--- 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