast
+
+lemma srewrite2:
+ shows "r1 \<leadsto> r2 \<Longrightarrow> True"
+ and "rs1 s\<leadsto> rs2 \<Longrightarrow> (rs1 @ rs) s\<leadsto>* (rs2 @ rs)"
+ apply(induct rule: rrewrite_srewrite.inducts)
+ apply(auto)
+ apply (metis append_Cons append_Nil srewrites1)
+ apply(meson srewrites.simps ss3)
+ apply (meson srewrites.simps ss4)
+ apply (meson srewrites.simps ss5)
+ by (metis append_Cons append_Nil srewrites.simps ss6)
+
+
+lemma srewrites3:
+ shows "rs1 s\<leadsto>* rs2 \<Longrightarrow> (rs1 @ rs) s\<leadsto>* (rs2 @ rs)"
+ apply(induct rs1 rs2 arbitrary: rs rule: srewrites.induct)
+ apply(auto)
+ by (meson srewrite2(2) srewrites_trans)
+
+(*
+lemma srewrites4:
+ assumes "rs3 s\<leadsto>* rs4" "rs1 s\<leadsto>* rs2"
+ shows "(rs1 @ rs3) s\<leadsto>* (rs2 @ rs4)"
+ using assms
+ apply(induct rs3 rs4 arbitrary: rs1 rs2 rule: srewrites.induct)
+ apply (simp add: srewrites3)
+ using srewrite1 by blast
+*)
+
+lemma srewrites6:
+ assumes "r1 \<leadsto>* r2"
+ shows "[r1] s\<leadsto>* [r2]"
+ using assms
+ apply(induct r1 r2 rule: rrewrites.induct)
+ apply(auto)
+ by (meson srewrites.simps srewrites_trans ss3)
+
+lemma srewrites7:
+ assumes "rs3 s\<leadsto>* rs4" "r1 \<leadsto>* r2"
+ shows "(r1 # rs3) s\<leadsto>* (r2 # rs4)"
+ using assms
+ by (smt (verit, del_insts) append.simps srewrites1 srewrites3 srewrites6 srewrites_trans)
+
+lemma ss6_stronger_aux:
+ shows "(rs1 @ rs2) s\<leadsto>* (rs1 @ distinctBy rs2 erase (set (map erase rs1)))"
+ apply(induct rs2 arbitrary: rs1)
+ apply(auto)
+ apply (smt (verit, best) append.assoc append.right_neutral append_Cons append_Nil split_list srewrite2(2) srewrites_trans ss6)
+ apply(drule_tac x="rs1 @ [a]" in meta_spec)
+ apply(simp)
+ done
+
+lemma ss6_stronger:
+ shows "rs1 s\<leadsto>* distinctBy rs1 erase {}"
+ using ss6_stronger_aux[of "[]" _] by auto
+
+lemma rewrite_preserves_fuse:
+ shows "r2 \<leadsto> r3 \<Longrightarrow> fuse bs r2 \<leadsto> fuse bs r3"
+ and "rs2 s\<leadsto> rs3 \<Longrightarrow> map (fuse bs) rs2 s\<leadsto> map (fuse bs) rs3"
+proof(induct rule: rrewrite_srewrite.inducts)
+ case (bs3 bs1 bs2 r)
+ then show "fuse bs (ASEQ bs1 (AONE bs2) r) \<leadsto> fuse bs (fuse (bs1 @ bs2) r)"
+ by (metis fuse.simps(5) fuse_append rrewrite_srewrite.bs3)
+next
+ case (bs7 bs1 r)
+ then show "fuse bs (AALTs bs1 [r]) \<leadsto> fuse bs (fuse bs1 r)"
+ by (metis fuse.simps(4) fuse_append rrewrite_srewrite.bs7)
+next
+ case (ss2 rs1 rs2 r)
+ then show "map (fuse bs) (r # rs1) s\<leadsto> map (fuse bs) (r # rs2)"
+ by (simp add: rrewrite_srewrite.ss2)
+next
+ case (ss3 r1 r2 rs)
+ then show "map (fuse bs) (r1 # rs) s\<leadsto> map (fuse bs) (r2 # rs)"
+ by (simp add: rrewrite_srewrite.ss3)
+next
+ case (ss5 bs1 rs1 rsb)
+ have "map (fuse bs) (AALTs bs1 rs1 # rsb) = AALTs (bs @ bs1) rs1 # (map (fuse bs) rsb)" by simp
+ also have "... s\<leadsto> ((map (fuse (bs @ bs1)) rs1) @ (map (fuse bs) rsb))"
+ by (simp add: rrewrite_srewrite.ss5)
+ finally show "map (fuse bs) (AALTs bs1 rs1 # rsb) s\<leadsto> map (fuse bs) (map (fuse bs1) rs1 @ rsb)"
+ by (simp add: comp_def fuse_append)
+next
+ case (ss6 a1 a2 rsa rsb rsc)
