
diff -r 8194086c2a8a -r 3954579ebdaf thys2/SizeBound2.thy
--- a/thys2/SizeBound2.thy	Thu Jan 20 01:48:18 2022 +0000
+++ b/thys2/SizeBound2.thy	Sat Jan 22 10:48:09 2022 +0000
@@ -183,6 +183,11 @@
 | "bmkeps(AALTs bs (r#rs)) = (if bnullable(r) then bs @ (bmkeps r) else (bmkeps (AALTs bs rs)))"
 | "bmkeps(ASTAR bs r) = bs @ [S]"
 
+fun 
+  bmkepss :: "arexp list \<Rightarrow> bit list"
+where
+  "bmkepss [] = []"
+| "bmkepss (r # rs) = (if bnullable(r) then (bmkeps r) else (bmkepss rs))"
 
 fun
  bder :: "char \<Rightarrow> arexp \<Rightarrow> arexp"
@@ -539,9 +544,7 @@
      (if (f x) \<in> acc then distinctBy xs f acc 
       else x # (distinctBy xs f ({f x} \<union> acc)))"
 
-lemma dB_single_step: 
-  shows "distinctBy (a#rs) f {} = a # distinctBy rs f {f a}"
-  by simp 
+ 
 
 fun flts :: "arexp list \<Rightarrow> arexp list"
   where 
@@ -559,6 +562,22 @@
 | "bsimp_ASEQ bs1 (AONE bs2) r2 = fuse (bs1 @ bs2) r2"
 | "bsimp_ASEQ bs1 r1 r2 = ASEQ  bs1 r1 r2"
 
+lemma bsimp_ASEQ0[simp]:
+  shows "bsimp_ASEQ bs r1 AZERO = AZERO"
+  by (case_tac r1)(simp_all)
+
+lemma bsimp_ASEQ1:
+  assumes "r1 \<noteq> AZERO" "r2 \<noteq> AZERO" "\<nexists>bs. r1 = AONE bs"
+  shows "bsimp_ASEQ bs r1 r2 = ASEQ bs r1 r2"
+  using assms
+  apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
+  apply(auto)
+  done
+
+lemma bsimp_ASEQ2[simp]:
+  shows "bsimp_ASEQ bs1 (AONE bs2) r2 = fuse (bs1 @ bs2) r2"
+  by (case_tac r2) (simp_all)
+
 
 fun bsimp_AALTs :: "bit list \<Rightarrow> arexp list \<Rightarrow> arexp"
   where
@@ -584,7 +603,7 @@
  "blexer_simp r s \<equiv> if bnullable (bders_simp (intern r) s) then 
                     decode (bmkeps (bders_simp (intern r) s)) r else None"
 
-export_code bders_simp in Scala module_name Example
+(*export_code bders_simp in Scala module_name Example*)
 
 lemma bders_simp_append:
   shows "bders_simp r (s1 @ s2) = bders_simp (bders_simp r s1) s2"
@@ -657,25 +676,7 @@
 
 
 
-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:
@@ -849,6 +850,7 @@
 | ss5:  "(AALTs bs1 rs1 # rsb) s\<leadsto> ((map (fuse bs1) rs1) @ rsb)"
 | ss6:  "erase a1 = erase a2 \<Longrightarrow> (rsa@[a1]@rsb@[a2]@rsc) s\<leadsto> (rsa@[a1]@rsb@rsc)"
 
+
 inductive 
   rrewrites:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto>* _" [100, 100] 100)
 where 
@@ -882,14 +884,6 @@
   done
 
