
diff -r cf7a5c863831 -r 6c13f76c070b thys3/BlexerSimp.thy
--- a/thys3/BlexerSimp.thy	Wed Feb 15 11:52:22 2023 +0000
+++ b/thys3/BlexerSimp.thy	Thu Feb 16 23:23:22 2023 +0000
@@ -2,6 +2,35 @@
   imports Blexer 
 begin
 
+fun distinctWith :: "'a list \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a list"
+  where
+  "distinctWith [] eq acc = []"
+| "distinctWith (x # xs) eq acc = 
+     (if (\<exists> y \<in> acc. eq x y) then distinctWith xs eq acc 
+      else x # (distinctWith xs eq ({x} \<union> acc)))"
+
+
+fun eq1 ("_ ~1 _" [80, 80] 80) where  
+  "AZERO ~1 AZERO = True"
+| "(AONE bs1) ~1 (AONE bs2) = True"
+| "(ACHAR bs1 c) ~1 (ACHAR bs2 d) = (if c = d then True else False)"
+| "(ASEQ bs1 ra1 ra2) ~1 (ASEQ bs2 rb1 rb2) = (ra1 ~1 rb1 \<and> ra2 ~1 rb2)"
+| "(AALTs bs1 []) ~1 (AALTs bs2 []) = True"
+| "(AALTs bs1 (r1 # rs1)) ~1 (AALTs bs2 (r2 # rs2)) = (r1 ~1 r2 \<and> (AALTs bs1 rs1) ~1 (AALTs bs2 rs2))"
+| "(ASTAR bs1 r1) ~1 (ASTAR bs2 r2) = r1 ~1 r2"
+| "(ANTIMES bs1 r1 n1) ~1 (ANTIMES bs2 r2 n2) = (r1 ~1 r2 \<and> n1 = n2)"
+| "_ ~1 _ = False"
+
+
+
+lemma eq1_L:
+  assumes "x ~1 y"
+  shows "L (erase x) = L (erase y)"
+  using assms
+  apply(induct rule: eq1.induct)
+  apply(auto elim: eq1.elims)
+  apply presburger
+  done
 
 fun flts :: "arexp list \<Rightarrow> arexp list"
   where 
@@ -42,11 +71,42 @@
 | "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 (distinctWith (flts (map bsimp rs)) eq1 {}) "
+| "bsimp r = r"
+
+
+fun 
+  bders_simp :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
+where
+  "bders_simp r [] = r"
+| "bders_simp r (c # s) = bders_simp (bsimp (bder c r)) s"
+
+definition blexer_simp where
+ "blexer_simp r s \<equiv> if bnullable (bders_simp (intern r) s) then 
+                    decode (bmkeps (bders_simp (intern r) s)) r else None"
+
+
+
+lemma bders_simp_append:
+  shows "bders_simp r (s1 @ s2) = bders_simp (bders_simp r s1) s2"
+  apply(induct s1 arbitrary: r s2)
+  apply(simp_all)
+  done
+
 lemma bmkeps_fuse:
   assumes "bnullable r"
   shows "bmkeps (fuse bs r) = bs @ bmkeps r"
   using assms
-  by (induct r rule: bnullable.induct) (auto)
+  apply(induct r rule: bnullable.induct) 
+        apply(auto)
+  apply (metis append.assoc bmkeps_retrieve bmkeps_retrieve_AALTs bnullable.simps(4))  
+  done
 
 lemma bmkepss_fuse: 
   assumes "bnullables rs"
@@ -65,84 +125,6 @@
 
 
 