+ then show "map (fuse bs) (rsa @ [a1] @ rsb @ [a2] @ rsc) s\<leadsto> map (fuse bs) (rsa @ [a1] @ rsb @ rsc)"
+ apply(simp)
+ apply(rule rrewrite_srewrite.ss6[simplified])
+ apply(simp add: erase_fuse)
+ done
+qed (auto intro: rrewrite_srewrite.intros)
+
+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_preserves_fuse)
+done
+
+
+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_srewrite.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_srewrite.intros)
+done
+
+lemma continuous_rewrite:
+ assumes "r1 \<leadsto>* AZERO"
+ shows "ASEQ bs1 r1 r2 \<leadsto>* AZERO"
+using assms bs1 star_seq by blast
+
+(*
+lemma continuous_rewrite2:
+ assumes "r1 \<leadsto>* AONE bs"
+ shows "ASEQ bs1 r1 r2 \<leadsto>* (fuse (bs1 @ bs) r2)"
+ using assms by (meson bs3 rrewrites.simps 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 bsimp_AALTs_rewrites:
+ shows "AALTs bs1 rs \<leadsto>* bsimp_AALTs bs1 rs"
+ by (smt (verit) bs6 bs7 bsimp_AALTs.elims rrewrites.simps)
+
+lemma trivialbsimp_srewrites:
+ assumes "\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* f x"
+ shows "rs s\<leadsto>* (map f rs)"
+using assms
+ apply(induction rs)
+ apply(simp_all add: srewrites7)
+ done
+
+lemma fltsfrewrites: "rs s\<leadsto>* flts rs"
+ apply(induction rs rule: flts.induct)
+ apply(auto intro: rrewrite_srewrite.intros)
+ apply (meson srewrites.simps srewrites1 ss5)
+ using rs1 srewrites7 apply presburger
+ using srewrites7 apply force
+ apply (simp add: srewrites7)
+ by (simp add: srewrites7)
+
+lemma bnullable0:
+shows "r1 \<leadsto> r2 \<Longrightarrow> bnullable r1 = bnullable r2"
+ and "rs1 s\<leadsto> rs2 \<Longrightarrow> bnullables rs1 = bnullables rs2"
+ apply(induct rule: rrewrite_srewrite.inducts)
+ apply(auto simp add: bnullable_fuse)
+ apply (meson UnCI bnullable_fuse imageI)
+ by (metis bnullable_correctness)
+
+
+lemma rewrites_bnullable_eq:
+ assumes "r1 \<leadsto>* r2"
+ shows "bnullable r1 = bnullable r2"
+using assms
+ apply(induction r1 r2 rule: rrewrites.induct)
+ apply simp
+ using bnullable0(1) by auto
+
+lemma rewrite_bmkeps_aux:
+ shows "r1 \<leadsto> r2 \<Longrightarrow> bnullable r1 \<Longrightarrow> bmkeps r1 = bmkeps r2"
+ and "rs1 s\<leadsto> rs2 \<Longrightarrow> bnullables rs1 \<Longrightarrow> bmkepss rs1 = bmkepss rs2"
+proof (induct rule: rrewrite_srewrite.inducts)
+ case (bs3 bs1 bs2 r)
+ have IH2: "bnullable (ASEQ bs1 (AONE bs2) r)" by fact
+ then show "bmkeps (ASEQ bs1 (AONE bs2) r) = bmkeps (fuse (bs1 @ bs2) r)"
+ by (simp add: bmkeps_fuse)
+next
+ case (bs7 bs r)
+ have IH2: "bnullable (AALTs bs [r])" by fact
+ then show "bmkeps (AALTs bs [r]) = bmkeps (fuse bs r)"
+ by (simp add: bmkeps_fuse)
+next
+ case (ss3 r1 r2 rs)
+ have IH1: "bnullable r1 \<Longrightarrow> bmkeps r1 = bmkeps r2" by fact
+ have as: "r1 \<leadsto> r2" by fact
+ from IH1 as show "bmkepss (r1 # rs) = bmkepss (r2 # rs)"
+ by (simp add: bnullable0)
+next
+ case (ss5 bs1 rs1 rsb)
+ have "bnullables (AALTs bs1 rs1 # rsb)" by fact
+ then show "bmkepss (AALTs bs1 rs1 # rsb) = bmkepss (map (fuse bs1) rs1 @ rsb)"