 
-lemma rewrite_fuse : 
-  assumes "r1 \<leadsto> r2"
-  shows "fuse bs r1 \<leadsto> fuse bs r2"
-  using assms
-  apply(induct rule: rrewrite_srewrite.inducts(1))
-  apply(auto intro: rrewrite_srewrite.intros)
-  apply (metis bs3 fuse_append)
-  by (metis bs7 fuse_append)
 
 lemma contextrewrites0: 
   "rs1 s\<leadsto>* rs2 \<Longrightarrow> AALTs bs rs1 \<leadsto>* AALTs bs rs2"
@@ -947,7 +941,6 @@
   assumes "r1 \<leadsto>* r2" 
   shows "[r1] s\<leadsto>* [r2]"
   using assms
-  
   apply(induct r1 r2 rule: rrewrites.induct)
    apply(auto)
   by (meson srewrites.simps srewrites_trans ss3)
@@ -958,6 +951,58 @@
   using assms
   by (smt (verit, best) append_Cons append_Nil srewrites1 srewrites3 srewrites6 srewrites_trans)
 
+lemma ss6_stronger_aux:
+  shows "(rs1 @ rs2) s\<leadsto>* (rs1 @ distinctBy rs2 erase (set (map erase rs1)))"
+  apply(induct rs2 arbitrary: rs1)
+   apply(auto)
+  apply (smt (verit, best) append.assoc append.right_neutral append_Cons append_Nil split_list srewrite2(2) srewrites_trans ss6)
+  apply(drule_tac x="rs1 @ [a]" in meta_spec)
+  apply(simp)
+  done
+
+lemma ss6_stronger:
+  shows "rs1 s\<leadsto>* distinctBy rs1 erase {}"
+  using ss6_stronger_aux[of "[]" _] by auto
+
+
+lemma rewrite_preserves_fuse: 
+  shows "r2 \<leadsto> r3 \<Longrightarrow> fuse bs r2 \<leadsto> fuse bs r3"
+  and   "rs2 s\<leadsto> rs3 \<Longrightarrow> map (fuse bs) rs2 s\<leadsto>* map (fuse bs) rs3"
+proof(induct rule: rrewrite_srewrite.inducts)
+  case (bs3 bs1 bs2 r)
+  then show ?case
+    by (metis fuse.simps(5) fuse_append rrewrite_srewrite.bs3) 
+next
+  case (bs7 bs r)
+  then show ?case
+    by (metis fuse.simps(4) fuse_append rrewrite_srewrite.bs7) 
+next
+  case (ss2 rs1 rs2 r)
+  then show ?case
+    using srewrites7 by force 
+next
+  case (ss3 r1 r2 rs)
+  then show ?case by (simp add: r_in_rstar srewrites7)
+next
+  case (ss5 bs1 rs1 rsb)
+  then show ?case 
+    apply(simp)
+    by (metis (mono_tags, lifting) comp_def fuse_append map_eq_conv rrewrite_srewrite.ss5 srewrites.simps)
+next
+  case (ss6 a1 a2 rsa rsb rsc)
+  then show ?case 
+    apply(simp only: map_append)
+    by (smt (verit, ccfv_threshold) erase_fuse list.simps(8) list.simps(9) rrewrite_srewrite.ss6 srewrites.simps)  
+qed (auto intro: rrewrite_srewrite.intros)
+
+
+lemma rewrites_fuse:  
+  assumes "r1 \<leadsto>* r2"
+  shows "fuse bs r1 \<leadsto>* fuse bs r2"
+using assms
+apply(induction r1 r2 arbitrary: bs rule: rrewrites.induct)
+apply(auto intro: rewrite_preserves_fuse rrewrites_trans)
+done
 
 
 lemma star_seq:  
@@ -981,6 +1026,12 @@
   shows "ASEQ bs1 r1 r2 \<leadsto>* AZERO"
 using assms bs1 star_seq by blast
 
+(*
+lemma continuous_rewrite2: 
+  assumes "r1 \<leadsto>* AONE bs"
+  shows "ASEQ bs1 r1 r2 \<leadsto>* (fuse (bs1 @ bs) r2)"
+  using assms  by (meson bs3 rrewrites.simps star_seq)
+*)
 
 lemma bsimp_aalts_simpcases: 
   shows "AONE bs \<leadsto>* bsimp (AONE bs)"  
@@ -988,6 +1039,9 @@
   and   "ACHAR bs c \<leadsto>* bsimp (ACHAR bs c)"
   by (simp_all)
 