-fun ABIncludedByC :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('c \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> bool" where
-  "ABIncludedByC a b c f subseteqPred = subseteqPred (f a b) c"
-
-fun furtherSEQ :: "rexp \<Rightarrow> rexp \<Rightarrow> rexp list" and 
-   turnIntoTerms :: "rexp \<Rightarrow> rexp list "
-   where
-  "furtherSEQ ONE r2 =  turnIntoTerms r2 "
-| "furtherSEQ r11 r2 = [SEQ r11 r2]"
-| "turnIntoTerms (SEQ ONE r2) =  turnIntoTerms r2"
-| "turnIntoTerms (SEQ r1 r2) = concat (map (\<lambda>r11. furtherSEQ r11 r2) (turnIntoTerms r1))"
-| "turnIntoTerms r = [r]"
-
-abbreviation "tint \<equiv> turnIntoTerms"
-
-fun seqFold :: "rexp \<Rightarrow> rexp list \<Rightarrow> rexp" where
-  "seqFold acc [] = acc"
-| "seqFold ONE (r # rs1) = seqFold r rs1"
-| "seqFold acc (r # rs1) = seqFold (SEQ acc r) rs1"
-
-
-fun attachCtx :: "arexp \<Rightarrow> rexp list \<Rightarrow> rexp set" where
-  "attachCtx r ctx = set (turnIntoTerms (seqFold (erase r) ctx))"
-
-
-fun rs1_subseteq_rs2 :: "rexp set \<Rightarrow> rexp set \<Rightarrow> bool" where
-  "rs1_subseteq_rs2 rs1 rs2 = (rs1 \<subseteq> rs2)"
-
-
-fun notZero :: "arexp \<Rightarrow> bool" where
-  "notZero AZERO = True"
-| "notZero _ = False"
-
-
-fun tset :: "arexp list \<Rightarrow> rexp set" where
-  "tset rs = set (concat (map turnIntoTerms (map erase rs)))"
-
-
-fun prune6 :: "rexp set \<Rightarrow> arexp \<Rightarrow> rexp list \<Rightarrow> arexp" where
-  "prune6 acc a ctx = (if (ABIncludedByC a ctx acc attachCtx rs1_subseteq_rs2) then AZERO else 
-                        (case a of (ASEQ bs r1 r2) \<Rightarrow> bsimp_ASEQ bs (prune6 acc r1 (erase r2 # ctx)) r2
-                                 | AALTs bs rs0 \<Rightarrow> bsimp_AALTs bs (filter notZero (map (\<lambda>r.(prune6 acc r ctx)) rs0))
-                                 | r \<Rightarrow> r
-                        )
-                      )
-            "
-
-abbreviation
-  "p6 acc r \<equiv> prune6 (tset acc) r Nil"
-
-
-fun dB6 :: "arexp list \<Rightarrow> rexp set \<Rightarrow> arexp list" where
-  "dB6 [] acc = []"
-| "dB6 (a # as) acc = (if (erase a \<in> acc) then (dB6 as acc) 
-                       else (let pruned = prune6 acc a [] in 
-                              (case pruned of
-                                 AZERO \<Rightarrow> dB6 as acc
-                               |xPrime \<Rightarrow> xPrime # (dB6 as ( (set (turnIntoTerms (erase pruned))) \<union> acc)  ) ) ))   "
-
-
-fun bsimpStrong6 :: "arexp \<Rightarrow> arexp" 
-  where
-  "bsimpStrong6 (ASEQ bs1 r1 r2) = bsimp_ASEQ bs1 (bsimpStrong6 r1) (bsimpStrong6 r2)"
-| "bsimpStrong6 (AALTs bs1 rs) = bsimp_AALTs bs1 (dB6 (flts (map bsimpStrong6 rs)) {}) "
-| "bsimpStrong6 r = r"
-
-
-fun 
-  bdersStrong6 :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
-where
-  "bdersStrong6 r [] = r"
-| "bdersStrong6 r (c # s) = bdersStrong6 (bsimpStrong6 (bder c r)) s"
-
-definition blexerStrong where
- "blexerStrong r s \<equiv> if bnullable (bdersStrong6 (intern r) s) then 
-                    decode (bmkeps (bdersStrong6 (intern r) s)) r else None"
-
-
-
 inductive 
   rrewrite:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto> _" [99, 99] 99)
 and 
@@ -162,9 +144,7 @@
 | ss4:  "(AZERO # rs) s\<leadsto> rs"
 | ss5:  "(AALTs bs1 rs1 # rsb) s\<leadsto> ((map (fuse bs1) rs1) @ rsb)"
 | ss6:  "L (erase a2) \<subseteq> L (erase a1) \<Longrightarrow> (rsa@[a1]@rsb@[a2]@rsc) s\<leadsto> (rsa@[a1]@rsb@rsc)"
-| ss7:  " (as @ [a] @ as1) s\<leadsto> (as @ [p6 as a] @ as1)"
 
-thm tset.simps
 
 inductive 
   rrewrites:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto>* _" [100, 100] 100)
@@ -203,6 +183,8 @@
    apply(auto)
   done
 
+
+
 lemma contextrewrites0: 
   "rs1 s\<leadsto>* rs2 \<Longrightarrow> AALTs bs rs1 \<leadsto>* AALTs bs rs2"
   apply(induct rs1 rs2 rule: srewrites.inducts)
@@ -227,24 +209,17 @@
    apply(auto)
   using srewrite1 by blast
 