+ by (simp add: bmkepss1 bmkepss2 bmkepss_fuse bnullable_fuse)
+next
+ case (ss6 a1 a2 rsa rsb rsc)
+ have as1: "erase a1 = erase a2" by fact
+ have as3: "bnullables (rsa @ [a1] @ rsb @ [a2] @ rsc)" by fact
+ show "bmkepss (rsa @ [a1] @ rsb @ [a2] @ rsc) = bmkepss (rsa @ [a1] @ rsb @ rsc)" using as1 as3
+ by (smt (verit, best) append_Cons bmkeps.simps(3) bmkepss.simps(2) bmkepss1 bmkepss2 bnullable_correctness)
+qed (auto)
+
+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: rewrites_bnullable_eq)
+ then have "bmkeps r2 = bmkeps r3"
+ using a3 bnullable0(1) rewrite_bmkeps_aux(1) by blast
+ then show "bmkeps r1 = bmkeps r3" using IH by simp
+qed
+
+
+lemma rewrites_to_bsimp:
+ shows "r \<leadsto>* bsimp r"
+proof (induction r rule: bsimp.induct)
+ case (1 bs1 r1 r2)
+ have IH1: "r1 \<leadsto>* bsimp r1" by fact
+ have IH2: "r2 \<leadsto>* bsimp r2" by fact
+ { assume as: "bsimp r1 = AZERO \<or> bsimp r2 = AZERO"
+ with IH1 IH2 have "r1 \<leadsto>* AZERO \<or> r2 \<leadsto>* AZERO" by auto
+ then have "ASEQ bs1 r1 r2 \<leadsto>* AZERO"
+ by (metis bs2 continuous_rewrite rrewrites.simps star_seq2)
+ then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" using as by auto
+ }
+ moreover
+ { assume "\<exists>bs. bsimp r1 = AONE bs"
+ then obtain bs where as: "bsimp r1 = AONE bs" by blast
+ with IH1 have "r1 \<leadsto>* AONE bs" by simp
+ then have "ASEQ bs1 r1 r2 \<leadsto>* fuse (bs1 @ bs) r2" using bs3 star_seq by blast
+ with IH2 have "ASEQ bs1 r1 r2 \<leadsto>* fuse (bs1 @ bs) (bsimp r2)"
+ using rewrites_fuse by (meson rrewrites_trans)
+ then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 (AONE bs) r2)" by simp
+ then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by (simp add: as)
+ }
+ moreover
+ { assume as1: "bsimp r1 \<noteq> AZERO" "bsimp r2 \<noteq> AZERO" and as2: "(\<nexists>bs. bsimp r1 = AONE bs)"
+ then have "bsimp_ASEQ bs1 (bsimp r1) (bsimp r2) = ASEQ bs1 (bsimp r1) (bsimp r2)"
+ by (simp add: bsimp_ASEQ1)
+ then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp_ASEQ bs1 (bsimp r1) (bsimp r2)" using as1 as2 IH1 IH2
+ by (metis rrewrites_trans star_seq star_seq2)
+ then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by simp
+ }
+ ultimately show "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by blast
+next
+ case (2 bs1 rs)
+ have IH: "\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* bsimp x" by fact
+ then have "rs s\<leadsto>* (map bsimp rs)" by (simp add: trivialbsimp_srewrites)
+ also have "... s\<leadsto>* flts (map bsimp rs)" by (simp add: fltsfrewrites)
+ also have "... s\<leadsto>* distinctBy (flts (map bsimp rs)) erase {}" by (simp add: ss6_stronger)
+ finally have "AALTs bs1 rs \<leadsto>* AALTs bs1 (distinctBy (flts (map bsimp rs)) erase {})"
+ using contextrewrites0 by blast
+ also have "... \<leadsto>* bsimp_AALTs bs1 (distinctBy (flts (map bsimp rs)) erase {})"
+ by (simp add: bsimp_AALTs_rewrites)
+ finally show "AALTs bs1 rs \<leadsto>* bsimp (AALTs bs1 rs)" by simp
+qed (simp_all)
+
+
+lemma to_zero_in_alt:
+ shows "AALT bs (ASEQ [] AZERO r) r2 \<leadsto> AALT bs AZERO r2"
+ by (simp add: bs1 bs8 ss3)
+
+
+
