--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/thys2/BitCodedCT.thy Sun Oct 10 18:35:21 2021 +0100
@@ -0,0 +1,3438 @@
+
+theory BitCodedCT
+ 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 (CHAR 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) = CHAR 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)"
+
+lemma decode_code_erase:
+ assumes "\<Turnstile> v : (erase a)"
+ shows "decode (code v) (erase a) = Some v"
+ using assms
+ by (simp add: decode_code)
+
+
+fun nonalt :: "arexp \<Rightarrow> bool"
+ where
+ "nonalt (AALTs bs2 rs) = False"
+| "nonalt r = True"
+
+
+fun good :: "arexp \<Rightarrow> bool" where
+ "good AZERO = False"
+| "good (AONE cs) = True"
+| "good (ACHAR cs c) = True"
+| "good (AALTs cs []) = False"
+| "good (AALTs cs [r]) = False"
+| "good (AALTs cs (r1#r2#rs)) = (\<forall>r' \<in> set (r1#r2#rs). good r' \<and> nonalt r')"
+| "good (ASEQ _ AZERO _) = False"
+| "good (ASEQ _ (AONE _) _) = False"
+| "good (ASEQ _ _ AZERO) = False"
+| "good (ASEQ cs r1 r2) = (good r1 \<and> good r2)"
+| "good (ASTAR cs 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
+
+
+fun intern :: "rexp \<Rightarrow> arexp" where
+ "intern ZERO = AZERO"
+| "intern ONE = AONE []"
+| "intern (CHAR 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 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 r:
+ assumes "bnullable (AALTs bs (a # rs))"
+ shows "bnullable a \<or> (\<not> bnullable a \<and> bnullable (AALTs bs rs))"
+ using assms
+ apply(induct rs)
+ apply(auto)
+ done
+
+lemma r0:
+ 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 r0)
+ 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)))"
+
+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 li :: "bit list \<Rightarrow> arexp list \<Rightarrow> arexp"
+ where
+ "li _ [] = AZERO"
+| "li bs [a] = fuse bs a"
+| "li bs as = AALTs bs as"
+
+
+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 :: "arexp \<Rightarrow> arexp"
+ where
+ "bsimp (ASEQ bs1 r1 r2) = bsimp_ASEQ bs1 (bsimp r1) (bsimp r2)"
+| "bsimp (AALTs bs1 rs) = bsimp_AALTs bs1 (flts (map bsimp rs))"
+| "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 asize0:
+ shows "0 < asize r"
+ apply(induct r)
+ apply(auto)
+ done
+
+
+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)
+ apply(simp)
+ done
+
+lemma bsimp_ASEQ_size:
+ shows "asize (bsimp_ASEQ bs r1 r2) \<le> Suc (asize r1 + asize r2)"
+ apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
+ apply(auto)
+ done
+
+lemma fuse_size:
+ shows "asize (fuse bs r) = asize r"
+ apply(induct r)
+ apply(auto)
+ done
+
+lemma flts_size:
+ shows "sum_list (map asize (flts rs)) \<le> sum_list (map asize rs)"
+ apply(induct rs rule: flts.induct)
+ apply(simp_all)
+ by (metis (mono_tags, lifting) add_mono comp_apply eq_imp_le fuse_size le_SucI map_eq_conv)
+
+
+lemma bsimp_AALTs_size:
+ shows "asize (bsimp_AALTs bs rs) \<le> Suc (sum_list (map asize rs))"
+ apply(induct rs rule: bsimp_AALTs.induct)
+ apply(auto simp add: fuse_size)
+ done
+
+
+lemma bsimp_size:
+ shows "asize (bsimp r) \<le> asize r"
+ apply(induct r)
+ apply(simp_all)
+ apply (meson Suc_le_mono add_mono_thms_linordered_semiring(1) bsimp_ASEQ_size le_trans)
+ apply(rule le_trans)
+ apply(rule bsimp_AALTs_size)
+ apply(simp)
+ apply(rule le_trans)
+ apply(rule flts_size)
+ by (simp add: sum_list_mono)
+
+lemma bsimp_asize0:
+ shows "(\<Sum>x\<leftarrow>rs. asize (bsimp x)) \<le> sum_list (map asize rs)"
+ apply(induct rs)
+ apply(auto)
+ by (simp add: add_mono bsimp_size)
+
+lemma bsimp_AALTs_size2:
+ assumes "\<forall>r \<in> set rs. nonalt r"
+ shows "asize (bsimp_AALTs bs rs) \<ge> sum_list (map asize rs)"
+ using assms
+ apply(induct rs rule: bsimp_AALTs.induct)
+ apply(simp_all add: fuse_size)
+ done
+
+
+lemma qq:
+ shows "map (asize \<circ> fuse bs) rs = map asize rs"
+ apply(induct rs)
+ apply(auto simp add: fuse_size)
+ done
+
+lemma flts_size2:
+ assumes "\<exists>bs rs'. AALTs bs rs' \<in> set rs"
+ shows "sum_list (map asize (flts rs)) < sum_list (map asize rs)"
+ using assms
+ apply(induct rs)
+ apply(auto simp add: qq)
+ apply (simp add: flts_size less_Suc_eq_le)
+ apply(case_tac a)
+ apply(auto simp add: qq)
+ prefer 2
+ apply (simp add: flts_size le_imp_less_Suc)
+ using less_Suc_eq by auto
+
+lemma bsimp_AALTs_size3:
+ assumes "\<exists>r \<in> set (map bsimp rs). \<not>nonalt r"
+ shows "asize (bsimp (AALTs bs rs)) < asize (AALTs bs rs)"
+ using assms flts_size2
+ apply -
+ apply(clarify)
+ apply(simp)
+ apply(drule_tac x="map bsimp rs" in meta_spec)
+ apply(drule meta_mp)
+ apply (metis list.set_map nonalt.elims(3))
+ apply(simp)
+ apply(rule order_class.order.strict_trans1)
+ apply(rule bsimp_AALTs_size)
+ apply(simp)
+ by (smt Suc_leI bsimp_asize0 comp_def le_imp_less_Suc le_trans map_eq_conv not_less_eq)
+
+
+
+
+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_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_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 b1:
+ "bsimp_ASEQ bs1 (AONE bs) r = fuse (bs1 @ bs) r"
+ apply(induct r)
+ apply(auto)
+ done
+
+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 b3:
+ shows "bnullable r = bnullable (bsimp r)"
+ using L_bsimp_erase bnullable_correctness nullable_correctness by auto
+
+
+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 q1:
+ assumes "\<forall>r \<in> set rs. bmkeps(bsimp r) = bmkeps r"
+ shows "map (\<lambda>r. bmkeps(bsimp r)) rs = map bmkeps rs"
+ using assms
+ apply(induct rs)
+ apply(simp)
+ apply(simp)
+ done
+
+lemma q3:
+ assumes "\<exists>r \<in> set rs. bnullable r"
+ shows "bmkeps (AALTs bs rs) = bmkeps (bsimp_AALTs bs rs)"
+ using assms
+ apply(induct bs rs rule: bsimp_AALTs.induct)
+ apply(simp)
+ apply(simp)
+ apply (simp add: b2)
+ apply(simp)
+ done
+
+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 r0 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:
+ shows "bnullable (AALTs bs rs) = (\<exists>r \<in> set rs. bnullable r)"
+ apply(induct rs arbitrary: bs)
+ apply(simp)
+ apply(simp)
+ done
+
+lemma fuse_empty:
+ shows "fuse [] r = r"
+ apply(induct r)
+ apply(auto)
+ done
+
+lemma flts_fuse:
+ shows "map (fuse bs) (flts rs) = flts (map (fuse bs) rs)"
+ apply(induct rs arbitrary: bs rule: flts.induct)
+ apply(auto simp add: fuse_append)
+ done
+
+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)
+ apply(simp)
+ apply(simp)
+ prefer 3
+ apply(simp)
+ apply(simp)
+ apply (simp add: bsimp_ASEQ_fuse)
+ apply(simp)
+ by (simp add: bsimp_AALTs_fuse fuse_append)
+
+lemma bsimp_fuse_AALTs:
+ shows "fuse bs (bsimp (AALTs [] rs)) = bsimp (AALTs bs rs)"
+ apply(subst bsimp_fuse)
+ apply(simp)
+ done
+
+lemma bsimp_fuse_AALTs2:
+ shows "fuse bs (bsimp_AALTs [] rs) = bsimp_AALTs bs rs"
+ using bsimp_AALTs_fuse fuse_append by auto
+
+
+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 (metis bnullable.elims(2) bnullable.elims(3) bsimp.simps(3) bsimp_ASEQ.simps(2) bsimp_ASEQ.simps(3) bsimp_ASEQ.simps(4) bsimp_ASEQ.simps(5) bsimp_ASEQ.simps(6))
+ apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
+ apply(auto)[1]
+ apply(subst bsimp_ASEQ2)
+ apply(subst bsimp_ASEQ2)
+ apply (metis assms(2) bsimp_fuse)
+ apply(subst bsimp_ASEQ1)
+ apply(auto)
+ done
+
+
+fun nonnested :: "arexp \<Rightarrow> bool"
+ where
+ "nonnested (AALTs bs2 []) = True"
+| "nonnested (AALTs bs2 ((AALTs bs1 rs1) # rs2)) = False"
+| "nonnested (AALTs bs2 (r # rs2)) = nonnested (AALTs bs2 rs2)"
+| "nonnested r = True"
+
+
+lemma k0:
+ shows "flts (r # rs1) = flts [r] @ flts rs1"
+ apply(induct r arbitrary: rs1)
+ apply(auto)
+ done
+
+lemma k00:
+ shows "flts (rs1 @ rs2) = flts rs1 @ flts rs2"
+ apply(induct rs1 arbitrary: rs2)
+ apply(auto)
+ by (metis append.assoc k0)
+
+lemma k0a:
+ shows "flts [AALTs bs rs] = map (fuse bs) rs"
+ apply(simp)
+ done
+
+
+lemma k0b:
+ assumes "nonalt r" "r \<noteq> AZERO"
+ shows "flts [r] = [r]"
+ using assms
+ apply(case_tac r)
+ apply(simp_all)
+ done
+
+lemma nn1:
+ assumes "nonnested (AALTs bs rs)"
+ shows "\<nexists>bs1 rs1. flts rs = [AALTs bs1 rs1]"
+ using assms
+ apply(induct rs rule: flts.induct)
+ apply(auto)
+ done
+
+lemma nn1q:
+ assumes "nonnested (AALTs bs rs)"
+ shows "\<nexists>bs1 rs1. AALTs bs1 rs1 \<in> set (flts rs)"
+ using assms
+ apply(induct rs rule: flts.induct)
+ apply(auto)
+ done
+
+lemma nn1qq:
+ assumes "nonnested (AALTs bs rs)"
+ shows "\<nexists>bs1 rs1. AALTs bs1 rs1 \<in> set rs"
+ using assms
+ apply(induct rs rule: flts.induct)
+ apply(auto)
+ done
+
+lemma nn10:
+ assumes "nonnested (AALTs cs rs)"
+ shows "nonnested (AALTs (bs @ cs) rs)"
+ using assms
+ apply(induct rs arbitrary: cs bs)
+ apply(simp_all)
+ apply(case_tac a)
+ apply(simp_all)
+ done
+
+lemma nn11a:
+ assumes "nonalt r"
+ shows "nonalt (fuse bs r)"
+ using assms
+ apply(induct r)
+ apply(auto)
+ done
+
+
+lemma nn1a:
+ assumes "nonnested r"
+ shows "nonnested (fuse bs r)"
+ using assms
+ apply(induct bs r arbitrary: rule: fuse.induct)
+ apply(simp_all add: nn10)
+ done
+
+lemma n0:
+ shows "nonnested (AALTs bs rs) \<longleftrightarrow> (\<forall>r \<in> set rs. nonalt r)"
+ apply(induct rs arbitrary: bs)
+ apply(auto)
+ apply (metis list.set_intros(1) nn1qq nonalt.elims(3))
+ apply (metis list.set_intros(2) nn1qq nonalt.elims(3))
+ by (metis nonalt.elims(2) nonnested.simps(3) nonnested.simps(4) nonnested.simps(5) nonnested.simps(6) nonnested.simps(7))
+
+
+
+
+lemma nn1c:
+ assumes "\<forall>r \<in> set rs. nonnested r"
+ shows "\<forall>r \<in> set (flts rs). nonalt r"
+ using assms
+ apply(induct rs rule: flts.induct)
+ apply(auto)
+ apply(rule nn11a)
+ by (metis nn1qq nonalt.elims(3))
+
+lemma nn1bb:
+ assumes "\<forall>r \<in> set rs. nonalt r"