-lemma srewrites_prepend:
-  shows "rs1 s\<leadsto>* rs2 \<Longrightarrow> (r # rs1) s\<leadsto>* (r # rs2)"
-  by (metis append_Cons append_Nil srewrites1)  
-
 lemma srewrite2: 
   shows  "r1 \<leadsto> r2 \<Longrightarrow> True"
   and "rs1 s\<leadsto> rs2 \<Longrightarrow> (rs1 @ rs) s\<leadsto>* (rs2 @ rs)"
-  apply(induct arbitrary: rs rule: rrewrite_srewrite.inducts)
-                apply simp+
-  using srewrites_prepend apply force
-      apply (simp add: rs_in_rstar ss3)
-  using ss4 apply force 
-  using rs_in_rstar ss5 apply auto[1]
-  using rs_in_rstar ss6 apply auto[1]
-  using rs_in_rstar ss7 by force
-
-
-
+  apply(induct rule: rrewrite_srewrite.inducts)
+  apply(auto)
+  apply (metis append_Cons append_Nil srewrites1)
+  apply(meson srewrites.simps ss3)
+  apply (meson srewrites.simps ss4)
+  apply (meson srewrites.simps ss5)
+  by (metis append_Cons append_Nil srewrites.simps ss6)
+  
 
 lemma srewrites3: 
   shows "rs1 s\<leadsto>* rs2 \<Longrightarrow> (rs1 @ rs) s\<leadsto>* (rs2 @ rs)"
@@ -252,6 +227,15 @@
    apply(auto)
   by (meson srewrite2(2) srewrites_trans)
 
+(*
+lemma srewrites4:
+  assumes "rs3 s\<leadsto>* rs4" "rs1 s\<leadsto>* rs2" 
+  shows "(rs1 @ rs3) s\<leadsto>* (rs2 @ rs4)"
+  using assms
+  apply(induct rs3 rs4 arbitrary: rs1 rs2 rule: srewrites.induct)
+  apply (simp add: srewrites3)
+  using srewrite1 by blast
+*)
 
 lemma srewrites6:
   assumes "r1 \<leadsto>* r2" 
@@ -267,178 +251,27 @@
   using assms
   by (smt (verit, best) append_Cons append_Nil srewrites1 srewrites3 srewrites6 srewrites_trans)
 
-(*harmless sorry*)
-lemma existing_preimage :
-  shows "f a \<in> f ` set rs1 \<Longrightarrow> \<exists>rs1a rs1b x. rs1 = rs1a @ [x] @ rs1b \<and> f x = f a "
-  apply(induct rs1)
-   apply simp
-  apply(case_tac "f a = f aa")
-  
-  sorry
-
-
-lemma deletes_dB:
-  shows " \<lbrakk>erase a \<in> erase ` set rs1\<rbrakk> \<Longrightarrow> (rs1 @ a # rs2) s\<leadsto>* (rs1 @ rs2)"
-  apply(subgoal_tac "\<exists>rs1a rs1b x. rs1 = rs1a @ [x] @ rs1b \<and> erase x = erase a")
+lemma ss6_stronger_aux:
+  shows "(rs1 @ rs2) s\<leadsto>* (rs1 @ distinctWith rs2 eq1 (set rs1))"
+  apply(induct rs2 arbitrary: rs1)
+  apply(auto)
   prefer 2
-   apply (meson existing_preimage)
+  apply(drule_tac x="rs1 @ [a]" in meta_spec)
+  apply(simp)
+  apply(drule_tac x="rs1" in meta_spec)
+  apply(subgoal_tac "(rs1 @ a # rs2) s\<leadsto>* (rs1 @ rs2)")
+  using srewrites_trans apply blast
+  apply(subgoal_tac "\<exists>rs1a rs1b. rs1 = rs1a @ [x] @ rs1b")
+  prefer 2
+  apply (simp add: split_list)
   apply(erule exE)+
-  apply simp
-  apply(subgoal_tac "(rs1a @ [x] @ rs1b @ [a] @ rs2) s\<leadsto> (rs1a @ [x] @ rs1b @ rs2)")  
-  apply (simp add: rs_in_rstar)
-  apply(subgoal_tac "L (erase a) \<subseteq> L (erase x)")
-  using ss6 apply presburger
-  by simp
-
-
-
-lemma ss6_realistic:
-  shows "(rs1 @ rs2) s\<leadsto>* (rs1 @ dB6 rs2 (tset rs1))"
-  apply(induct rs2 arbitrary: rs1)
-   apply simp
-  apply(rename_tac cc' cc)
-  apply(subgoal_tac "(cc @ a # cc') s\<leadsto>* (cc @ a # dB6 cc' (tset (cc @ [a])))")
-   prefer 2
-  apply (metis append.assoc append.left_neutral append_Cons)
-  apply(subgoal_tac "(cc @ a # dB6 cc' (tset (cc @ [a]))) s\<leadsto>* (cc @ (p6 cc a) # dB6 cc' (tset (cc @ [a])))") 
-  sorry
-
-
-
+  apply(simp)
+  using eq1_L rs_in_rstar ss6 by force
 