+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_preserves_bder:
+ shows "r1 \<leadsto> r2 \<Longrightarrow> bder c r1 \<leadsto>* bder c r2"
+ and "rs1 s\<leadsto> rs2 \<Longrightarrow> map (bder c) rs1 s\<leadsto>* map (bder c) rs2"
+proof(induction rule: rrewrite_srewrite.inducts)
+ case (bs1 bs r2)
+ show "bder c (ASEQ bs AZERO r2) \<leadsto>* bder c AZERO"
+ by (simp add: continuous_rewrite)
+next
+ case (bs2 bs r1)
+ show "bder c (ASEQ bs r1 AZERO) \<leadsto>* bder c AZERO"
+ apply(auto)
+ apply (meson bs6 contextrewrites0 rrewrite_srewrite.bs2 rs2 ss3 ss4 sss1 sss2)
+ by (simp add: r_in_rstar rrewrite_srewrite.bs2)
+next
+ case (bs3 bs1 bs2 r)
+ show "bder c (ASEQ bs1 (AONE bs2) r) \<leadsto>* bder c (fuse (bs1 @ bs2) r)"
+ apply(simp)
+ by (metis (no_types, lifting) bder_fuse bs8 bs7 fuse_append rrewrites.simps ss4 to_zero_in_alt)
+next
+ case (bs4 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 (metis bder.simps(5) bnullable0(1) contextrewrites1 rewrite_bmkeps_aux(1) star_seq)
+next
+ case (bs5 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)"
+ apply(simp)
+ apply(auto)
+ using contextrewrites0 r_in_rstar rewrites_fuse srewrites6 srewrites7 star_seq2 apply presburger
+ using star_seq2 by blast
+next
+ case (bs6 bs)
+ show "bder c (AALTs bs []) \<leadsto>* bder c AZERO"
+ using rrewrite_srewrite.bs6 by force
+next
+ case (bs7 bs r)
+ show "bder c (AALTs bs [r]) \<leadsto>* bder c (fuse bs r)"
+ by (simp add: bder_fuse r_in_rstar rrewrite_srewrite.bs7)
+next
+ case (bs8 rs1 rs2 bs)
+ have IH1: "map (bder c) rs1 s\<leadsto>* map (bder c) rs2" by fact
+ then show "bder c (AALTs bs rs1) \<leadsto>* bder c (AALTs bs rs2)"
+ using contextrewrites0 by force
+next
+ case ss1
+ show "map (bder c) [] s\<leadsto>* map (bder c) []" by simp
+next
+ case (ss2 rs1 rs2 r)
+ have IH1: "map (bder c) rs1 s\<leadsto>* map (bder c) rs2" by fact
+ then show "map (bder c) (r # rs1) s\<leadsto>* map (bder c) (r # rs2)"
+ by (simp add: srewrites7)
+next
+ case (ss3 r1 r2 rs)
+ have IH: "bder c r1 \<leadsto>* bder c r2" by fact
+ then show "map (bder c) (r1 # rs) s\<leadsto>* map (bder c) (r2 # rs)"
+ by (simp add: srewrites7)
+next
+ case (ss4 rs)
+ show "map (bder c) (AZERO # rs) s\<leadsto>* map (bder c) rs"
+ using rrewrite_srewrite.ss4 by fastforce
+next
+ case (ss5 bs1 rs1 rsb)
+ show "map (bder c) (AALTs bs1 rs1 # rsb) s\<leadsto>* map (bder c) (map (fuse bs1) rs1 @ rsb)"
+ apply(simp)
+ using bder_fuse_list map_map rrewrite_srewrite.ss5 srewrites.simps by blast
+next
+ case (ss6 a1 a2 bs rsa rsb)
+ have as: "erase a1 = erase a2" by fact
+ show "map (bder c) (bs @ [a1] @ rsa @ [a2] @ rsb) s\<leadsto>* map (bder c) (bs @ [a1] @ rsa @ rsb)"
+ apply(simp only: map_append)
+ by (smt (verit, best) erase_bder list.simps(8) list.simps(9) as rrewrite_srewrite.ss6 srewrites.simps)
+qed
+
+lemma rewrites_preserves_bder:
+ 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_preserves_bder rrewrites_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_preserves_bder)
+ also have "... \<leadsto>* bders_simp (bders_simp r xs) [x]" using IH
+ by (simp add: rewrites_to_bsimp)