+ shows "nonnested (bsimp_AALTs bs rs)"
+ using assms
+ apply(induct bs rs rule: bsimp_AALTs.induct)
+ apply(auto)
+ apply (metis nn11a nonalt.simps(1) nonnested.elims(3))
+ using n0 by auto
+
+lemma nn1b:
+ shows "nonnested (bsimp r)"
+ apply(induct r)
+ apply(simp_all)
+ apply(case_tac "bsimp r1 = AZERO")
+ apply(simp)
+ apply(case_tac "bsimp r2 = AZERO")
+ apply(simp)
+ apply(subst bsimp_ASEQ0)
+ apply(simp)
+ apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
+ apply(auto)[1]
+ apply(subst bsimp_ASEQ2)
+ apply (simp add: nn1a)
+ apply(subst bsimp_ASEQ1)
+ apply(auto)
+ apply(rule nn1bb)
+ apply(auto)
+ by (metis (mono_tags, hide_lams) imageE nn1c set_map)
+
+lemma nn1d:
+ assumes "bsimp r = AALTs bs rs"
+ shows "\<forall>r1 \<in> set rs. \<forall> bs. r1 \<noteq> AALTs bs rs2"
+ using nn1b assms
+ by (metis nn1qq)
+
+lemma nn_flts:
+ assumes "nonnested (AALTs bs rs)"
+ shows "\<forall>r \<in> set (flts rs). nonalt r"
+ using assms
+ apply(induct rs arbitrary: bs rule: flts.induct)
+ apply(auto)
+ done
+
+lemma rt:
+ shows "sum_list (map asize (flts (map bsimp rs))) \<le> sum_list (map asize rs)"
+ apply(induct rs)
+ apply(simp)
+ apply(simp)
+ apply(subst k0)
+ apply(simp)
+ by (smt add_le_cancel_right add_mono bsimp_size flts.simps(1) flts_size k0 le_iff_add list.simps(9) map_append sum_list.Cons sum_list.append trans_le_add1)
+
+lemma bsimp_AALTs_qq:
+ assumes "1 < length rs"
+ shows "bsimp_AALTs bs rs = AALTs bs rs"
+ using assms
+ apply(case_tac rs)
+ apply(simp)
+ apply(case_tac list)
+ apply(simp_all)
+ done
+
+
+lemma bsimp_AALTs1:
+ assumes "nonalt r"
+ shows "bsimp_AALTs bs (flts [r]) = fuse bs r"
+ using assms
+ apply(case_tac r)
+ apply(simp_all)
+ done
+
+lemma bbbbs:
+ assumes "good r" "r = AALTs bs1 rs"
+ shows "bsimp_AALTs bs (flts [r]) = AALTs bs (map (fuse bs1) rs)"
+ using assms
+ by (metis (no_types, lifting) Nil_is_map_conv append.left_neutral append_butlast_last_id bsimp_AALTs.elims butlast.simps(2) good.simps(4) good.simps(5) k0a map_butlast)
+
+lemma bbbbs1:
+ shows "nonalt r \<or> (\<exists>bs rs. r = AALTs bs rs)"
+ using nonalt.elims(3) by auto
+
+
+lemma good_fuse:
+ shows "good (fuse bs r) = good r"
+ apply(induct r arbitrary: bs)
+ apply(auto)
+ apply(case_tac r1)
+ apply(simp_all)
+ apply(case_tac r2)
+ apply(simp_all)
+ apply(case_tac r2)
+ apply(simp_all)
+ apply(case_tac r2)
+ apply(simp_all)
+ apply(case_tac r2)
+ apply(simp_all)
+ apply(case_tac r1)
+ apply(simp_all)
+ apply(case_tac r2)
+ apply(simp_all)
+ apply(case_tac r2)
+ apply(simp_all)
+ apply(case_tac r2)
+ apply(simp_all)
+ apply(case_tac r2)
+ apply(simp_all)
+ apply(case_tac x2a)
+ apply(simp_all)
+ apply(case_tac list)
+ apply(simp_all)
+ apply(case_tac x2a)
+ apply(simp_all)
+ apply(case_tac list)
+ apply(simp_all)
+ done
+
+lemma good0:
+ assumes "rs \<noteq> Nil" "\<forall>r \<in> set rs. nonalt r"
+ shows "good (bsimp_AALTs bs rs) \<longleftrightarrow> (\<forall>r \<in> set rs. good r)"
+ using assms
+ apply(induct bs rs rule: bsimp_AALTs.induct)
+ apply(auto simp add: good_fuse)
+ done
+
+lemma good0a:
+ assumes "flts (map bsimp rs) \<noteq> Nil" "\<forall>r \<in> set (flts (map bsimp rs)). nonalt r"
+ shows "good (bsimp (AALTs bs rs)) \<longleftrightarrow> (\<forall>r \<in> set (flts (map bsimp rs)). good r)"
+ using assms
+ apply(simp)
+ apply(auto)
+ apply(subst (asm) good0)
+ apply(simp)
+ apply(auto)
+ apply(subst good0)
+ apply(simp)
+ apply(auto)
+ done
+
+lemma flts0:
+ assumes "r \<noteq> AZERO" "nonalt r"
+ shows "flts [r] \<noteq> []"
+ using assms
+ apply(induct r)
+ apply(simp_all)
+ done
+
+lemma flts1:
+ assumes "good r"
+ shows "flts [r] \<noteq> []"
+ using assms
+ apply(induct r)
+ apply(simp_all)
+ apply(case_tac x2a)
+ apply(simp)
+ apply(simp)
+ done
+
+lemma flts2:
+ assumes "good r"
+ shows "\<forall>r' \<in> set (flts [r]). good r' \<and> nonalt r'"
+ using assms
+ apply(induct r)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ prefer 2
+ apply(simp)
+ apply(auto)[1]
+ apply (metis bsimp_AALTs.elims good.simps(4) good.simps(5) good.simps(6) good_fuse)
+ apply (metis bsimp_AALTs.elims good.simps(4) good.simps(5) good.simps(6) nn11a)
+ apply fastforce
+ apply(simp)
+ done
+
+
+lemma flts3:
+ assumes "\<forall>r \<in> set rs. good r \<or> r = AZERO"
+ shows "\<forall>r \<in> set (flts rs). good r"
+ using assms
+ apply(induct rs arbitrary: rule: flts.induct)
+ apply(simp_all)
+ by (metis UnE flts2 k0a set_map)
+
+lemma flts3b:
+ assumes "\<exists>r\<in>set rs. good r"
+ shows "flts rs \<noteq> []"
+ using assms
+ apply(induct rs arbitrary: rule: flts.induct)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ apply(auto)
+ done
+
+lemma flts4:
+ assumes "bsimp_AALTs bs (flts rs) = AZERO"
+ shows "\<forall>r \<in> set rs. \<not> good r"
+ using assms
+ apply(induct rs arbitrary: bs rule: flts.induct)
+ apply(auto)
+ defer
+ apply (metis (no_types, lifting) Nil_is_append_conv append_self_conv2 bsimp_AALTs.elims butlast.simps(2) butlast_append flts3b nonalt.simps(1) nonalt.simps(2))
+ apply (metis arexp.distinct(7) bsimp_AALTs.elims flts2 good.simps(1) good.simps(2) good0 k0b list.distinct(1) list.inject nonalt.simps(3))
+ apply (metis arexp.distinct(3) arexp.distinct(7) bsimp_AALTs.elims fuse.simps(3) list.distinct(1) list.inject)
+ apply (metis arexp.distinct(7) bsimp_AALTs.elims good.simps(1) good_fuse list.distinct(1) list.inject)
+ apply (metis arexp.distinct(7) bsimp_AALTs.elims list.distinct(1) list.inject)
+ apply (metis arexp.distinct(7) bsimp_AALTs.elims flts2 good.simps(1) good.simps(33) good0 k0b list.distinct(1) list.inject nonalt.simps(6))
+ by (metis (no_types, lifting) Nil_is_append_conv append_Nil2 arexp.distinct(7) bsimp_AALTs.elims butlast.simps(2) butlast_append flts1 flts2 good.simps(1) good0 k0a)
+
+
+lemma flts_nil:
+ assumes "\<forall>y. asize y < Suc (sum_list (map asize rs)) \<longrightarrow>
+ good (bsimp y) \<or> bsimp y = AZERO"
+ and "\<forall>r\<in>set rs. \<not> good (bsimp r)"
+ shows "flts (map bsimp rs) = []"
+ using assms
+ apply(induct rs)
+ apply(simp)
+ apply(simp)
+ apply(subst k0)
+ apply(simp)
+ by force
+
+lemma flts_nil2:
+ assumes "\<forall>y. asize y < Suc (sum_list (map asize rs)) \<longrightarrow>
+ good (bsimp y) \<or> bsimp y = AZERO"
+ and "bsimp_AALTs bs (flts (map bsimp rs)) = AZERO"
+ shows "flts (map bsimp rs) = []"
+ using assms
+ apply(induct rs arbitrary: bs)
+ apply(simp)
+ apply(simp)
+ apply(subst k0)
+ apply(simp)
+ apply(subst (asm) k0)
+ apply(auto)
+ apply (metis flts.simps(1) flts.simps(2) flts4 k0 less_add_Suc1 list.set_intros(1))
+ by (metis flts.simps(2) flts4 k0 less_add_Suc1 list.set_intros(1))
+
+
+
+lemma good_SEQ:
+ assumes "r1 \<noteq> AZERO" "r2 \<noteq> AZERO" "\<forall>bs. r1 \<noteq> AONE bs"
+ shows "good (ASEQ bs r1 r2) = (good r1 \<and> good r2)"
+ using assms
+ apply(case_tac r1)
+ apply(simp_all)
+ apply(case_tac r2)
+ apply(simp_all)
+ apply(case_tac r2)
+ apply(simp_all)
+ apply(case_tac r2)
+ apply(simp_all)
+ apply(case_tac r2)
+ apply(simp_all)
+ done
+
+lemma good1:
+ shows "good (bsimp a) \<or> bsimp a = AZERO"
+ apply(induct a taking: asize rule: measure_induct)
+ apply(case_tac x)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ prefer 3
+ apply(simp)
+ prefer 2
+ (* AALTs case *)
+ apply(simp only:)
+ apply(case_tac "x52")
+ apply(simp)
+ thm good0a
+ (* AALTs list at least one - case *)
+ apply(simp only: )
+ apply(frule_tac x="a" in spec)
+ apply(drule mp)
+ apply(simp)
+ (* either first element is good, or AZERO *)
+ apply(erule disjE)
+ prefer 2
+ apply(simp)
+ (* in the AZERO case, the size is smaller *)
+ apply(drule_tac x="AALTs x51 list" in spec)
+ apply(drule mp)
+ apply(simp add: asize0)
+ apply(subst (asm) bsimp.simps)
+ apply(subst (asm) bsimp.simps)
+ apply(assumption)
+ (* in the good case *)
+ apply(frule_tac x="AALTs x51 list" in spec)
+ apply(drule mp)
+ apply(simp add: asize0)
+ apply(erule disjE)
+ apply(rule disjI1)
+ apply(simp add: good0)
+ apply(subst good0)
+ apply (metis Nil_is_append_conv flts1 k0)
+ apply (metis ex_map_conv list.simps(9) nn1b nn1c)
+ apply(simp)
+ apply(subst k0)
+ apply(simp)
+ apply(auto)[1]
+ using flts2 apply blast
+ apply(subst (asm) good0)
+ prefer 3
+ apply(auto)[1]
+ apply auto[1]
+ apply (metis ex_map_conv nn1b nn1c)
+ (* in the AZERO case *)
+ apply(simp)
+ apply(frule_tac x="a" in spec)
+ apply(drule mp)
+ apply(simp)
+ apply(erule disjE)
+ apply(rule disjI1)
+ apply(subst good0)
+ apply(subst k0)
+ using flts1 apply blast
+ apply(auto)[1]
+ apply (metis (no_types, hide_lams) ex_map_conv list.simps(9) nn1b nn1c)
+ apply(auto)[1]
+ apply(subst (asm) k0)
+ apply(auto)[1]
+ using flts2 apply blast
+ apply(frule_tac x="AALTs x51 list" in spec)
+ apply(drule mp)
+ apply(simp add: asize0)
+ apply(erule disjE)
+ apply(simp)
+ apply(simp)
+ apply (metis add.left_commute flts_nil2 less_add_Suc1 less_imp_Suc_add list.distinct(1) list.set_cases nat.inject)
+ apply(subst (2) k0)
+ apply(simp)
+ (* SEQ case *)
+ apply(simp)
+ apply(case_tac "bsimp x42 = AZERO")
+ apply(simp)
+ apply(case_tac "bsimp x43 = AZERO")
+ apply(simp)
+ apply(subst (2) bsimp_ASEQ0)
+ apply(simp)
+ apply(case_tac "\<exists>bs. bsimp x42 = AONE bs")
+ apply(auto)[1]
+ apply(subst bsimp_ASEQ2)
+ using good_fuse apply force
+ apply(subst bsimp_ASEQ1)
+ apply(auto)
+ apply(subst good_SEQ)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ using less_add_Suc1 less_add_Suc2 by blast
+
+lemma good1a:
+ assumes "L(erase a) \<noteq> {}"
+ shows "good (bsimp a)"
+ using good1 assms
+ using L_bsimp_erase by force
+
+
+
+lemma flts_append:
+ "flts (xs1 @ xs2) = flts xs1 @ flts xs2"
