theory BitCoded
imports "Lexer"
begin
section {* Bit-Encodings *}
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)"
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#rs)) = (\<forall>r' \<in> set (r#rs). good r')"
| "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 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"
value "good (AALTs [] [AALTs [] [AONE []]])"
value "bsimp (AALTs [] [AONE [], AALTs [] [AONE []]])"
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_thms_linordered_semiring(1) comp_apply fuse_size le_SucI order_refl sum_list_cong)
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 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
fun spill where
"spill (AALTs bs rs) = map (fuse bs) rs"
lemma k0a2:
assumes "\<not> nonalt r"
shows "flts [r] = spill r"
using assms
apply(case_tac r)
apply(simp_all)
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 good_fuse:
shows "good (fuse bs r) = good r"
apply(induct r)
apply(auto)
apply (metis arexp.distinct(25) arexp.distinct(7) arexp.inject(4) good.elims(3) good.simps(4) good.simps(5))
by (metis good.simps(4) good.simps(5) neq_Nil_conv)
lemma good0:
assumes "rs \<noteq> Nil"
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"
shows "good (bsimp (AALTs bs rs)) \<longleftrightarrow> (\<forall>r \<in> set (flts (map bsimp rs)). good r)"
using assms
apply(simp)
apply(rule good0)
apply(simp)
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'"
using assms
apply(induct r)
apply(simp)
apply(simp)
apply(simp)
prefer 2
apply(simp)
apply(auto)[1]
apply (metis good.simps(5) good_fuse in_set_insert insert_Nil list.exhaust)
prefer 2
apply(simp)
by fastforce
lemma flts3a:
assumes "good r"
shows "good (AALTs bs (flts [r]))"
using assms
by (metis flts1 flts2 good.simps(5) neq_Nil_conv)
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.simps(6) 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.simps(2) bsimp_AALTs.simps(3) flts.simps(1) flts.simps(2) flts1 good.simps(7) good_fuse neq_Nil_conv)
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 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
apply(simp only:)
apply(case_tac "x52")
apply(simp)
apply(simp only: good0a)
apply(frule_tac x="a" in spec)
apply(drule mp)
apply(simp)
apply(erule disjE)
prefer 2
apply(simp)
apply(frule_tac x="AALTs x51 list" in spec)
apply(drule mp)
apply(simp add: asize0)
apply(auto)[1]
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(simp)
apply(subst k0)
apply(simp)
apply(auto)[1]
using flts2 apply blast
apply (metis good0 in_set_member member_rec(2))
apply(simp)
apply(rule disjI1)
apply(drule flts4)
apply(subst k0)
apply(subst good0)
apply (metis append_is_Nil_conv flts1 k0)
apply(auto)[1]
using flts2 apply blast
apply (metis add.commute add_lessD1 flts_nil list.distinct(1) list.set_cases not_less_eq)
(* 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)
using less_add_Suc1 apply blast
using less_add_Suc2 by blast
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_idem:
assumes "\<forall>y. asize y < Suc (sum_list (map asize rs)) \<longrightarrow>
bsimp (bsimp y) = bsimp y"
shows "map bsimp (flts (map bsimp rs)) = flts (map bsimp rs)"
using assms
apply(induct rs)
apply(simp)
apply(simp)
apply(subst k0)
apply(subst (2) k0)
apply(simp add: flts_append)
using good1
apply -
apply(drule_tac x="a" in meta_spec)
apply(erule disjE)
prefer 2
apply(simp)
using flts.simps
apply(case_tac a)
apply(simp_all)
defer
apply(drule g1)
apply(erule disjE)
apply(simp)
defer
apply(auto)[1]
lemma flts_idem:
assumes "\<forall>y. asize y < Suc (sum_list (map asize rs)) \<longrightarrow>
bsimp (bsimp y) = bsimp y"
shows "flts (map bsimp (flts (map bsimp rs))) = flts (map bsimp rs)"
using assms
apply(induct rs)
apply(simp)
apply(simp)
apply(subst k0)
apply(subst (2) k0)
apply(simp add: flts_append)
using good1
apply -
apply(drule_tac x="a" in meta_spec)
apply(erule disjE)
prefer 2
apply(simp)
using flts.simps