+lemma bsimp_AALTs_rewrites: 
+  shows "AALTs bs1 rs \<leadsto>* bsimp_AALTs bs1 rs"
+  by (smt (verit) bs6 bs7 bsimp_AALTs.elims rrewrites.simps)
 
 lemma trivialbsimp_srewrites: 
   "\<lbrakk>\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* f x \<rbrakk> \<Longrightarrow> rs s\<leadsto>* (map f rs)"
@@ -996,129 +1050,17 @@
   apply(simp)
   using srewrites7 by auto
 
-lemma alts_simpalts: 
-  "(\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* bsimp x) \<Longrightarrow> 
-  AALTs bs1 rs \<leadsto>* AALTs bs1 (map bsimp rs)"
-  apply(induct rs)
-   apply(auto)[1]
-  using trivialbsimp_srewrites apply auto[1]
-  by (simp add: contextrewrites0 srewrites7)
-
-
-lemma bsimp_AALTs_rewrites: 
-  shows "AALTs bs1 rs \<leadsto>* bsimp_AALTs bs1 rs"
-  by (smt (verit) bs6 bs7 bsimp_AALTs.elims rrewrites.simps)
-
-lemma fltsfrewrites: "rs s\<leadsto>* (flts rs)"
-  
-  apply(induction rs)
-  apply simp
-  apply(case_tac a)
-       apply(auto)
-  using ss4 apply blast
-  using srewrites7 apply force
-  using rs1 srewrites7 apply presburger
-  using srewrites7 apply force
-  apply (meson srewrites.simps srewrites1 ss5)
-  by (simp add: srewrites7)
-
-
-lemma flts_rewrites: "AALTs bs1 rs \<leadsto>* AALTs bs1 (flts rs)"
-  by (simp add: contextrewrites0 fltsfrewrites)
 
 
-(* delete*)
-lemma threelistsappend: "rsa@a#rsb = (rsa@[a])@rsb"
-  apply auto
-  done
-
-lemma somewhereInside: "r \<in> set rs \<Longrightarrow> \<exists>rs1 rs2. rs = rs1@[r]@rs2"
-  using split_list by fastforce
-
-lemma somewhereMapInside: "f r \<in> f ` set rs \<Longrightarrow> \<exists>rs1 rs2 a. rs = rs1@[a]@rs2 \<and> f a = f r"
-  apply auto
-  by (metis split_list)
-
-lemma alts_dBrewrites_withFront: 
-  "AALTs bs (rsa @ rs) \<leadsto>* AALTs bs (rsa @ distinctBy rs erase (erase ` set rsa))"
-  
-  apply(induction rs arbitrary: rsa)
-   apply simp
-  
-  apply(drule_tac x = "rsa@[a]" in meta_spec)
-  
-  apply(subst threelistsappend)
-  apply(rule rrewrites_trans)
-   apply simp
-  
-  apply(case_tac "a \<in> set rsa")
-   apply simp
-   apply(drule somewhereInside)
-   apply(erule exE)+
-   apply simp
-  using bs10 ss6 apply auto[1]
-  
-  apply(subgoal_tac "erase ` set (rsa @ [a]) = insert (erase a) (erase ` set rsa)")
-  prefer 2
-    
-   apply auto[1]
-  apply(case_tac "erase a \<in> erase `set rsa")
-
-   apply simp
-  apply(subgoal_tac "AALTs bs (rsa @ a # distinctBy rs erase (insert (erase a) (erase ` set rsa))) \<leadsto>
-                     AALTs bs (rsa @ distinctBy rs erase (insert (erase a) (erase ` set rsa)))")
-    apply force
-  apply (smt (verit, ccfv_threshold) append.assoc append.left_neutral append_Cons append_Nil bs10 imageE insertCI insert_image somewhereMapInside ss6)
-  by simp
-
- 
+lemma fltsfrewrites: "rs s\<leadsto>* (flts rs)"
+  apply(induction rs rule: flts.induct)
+  apply(auto intro: rrewrite_srewrite.intros)
+  apply (meson srewrites.simps srewrites1 ss5)
+  using rs1 srewrites7 apply presburger
+  using srewrites7 apply force
+  apply (simp add: srewrites7)
+  by (simp add: srewrites7)
 