+ apply(induct xs1 arbitrary: xs2 rule: rev_induct)
+ apply(auto)
+ apply(case_tac xs)
+ apply(auto)
+ apply(case_tac x)
+ apply(auto)
+ apply(case_tac x)
+ apply(auto)
+ done
+
+lemma g1:
+ assumes "good (bsimp_AALTs bs rs)"
+ shows "bsimp_AALTs bs rs = AALTs bs rs \<or> (\<exists>r. rs = [r] \<and> bsimp_AALTs bs [r] = fuse bs r)"
+using assms
+ apply(induct rs arbitrary: bs)
+ apply(simp)
+ apply(case_tac rs)
+ apply(simp only:)
+ apply(simp)
+ apply(case_tac list)
+ apply(simp)
+ by simp
+
+lemma flts_0:
+ assumes "nonnested (AALTs bs rs)"
+ shows "\<forall>r \<in> set (flts rs). r \<noteq> AZERO"
+ using assms
+ apply(induct rs arbitrary: bs rule: flts.induct)
+ apply(simp)
+ apply(simp)
+ defer
+ apply(simp)
+ apply(simp)
+ apply(simp)
+apply(simp)
+ apply(rule ballI)
+ apply(simp)
+ done
+
+lemma flts_0a:
+ assumes "nonnested (AALTs bs rs)"
+ shows "AZERO \<notin> set (flts rs)"
+ using assms
+ using flts_0 by blast
+
+lemma qqq1:
+ shows "AZERO \<notin> set (flts (map bsimp rs))"
+ by (metis ex_map_conv flts3 good.simps(1) good1)
+
+
+fun nonazero :: "arexp \<Rightarrow> bool"
+ where
+ "nonazero AZERO = False"
+| "nonazero r = True"
+
+lemma flts_concat:
+ shows "flts rs = concat (map (\<lambda>r. flts [r]) rs)"
+ apply(induct rs)
+ apply(auto)
+ apply(subst k0)
+ apply(simp)
+ done
+
+lemma flts_single1:
+ assumes "nonalt r" "nonazero r"
+ shows "flts [r] = [r]"
+ using assms
+ apply(induct r)
+ apply(auto)
+ done
+
+lemma flts_qq:
+ assumes "\<forall>y. asize y < Suc (sum_list (map asize rs)) \<longrightarrow> good y \<longrightarrow> bsimp y = y"
+ "\<forall>r'\<in>set rs. good r' \<and> nonalt r'"
+ shows "flts (map bsimp rs) = rs"
+ using assms
+ apply(induct rs)
+ apply(simp)
+ apply(simp)
+ apply(subst k0)
+ apply(subgoal_tac "flts [bsimp a] = [a]")
+ prefer 2
+ apply(drule_tac x="a" in spec)
+ apply(drule mp)
+ apply(simp)
+ apply(auto)[1]
+ using good.simps(1) k0b apply blast
+ apply(auto)[1]
+ done
+
+lemma test:
+ assumes "good r"
+ shows "bsimp r = r"
+ using assms
+ apply(induct r taking: "asize" rule: measure_induct)
+ apply(erule good.elims)
+ apply(simp_all)
+ apply(subst k0)
+ apply(subst (2) k0)
+ apply(subst flts_qq)
+ apply(auto)[1]
+ apply(auto)[1]
+ apply (metis append_Cons append_Nil bsimp_AALTs.simps(3) good.simps(1) k0b)
+ apply force+
+ apply (metis (no_types, lifting) add_Suc add_Suc_right asize.simps(5) bsimp.simps(1) bsimp_ASEQ.simps(19) less_add_Suc1 less_add_Suc2)
+ apply (metis add_Suc add_Suc_right arexp.distinct(5) arexp.distinct(7) asize.simps(4) asize.simps(5) bsimp.simps(1) bsimp.simps(2) bsimp_ASEQ1 good.simps(21) good.simps(8) less_add_Suc1 less_add_Suc2)
+ apply force+
+ apply (metis (no_types, lifting) add_Suc add_Suc_right arexp.distinct(5) arexp.distinct(7) asize.simps(4) asize.simps(5) bsimp.simps(1) bsimp.simps(2) bsimp_ASEQ1 good.simps(25) good.simps(8) less_add_Suc1 less_add_Suc2)
+ apply (metis add_Suc add_Suc_right arexp.distinct(7) asize.simps(4) bsimp.simps(2) bsimp_ASEQ1 good.simps(26) good.simps(8) less_add_Suc1 less_add_Suc2)
+ apply force+
+ done
+
+lemma test2:
+ assumes "good r"
+ shows "bsimp r = r"
+ using assms
+ apply(induct r taking: "asize" rule: measure_induct)
+ apply(case_tac x)
+ apply(simp_all)
+ defer
+ (* AALT case *)
+ apply(subgoal_tac "1 < length x52")
+ prefer 2
+ apply(case_tac x52)
+ apply(simp)
+ apply(simp)
+ apply(case_tac list)
+ apply(simp)
+ apply(simp)
+ apply(subst bsimp_AALTs_qq)
+ prefer 2
+ apply(subst flts_qq)
+ apply(auto)[1]
+ apply(auto)[1]
+ apply(case_tac x52)
+ apply(simp)
+ apply(simp)
+ apply(case_tac list)
+ apply(simp)
+ apply(simp)
+ apply(auto)[1]
+ apply (metis (no_types, lifting) bsimp_AALTs.elims good.simps(6) length_Cons length_pos_if_in_set list.size(3) nat_neq_iff)
+ apply(simp)
+ apply(case_tac x52)
+ apply(simp)
+ apply(simp)
+ apply(case_tac list)
+ apply(simp)
+ apply(simp)
+ apply(subst k0)
+ apply(simp)
+ apply(subst (2) k0)
+ apply(simp)
+ apply (simp add: Suc_lessI flts1 one_is_add)
+ (* SEQ case *)
+ apply(case_tac "bsimp x42 = AZERO")
+ apply simp
+ apply (metis asize.elims good.simps(10) good.simps(11) good.simps(12) good.simps(2) good.simps(7) good.simps(9) good_SEQ less_add_Suc1)
+ apply(case_tac "\<exists>bs'. bsimp x42 = AONE bs'")
+ apply(auto)[1]
+ defer
+ apply(case_tac "bsimp x43 = AZERO")
+ apply(simp)
+ apply (metis bsimp.elims bsimp.simps(3) good.simps(10) good.simps(11) good.simps(12) good.simps(8) good.simps(9) good_SEQ less_add_Suc2)
+ apply(auto)
+ apply (subst bsimp_ASEQ1)
+ apply(auto)[3]
+ apply(auto)[1]
+ apply (metis bsimp.simps(3) good.simps(2) good_SEQ less_add_Suc1)
+ apply (metis bsimp.simps(3) good.simps(2) good_SEQ less_add_Suc1 less_add_Suc2)
+ apply (subst bsimp_ASEQ2)
+ apply(drule_tac x="x42" in spec)
+ apply(drule mp)
+ apply(simp)
+ apply(drule mp)
+ apply (metis bsimp.elims bsimp.simps(3) good.simps(10) good.simps(11) good.simps(2) good_SEQ)
+ apply(simp)
+ done
+
+
+lemma bsimp_idem:
+ shows "bsimp (bsimp r) = bsimp r"
+ using test good1
+ by force
+
+
+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 r0)
+ apply(case_tac "bnullable a")
+ apply (metis append.assoc b2 bnullable_correctness erase_fuse r0)
+ 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: r0)
+ 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: r0)
+ 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: r0)
+ apply(case_tac "bnullable (ASEQ x41 x42 x43)")
+ apply(case_tac list)
+ apply(simp)
+ apply(simp)
+ apply (simp add: r0)
+ apply(simp)
+ using qq4 r1 r2 by auto
+
+
+
+lemma k1:
+ assumes "\<And>x2aa. \<lbrakk>x2aa \<in> set x2a; bnullable x2aa\<rbrakk> \<Longrightarrow> bmkeps x2aa = bmkeps (bsimp x2aa)"
+ "\<exists>x\<in>set x2a. bnullable x"
+ shows "bmkeps (AALTs x1 (flts x2a)) = bmkeps (AALTs x1 (flts (map bsimp x2a)))"
+ using assms
+ apply(induct x2a)
+ apply fastforce
+ apply(simp)
+ apply(subst k0)
+ apply(subst (2) k0)
+ apply(auto)[1]
+ apply (metis b3 k0 list.set_intros(1) qs3 r0)
+ by (smt b3 imageI insert_iff k0 list.set(2) qq3 qs3 r0 r1 set_map)
+
+
+
+lemma bmkeps_simp:
+ assumes "bnullable r"
+ shows "bmkeps r = bmkeps (bsimp r)"
+ using assms
+ apply(induct r)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ prefer 3
+ apply(simp)
+ apply(case_tac "bsimp r1 = AZERO")
+ apply(simp)
+ apply(auto)[1]
+ apply (metis L_bsimp_erase L_flat_Prf1 L_flat_Prf2 Prf_elims(1) bnullable_correctness erase.simps(1) mkeps_nullable)
+ apply(case_tac "bsimp r2 = AZERO")
+ apply(simp)
+ apply(auto)[1]
+ apply (metis L_bsimp_erase L_flat_Prf1 L_flat_Prf2 Prf_elims(1) bnullable_correctness erase.simps(1) mkeps_nullable)
+ apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
+ apply(auto)[1]
+ apply(subst b1)
+ apply(subst b2)
+ apply(simp add: b3[symmetric])
+ apply(simp)
+ apply(subgoal_tac "bsimp_ASEQ x1 (bsimp r1) (bsimp r2) = ASEQ x1 (bsimp r1) (bsimp r2)")
+ prefer 2
+ apply (smt b3 bnullable.elims(2) bsimp_ASEQ.simps(17) bsimp_ASEQ.simps(19) bsimp_ASEQ.simps(20) bsimp_ASEQ.simps(21) bsimp_ASEQ.simps(22) bsimp_ASEQ.simps(24) bsimp_ASEQ.simps(25) bsimp_ASEQ.simps(26) bsimp_ASEQ.simps(27) bsimp_ASEQ.simps(29) bsimp_ASEQ.simps(30) bsimp_ASEQ.simps(31))
+ apply(simp)
+ apply(simp)
+ thm q3
+ apply(subst q3[symmetric])
+ apply simp
+ using b3 qq4 apply auto[1]
+ apply(subst qs3)
+ apply simp
+ using k1 by blast
+
+thm bmkeps_retrieve bmkeps_simp bder_retrieve
+
+lemma bmkeps_bder_AALTs:
+ assumes "\<exists>r \<in> set rs. bnullable (bder c r)"
+ shows "bmkeps (bder c (bsimp_AALTs bs rs)) = bmkeps (bsimp_AALTs bs (map (bder c) rs))"
+ using assms
+ apply(induct rs)
+ apply(simp)
+ apply(simp)
+ apply(auto)
+ apply(case_tac rs)
+ apply(simp)
+ apply (metis (full_types) Prf_injval bder_retrieve bmkeps_retrieve bnullable_correctness erase_bder erase_fuse mkeps_nullable retrieve_fuse2)
+ apply(simp)
+ apply(case_tac rs)
+ apply(simp_all)
+ done
+
+lemma bbs0:
+ shows "blexer_simp r [] = blexer r []"
+ apply(simp add: blexer_def blexer_simp_def)
+ done
+
+lemma bbs1:
+ shows "blexer_simp r [c] = blexer r [c]"
+ apply(simp add: blexer_def blexer_simp_def)
+ apply(auto)
+ defer
+ using b3 apply auto[1]
+ using b3 apply auto[1]
+ apply(subst bmkeps_simp[symmetric])
+ apply(simp)
+ apply(simp)
+ done
+
+lemma oo:
+ shows "(case (blexer (der c r) s) of None \<Rightarrow> None | Some v \<Rightarrow> Some (injval r c v)) = blexer r (c # s)"
+ apply(simp add: blexer_correctness)
+ 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
+
+lemma XXX2_helper:
+ assumes "\<forall>y. asize y < Suc (sum_list (map asize rs)) \<longrightarrow> good y \<longrightarrow> bsimp y = y"
+ "\<forall>r'\<in>set rs. good r' \<and> nonalt r'"
+ shows "flts (map (bsimp \<circ> bder c) (flts (map bsimp rs))) = flts (map (bsimp \<circ> bder c) rs)"
+ using assms
+ apply(induct rs arbitrary: c)
+ apply(simp)
+ apply(simp)
+ apply(subst k0)
+ apply(simp add: flts_append)
+ apply(subst (2) k0)
+ apply(simp add: flts_append)
+ apply(subgoal_tac "flts [a] = [a]")
+ prefer 2
+ using good.simps(1) k0b apply blast
+ apply(simp)
+ done
+
+lemma bmkeps_good:
+ assumes "good a"
+ shows "bmkeps (bsimp a) = bmkeps a"
+ using assms
+ using test2 by auto
+
+
+lemma xxx_bder:
+ assumes "good r"
+ shows "L (erase r) \<noteq> {}"
+ using assms
+ apply(induct r rule: good.induct)
+ apply(auto simp add: Sequ_def)
+ done
+
+lemma xxx_bder2:
+ assumes "L (erase (bsimp r)) = {}"
+ shows "bsimp r = AZERO"
+ using assms xxx_bder test2 good1
+ by blast
+
+lemma XXX2aa:
+ assumes "good a"
+ shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
+ using assms
+ by (simp add: test2)
+
+lemma XXX2aa_ders:
+ assumes "good a"
+ shows "bsimp (bders (bsimp a) s) = bsimp (bders a s)"