 lemma ss6_stronger:
-  shows "rs1 s\<leadsto>* dB6 rs1 {}"
-  using ss6
-  by (metis append_Nil concat.simps(1) list.set(1) list.simps(8) ss6_realistic tset.simps)
-
- 
-lemma tint_fuse:
-  shows "tint (erase a) = tint (erase (fuse bs a))"
-  by (simp add: erase_fuse)
-
-lemma tint_fuse2:
-  shows " set (tint (erase a)) =
-     set (tint (erase (fuse bs a)))"
-  using tint_fuse by auto
-
-lemma fused_bits_at_head:
-  shows "fuse bs a = ASEQ bs1 a1 a2 \<Longrightarrow> \<exists>bs2. bs1 = bs @ bs2"
-  
-(* by (smt (verit) arexp.distinct(13) arexp.distinct(19) arexp.distinct(25) arexp.distinct(27) arexp.distinct(5) arexp.inject(3) fuse.elims)
-harmless sorry
-*)
-
-
-  sorry
-
-thm seqFold.simps
-
-lemma seqFold_concats:
-  shows "r \<noteq> ONE \<Longrightarrow> seqFold r [r1] = SEQ r r1"
-  apply(case_tac r)
-       apply simp+
-done
-
-
-lemma tint_seqFold_eq: shows
-"set (tint (seqFold (erase x42) [erase x43])) = 
-           set (tint (SEQ (erase x42) (erase x43)))"
-  apply(case_tac "erase x42 = ONE")
-   apply simp
-  using seqFold_concats
-  apply simp
-  done
-
-fun top :: "arexp \<Rightarrow> bit list" where
-  "top AZERO = []"
-| "top (AONE bs) = bs"
-| "top (ASEQ bs r1 r2) = bs"
-| "top (ACHAR v va) = v"
-| "top (AALTs v va) = v"
-| "top (ASTAR v va) = v "
-
-
-
-
-lemma p6fa_aux:
-  shows " fuse bs
-            (bsimp_AALTs bs\<^sub>0 as) = 
-           
-            (bsimp_AALTs (bs @ bs\<^sub>0) as)"
-  by (metis bsimp_AALTs.simps(1) bsimp_AALTs.simps(2) bsimp_AALTs.simps(3) fuse.simps(1) fuse.simps(4) fuse_append neq_Nil_conv)
-
+  shows "rs1 s\<leadsto>* distinctWith rs1 eq1 {}"
+  by (metis append_Nil list.set(1) ss6_stronger_aux)
 
-lemma p6pfuse_alts:
-  shows 
-" \<And>bs\<^sub>0 as\<^sub>0.     
-       \<lbrakk>\<And>a bs. set (tint (erase a)) = set (tint (erase (fuse bs a))); a = AALTs bs\<^sub>0 as\<^sub>0\<rbrakk>
-       \<Longrightarrow> \<not> set (tint (erase a)) \<subseteq> (\<Union>x\<in>set as. set (tint (erase x))) \<longrightarrow>
-           fuse bs
-            (case a of AZERO \<Rightarrow> AZERO | AONE x \<Rightarrow> AONE x | ACHAR x xa \<Rightarrow> ACHAR x xa
-             | ASEQ bs r1 r2 \<Rightarrow> bsimp_ASEQ bs (prune6 (\<Union>x\<in>set as. set (tint (erase x))) r1 [erase r2]) r2
-             | AALTs bs rs0 \<Rightarrow> bsimp_AALTs bs (filter notZero (map (\<lambda>r. prune6 (\<Union>x\<in>set as. set (tint (erase x))) r []) rs0)) | ASTAR x xa \<Rightarrow> ASTAR x xa) 
-              =
-           (case fuse bs a of AZERO \<Rightarrow> AZERO | AONE x \<Rightarrow> AONE x | ACHAR x xa \<Rightarrow> ACHAR x xa
-            | ASEQ bs r1 r2 \<Rightarrow> bsimp_ASEQ bs (prune6 (\<Union>x\<in>set as. set (tint (erase x))) r1 [erase r2]) r2
-            | AALTs bs rs0 \<Rightarrow> bsimp_AALTs bs (filter notZero (map (\<lambda>r. prune6 (\<Union>x\<in>set as. set (tint (erase x))) r []) rs0)) | ASTAR x xa \<Rightarrow> ASTAR x xa)"
-  apply simp
-  using p6fa_aux by presburger
-
-
-
-
-lemma prune6_preserves_fuse:
-  shows "fuse bs (p6 as a) = p6 as (fuse bs a)"
-  using tint_fuse2
-  apply simp
-  apply(case_tac a)
-       apply simp
-  apply simp
-     apply simp
-
-  using fused_bits_at_head
-
-    apply simp
-  using tint_seqFold_eq
-  apply simp
-    apply (smt (z3) bsimp_ASEQ.simps(1) bsimp_ASEQ0 bsimp_ASEQ1 bsimp_ASEQ2 fuse.simps(1) fuse.simps(5) fuse_append)
-  using p6pfuse_alts apply presburger
-  by simp
-
-
-(*
-The top-level bitlist stays the same:
-\<^sub>b\<^sub>sa ------pruning----->  \<^sub>b\<^sub>s\<^sub>' b  \<Longrightarrow>        \<exists>bs3. bs' = bs @ bs3 
-*)
-lemma top_bitcodes_preserved_p6:
-  shows "top r = bs \<Longrightarrow> p6 as r = AZERO \<or> (\<exists>bs3. top (p6 as r) = bs @ bs3)"
-  
-
-  apply(induct r arbitrary: as)
-
-(*this sorry requires more care *)
-  
-  sorry
-
-
-
-corollary prune6_preserves_fuse_srewrite:
-  shows "(as @ map (fuse bs) [a] @ as2) s\<leadsto>* (as @ map (fuse bs) [p6 as a] @ as2)"
-  apply(subgoal_tac "map (fuse bs) [a] = [fuse bs a]")
-  apply(subgoal_tac "(as @ [fuse bs a] @ as2) s\<leadsto>* (as @ [ (p6 as (fuse bs a))] @ as2)")
-  using prune6_preserves_fuse apply auto[1]
-  using rs_in_rstar ss7 apply presburger
-  by simp
-
-lemma prune6_invariant_fuse:
-  shows "p6 as a = p6 (map (fuse bs) as) a"
-  apply simp
-  using erase_fuse by presburger
-
-
-lemma p6pfs_cor:
-  shows "(map (fuse bs) as @ map (fuse bs) [a] @ map (fuse bs) as1) s\<leadsto>* (map (fuse bs) as @ map (fuse bs) [p6 as a] @ map (fuse bs) as1)"
-  by (metis prune6_invariant_fuse prune6_preserves_fuse_srewrite)
 