-lemma alts_dBrewrites: 
-  shows "AALTs bs rs \<leadsto>* AALTs bs (distinctBy rs erase {})"
-  
-  apply(induction rs)
-   apply simp
-  apply simp
-  using alts_dBrewrites_withFront
-  by (metis append_Nil dB_single_step empty_set image_empty)
-
-lemma bsimp_rewrite: 
-  shows "r \<leadsto>* bsimp r"
-proof (induction r rule: bsimp.induct)
-  case (1 bs1 r1 r2)
-  then show "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)"
-    apply(simp)
-    apply(case_tac "bsimp r1 = AZERO")
-        apply simp
-  using continuous_rewrite apply blast
-       apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
-        apply(erule exE)
-        apply simp
-        apply(subst bsimp_ASEQ2)
-        apply (meson rrewrites_trans rrewrite_srewrite.intros(3) rrewrites.intros(2) star_seq star_seq2)
-       apply (smt (verit, best) bsimp_ASEQ0 bsimp_ASEQ1 rrewrites_trans rrewrite_srewrite.intros(2) rs2 star_seq star_seq2)
-  done
-next
-  case (2 bs1 rs)
-  then show "AALTs bs1 rs \<leadsto>* bsimp (AALTs bs1 rs)"
-    by (metis alts_dBrewrites alts_simpalts bsimp.simps(2) bsimp_AALTs_rewrites flts_rewrites rrewrites_trans)  
-next
-  case "3_1"
-  then show "AZERO \<leadsto>* bsimp AZERO"
-    by simp
-next
-  case ("3_2" v)
-  then show "AONE v \<leadsto>* bsimp (AONE v)" 
-    by simp
-next
-  case ("3_3" v va)
-  then show "ACHAR v va \<leadsto>* bsimp (ACHAR v va)" 
-    by simp
-next
-  case ("3_4" v va)
-  then show "ASTAR v va \<leadsto>* bsimp (ASTAR v va)" 
-    by simp
-qed
 
 lemma bnullable1:
 shows "r1 \<leadsto> r2 \<Longrightarrow> (bnullable r1 \<Longrightarrow> bnullable r2)" 
@@ -1154,7 +1096,7 @@
 lemma rewritesnullable: 
   assumes "r1 \<leadsto>* r2" "bnullable r1"
   shows "bnullable r2"
-using assms
+using assms 
   apply(induction r1 r2 rule: rrewrites.induct)
   apply simp
   using rewrite_non_nullable_strong by blast
@@ -1240,50 +1182,70 @@
   then show "bmkeps r1 = bmkeps r3" using IH by simp
 qed
 