+ using assms
+ by (simp add: test2)
+
+lemma XXX4a:
+ shows "good (bders_simp (bsimp r) s) \<or> bders_simp (bsimp r) s = AZERO"
+ apply(induct s arbitrary: r rule: rev_induct)
+ apply(simp)
+ apply (simp add: good1)
+ apply(simp add: bders_simp_append)
+ apply (simp add: good1)
+ done
+
+lemma XXX4a_good:
+ assumes "good a"
+ shows "good (bders_simp a s) \<or> bders_simp a s = AZERO"
+ using assms
+ apply(induct s arbitrary: a rule: rev_induct)
+ apply(simp)
+ apply(simp add: bders_simp_append)
+ apply (simp add: good1)
+ done
+
+lemma XXX4a_good_cons:
+ assumes "s \<noteq> []"
+ shows "good (bders_simp a s) \<or> bders_simp a s = AZERO"
+ using assms
+ apply(case_tac s)
+ apply(auto)
+ using XXX4a by blast
+
+lemma XXX4b:
+ assumes "good a" "L (erase (bders_simp a s)) \<noteq> {}"
+ shows "good (bders_simp a s)"
+ using assms
+ apply(induct s arbitrary: a)
+ apply(simp)
+ apply(simp)
+ apply(subgoal_tac "L (erase (bder a aa)) = {} \<or> L (erase (bder a aa)) \<noteq> {}")
+ prefer 2
+ apply(auto)[1]
+ apply(erule disjE)
+ apply(subgoal_tac "bsimp (bder a aa) = AZERO")
+ prefer 2
+ using L_bsimp_erase xxx_bder2 apply auto[1]
+ apply(simp)
+ apply (metis L.simps(1) XXX4a erase.simps(1))
+ apply(drule_tac x="bsimp (bder a aa)" in meta_spec)
+ apply(drule meta_mp)
+ apply simp
+ apply(rule good1a)
+ apply(auto)
+ done
+
+lemma bders_AZERO:
+ shows "bders AZERO s = AZERO"
+ and "bders_simp AZERO s = AZERO"
+ apply (induct s)
+ apply(auto)
+ done
+
+lemma LA:
+ assumes "\<Turnstile> v : ders s (erase r)"
+ shows "retrieve (bders r s) v = retrieve r (flex (erase r) id s v)"
+ using assms
+ apply(induct s arbitrary: r v rule: rev_induct)
+ apply(simp)
+ apply(simp add: bders_append ders_append)
+ apply(subst bder_retrieve)
+ apply(simp)
+ apply(drule Prf_injval)
+ by (simp add: flex_append)
+
+
+lemma LB:
+ assumes "s \<in> (erase r) \<rightarrow> v"
+ shows "retrieve r v = retrieve r (flex (erase r) id s (mkeps (ders s (erase r))))"
+ using assms
+ apply(induct s arbitrary: r v rule: rev_induct)
+ apply(simp)
+ apply(subgoal_tac "v = mkeps (erase r)")
+ prefer 2
+ apply (simp add: Posix1(1) Posix_determ Posix_mkeps nullable_correctness)
+ apply(simp)
+ apply(simp add: flex_append ders_append)
+ by (metis Posix_determ Posix_flex Posix_injval Posix_mkeps ders_snoc lexer_correctness(2) lexer_flex)
+
+lemma LB_sym:
+ assumes "s \<in> (erase r) \<rightarrow> v"
+ shows "retrieve r v = retrieve r (flex (erase r) id s (mkeps (erase (bders r s))))"
+ using assms
+ by (simp add: LB)
+
+
+lemma LC:
+ assumes "s \<in> (erase r) \<rightarrow> v"
+ shows "retrieve r v = retrieve (bders r s) (mkeps (erase (bders r s)))"
+ apply(simp)
+ by (metis LA LB Posix1(1) assms lexer_correct_None lexer_flex mkeps_nullable)
+
+
+lemma L0:
+ assumes "bnullable a"
+ shows "retrieve (bsimp a) (mkeps (erase (bsimp a))) = retrieve a (mkeps (erase a))"
+ using assms
+ by (metis b3 bmkeps_retrieve bmkeps_simp bnullable_correctness)
+
+thm bmkeps_retrieve
+
+lemma L0a:
+ assumes "s \<in> L(erase a)"
+ shows "retrieve (bsimp (bders a s)) (mkeps (erase (bsimp (bders a s)))) =
+ retrieve (bders a s) (mkeps (erase (bders a s)))"
+ using assms
+ by (metis L0 bnullable_correctness erase_bders lexer_correct_None lexer_flex)
+
+lemma L0aa:
+ assumes "s \<in> L (erase a)"
+ shows "[] \<in> erase (bsimp (bders a s)) \<rightarrow> mkeps (erase (bsimp (bders a s)))"
+ using assms
+ by (metis Posix_mkeps b3 bnullable_correctness erase_bders lexer_correct_None lexer_flex)
+
+lemma L0aaa:
+ assumes "[c] \<in> L (erase a)"
+ shows "[c] \<in> (erase a) \<rightarrow> flex (erase a) id [c] (mkeps (erase (bder c a)))"
+ using assms
+ by (metis bders.simps(1) bders.simps(2) erase_bders lexer_correct_None lexer_correct_Some lexer_flex option.inject)
+
+lemma L0aaaa:
+ assumes "[c] \<in> L (erase a)"
+ shows "[c] \<in> (erase a) \<rightarrow> flex (erase a) id [c] (mkeps (erase (bders a [c])))"
+ using assms
+ using L0aaa by auto
+
+
+lemma L02:
+ assumes "bnullable (bder c a)"
+ shows "retrieve (bsimp a) (flex (erase (bsimp a)) id [c] (mkeps (erase (bder c (bsimp a))))) =
+ retrieve (bder c (bsimp a)) (mkeps (erase (bder c (bsimp a))))"
+ using assms
+ apply(simp)
+ using bder_retrieve L0 bmkeps_simp bmkeps_retrieve L0 LA LB
+ apply(subst bder_retrieve[symmetric])
+ apply (metis L_bsimp_erase bnullable_correctness der_correctness erase_bder mkeps_nullable nullable_correctness)
+ apply(simp)
+ done
+
+lemma L02_bders:
+ assumes "bnullable (bders a s)"
+ shows "retrieve (bsimp a) (flex (erase (bsimp a)) id s (mkeps (erase (bders (bsimp a) s)))) =
+ retrieve (bders (bsimp a) s) (mkeps (erase (bders (bsimp a) s)))"
+ using assms
+ by (metis LA L_bsimp_erase bnullable_correctness ders_correctness erase_bders mkeps_nullable nullable_correctness)
+
+
+
+
+lemma L03:
+ assumes "bnullable (bder c a)"
+ shows "retrieve (bder c (bsimp a)) (mkeps (erase (bder c (bsimp a)))) =
+ bmkeps (bsimp (bder c (bsimp a)))"
+ using assms
+ by (metis L0 L_bsimp_erase bmkeps_retrieve bnullable_correctness der_correctness erase_bder nullable_correctness)
+
+lemma L04:
+ assumes "bnullable (bder c a)"
+ shows "retrieve (bder c (bsimp a)) (mkeps (erase (bder c (bsimp a)))) =
+ retrieve (bsimp (bder c (bsimp a))) (mkeps (erase (bsimp (bder c (bsimp a)))))"
+ using assms
+ by (metis L0 L_bsimp_erase bnullable_correctness der_correctness erase_bder nullable_correctness)
+
+lemma L05:
+ assumes "bnullable (bder c a)"
+ shows "retrieve (bder c (bsimp a)) (mkeps (erase (bder c (bsimp a)))) =
+ retrieve (bsimp (bder c (bsimp a))) (mkeps (erase (bsimp (bder c (bsimp a)))))"
+ using assms
+ using L04 by auto
+
+lemma L06:
+ assumes "bnullable (bder c a)"
+ shows "bmkeps (bder c (bsimp a)) = bmkeps (bsimp (bder c (bsimp a)))"
+ using assms
+ by (metis L03 L_bsimp_erase bmkeps_retrieve bnullable_correctness der_correctness erase_bder nullable_correctness)
+
+lemma L07:
+ assumes "s \<in> L (erase r)"
+ shows "retrieve r (flex (erase r) id s (mkeps (ders s (erase r))))
+ = retrieve (bders r s) (mkeps (erase (bders r s)))"
+ using assms
+ using LB LC lexer_correct_Some by auto
+
+lemma LXXX:
+ assumes "s \<in> (erase r) \<rightarrow> v" "s \<in> (erase (bsimp r)) \<rightarrow> v'"
+ shows "retrieve r v = retrieve (bsimp r) v'"
+ using assms
+ apply -
+ thm LC
+ apply(subst LC)
+ apply(assumption)
+ apply(subst L0[symmetric])
+ using bnullable_correctness lexer_correctness(2) lexer_flex apply fastforce
+ apply(subst (2) LC)
+ apply(assumption)
+ apply(subst (2) L0[symmetric])
+ using bnullable_correctness lexer_correctness(2) lexer_flex apply fastforce
+
+ oops
+
+
+lemma L07a:
+ assumes "s \<in> L (erase r)"
+ shows "retrieve (bsimp r) (flex (erase (bsimp r)) id s (mkeps (ders s (erase (bsimp r)))))
+ = retrieve r (flex (erase r) id s (mkeps (ders s (erase r))))"
+ using assms
+ apply(induct s arbitrary: r)
+ apply(simp)
+ using L0a apply force
+ apply(drule_tac x="(bder a r)" in meta_spec)
+ apply(drule meta_mp)
+ apply (metis L_bsimp_erase erase_bder lexer.simps(2) lexer_correct_None option.case(1))
+ apply(drule sym)
+ apply(simp)
+ apply(subst (asm) bder_retrieve)
+ apply (metis Posix_Prf Posix_flex Posix_mkeps ders.simps(2) lexer_correct_None lexer_flex)
+ apply(simp only: flex_fun_apply)
+ apply(simp)
+ using L0[no_vars] bder_retrieve[no_vars] LA[no_vars] LC[no_vars] L07[no_vars]
+ oops
+
+lemma L08:
+ assumes "s \<in> L (erase r)"
+ shows "retrieve (bders (bsimp r) s) (mkeps (erase (bders (bsimp r) s)))
+ = retrieve (bders r s) (mkeps (erase (bders r s)))"
+ using assms
+ apply(induct s arbitrary: r)
+ apply(simp)
+ using L0 bnullable_correctness nullable_correctness apply blast
+ apply(simp add: bders_append)
+ apply(drule_tac x="(bder a (bsimp r))" in meta_spec)
+ apply(drule meta_mp)
+ apply (metis L_bsimp_erase erase_bder lexer.simps(2) lexer_correct_None option.case(1))
+ apply(drule sym)
+ apply(simp)
+ apply(subst LA)
+ apply (metis L0aa L_bsimp_erase Posix1(1) ders.simps(2) ders_correctness erase_bder erase_bders mkeps_nullable nullable_correctness)
+ apply(subst LA)
+ using lexer_correct_None lexer_flex mkeps_nullable apply force
+
+ using L0[no_vars] bder_retrieve[no_vars] LA[no_vars] LC[no_vars] L07[no_vars]
+
+thm L0[no_vars] bder_retrieve[no_vars] LA[no_vars] LC[no_vars] L07[no_vars]
+ oops
+
+lemma test:
+ assumes "s = [c]"
+ shows "retrieve (bders r s) v = XXX" and "YYY = retrieve r (flex (erase r) id s v)"
+ using assms
+ apply(simp only: bders.simps)
+ defer
+ using assms
+ apply(simp only: flex.simps id_simps)
+ using L0[no_vars] bder_retrieve[no_vars] LA[no_vars] LC[no_vars]
+ find_theorems "retrieve (bders _ _) _"
+ find_theorems "retrieve _ (mkeps _)"
+ oops
+
+lemma L06X:
+ assumes "bnullable (bder c a)"
+ shows "bmkeps (bder c (bsimp a)) = bmkeps (bder c a)"
+ using assms
+ apply(induct a arbitrary: c)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ prefer 3
+ apply(simp)
+ prefer 2
+ apply(simp)
+
+ defer
+ oops
+
+lemma L06_2:
+ assumes "bnullable (bders a [c,d])"
+ shows "bmkeps (bders (bsimp a) [c,d]) = bmkeps (bsimp (bders (bsimp a) [c,d]))"
+ using assms
+ apply(simp)
+ by (metis L_bsimp_erase bmkeps_simp bnullable_correctness der_correctness erase_bder nullable_correctness)
+
+lemma L06_bders:
+ assumes "bnullable (bders a s)"
+ shows "bmkeps (bders (bsimp a) s) = bmkeps (bsimp (bders (bsimp a) s))"
+ using assms
+ by (metis L_bsimp_erase bmkeps_simp bnullable_correctness ders_correctness erase_bders nullable_correctness)
+
+lemma LLLL:
+ shows "L (erase a) = L (erase (bsimp a))"
+ and "L (erase a) = {flat v | v. \<Turnstile> v: (erase a)}"
+ and "L (erase a) = {flat v | v. \<Turnstile> v: (erase (bsimp a))}"