 lemma rewrite_preserves_fuse: 
   shows "r2 \<leadsto> r3 \<Longrightarrow> fuse bs r2 \<leadsto> fuse bs r3"
@@ -468,13 +301,7 @@
   then show ?case 
     apply(simp only: map_append)
     by (smt (verit, best) erase_fuse list.simps(8) list.simps(9) rrewrite_srewrite.ss6 srewrites.simps)
-next
-  case (ss7 as a as1)
-  then show ?case
-    apply(simp only: map_append)
-    using p6pfs_cor by presburger
 qed (auto intro: rrewrite_srewrite.intros)
-  
 
 
 lemma rewrites_fuse:  
@@ -507,12 +334,17 @@
   shows "ASEQ bs1 r1 r2 \<leadsto>* AZERO"
 using assms bs1 star_seq by blast
 
-
+(*
+lemma continuous_rewrite2: 
+  assumes "r1 \<leadsto>* AONE bs"
+  shows "ASEQ bs1 r1 r2 \<leadsto>* (fuse (bs1 @ bs) r2)"
+  using assms  by (meson bs3 rrewrites.simps star_seq)
+*)
 
 lemma bsimp_aalts_simpcases: 
-  shows "AONE bs \<leadsto>* bsimpStrong6 (AONE bs)"  
-  and   "AZERO \<leadsto>* bsimpStrong6 AZERO" 
-  and   "ACHAR bs c \<leadsto>* bsimpStrong6 (ACHAR bs c)"
+  shows "AONE bs \<leadsto>* bsimp (AONE bs)"  
+  and   "AZERO \<leadsto>* bsimp AZERO" 
+  and   "ACHAR bs c \<leadsto>* bsimp (ACHAR bs c)"
   by (simp_all)
 
 lemma bsimp_AALTs_rewrites: 
@@ -535,20 +367,17 @@
   using rs1 srewrites7 apply presburger
   using srewrites7 apply force
   apply (simp add: srewrites7)
+   apply(simp add: srewrites7)
   by (simp add: srewrites7)
+  
 
 lemma bnullable0:
 shows "r1 \<leadsto> r2 \<Longrightarrow> bnullable r1 = bnullable r2" 
   and "rs1 s\<leadsto> rs2 \<Longrightarrow> bnullables rs1 = bnullables rs2" 
-   apply(induct rule: rrewrite_srewrite.inducts)
-              apply simp
-             apply simp
-              apply (simp add: bnullable_fuse)
-  using bnullable.simps(5) apply presburger
-          apply simp
-         apply simp
-  sorry   
-
+  apply(induct rule: rrewrite_srewrite.inducts)
+  apply(auto simp add:  bnullable_fuse)
+   apply (meson UnCI bnullable_fuse imageI)
+  using bnullable_correctness nullable_correctness by blast 
 