+
+lemma rewrites_to_bsimp: 
+  shows "r \<leadsto>* bsimp r"
+proof (induction r rule: bsimp.induct)
+  case (1 bs1 r1 r2)
+  have IH1: "r1 \<leadsto>* bsimp r1" by fact
+  have IH2: "r2 \<leadsto>* bsimp r2" by fact
+  { assume as: "bsimp r1 = AZERO \<or> bsimp r2 = AZERO"
+    with IH1 IH2 have "r1 \<leadsto>* AZERO \<or> r2 \<leadsto>* AZERO" by auto
+    then have "ASEQ bs1 r1 r2 \<leadsto>* AZERO"
+      by (metis bs2 continuous_rewrite rrewrites.simps star_seq2)  
+    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" using as by auto
+  }
+  moreover
+  { assume "\<exists>bs. bsimp r1 = AONE bs"
+    then obtain bs where as: "bsimp r1 = AONE bs" by blast
+    with IH1 have "r1 \<leadsto>* AONE bs" by simp
+    then have "ASEQ bs1 r1 r2 \<leadsto>* fuse (bs1 @ bs) r2" using bs3 star_seq by blast
+    with IH2 have "ASEQ bs1 r1 r2 \<leadsto>* fuse (bs1 @ bs) (bsimp r2)"
+      using rewrites_fuse by (meson rrewrites_trans) 
+    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 (AONE bs) r2)" by simp
+    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by (simp add: as) 
+  } 
+  moreover
+  { assume as1: "bsimp r1 \<noteq> AZERO" "bsimp r2 \<noteq> AZERO" and as2: "(\<nexists>bs. bsimp r1 = AONE bs)" 
+    then have "bsimp_ASEQ bs1 (bsimp r1) (bsimp r2) = ASEQ bs1 (bsimp r1) (bsimp r2)" 
+      by (simp add: bsimp_ASEQ1) 
+    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp_ASEQ bs1 (bsimp r1) (bsimp r2)" using as1 as2 IH1 IH2
+      by (metis rrewrites_trans star_seq star_seq2) 
+    then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by simp
+  } 
+  ultimately show "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by blast
+next
+  case (2 bs1 rs)
+  have IH: "\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* bsimp x" by fact
+  then have "rs s\<leadsto>* (map bsimp rs)" by (simp add: trivialbsimp_srewrites)
+  also have "... s\<leadsto>* flts (map bsimp rs)" by (simp add: fltsfrewrites) 
+  also have "... s\<leadsto>* distinctBy (flts (map bsimp rs)) erase {}" by (simp add: ss6_stronger) 
+  finally have "AALTs bs1 rs \<leadsto>* AALTs bs1 (distinctBy (flts (map bsimp rs)) erase {})"
+    using contextrewrites0 by blast
+  also have "... \<leadsto>* bsimp_AALTs  bs1 (distinctBy (flts (map bsimp rs)) erase {})"
+    by (simp add: bsimp_AALTs_rewrites)     
+  finally show "AALTs bs1 rs \<leadsto>* bsimp (AALTs bs1 rs)" by simp
+next
+  case "3_1"
+  then show "AZERO \<leadsto>* bsimp AZERO" by simp
+next
+  case ("3_2" v)
+  then show "AONE v \<leadsto>* bsimp (AONE v)" by simp
+next
+  case ("3_3" v va)
+  then show "ACHAR v va \<leadsto>* bsimp (ACHAR v va)" by simp
+next
+  case ("3_4" v va)
+  then show "ASTAR v va \<leadsto>* bsimp (ASTAR v va)" by simp
+qed
+
+
+
 lemma to_zero_in_alt: 
-  shows "AALT bs (ASEQ [] AZERO r) r2 \<leadsto>  AALT bs AZERO r2"
+  shows "AALT bs (ASEQ [] AZERO r) r2 \<leadsto> AALT bs AZERO r2"
   by (simp add: bs1 bs10 ss3)
 