+ using L_bsimp_erase apply(blast)
+ apply (simp add: L_flat_Prf)
+ using L_bsimp_erase L_flat_Prf apply(auto)[1]
+ done
+
+
+
+lemma L07XX:
+ assumes "s \<in> L (erase a)"
+ shows "s \<in> erase a \<rightarrow> flex (erase a) id s (mkeps (ders s (erase a)))"
+ using assms
+ by (meson lexer_correct_None lexer_correctness(1) lexer_flex)
+
+lemma LX0:
+ assumes "s \<in> L r"
+ shows "decode (bmkeps (bders (intern r) s)) r = Some(flex r id s (mkeps (ders s r)))"
+ by (metis assms blexer_correctness blexer_def lexer_correct_None lexer_flex)
+
+
+lemma L02_bders2:
+ assumes "bnullable (bders a s)" "s = [c]"
+ shows "retrieve (bders (bsimp a) s) (mkeps (erase (bders (bsimp a) s))) =
+ retrieve (bders a s) (mkeps (erase (bders a s)))"
+ using assms
+ apply(simp)
+
+ apply(induct s arbitrary: a)
+ apply(simp)
+ using L0 apply auto[1]
+ oops
+
+thm bmkeps_retrieve bmkeps_simp Posix_mkeps
+
+lemma WQ1:
+ assumes "s \<in> L (der c r)"
+ shows "s \<in> der c r \<rightarrow> mkeps (ders s (der c r))"
+ using assms
+ oops
+
+lemma L02_bsimp:
+ assumes "bnullable (bders a s)"
+ shows "retrieve (bsimp a) (flex (erase (bsimp a)) id s (mkeps (erase (bders (bsimp a) s)))) =
+ retrieve a (flex (erase a) id s (mkeps (erase (bders a s))))"
+ using assms
+ apply(induct s arbitrary: a)
+ apply(simp)
+ apply (simp add: L0)
+ apply(simp)
+ apply(drule_tac x="bder a aa" in meta_spec)
+ apply(simp)
+ apply(subst (asm) bder_retrieve)
+ using Posix_Prf Posix_flex Posix_mkeps bnullable_correctness apply fastforce
+ apply(simp add: flex_fun_apply)
+ apply(drule sym)
+ apply(simp)
+ apply(subst flex_injval)
+ apply(subst bder_retrieve[symmetric])
+ apply (metis L_bsimp_erase Posix_Prf Posix_flex Posix_mkeps bders.simps(2) bnullable_correctness ders.simps(2) erase_bders lexer_correct_None lexer_flex option.distinct(1))
+ apply(simp only: erase_bder[symmetric] erase_bders[symmetric])
+ apply(subst LB_sym[symmetric])
+ apply(simp)
+ oops
+
+lemma L1:
+ assumes "s \<in> r \<rightarrow> v"
+ shows "decode (bmkeps (bders (intern r) s)) r = Some v"
+ using assms
+ by (metis blexer_correctness blexer_def lexer_correctness(1) option.distinct(1))
+
+lemma L2:
+ assumes "s \<in> (der c r) \<rightarrow> v"
+ shows "decode (bmkeps (bders (intern r) (c # s))) r = Some (injval r c v)"
+ using assms
+ apply(subst bmkeps_retrieve)
+ using Posix1(1) lexer_correct_None lexer_flex apply fastforce
+ using MAIN_decode
+ apply(subst MAIN_decode[symmetric])
+ apply(simp)
+ apply (meson Posix1(1) lexer_correct_None lexer_flex mkeps_nullable)
+ apply(simp)
+ apply(subgoal_tac "v = flex (der c r) id s (mkeps (ders s (der c r)))")
+ prefer 2
+ apply (metis Posix_determ lexer_correctness(1) lexer_flex option.distinct(1))
+ apply(simp)
+ apply(subgoal_tac "injval r c (flex (der c r) id s (mkeps (ders s (der c r)))) =
+ (flex (der c r) ((\<lambda>v. injval r c v) o id) s (mkeps (ders s (der c r))))")
+ apply(simp)
+ using flex_fun_apply by blast
+
+lemma L3:
+ assumes "s2 \<in> (ders s1 r) \<rightarrow> v"
+ shows "decode (bmkeps (bders (intern r) (s1 @ s2))) r = Some (flex r id s1 v)"
+ using assms
+ apply(induct s1 arbitrary: r s2 v rule: rev_induct)
+ apply(simp)
+ using L1 apply blast
+ apply(simp add: ders_append)
+ apply(drule_tac x="r" in meta_spec)
+ apply(drule_tac x="x # s2" in meta_spec)
+ apply(drule_tac x="injval (ders xs r) x v" in meta_spec)
+ apply(drule meta_mp)
+ defer
+ apply(simp)
+ apply(simp add: flex_append)
+ by (simp add: Posix_injval)
+
+
+
+lemma bders_snoc:
+ "bder c (bders a s) = bders a (s @ [c])"
+ apply(simp add: bders_append)
+ done
+
+
+lemma QQ1:
+ shows "bsimp (bders (bsimp a) []) = bders_simp (bsimp a) []"
+ apply(simp)
+ apply(simp add: bsimp_idem)
+ done
+
+lemma QQ2:
+ shows "bsimp (bders (bsimp a) [c]) = bders_simp (bsimp a) [c]"
+ apply(simp)
+ done
+
+lemma XXX2a_long:
+ assumes "good a"
+ shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
+ using assms
+ apply(induct a arbitrary: c taking: asize rule: measure_induct)
+ apply(case_tac x)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ prefer 3
+ apply(simp)
+ apply(simp)
+ apply(auto)[1]
+apply(case_tac "x42 = AZERO")
+ apply(simp)
+ apply(case_tac "x43 = AZERO")
+ apply(simp)
+ using test2 apply force
+ apply(case_tac "\<exists>bs. x42 = AONE bs")
+ apply(clarify)
+ apply(simp)
+ apply(subst bsimp_ASEQ1)
+ apply(simp)
+ using b3 apply force
+ using bsimp_ASEQ0 test2 apply force
+ thm good_SEQ test2
+ apply (simp add: good_SEQ test2)
+ apply (simp add: good_SEQ test2)
+ apply(case_tac "x42 = AZERO")
+ apply(simp)
+ apply(case_tac "x43 = AZERO")
+ apply(simp)
+ apply (simp add: bsimp_ASEQ0)
+ apply(case_tac "\<exists>bs. x42 = AONE bs")
+ apply(clarify)
+ apply(simp)
+ apply(subst bsimp_ASEQ1)
+ apply(simp)
+ using bsimp_ASEQ0 test2 apply force
+ apply (simp add: good_SEQ test2)
+ apply (simp add: good_SEQ test2)
+ apply (simp add: good_SEQ test2)
+ (* AALTs case *)
+ apply(simp)
+ using test2 by fastforce
+
+lemma XXX2a_long_without_good:
+ assumes "a = AALTs bs0 [AALTs bs1 [AALTs bs2 [ASTAR [] (AONE bs7), AONE bs6, ASEQ bs3 (ACHAR bs4 d) (AONE bs5)]]]"
+ shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
+ "bsimp (bder c (bsimp a)) = XXX"
+ "bsimp (bder c a) = YYY"
+ using assms
+ apply(simp)
+ using assms
+ apply(simp)
+ prefer 2
+ using assms
+ apply(simp)
+ oops
+
+lemma bder_bsimp_AALTs:
+ shows "bder c (bsimp_AALTs bs rs) = bsimp_AALTs bs (map (bder c) rs)"
+ apply(induct bs rs rule: bsimp_AALTs.induct)
+ apply(simp)
+ apply(simp)
+ apply (simp add: bder_fuse)
+ apply(simp)
+ done
+
+lemma flts_nothing:
+ assumes "\<forall>r \<in> set rs. r \<noteq> AZERO" "\<forall>r \<in> set rs. nonalt r"
+ shows "flts rs = rs"
+ using assms
+ apply(induct rs rule: flts.induct)
+ apply(auto)
+ done
+
+lemma flts_flts:
+ assumes "\<forall>r \<in> set rs. good r"
+ shows "flts (flts rs) = flts rs"
+ using assms
+ apply(induct rs taking: "\<lambda>rs. sum_list (map asize rs)" rule: measure_induct)
+ apply(case_tac x)
+ apply(simp)
+ apply(simp)
+ apply(case_tac a)
+ apply(simp_all add: bder_fuse flts_append)
+ apply(subgoal_tac "\<forall>r \<in> set x52. r \<noteq> AZERO")
+ prefer 2
+ apply (metis Nil_is_append_conv bsimp_AALTs.elims good.simps(1) good.simps(5) good0 list.distinct(1) n0 nn1b split_list_last test2)
+ apply(subgoal_tac "\<forall>r \<in> set x52. nonalt r")
+ prefer 2
+ apply (metis n0 nn1b test2)
+ by (metis flts_fuse flts_nothing)
+
+
+lemma PP:
+ assumes "bnullable (bders r s)"
+ shows "bmkeps (bders (bsimp r) s) = bmkeps (bders r s)"
+ using assms
+ apply(induct s arbitrary: r)
+ apply(simp)
+ using bmkeps_simp apply auto[1]
+ apply(simp add: bders_append bders_simp_append)
+ oops
+
+lemma PP:
+ assumes "bnullable (bders r s)"
+ shows "bmkeps (bders_simp (bsimp r) s) = bmkeps (bders r s)"
+ using assms
+ apply(induct s arbitrary: r rule: rev_induct)
+ apply(simp)
+ using bmkeps_simp apply auto[1]
+ apply(simp add: bders_append bders_simp_append)
+ apply(drule_tac x="bder a (bsimp r)" in meta_spec)
+ apply(drule_tac meta_mp)
+ defer
+ oops
+
+
+lemma
+ assumes "asize (bsimp a) = asize a" "a = AALTs bs [AALTs bs2 [], AZERO, AONE bs3]"
+ shows "bsimp a = a"
+ using assms
+ apply(simp)
+ oops
+
+
+lemma iii:
+ assumes "bsimp_AALTs bs rs \<noteq> AZERO"
+ shows "rs \<noteq> []"
+ using assms
+ apply(induct bs rs rule: bsimp_AALTs.induct)
+ apply(auto)
+ done
+
+lemma
+ assumes "\<forall>y. asize y < Suc (sum_list (map asize x52)) \<longrightarrow> asize (bsimp y) = asize y \<longrightarrow> bsimp y \<noteq> AZERO \<longrightarrow> bsimp y = y"
+ "asize (bsimp_AALTs x51 (flts (map bsimp x52))) = Suc (sum_list (map asize x52))"
+ "bsimp_AALTs x51 (flts (map bsimp x52)) \<noteq> AZERO"
+ shows "bsimp_AALTs x51 (flts (map bsimp x52)) = AALTs x51 x52"
+ using assms
+ apply(induct x52 arbitrary: x51)
+ apply(simp)
+ oops
+
+
+lemma
+ assumes "asize (bsimp a) = asize a" "bsimp a \<noteq> AZERO"
+ shows "bsimp a = a"
+ using assms
+ apply(induct a taking: asize rule: measure_induct)
+ apply(case_tac x)
+ apply(simp_all)
+ apply(case_tac "(bsimp x42) = AZERO")
+ apply(simp add: asize0)
+ apply(case_tac "(bsimp x43) = AZERO")
+ apply(simp add: asize0)
+ apply (metis bsimp_ASEQ0)
+ apply(case_tac "\<exists>bs. (bsimp x42) = AONE bs")
+ apply(auto)[1]
+ apply (metis b1 bsimp_size fuse_size less_add_Suc2 not_less)
+ apply (metis Suc_inject add.commute asize.simps(5) bsimp_ASEQ1 bsimp_size leD le_neq_implies_less less_add_Suc2 less_add_eq_less)
+ (* ALT case *)
+ apply(frule iii)
+ apply(case_tac x52)
+ apply(simp)
+ apply(simp)
+ apply(subst k0)
+ apply(subst (asm) k0)
+ apply(subst (asm) (2) k0)
+ apply(subst (asm) (3) k0)
+ apply(case_tac "(bsimp a) = AZERO")
+ apply(simp)
+ apply (metis (no_types, lifting) Suc_le_lessD asize0 bsimp_AALTs_size le_less_trans less_add_same_cancel2 not_less_eq rt)
+ apply(simp)
+ apply(case_tac "nonalt (bsimp a)")
+ prefer 2
+ apply(drule_tac x="AALTs x51 (bsimp a # list)" in spec)
+ apply(drule mp)
+ apply (metis asize.simps(4) bsimp.simps(2) bsimp_AALTs_size3 k0 less_not_refl list.set_intros(1) list.simps(9) sum_list.Cons)
+ apply(drule mp)
+ apply(simp)
+ apply (metis asize.simps(4) bsimp.simps(2) bsimp_AALTs_size3 k0 lessI list.set_intros(1) list.simps(9) not_less_eq sum_list.Cons)
+ apply(drule mp)
+ apply(simp)
+ using bsimp_idem apply auto[1]
+ apply(simp add: bsimp_idem)
+ apply (metis append.left_neutral append_Cons asize.simps(4) bsimp.simps(2) bsimp_AALTs_size3 k00 less_not_refl list.set_intros(1) list.simps(9) sum_list.Cons)
+ apply (metis bsimp.simps(2) bsimp_idem k0 list.simps(9) nn1b nonalt.elims(3) nonnested.simps(2))