 
 lemma rewritesnullable: 
@@ -557,7 +386,7 @@
 using assms 
   apply(induction r1 r2 rule: rrewrites.induct)
   apply simp
-  using bnullable0(1) by presburger
+  using bnullable0(1) by auto
 
 lemma rewrite_bmkeps_aux: 
   shows "r1 \<leadsto> r2 \<Longrightarrow> (bnullable r1 \<and> bnullable r2 \<Longrightarrow> bmkeps r1 = bmkeps r2)"
@@ -575,19 +404,22 @@
 next
   case (ss5 bs1 rs1 rsb)
   then show ?case
-    by (simp add: bmkepss1 bmkepss2 bmkepss_fuse bnullable_fuse)
+    apply (simp add: bmkepss1 bmkepss2 bmkepss_fuse bnullable_fuse)
+    apply(case_tac rs1)
+     apply(auto simp add: bnullable_fuse)
+    apply (metis bmkeps_retrieve bmkeps_retrieve_AALTs bnullable.simps(4))
+    apply (metis bmkeps_retrieve bmkeps_retrieve_AALTs bnullable.simps(4))
+    apply (metis bmkeps_retrieve bmkeps_retrieve_AALTs bnullable.simps(4))
+    by (metis bmkeps_retrieve bmkeps_retrieve_AALTs bnullable.simps(4))    
 next
   case (ss6 a1 a2 rsa rsb rsc)
   then show ?case
     by (smt (verit, best) Nil_is_append_conv bmkepss1 bmkepss2 bnullable_correctness in_set_conv_decomp list.distinct(1) list.set_intros(1) nullable_correctness set_ConsD subsetD)
 next
-           prefer 10
-  case (ss7 as a as1)
+  case (bs10 rs1 rs2 bs)
   then show ?case
-    
-(*this sorry requires more effort*)
-  sorry
-qed(auto)
+    by (metis bmkeps_retrieve bmkeps_retrieve_AALTs bnullable.simps(4)) 
+qed (auto)
 
 lemma rewrites_bmkeps: 
   assumes "r1 \<leadsto>* r2" "bnullable r1" 
@@ -609,6 +441,50 @@
 qed
 
 
+lemma rewrites_to_bsimp: 
+  shows "r \<leadsto>* bsimp r"
+proof (induction r rule: bsimp.induct)
+  case (1 bs1 r1 r2)
+  have IH1: "r1 \<leadsto>* bsimp r1" by fact
+  have IH2: "r2 \<leadsto>* bsimp r2" by fact
+  { assume as: "bsimp r1 = AZERO \<or> bsimp r2 = AZERO"
+    with IH1 IH2 have "r1 \<leadsto>* AZERO \<or> r2 \<leadsto>* AZERO" by auto
+    then have "ASEQ bs1 r1 r2 \<leadsto>* AZERO"
+      by (metis bs2 continuous_rewrite rrewrites.simps star_seq2)  
+    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" using as by auto
+  }
+  moreover
+  { assume "\<exists>bs. bsimp r1 = AONE bs"
+    then obtain bs where as: "bsimp r1 = AONE bs" by blast
+    with IH1 have "r1 \<leadsto>* AONE bs" by simp
+    then have "ASEQ bs1 r1 r2 \<leadsto>* fuse (bs1 @ bs) r2" using bs3 star_seq by blast
+    with IH2 have "ASEQ bs1 r1 r2 \<leadsto>* fuse (bs1 @ bs) (bsimp r2)"
+      using rewrites_fuse by (meson rrewrites_trans) 
+    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 (AONE bs) r2)" by simp
+    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by (simp add: as) 
+  } 
+  moreover
+  { assume as1: "bsimp r1 \<noteq> AZERO" "bsimp r2 \<noteq> AZERO" and as2: "(\<nexists>bs. bsimp r1 = AONE bs)" 
+    then have "bsimp_ASEQ bs1 (bsimp r1) (bsimp r2) = ASEQ bs1 (bsimp r1) (bsimp r2)" 
+      by (simp add: bsimp_ASEQ1) 
+    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp_ASEQ bs1 (bsimp r1) (bsimp r2)" using as1 as2 IH1 IH2
+      by (metis rrewrites_trans star_seq star_seq2) 
+    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by simp
+  } 
+  ultimately show "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by blast
+next
+  case (2 bs1 rs)
+  have IH: "\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* bsimp x" by fact
+  then have "rs s\<leadsto>* (map bsimp rs)" by (simp add: trivialbsimp_srewrites)
+  also have "... s\<leadsto>* flts (map bsimp rs)" by (simp add: fltsfrewrites) 
+  also have "... s\<leadsto>* distinctWith (flts (map bsimp rs)) eq1 {}" by (simp add: ss6_stronger)
+  finally have "AALTs bs1 rs \<leadsto>* AALTs bs1 (distinctWith (flts (map bsimp rs)) eq1 {})"
+    using contextrewrites0 by auto
+  also have "... \<leadsto>* bsimp_AALTs  bs1 (distinctWith (flts (map bsimp rs)) eq1 {})"
+    by (simp add: bsimp_AALTs_rewrites)     
+  finally show "AALTs bs1 rs \<leadsto>* bsimp (AALTs bs1 rs)" by simp
+qed (simp_all)
+
 