+ finally show "bders r (xs @ [x]) \<leadsto>* bders_simp r (xs @ [x])"
+ by (simp add: bders_simp_append)
+qed
+
+lemma main_aux:
+ 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_blexer_simp:
+ shows "blexer r s = blexer_simp r s"
+ unfolding blexer_def blexer_simp_def
+ by (metis central main_aux rewrites_bnullable_eq)
+
+
+theorem blexersimp_correctness:
+ shows "lexer r s = blexer_simp r s"
+ using blexer_correctness main_blexer_simp by simp
+
+
+(* below is the idempotency of bsimp *)
+
+lemma bsimp_ASEQ_fuse:
+ shows "fuse bs1 (bsimp_ASEQ bs2 r1 r2) = bsimp_ASEQ (bs1 @ bs2) r1 r2"
+ apply(induct r1 r2 arbitrary: bs1 bs2 rule: bsimp_ASEQ.induct)
+ apply(auto)
+ done
+
+lemma bsimp_AALTs_fuse:
+ assumes "\<forall>r \<in> set rs. fuse bs1 (fuse bs2 r) = fuse (bs1 @ bs2) r"
+ shows "fuse bs1 (bsimp_AALTs bs2 rs) = bsimp_AALTs (bs1 @ bs2) rs"
+ using assms
+ apply(induct bs2 rs arbitrary: bs1 rule: bsimp_AALTs.induct)
+ apply(auto)
+ done
+
+lemma bsimp_fuse:
+ shows "fuse bs (bsimp r) = bsimp (fuse bs r)"
+ apply(induct r arbitrary: bs)
+ apply(simp_all add: bsimp_ASEQ_fuse bsimp_AALTs_fuse fuse_append)
+ done
+
+lemma bsimp_ASEQ_idem:
+ assumes "bsimp (bsimp r1) = bsimp r1" "bsimp (bsimp r2) = bsimp r2"
+ shows "bsimp (bsimp_ASEQ x1 (bsimp r1) (bsimp r2)) = bsimp_ASEQ x1 (bsimp r1) (bsimp r2)"
+ using assms
+ apply(case_tac "bsimp r1 = AZERO")
+ apply(simp)
+ apply(case_tac "bsimp r2 = AZERO")
+ apply(simp)
+ apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
+ apply(auto)[1]
+ apply (metis bsimp_fuse)
+ apply(simp add: bsimp_ASEQ1)
+ done
+
+lemma bsimp_AALTs_idem:
+ assumes "\<forall>r \<in> set rs. bsimp (bsimp r) = bsimp r"
+ shows "bsimp (bsimp_AALTs bs rs) = bsimp_AALTs bs (map bsimp rs)"
+ using assms
+ apply(induct bs rs rule: bsimp_AALTs.induct)
+ apply(simp)
+ apply(simp)
+ using bsimp_fuse apply presburger
+ oops
+
+lemma bsimp_idem_rev:
+ shows "\<nexists>r2. bsimp r1 \<leadsto> r2"
+ apply(induct r1 rule: bsimp.induct)
+ apply(auto)
+ defer
+ defer
+ using rrewrite.simps apply blast
+ using rrewrite.cases apply blast
+ using rrewrite.simps apply blast
+ using rrewrite.cases apply blast
+ apply(case_tac "bsimp r1 = AZERO")
+ apply(simp)
+ apply(case_tac "bsimp r2 = AZERO")
+ apply(simp)
+ apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
+ apply(auto)[1]
+ prefer 2
+ apply (smt (verit, best) arexp.distinct(25) arexp.inject(3) bsimp_ASEQ1 rrewrite.simps)
+ defer
+ oops
+
+lemma bsimp_idem:
+ shows "bsimp (bsimp r) = bsimp r"
+ apply(induct r rule: bsimp.induct)
+ apply(auto)
+ using bsimp_ASEQ_idem apply presburger
+ oops
+
+export_code blexer_simp blexer lexer bders bders_simp in Scala module_name VerifiedLexers
+
+
+unused_thms
+
+
+fun concatLen:: "arexp \<Rightarrow> int"
+ where
+"concatLen AZERO = 0"
+| "concatLen (AONE bs) = 0"
+| "concatLen (ACHAR bs a) = 0"
+| "concatLen (ASEQ bs a1 a2) = 1 + (max (concatLen a1) ( concatLen a2))"
+| "concatLen (ASTAR bs r0) = 1 + (concatLen r0)"
+| "concatLen (AALTS bs as) = foldl max (map concatLen as)"
+
+
+
+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