+ apply(subgoal_tac "flts [bsimp a] = [bsimp a]")
+ prefer 2
+ using k0b apply blast
+ apply(clarify)
+ apply(simp only:)
+ apply(simp)
+ apply(case_tac "flts (map bsimp list) = Nil")
+ apply (metis bsimp_AALTs1 bsimp_size fuse_size less_add_Suc1 not_less)
+ apply (subgoal_tac "bsimp_AALTs x51 (bsimp a # flts (map bsimp list)) = AALTs x51 (bsimp a # flts (map bsimp list))")
+ prefer 2
+ apply (metis bsimp_AALTs.simps(3) neq_Nil_conv)
+ apply(auto)
+ apply (metis add.commute bsimp_size leD le_neq_implies_less less_add_Suc1 less_add_eq_less rt)
+ oops
+
+
+
+
+lemma OOO:
+ shows "bsimp (bsimp_AALTs bs rs) = bsimp_AALTs bs (flts (map bsimp rs))"
+ apply(induct rs arbitrary: bs taking: "\<lambda>rs. sum_list (map asize rs)" rule: measure_induct)
+ apply(case_tac x)
+ apply(simp)
+ apply(simp)
+ apply(case_tac "a = AZERO")
+ apply(simp)
+ apply(case_tac "list")
+ apply(simp)
+ apply(simp)
+ apply(case_tac "bsimp a = AZERO")
+ apply(simp)
+ apply(case_tac "list")
+ apply(simp)
+ apply(simp add: bsimp_fuse[symmetric])
+ apply(simp)
+ apply(case_tac "nonalt (bsimp a)")
+ apply(case_tac list)
+ apply(simp)
+ apply(subst k0b)
+ apply(simp)
+ apply(simp)
+ apply(simp add: bsimp_fuse)
+ apply(simp)
+ apply(subgoal_tac "asize (bsimp a) < asize a \<or> asize (bsimp a) = asize a")
+ prefer 2
+ using bsimp_size le_neq_implies_less apply blast
+ apply(erule disjE)
+ apply(drule_tac x="(bsimp a) # list" in spec)
+ apply(drule mp)
+ apply(simp)
+ apply(simp)
+ apply (metis bsimp.simps(2) bsimp_AALTs.elims bsimp_AALTs.simps(2) bsimp_fuse bsimp_idem list.distinct(1) list.inject list.simps(9))
+ apply(subgoal_tac "\<exists>bs rs. bsimp a = AALTs bs rs \<and> rs \<noteq> Nil \<and> length rs > 1")
+ prefer 2
+ apply (metis bbbbs1 bsimp.simps(2) bsimp_AALTs.simps(1) bsimp_idem flts.simps(1) good.simps(5) good1 length_0_conv length_Suc_conv less_one list.simps(8) nat_neq_iff not_less_eq)
+ apply(auto)
+ oops
+
+
+lemma
+ assumes "rs = [AALTs bsa [AONE bsb, AONE bsb]]"
+ shows "bsimp (bsimp_AALTs bs rs) = bsimp_AALTs bs (flts (map bsimp rs))"
+ using assms
+ apply(simp)
+ oops
+
+
+
+lemma CT1:
+ shows "bsimp (AALTs bs as) = bsimp(AALTs bs (map bsimp as))"
+ apply(induct as arbitrary: bs)
+ apply(simp)
+ apply(simp)
+ by (simp add: bsimp_idem comp_def)
+
+lemma CT1a:
+ shows "bsimp (AALT bs a1 a2) = bsimp(AALT bs (bsimp a1) (bsimp a2))"
+ by (metis CT1 list.simps(8) list.simps(9))
+
+(* CT *)
+
+lemma CTU:
+ shows "bsimp_AALTs bs as = li bs as"
+ apply(induct bs as rule: li.induct)
+ apply(auto)
+ done
+
+
+
+lemma CTa:
+ assumes "\<forall>r \<in> set as. nonalt r \<and> r \<noteq> AZERO"
+ shows "flts as = as"
+ using assms
+ apply(induct as)
+ apply(simp)
+ apply(case_tac as)
+ apply(simp)
+ apply (simp add: k0b)
+ using flts_nothing by auto
+
+lemma CT0:
+ assumes "\<forall>r \<in> set as1. nonalt r \<and> r \<noteq> AZERO"
+ shows "flts [bsimp_AALTs bs1 as1] = flts (map (fuse bs1) as1)"
+ using assms CTa
+ apply(induct as1 arbitrary: bs1)
+ apply(simp)
+ apply(simp)
+ apply(case_tac as1)
+ apply(simp)
+ apply(simp)
+proof -
+fix a :: arexp and as1a :: "arexp list" and bs1a :: "bit list" and aa :: arexp and list :: "arexp list"
+ assume a1: "nonalt a \<and> a \<noteq> AZERO \<and> nonalt aa \<and> aa \<noteq> AZERO \<and> (\<forall>r\<in>set list. nonalt r \<and> r \<noteq> AZERO)"
+ assume a2: "\<And>as. \<forall>r\<in>set as. nonalt r \<and> r \<noteq> AZERO \<Longrightarrow> flts as = as"
+ assume a3: "as1a = aa # list"
+ have "flts [a] = [a]"
+using a1 k0b by blast
+then show "fuse bs1a a # fuse bs1a aa # map (fuse bs1a) list = flts (fuse bs1a a # fuse bs1a aa # map (fuse bs1a) list)"
+ using a3 a2 a1 by (metis (no_types) append.left_neutral append_Cons flts_fuse k00 k0b list.simps(9))
+qed
+
+
+lemma CT01:
+ assumes "\<forall>r \<in> set as1. nonalt r \<and> r \<noteq> AZERO" "\<forall>r \<in> set as2. nonalt r \<and> r \<noteq> AZERO"
+ shows "flts [bsimp_AALTs bs1 as1, bsimp_AALTs bs2 as2] = flts ((map (fuse bs1) as1) @ (map (fuse bs2) as2))"
+ using assms CT0
+ by (metis k0 k00)
+
+
+
+
+lemma
+ shows "bsimp (AALT bs (AALTs bs1 (map (bder c) as1)) (AALTs bs2 (map (bder c) as2)))
+ = bsimp (AALTs bs ((map (fuse bs1) (map (bder c) as1)) @
+ (map (fuse bs2) (map (bder c) as2))))"
+ apply(subst bsimp_idem[symmetric])
+ apply(simp)
+ oops
+
+lemma CT_exp:
+ assumes "\<forall>a \<in> set as. bsimp (bder c (bsimp a)) = bsimp (bder c a)"
+ shows "map bsimp (map (bder c) as) = map bsimp (map (bder c) (map bsimp as))"
+ using assms
+ apply(induct as)
+ apply(auto)
+ done
+
+lemma asize_set:
+ assumes "a \<in> set as"
+ shows "asize a < Suc (sum_list (map asize as))"
+ using assms
+ apply(induct as arbitrary: a)
+ apply(auto)
+ using le_add2 le_less_trans not_less_eq by blast
+
+
+lemma XXX2a_long_without_good:
+ shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
+ apply(induct a arbitrary: c taking: "\<lambda>a. asize a" rule: measure_induct)
+ apply(case_tac x)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ prefer 3
+ apply(simp)
+ (* AALT case *)
+ prefer 2
+ apply(simp del: bsimp.simps)
+ apply(subst (2) CT1)
+ apply(subst CT_exp)
+ apply(auto)[1]
+ using asize_set apply blast
+ apply(subst CT1[symmetric])
+ apply(simp)
+ oops
+
+lemma YY:
+ assumes "flts (map bsimp as1) = xs"
+ shows "flts (map bsimp (map (fuse bs1) as1)) = map (fuse bs1) xs"
+ using assms
+ apply(induct as1 arbitrary: bs1 xs)
+ apply(simp)
+ apply(auto)
+ by (metis bsimp_fuse flts_fuse k0 list.simps(9))
+
+
+lemma flts_nonalt:
+ assumes "flts (map bsimp xs) = ys"
+ shows "\<forall>y \<in> set ys. nonalt y"
+ using assms
+ apply(induct xs arbitrary: ys)
+ apply(auto)
+ apply(case_tac xs)
+ apply(auto)
+ using flts2 good1 apply fastforce
+ by (smt ex_map_conv list.simps(9) nn1b nn1c)
+
+lemma WWW2:
+ shows "bsimp (bsimp_AALTs bs1 (flts (map bsimp as1))) =
+ bsimp_AALTs bs1 (flts (map bsimp as1))"
+ by (metis bsimp.simps(2) bsimp_idem)
+
+lemma WWW3:
+ shows "flts [bsimp_AALTs bs1 (flts (map bsimp as1))] =
+ flts (map bsimp (map (fuse bs1) as1))"
+ by (metis CT0 YY flts_nonalt flts_nothing qqq1)
+
+lemma WWW4:
+ shows "map (bder c \<circ> fuse bs1) as1 = map (fuse bs1) (map (bder c) as1)"
+ apply(induct as1)
+ apply(auto)
+ using bder_fuse by blast
+
+lemma WWW5:
+ shows "map (bsimp \<circ> bder c) as1 = map bsimp (map (bder c) as1)"
+ apply(induct as1)
+ apply(auto)
+ done
+
+lemma WWW6:
+ shows "bsimp (bder c (bsimp_AALTs x51 (flts [bsimp a1, bsimp a2]) ) ) =
+ bsimp(bsimp_AALTs x51 (map (bder c) (flts [bsimp a1, bsimp a2]))) "
+ using bder_bsimp_AALTs by auto
+
+lemma WWW7:
+ shows "bsimp (bsimp_AALTs x51 (map (bder c) (flts [bsimp a1, bsimp a2]))) =
+ bsimp(bsimp_AALTs x51 (flts (map (bder c) [bsimp a1, bsimp a2])))"
+ sorry
+
+
+lemma stupid:
+ assumes "a = b"
+ shows "bsimp(a) = bsimp(b)"
+ using assms
+ apply(auto)
+ done
+(*
+proving idea:
+bsimp_AALTs x51 (map (bder c) (flts [a1, a2])) = bsimp_AALTs x51 (map (bder c) (flts [a1]++[a2]))
+= bsimp_AALTs x51 (map (bder c) ((flts [a1])++(flts [a2]))) =
+bsimp_AALTs x51 (map (bder c) (flts [a1]))++(map (bder c) (flts [a2])) = A
+and then want to prove that
+map (bder c) (flts [a]) = flts [bder c a] under the condition
+that a is either a seq with the first elem being not nullable, or a character equal to c,
+or an AALTs, or a star
+Then, A = bsimp_AALTs x51 (flts [bder c a]) ++ (map (bder c) (flts [a2])) = A1
+Using the same condition for a2, we get
+A1 = bsimp_AALTs x51 (flts [bder c a1]) ++ (flts [bder c a2])
+=bsimp_AALTs x51 flts ([bder c a1] ++ [bder c a2])
+=bsimp_AALTs x51 flts ([bder c a1, bder c a2])
+ *)
+lemma manipulate_flts:
+ shows "bsimp_AALTs x51 (map (bder c) (flts [a1, a2])) =
+bsimp_AALTs x51 ((map (bder c) (flts [a1])) @ (map (bder c) (flts [a2])))"
+ by (metis k0 map_append)
+
+lemma go_inside_flts:
+ assumes " (bder c a1 \<noteq> AZERO) "
+ "\<not>(\<exists> a01 a02 x02. ( (a1 = ASEQ x02 a01 a02) \<and> bnullable(a01) ) )"
+shows "map (bder c) (flts [a1]) = flts [bder c a1]"
+ using assms
+ apply -
+ apply(case_tac a1)
+ apply(simp)
+ apply(simp)
+ apply(case_tac "x32 = c")
+ prefer 2
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ apply (simp add: WWW4)
+ apply(simp add: bder_fuse)
+ done
+
+lemma medium010:
+ assumes " (bder c a1 = AZERO) "
+ shows "map (bder c) (flts [a1]) = [AZERO] \<or> map (bder c) (flts [a1]) = []"
+ using assms
+ apply -
+ apply(case_tac a1)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ done
+
+lemma medium011:
+ assumes " (bder c a1 = AZERO) "
+ shows "flts (map (bder c) [a1, a2]) = flts [bder c a2]"
+ using assms
+ apply -
+ apply(simp)
+ done
+
+lemma medium01central:
+ shows "bsimp(bsimp_AALTs x51 (map (bder c) (flts [a2])) ) = bsimp(bsimp_AALTs x51 (flts [bder c a2]))"
+ sorry
+
+
+lemma plus_bsimp:
+ assumes "bsimp( bsimp a) = bsimp (bsimp b)"
+ shows "bsimp a = bsimp b"
+ using assms
+ apply -
+ by (simp add: bsimp_idem)
+lemma patience_good5:
+ assumes "bsimp r = AALTs x y"
+ shows " \<exists> a aa list. y = a#aa#list"
+ by (metis Nil_is_map_conv arexp.simps(13) assms bsimp_AALTs.elims flts1 good.simps(5) good1 k0a)
+
+(*SAD*)
+(*this does not hold actually
+lemma bsimp_equiv0:
+ shows "bsimp(bsimp r) = bsimp(bsimp (AALTs [] [r]))"
+ apply(simp)
+ apply(case_tac "bsimp r")
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ thm good1
+ using good1
+ apply -
+ apply(drule_tac x="r" in meta_spec)
+ apply(erule disjE)
+
+ apply(simp only: bsimp_AALTs.simps)
+ apply(simp only:flts.simps)
+ apply(drule patience_good5)
+ apply(clarify)
+ apply(subst bsimp_AALTs_qq)
+ apply simp
+ prefer 2
+ sorry*)
+
+(*exercise: try multiple ways of proving this*)
+(*this lemma does not hold.........