 lemma to_zero_in_alt: 
   shows "AALT bs (ASEQ [] AZERO r) r2 \<leadsto> AALT bs AZERO r2"
@@ -626,12 +502,6 @@
   shows "(map f [a]) = [f a]"
   by (simp)
 
-lemma "set (tint (erase a)) \<subseteq> (\<Union>x\<in>set as. set (tint (erase x))) \<Longrightarrow>
-       set (tint (erase (bder c a))) \<subseteq> (\<Union>x\<in>set (map (bder c) as). set (tint (erase x)))"
-  
-  sorry
-
-
 lemma rewrite_preserves_bder: 
   shows "r1 \<leadsto> r2 \<Longrightarrow> (bder c r1) \<leadsto>* (bder c r2)"
   and   "rs1 s\<leadsto> rs2 \<Longrightarrow> map (bder c) rs1 s\<leadsto>* map (bder c) rs2"
@@ -707,11 +577,6 @@
     apply(rule_tac rrewrite_srewrite.ss6)
     using Der_def der_correctness apply auto[1]
     by blast
-next
-  case (ss7 as a as1)
-  then show ?case
-    apply simp
-    sorry
 qed
 
 lemma rewrites_preserves_bder: 
@@ -720,105 +585,49 @@
 using assms  
 apply(induction r1 r2 rule: rrewrites.induct)
 apply(simp_all add: rewrite_preserves_bder rrewrites_trans)
-  done
-
+done
 
 
-lemma bders_simp_appendStrong :
-  shows "bdersStrong6 r (s1 @ s2) = bdersStrong6 (bdersStrong6 r s1) s2"
-  apply(induct s1 arbitrary: s2 rule: rev_induct)
-   apply simp
-  apply auto
-  done
+lemma central:  
+  shows "bders r s \<leadsto>* bders_simp r s"
+proof(induct s arbitrary: r rule: rev_induct)
+  case Nil
+  then show "bders r [] \<leadsto>* bders_simp r []" by simp
+next
+  case (snoc x xs)
+  have IH: "\<And>r. bders r xs \<leadsto>* bders_simp r xs" by fact
+  have "bders r (xs @ [x]) = bders (bders r xs) [x]" by (simp add: bders_append)
+  also have "... \<leadsto>* bders (bders_simp r xs) [x]" using IH
+    by (simp add: rewrites_preserves_bder)
+  also have "... \<leadsto>* bders_simp (bders_simp r xs) [x]" using IH
+    by (simp add: rewrites_to_bsimp)
+  finally show "bders r (xs @ [x]) \<leadsto>* bders_simp r (xs @ [x])" 
+    by (simp add: bders_simp_append)
+qed
+
+lemma main_aux: 
+  assumes "bnullable (bders r s)"
+  shows "bmkeps (bders r s) = bmkeps (bders_simp r s)"
+proof -
+  have "bders r s \<leadsto>* bders_simp r s" by (rule central)
+  then 
+  show "bmkeps (bders r s) = bmkeps (bders_simp r s)" using assms
+    by (rule rewrites_bmkeps)
+qed  
 
 
 