 
-lemma rewrite_fuse2: 
-  shows "r2 \<leadsto> r3 \<Longrightarrow> True"
-  and   "rs2 s\<leadsto> rs3 \<Longrightarrow> (\<And>bs. map (fuse bs) rs2 s\<leadsto>* map (fuse bs) rs3)"
-proof(induct rule: rrewrite_srewrite.inducts)
-  case ss1
-  then show ?case
-    by simp 
-next
-  case (ss2 rs1 rs2 r)
-  then show ?case
-    using srewrites7 by force 
-next
-  case (ss3 r1 r2 rs)
-  then show ?case
-    by (simp add: r_in_rstar rewrite_fuse srewrites7)
-next
-  case (ss4 rs)
-  then show ?case
-    by (metis fuse.simps(1) list.simps(9) rrewrite_srewrite.ss4 srewrites.simps) 
-next
-  case (ss5 bs1 rs1 rsb)
-  then show ?case 
-    apply(simp)
-    by (metis (mono_tags, lifting) comp_def fuse_append map_eq_conv rrewrite_srewrite.ss5 srewrites.simps)
-next
-  case (ss6 a1 a2 rsa rsb rsc)
-  then show ?case 
-    apply(simp only: map_append)
-    by (smt (verit, ccfv_threshold) erase_fuse list.simps(8) list.simps(9) rrewrite_srewrite.ss6 srewrites.simps)  
-qed (auto)
-
-
-lemma rewrites_fuse:  
-  assumes "r1 \<leadsto>* r2"
-  shows "fuse bs r1 \<leadsto>* fuse bs r2"
-using assms
-apply(induction r1 r2 arbitrary: bs rule: rrewrites.induct)
-apply(auto intro: rewrite_fuse rrewrites_trans)
-done
 
 lemma  bder_fuse_list: 
   shows "map (bder c \<circ> fuse bs1) rs1 = map (fuse bs1 \<circ> bder c) rs1"
@@ -1292,7 +1254,7 @@
   done
 
 
-lemma rewrite_after_der: 
+lemma rewrite_preserves_bder: 
   shows "r1 \<leadsto> r2 \<Longrightarrow> (bder c r1) \<leadsto>* (bder c r2)"
   and   "rs1 s\<leadsto> rs2 \<Longrightarrow> map (bder c) rs1 s\<leadsto>* map (bder c) rs2"
 proof(induction rule: rrewrite_srewrite.inducts)
@@ -1309,6 +1271,7 @@
   case (bs3 bs1 bs2 r)
   then show ?case 
     apply(simp)
+    
     by (metis (no_types, lifting) bder_fuse bs10 bs7 fuse_append rrewrites.simps ss4 to_zero_in_alt)
 next
   case (bs4 r1 r2 bs r3)
@@ -1364,12 +1327,12 @@
     by (smt (verit, best) erase_bder list.simps(8) list.simps(9) local.ss6 rrewrite_srewrite.ss6 srewrites.simps)
 qed
 
-lemma rewrites_after_der: 
+lemma rewrites_preserves_bder: 
   assumes "r1 \<leadsto>* r2"
   shows "bder c r1 \<leadsto>* bder c r2"
 using assms  
 apply(induction r1 r2 rule: rrewrites.induct)
-apply(simp_all add: rewrite_after_der rrewrites_trans)
+apply(simp_all add: rewrite_preserves_bder rrewrites_trans)
 done
 
 
@@ -1383,18 +1346,14 @@
   have IH: "\<And>r. bders r xs \<leadsto>* bders_simp r xs" by fact
   have "bders r (xs @ [x]) = bders (bders r xs) [x]" by (simp add: bders_append)
   also have "... \<leadsto>* bders (bders_simp r xs) [x]" using IH
-    by (simp add: rewrites_after_der)
+    by (simp add: rewrites_preserves_bder)
   also have "... \<leadsto>* bders_simp (bders_simp r xs) [x]" using IH
-    by (simp add: bsimp_rewrite)
+    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 quasi_main: 
+lemma main_aux: 
   assumes "bnullable (bders r s)"
   shows "bmkeps (bders r s) = bmkeps (bders_simp r s)"
 proof -
@@ -1407,15 +1366,15 @@
 
 
 
-theorem main_main: 
+theorem main_blexer_simp: 
   shows "blexer r s = blexer_simp r s"
   unfolding blexer_def blexer_simp_def
-  using b4 quasi_main by simp
+  using b4 main_aux by simp
 
 
 theorem blexersimp_correctness: 
   shows "lexer r s = blexer_simp r s"
-  using blexer_correctness main_main by simp
+  using blexer_correctness main_blexer_simp by simp
 
 
 