+lemma bsimp_equiv1:
+ shows "bsimp r = bsimp (AALTs [] [r])"
+ using plus_bsimp
+ apply -
+ using bsimp_equiv0 by blast
+ (*apply(simp)
+ apply(case_tac "bsimp r")
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+(*use lemma good1*)
+ thm good1
+ using good1
+ apply -
+ apply(drule_tac x="r" in meta_spec)
+ apply(erule disjE)
+
+ apply(subst flts_single1)
+ apply(simp only: bsimp_AALTs.simps)
+ prefer 2
+
+ thm flts_single1
+
+ find_theorems "flts _ = _"*)
+*)
+lemma bsimp_equiv2:
+ shows "bsimp (AALTs x51 [r]) = bsimp (AALT x51 AZERO r)"
+ sorry
+
+lemma medium_stupid_isabelle:
+ assumes "rs = a # list"
+ shows "bsimp_AALTs x51 (AZERO # rs) = AALTs x51 (AZERO#rs)"
+ using assms
+ apply -
+ apply(simp)
+ done
+(*
+lemma mediumlittle:
+ shows "bsimp(bsimp_AALTs x51 rs) = bsimp(bsimp_AALTs x51 (AZERO # rs))"
+ apply(case_tac rs)
+ apply(simp)
+ apply(case_tac list)
+ apply(subst medium_stupid_isabelle)
+ apply(simp)
+ prefer 2
+ apply simp
+ apply(rule_tac s="a#list" and t="rs" in subst)
+ apply(simp)
+ apply(rule_tac t="list" and s= "[]" in subst)
+ apply(simp)
+ (*dunno what is the rule for x#nil = x*)
+ apply(case_tac a)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ prefer 3
+ apply simp
+ apply(simp only:bsimp_AALTs.simps)
+
+ apply simp
+ apply(case_tac "bsimp x42")
+ apply(simp)
+ apply simp
+ apply(case_tac "bsimp x43")
+ apply simp
+ apply simp
+ apply simp
+ apply simp
+ apply(simp only:bsimp_ASEQ.simps)
+ using good1
+ apply -
+ apply(drule_tac x="x43" in meta_spec)
+ apply(erule disjE)
+ apply(subst bsimp_AALTs_qq)
+ using patience_good5 apply force
+ apply(simp only:bsimp_AALTs.simps)
+ apply(simp only:fuse.simps)
+ apply(simp only:flts.simps)
+(*OK from here you actually realize this lemma doesnt hold*)
+ apply(simp)
+ apply(simp)
+ apply(rule_tac t="rs" and s="a#list" in subst)
+ apply(simp)
+ apply(rule_tac t="list" and s="[]" in subst)
+ apply(simp)
+ (*apply(simp only:bsimp_AALTs.simps)*)
+ (*apply(simp only:fuse.simps)*)
+ sorry
+*)
+lemma singleton_list_map:
+ shows"map f [a] = [f a]"
+ apply simp
+ done
+lemma map_application2:
+ shows"map f [a,b] = [f a, f b]"
+ apply simp
+ done
+(*SAD*)
+(* bsimp (bder c (bsimp_AALTs x51 (flts [bsimp a1, bsimp a2]))) =
+ bsimp (AALT x51 (bder c (bsimp a1)) (bder c (bsimp a2)))*)
+(*This equality does not hold*)
+lemma medium01:
+ assumes " (bder c a1 = AZERO) "
+ shows "bsimp(bsimp_AALTs x51 (map (bder c) (flts [ a1, a2]))) =
+ bsimp(bsimp_AALTs x51 (flts (map (bder c) [ a1, a2])))"
+ apply(subst manipulate_flts)
+ using assms
+ apply -
+ apply(subst medium011)
+ apply(simp)
+ apply(case_tac "map (bder c) (flts [a1]) = []")
+ apply(simp)
+ using medium01central apply blast
+apply(frule medium010)
+ apply(erule disjE)
+ prefer 2
+ apply(simp)
+ apply(simp)
+ apply(case_tac a2)
+ apply simp
+ apply simp
+ apply simp
+ apply(simp only:flts.simps)
+(*HOW do i say here to replace ASEQ ..... back into a2*)
+(*how do i say here to use the definition of map function
+without lemma, of course*)
+(*how do i say here that AZERO#map (bder c) [ASEQ x41 x42 x43]'s list.len >1
+without a lemma, of course*)
+ apply(subst singleton_list_map)
+ apply(simp only: bsimp_AALTs.simps)
+ apply(case_tac "bder c (ASEQ x41 x42 x43)")
+ apply simp
+ apply simp
+ apply simp
+ prefer 3
+ apply simp
+ apply(rule_tac t="bder c (ASEQ x41 x42 x43)"
+and s="ASEQ x41a x42a x43a" in subst)
+ apply simp
+ apply(simp only: flts.simps)
+ apply(simp only: bsimp_AALTs.simps)
+ apply(simp only: fuse.simps)
+ apply(subst (2) bsimp_idem[symmetric])
+ apply(subst (1) bsimp_idem[symmetric])
+ apply(simp only:bsimp.simps)
+ apply(subst map_application2)
+ apply(simp only: bsimp.simps)
+ apply(simp only:flts.simps)
+(*want to happily change between a2 and ASEQ x41 42 43, and eliminate now
+redundant conditions such as map (bder c) (flts [a1]) = [AZERO] *)
+ apply(case_tac "bsimp x42a")
+ apply(simp)
+ apply(case_tac "bsimp x43a")
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ prefer 2
+ apply(simp)
+ apply(rule_tac t="bsimp x43a"
+and s="AALTs x51a x52" in subst)
+ apply simp
+ apply(simp only:bsimp_ASEQ.simps)
+ apply(simp only:fuse.simps)
+ apply(simp only:flts.simps)
+
+ using medium01central mediumlittle by auto
+
+
+
+lemma medium1:
+ assumes " (bder c a1 \<noteq> AZERO) "
+ "\<not>(\<exists> a01 a02 x02. ( (a1 = ASEQ x02 a01 a02) \<and> bnullable(a01) ) )"
+" (bder c a2 \<noteq> AZERO)"
+ "\<not>(\<exists> a11 a12 x12. ( (a2 = ASEQ x12 a11 a12) \<and> bnullable(a11) ) )"
+ shows "bsimp_AALTs x51 (map (bder c) (flts [ a1, a2])) =
+ bsimp_AALTs x51 (flts (map (bder c) [ a1, a2]))"
+ using assms
+ apply -
+ apply(subst manipulate_flts)
+ apply(case_tac "a1")
+ apply(simp)
+ apply(simp)
+ apply(case_tac "x32 = c")
+ prefer 2
+ apply(simp)
+ prefer 2
+ apply(case_tac "bnullable x42")
+ apply(simp)
+ apply(simp)
+
+ apply(case_tac "a2")
+ apply(simp)
+ apply(simp)
+ apply(case_tac "x32 = c")
+ prefer 2
+ apply(simp)
+ apply(simp)
+ apply(case_tac "bnullable x42a")
+ apply(simp)
+ apply(subst go_inside_flts)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ apply (simp add: WWW4)
+ apply(simp)
+ apply (simp add: WWW4)
+ apply (simp add: go_inside_flts)
+ apply (metis (no_types, lifting) go_inside_flts k0 list.simps(8) list.simps(9))
+ by (smt bder.simps(6) flts.simps(1) flts.simps(6) flts.simps(7) go_inside_flts k0 list.inject list.simps(9))
+
+lemma big0:
+ shows "bsimp (AALT x51 (AALTs bs1 as1) (AALTs bs2 as2)) =
+ bsimp (AALTs x51 ((map (fuse bs1) as1) @ (map (fuse bs2) as2)))"
+ by (smt WWW3 bsimp.simps(2) k0 k00 list.simps(8) list.simps(9) map_append)
+
+lemma bignA:
+ shows "bsimp (AALTs x51 (AALTs bs1 as1 # as2)) =
+ bsimp (AALTs x51 ((map (fuse bs1) as1) @ as2))"
+ apply(simp)
+ apply(subst k0)
+ apply(subst WWW3)
+ apply(simp add: flts_append)
+ done
+
+lemma hardest:
+ shows "bsimp (bder c (bsimp_AALTs x51 (flts [bsimp a1, bsimp a2]))) =
+ bsimp (AALT x51 (bder c (bsimp a1)) (bder c (bsimp a2)))"
+ apply(case_tac "bsimp a1")
+ apply(case_tac "bsimp a2")
+ apply simp
+ apply simp
+ apply(rule_tac t="bsimp a1"
+and s="AZERO" in subst)
+ apply simp
+ apply(rule_tac t="bsimp a2"
+and s="ACHAR x31 x32" in subst)
+ apply simp
+ apply simp
+ apply(rule_tac t="bsimp a1"
+and s="AZERO" in subst)
+ apply simp
+ apply(rule_tac t="bsimp a2"
+and s="ASEQ x41 x42 x43" in subst)
+ apply simp
+ apply(case_tac "bnullable x42")
+ apply(simp only: bder.simps)
+ apply(simp)
+ apply(case_tac "flts
+ [bsimp_ASEQ [] (bsimp (bder c x42)) (bsimp x43),
+ bsimp (fuse (bmkeps x42) (bder c x43))]")
+ apply(simp)
+ apply simp
+(*counterexample finder*)
+
+
+lemma XXX2a_long_without_good:
+ shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
+ apply(induct a arbitrary: c taking: "\<lambda>a. asize a" rule: measure_induct)
+ apply(case_tac x)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ prefer 3
+ apply(simp)
+ (* AALT case *)
+ prefer 2
+ apply(simp only:)
+ apply(case_tac "\<exists>a1 a2. x52 = [a1, a2]")
+ apply(clarify)
+ apply(simp del: bsimp.simps)
+ apply(subst (2) CT1)
+ apply(simp del: bsimp.simps)
+ apply(rule_tac t="bsimp (bder c a1)" and s="bsimp (bder c (bsimp a1))" in subst)
+ apply(simp del: bsimp.simps)
+ apply(rule_tac t="bsimp (bder c a2)" and s="bsimp (bder c (bsimp a2))" in subst)
+ apply(simp del: bsimp.simps)
+ apply(subst CT1a[symmetric])
+ apply(subst bsimp.simps)
+ apply(simp del: bsimp.simps)
+(*bsimp_AALTs x51 (map (bder c) (flts [a1, a2])) =
+ bsimp_AALTs x51 (flts (map (bder c) [a1, a2]))*)
+ apply(case_tac "\<exists>bs1 as1. bsimp a1 = AALTs bs1 as1")
+ apply(case_tac "\<exists>bs2 as2. bsimp a2 = AALTs bs2 as2")
+ apply(clarify)
+ apply(simp only:)
+ apply(simp del: bsimp.simps bder.simps)
+ apply(subst bsimp_AALTs_qq)
+ prefer 2
+ apply(simp del: bsimp.simps)
+ apply(subst big0)
+ apply(simp add: WWW4)
+ apply (metis One_nat_def Suc_eq_plus1 Suc_lessI arexp.distinct(7) bsimp.simps(2) bsimp_AALTs.simps(1) bsimp_idem flts.simps(1) length_append length_greater_0_conv length_map not_add_less2 not_less_eq)
+ oops
+
+lemma XXX2a_long_without_good:
+ shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
+ apply(induct a arbitrary: c taking: "\<lambda>a. asize a" rule: measure_induct)
+ apply(case_tac x)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ prefer 3
+ apply(simp)
+ (* AALT case *)
+ prefer 2
+ apply(subgoal_tac "nonnested (bsimp x)")
+ prefer 2
+ using nn1b apply blast
+ apply(simp only:)
+ apply(drule_tac x="AALTs x51 (flts x52)" in spec)
+ apply(drule mp)
+ defer
+ apply(drule_tac x="c" in spec)
+ apply(simp)
+ apply(rotate_tac 2)
+
+ apply(drule sym)
+ apply(simp)