 
-lemma rewrites_to_bsimpStrong: 
-  shows "r \<leadsto>* bsimpStrong6 r"
-proof (induction r rule: bsimpStrong6.induct)
-  case (1 bs1 r1 r2)
-  have IH1: "r1 \<leadsto>* bsimpStrong6 r1" by fact
-  have IH2: "r2 \<leadsto>* bsimpStrong6 r2" by fact
-  { assume as: "bsimpStrong6 r1 = AZERO \<or> bsimpStrong6 r2 = AZERO"
-    with IH1 IH2 have "r1 \<leadsto>* AZERO \<or> r2 \<leadsto>* AZERO" by auto
-    then have "ASEQ bs1 r1 r2 \<leadsto>* AZERO"
-      by (metis bs2 continuous_rewrite rrewrites.simps star_seq2)  
-    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimpStrong6 (ASEQ bs1 r1 r2)" using as by auto
-  }
-  moreover
-  { assume "\<exists>bs. bsimpStrong6 r1 = AONE bs"
-    then obtain bs where as: "bsimpStrong6 r1 = AONE bs" by blast
-    with IH1 have "r1 \<leadsto>* AONE bs" by simp
-    then have "ASEQ bs1 r1 r2 \<leadsto>* fuse (bs1 @ bs) r2" using bs3 star_seq by blast
-    with IH2 have "ASEQ bs1 r1 r2 \<leadsto>* fuse (bs1 @ bs) (bsimpStrong6 r2)"
-      using rewrites_fuse by (meson rrewrites_trans) 
-    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimpStrong6 (ASEQ bs1 (AONE bs) r2)" by simp
-    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimpStrong6 (ASEQ bs1 r1 r2)" by (simp add: as) 
-  } 
-  moreover
-  { assume as1: "bsimpStrong6 r1 \<noteq> AZERO" "bsimpStrong6 r2 \<noteq> AZERO" and as2: "(\<nexists>bs. bsimpStrong6 r1 = AONE bs)" 
-    then have "bsimp_ASEQ bs1 (bsimpStrong6 r1) (bsimpStrong6 r2) = ASEQ bs1 (bsimpStrong6 r1) (bsimpStrong6 r2)" 
-      by (simp add: bsimp_ASEQ1) 
-    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp_ASEQ bs1 (bsimpStrong6 r1) (bsimpStrong6 r2)" using as1 as2 IH1 IH2
-      by (metis rrewrites_trans star_seq star_seq2) 
-    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimpStrong6 (ASEQ bs1 r1 r2)" by simp
-  } 
-  ultimately show "ASEQ bs1 r1 r2 \<leadsto>* bsimpStrong6 (ASEQ bs1 r1 r2)" 
-    by blast
-next
-  case (2 bs1 rs)
-  have IH: "\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* bsimpStrong6 x" by fact
-  then have "rs s\<leadsto>* (map bsimpStrong6 rs)" by (simp add: trivialbsimp_srewrites)
-  also have "... s\<leadsto>* flts (map bsimpStrong6 rs)" by (simp add: fltsfrewrites) 
-  also have "... s\<leadsto>* dB6 (flts (map bsimpStrong6 rs))  {}" by (simp add: ss6_stronger)
-  finally have "AALTs bs1 rs \<leadsto>* AALTs bs1 (dB6 (flts (map bsimpStrong6 rs))  {})"
-    using contextrewrites0 by auto
-  also have "... \<leadsto>* bsimp_AALTs  bs1 (dB6 (flts (map bsimpStrong6 rs))  {})"
-    by (simp add: bsimp_AALTs_rewrites)     
-  finally show "AALTs bs1 rs \<leadsto>* bsimpStrong6 (AALTs bs1 rs)" 
-    using bsimpStrong6.simps(2) by presburger
-qed (simp_all)
+theorem main_blexer_simp: 
+  shows "blexer r s = blexer_simp r s"
+  unfolding blexer_def blexer_simp_def
+  by (metis central main_aux rewritesnullable)
+
+theorem blexersimp_correctness: 
+  shows "lexer r s = blexer_simp r s"
+  using blexer_correctness main_blexer_simp by simp
 
 
-
-
-lemma centralStrong:  
-  shows "bders r s \<leadsto>* bdersStrong6 r s"
-proof(induct s arbitrary: r rule: rev_induct)
-  case Nil
-  then show "bders r [] \<leadsto>* bdersStrong6 r []" by simp
-next
-  case (snoc x xs)
-  have IH: "\<And>r. bders r xs \<leadsto>* bdersStrong6 r xs" by fact
-  have "bders r (xs @ [x]) = bders (bders r xs) [x]" by (simp add: bders_append)
-  also have "... \<leadsto>* bders (bdersStrong6 r xs) [x]" using IH
-    by (simp add: rewrites_preserves_bder)
-  also have "... \<leadsto>* bdersStrong6 (bdersStrong6 r xs) [x]" using IH
-    by (simp add: rewrites_to_bsimpStrong)
-  finally show "bders r (xs @ [x]) \<leadsto>* bdersStrong6 r (xs @ [x])" 
-    by (simp add: bders_simp_appendStrong)
-qed
-
-lemma mainStrong: 
-  assumes "bnullable (bders r s)"
-  shows "bmkeps (bders r s) = bmkeps (bdersStrong6 r s)"
-proof -
-  have "bders r s \<leadsto>* bdersStrong6 r s" 
-    using centralStrong by force
-  then 
-  show "bmkeps (bders r s) = bmkeps (bdersStrong6 r s)" 
-    using assms rewrites_bmkeps by blast
-qed
-
-
-
-
-theorem blexerStrong_correct :
-  shows "blexerStrong r s = blexer r s"
-  unfolding blexerStrong_def blexer_def
-  by (metis centralStrong mainStrong rewritesnullable)
-
-
+unused_thms
 
 end