+
+ apply(simp only: bder.simps)
+ apply(simp only: bsimp.simps)
+ apply(subst bder_bsimp_AALTs)
+ apply(case_tac x52)
+ apply(simp)
+ apply(simp)
+ apply(case_tac list)
+ apply(simp)
+ apply(case_tac a)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ defer
+ apply(simp)
+
+
+ (* case AALTs list is not empty *)
+ apply(simp)
+ apply(subst k0)
+ apply(subst (2) k0)
+ apply(simp)
+ apply(case_tac "bsimp a = AZERO")
+ apply(subgoal_tac "bsimp (bder c a) = AZERO")
+ prefer 2
+ using less_iff_Suc_add apply auto[1]
+ apply(simp)
+ apply(drule_tac x="AALTs x51 list" in spec)
+ apply(drule mp)
+ apply(simp add: asize0)
+ apply(drule_tac x="c" in spec)
+ apply(simp add: bder_bsimp_AALTs)
+ apply(case_tac "nonalt (bsimp a)")
+ prefer 2
+ apply(drule_tac x="bsimp (AALTs x51 (a#list))" in spec)
+ apply(drule mp)
+ apply(rule order_class.order.strict_trans2)
+ apply(rule bsimp_AALTs_size3)
+ apply(auto)[1]
+ apply(simp)
+ apply(subst (asm) bsimp_idem)
+ apply(drule_tac x="c" in spec)
+ apply(simp)
+ find_theorems "_ < _ \<Longrightarrow> _ \<le> _ \<Longrightarrow>_ < _"
+ apply(rule le_trans)
+ apply(subgoal_tac "flts [bsimp a] = [bsimp a]")
+ prefer 2
+ using k0b apply blast
+ apply(simp)
+ find_theorems "asize _ < asize _"
+
+ using bder_bsimp_AALTs
+ apply(case_tac list)
+ apply(simp)
+ sledgeha mmer [timeout=6000]
+
+ apply(case_tac "\<exists>r \<in> set (map bsimp x52). \<not>nonalt r")
+ apply(drule_tac x="bsimp (AALTs x51 x52)" in spec)
+ apply(drule mp)
+ using bsimp_AALTs_size3 apply blast
+ apply(drule_tac x="c" in spec)
+ apply(subst (asm) (2) test)
+
+ apply(case_tac x52)
+ apply(simp)
+ apply(simp)
+ apply(case_tac "bsimp a = AZERO")
+ apply(simp)
+ apply(subgoal_tac "bsimp (bder c a) = AZERO")
+ prefer 2
+ apply auto[1]
+ apply (metis L.simps(1) L_bsimp_erase der.simps(1) der_correctness erase.simps(1) erase_bder xxx_bder2)
+ apply(simp)
+ apply(drule_tac x="AALTs x51 list" in spec)
+ apply(drule mp)
+ apply(simp add: asize0)
+ apply(simp)
+ apply(case_tac list)
+ prefer 2
+ apply(simp)
+ apply(case_tac "bsimp aa = AZERO")
+ apply(simp)
+ apply(subgoal_tac "bsimp (bder c aa) = AZERO")
+ prefer 2
+ apply auto[1]
+ apply (metis add.left_commute bder.simps(1) bsimp.simps(3) less_add_Suc1)
+ apply(simp)
+ apply(drule_tac x="AALTs x51 (a#lista)" in spec)
+ apply(drule mp)
+ apply(simp add: asize0)
+ apply(simp)
+ apply (metis flts.simps(2) k0)
+ apply(subst k0)
+ apply(subst (2) k0)
+
+
+ using less_add_Suc1 apply fastforce
+ apply(subst k0)
+
+
+ apply(simp)
+ apply(case_tac "bsimp a = AZERO")
+ apply(simp)
+ apply(subgoal_tac "bsimp (bder c a) = AZERO")
+ prefer 2
+ apply auto[1]
+ apply(simp)
+ apply(case_tac "nonalt (bsimp a)")
+ apply(subst bsimp_AALTs1)
+ apply(simp)
+ using less_add_Suc1 apply fastforce
+
+ apply(subst bsimp_AALTs1)
+
+ using nn11a apply b last
+
+ (* SEQ case *)
+ apply(clarify)
+ apply(subst bsimp.simps)
+ apply(simp del: bsimp.simps)
+ apply(auto simp del: bsimp.simps)[1]
+ apply(subgoal_tac "bsimp x42 \<noteq> AZERO")
+ prefer 2
+ using b3 apply force
+ apply(case_tac "bsimp x43 = AZERO")
+ apply(simp)
+ apply (simp add: bsimp_ASEQ0)
+ apply (metis bder.simps(1) bsimp.simps(3) bsimp_AALTs.simps(1) bsimp_fuse flts.simps(1) flts.simps(2) fuse.simps(1) less_add_Suc2)
+ apply(case_tac "\<exists>bs. bsimp x42 = AONE bs")
+ apply(clarify)
+ apply(simp)
+ apply(subst bsimp_ASEQ2)
+ apply(subgoal_tac "bsimp (bder c x42) = AZERO")
+ prefer 2
+ using less_add_Suc1 apply fastforce
+ apply(simp)
+ apply(frule_tac x="x43" in spec)
+ apply(drule mp)
+ apply(simp)
+ apply(drule_tac x="c" in spec)
+ apply(subst bder_fuse)
+ apply(subst bsimp_fuse[symmetric])
+ apply(simp)
+ apply(subgoal_tac "bmkeps x42 = bs")
+ prefer 2
+ apply (simp add: bmkeps_simp)
+ apply(simp)
+ apply(subst bsimp_fuse[symmetric])
+ apply(case_tac "nonalt (bsimp (bder c x43))")
+ apply(subst bsimp_AALTs1)
+ using nn11a apply blast
+ using fuse_append apply blast
+ apply(subgoal_tac "\<exists>bs rs. bsimp (bder c x43) = AALTs bs rs")
+ prefer 2
+ using bbbbs1 apply blast
+ apply(clarify)
+ apply(simp)
+ apply(case_tac rs)
+ apply(simp)
+ apply (metis arexp.distinct(7) good.simps(4) good1)
+ apply(simp)
+ apply(case_tac list)
+ apply(simp)
+ apply (metis arexp.distinct(7) good.simps(5) good1)
+ apply(simp del: bsimp_AALTs.simps)
+ apply(simp only: bsimp_AALTs.simps)
+ apply(simp)
+
+
+
+
+(* HERE *)
+apply(case_tac "x42 = AZERO")
+ apply(simp)
+ apply(case_tac "bsimp x43 = AZERO")
+ apply(simp)
+ apply (simp add: bsimp_ASEQ0)
+ apply(subgoal_tac "bsimp (fuse (bmkeps x42) (bder c x43)) = AZERO")
+ apply(simp)
+ apply (met is bder.simps(1) bsimp.simps(3) bsimp_fuse fuse.simps(1) less_add_Suc2)
+ apply(case_tac "\<exists>bs. bsimp x42 = AONE bs")
+ apply(clarify)
+ apply(simp)
+ apply(subst bsimp_ASEQ2)
+ apply(subgoal_tac "bsimp (bder c x42) = AZERO")
+ apply(simp)
+ prefer 2
+ using less_add_Suc1 apply fastforce
+ apply(subgoal_tac "bmkeps x42 = bs")
+ prefer 2
+ apply (simp add: bmkeps_simp)
+ apply(simp)
+ apply(case_tac "nonalt (bsimp (bder c x43))")
+ apply (metis bder_fuse bsimp_AALTs.simps(1) bsimp_AALTs.simps(2) bsimp_fuse flts.simps(1) flts.simps(2) fuse.simps(1) fuse_append k0b less_add_Suc2 nn11a)
+ apply(subgoal_tac "nonnested (bsimp (bder c x43))")
+ prefer 2
+ using nn1b apply blast
+ apply(case_tac x43)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ prefer 3
+ apply(simp)
+ apply (metis arexp.distinct(25) arexp.distinct(7) arexp.distinct(9) bsimp_ASEQ.simps(1) bsimp_ASEQ.simps(11) bsimp_ASEQ1 nn11a nonalt.elims(3) nonalt.simps(6))
+ apply(simp)
+ apply(auto)[1]
+ apply(case_tac "(bsimp (bder c x42a)) = AZERO")
+ apply(simp)
+
+ apply(simp)
+
+
+
+ apply(subgoal_tac "(\<exists>bs1 rs1. 1 < length rs1 \<and> bsimp (bder c x43) = AALTs bs1 rs1 ) \<or>
+ (\<exists>bs1 r. bsimp (bder c x43) = fuse bs1 r)")
+ prefer 2
+ apply (metis fuse_empty)
+ apply(erule disjE)
+ prefer 2
+ apply(clarify)
+ apply(simp only:)
+ apply(simp)
+ apply(simp add: fuse_append)
+ apply(subst bder_fuse)
+ apply(subst bsimp_fuse[symmetric])
+ apply(subst bder_fuse)
+ apply(subst bsimp_fuse[symmetric])
+ apply(subgoal_tac "bsimp (bder c (bsimp x43)) = bsimp (bder c x43)")
+ prefer 2
+ using less_add_Suc2 apply bl ast
+ apply(simp only: )
+ apply(subst bsimp_fuse[symmetric])
+ apply(simp only: )
+
+ apply(simp only: fuse.simps)
+ apply(simp)
+ apply(case_tac rs1)
+ apply(simp)
+ apply (me tis arexp.distinct(7) fuse.simps(1) good.simps(4) good1 good_fuse)
+ apply(simp)
+ apply(case_tac list)
+ apply(simp)
+ apply (me tis arexp.distinct(7) fuse.simps(1) good.simps(5) good1 good_fuse)
+ apply(simp only: bsimp_AALTs.simps map_cons.simps)
+ apply(auto)[1]
+
+
+
+ apply(subst bsimp_fuse[symmetric])
+ apply(subgoal_tac "bmkeps x42 = bs")
+ prefer 2
+ apply (simp add: bmkeps_simp)
+
+
+ apply(simp)
+
+ using b3 apply force
+ using bsimp_ASEQ0 test2 apply fo rce
+ thm good_SEQ test2
+ apply (simp add: good_SEQ test2)
+ apply (simp add: good_SEQ test2)
+ apply(case_tac "x42 = AZERO")
+ apply(simp)
+ apply(case_tac "x43 = AZERO")
+ apply(simp)
+ apply (simp add: bsimp_ASEQ0)
+ apply(case_tac "\<exists>bs. x42 = AONE bs")
+ apply(clarify)
+ apply(simp)
+ apply(subst bsimp_ASEQ1)
+ apply(simp)
+ using bsimp_ASEQ0 test2 apply fo rce
+ apply (simp add: good_SEQ test2)
+ apply (simp add: good_SEQ test2)
+ apply (simp add: good_SEQ test2)
+ (* AALTs case *)
+ apply(simp)
+ using test2 by fa st force
+
+
+lemma XXX4ab:
+ shows "good (bders_simp (bsimp r) s) \<or> bders_simp (bsimp r) s = AZERO"
+ apply(induct s arbitrary: r rule: rev_induct)
+ apply(simp)
+ apply (simp add: good1)
+ apply(simp add: bders_simp_append)
+ apply (simp add: good1)
+ done
+
+lemma XXX4:
+ assumes "good a"
+ shows "bders_simp a s = bsimp (bders a s)"
+ using assms
+ apply(induct s arbitrary: a rule: rev_induct)
+ apply(simp)
+ apply (simp add: test2)
+ apply(simp add: bders_append bders_simp_append)
+ oops
+
+
+lemma MAINMAIN:
+ "blexer r s = blexer_simp r s"
+ apply(induct s arbitrary: r)
+ apply(simp add: blexer_def blexer_simp_def)
+ apply(simp add: blexer_def blexer_simp_def del: bders.simps bders_simp.simps)
+ apply(auto simp del: bders.simps bders_simp.simps)
+ prefer 2
+ apply (metis b4 bders.simps(2) bders_simp.simps(2))
+ prefer 2
+ apply (metis b4 bders.simps(2))
+ apply(subst bmkeps_simp)
+ apply(simp)
+ apply(case_tac s)
+ apply(simp only: bders.simps)
+ apply(subst bders_simp.simps)
+ apply(simp)
+ oops
+
+
+lemma
+ fixes n :: nat
+ shows "(\<Sum>i \<in> {0..n}. i) = n * (n + 1) div 2"
+ apply(induct n)
+ apply(simp)
+ apply(simp)
+ done
+
+
+
+
+
+end
\ No newline at end of file