apply(case_tac a)
apply(simp_all)
defer
apply(drule g1)
apply(erule disjE)
apply(simp)
defer
apply(auto)[1]
apply(subst g1)
apply(simp)
apply(simp)
apply (me tis (full_types) arexp.inject(4) bsimp_AALTs.simps(2) flts3a fuse_empty g1 list.distinct(1))
apply(case_tac "bsimp a = AZERO")
apply(simp)
apply(case_tac "nonalt (bsimp a)")
thm k0 k0a k0b
apply(subst k0b)
apply(simp)
apply(simp)
apply(simp)
apply(subst k0b)
apply(simp)
apply(simp)
apply(simp)
apply(subst k0)
apply(subst k0b)
apply(simp)
apply(simp)
apply(simp)
apply(simp)
apply(simp add: k00)
apply(subst k0a2)
apply(simp)
apply(subst k0a2)
apply(simp)
apply(case_tac a)
apply(simp_all)
oops
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)
lemma cc:
assumes "bsimp (fuse bs' r) = (AALTs bs rs)"
shows "\<forall>r \<in> set rs. r \<noteq> AZERO"
using assms
apply(induct r arbitrary: rs bs bs' rule: bsimp.induct)
apply(simp)
apply(case_tac "bsimp r1 = AZERO")
apply simp
apply(case_tac "bsimp r2 = AZERO")
apply(simp)
apply(case_tac "\<exists>bs'. bsimp r1 = AONE bs'")
apply(auto)[1]
apply (simp add: bsimp_ASEQ0)
apply(case_tac "\<exists>bs'. bsimp r1 = AONE bs'")
apply(auto)[2]
apply (simp add: bsimp_ASEQ2)
using bsimp_fuse apply fastforce
apply (simp add: bsimp_ASEQ1)
prefer 2
apply(simp)
defer
apply(simp)
apply(simp)
apply(simp)
(* AALT case *)
apply(simp only: fuse.simps)
apply(simp)
apply(case_tac "flts (map bsimp rs)")
apply(simp)
apply(simp)
apply(case_tac list)
apply(simp)
apply(case_tac a)
apply(simp_all)
apply(auto)
apply (metis ex_map_conv list.set_intros(1) nn1b nn1c nonalt.simps(1))
apply(case_tac rs)
apply(simp)
apply(simp)
apply(case_tac list)
apply(simp)
apply(subgoal_tac "\<forall>r \<in> set (flts (map bsimp rs)). r \<noteq> AZERO")
prefer 2
apply(rule_tac bs="bs' @ bs1" in flts_0)
thm bsimp_AALTs_qq
apply(case_tac "1 < length rs")
apply(drule_tac bsimp_AALTs_qq)
apply(subgoal_tac "nonnested (AALTs bs rsa)")
prefer 2
apply (metis nn1b)
apply(rule ballI)
apply(simp)
apply(drule_tac x="r" in meta_spec)
apply(simp)
(* HERE *)
apply(drule flts_0)
apply(simp)
apply(subst
apply (sm t arexp.distinct(15) arexp.distinct(21) arexp.distinct(25) arexp.distinct(29) arexp.inject(4) b1 fuse.elims)
prefer 2
apply(induct r arbitrary: rs bs bs' rule: bsimp.induct)
apply(auto)
apply(case_tac "bsimp r1 = AZERO")
apply simp
apply(case_tac "bsimp r2 = AZERO")
apply(simp)
apply(case_tac "\<exists>bs'. bsimp r1 = AONE bs'")
apply(auto)
apply (simp add: bsimp_ASEQ0)
apply(case_tac "\<exists>bs'. bsimp r1 = AONE bs'")
apply(auto)
apply (simp add: bsimp_ASEQ2)
using bsimp_fuse apply fast force
apply (simp add: bsimp_ASEQ1)
apply(subst
apply (sm t arexp.distinct(15) arexp.distinct(21) arexp.distinct(25) arexp.distinct(29) arexp.inject(4) b1 fuse.elims)
prefer 2
lemma ww1:
assumes "flts [r1] = [r2]" "r1 \<noteq> AZERO"
shows "r1 = r2"
using assms
apply(case_tac r1)
apply(simp)
apply(simp)
apply(simp)
apply(simp)
prefer 2
apply(simp)
apply(simp)
apply(auto)
oops
lemma bsimp_idem:
shows "bsimp (bsimp r) = bsimp r"
apply(induct r taking: "asize" rule: measure_induct)
apply(case_tac x)
apply(simp)
apply(simp)
apply(simp)
prefer 3
apply(simp)
apply(simp)
apply (simp add: bsimp_ASEQ_idem)
apply(clarify)
apply(case_tac x52)
apply(simp)
(* AALT case where rs is of the form _ # _ *)
apply(clarify)
apply(simp)
apply(case_tac "length (flts (bsimp a # map bsimp list)) \<le> 1")
prefer 2
apply(subst bsimp_AALTs_qq)
apply(auto)[1]
apply(simp)
apply(subst k0)
apply(simp)
apply(simp add: flts_append)
apply(subst (2) k0)
apply(simp add: flts_append)
prefer 2
apply(subgoal_tac "length (flts (bsimp a # map bsimp list)) = 0 \<or>
length (flts (bsimp a # map bsimp list)) = 1")
prefer 2
apply(auto)[1]
using le_SucE apply blast
apply(erule disjE)
apply(simp)
apply(simp)
apply(subst k0)
apply(subst (2) k0)
apply(subst (asm) k0)
apply(simp)
apply(subgoal_tac "length (flts [bsimp a]) = 1 \<or>
length (flts (map bsimp list)) = 1")
prefer 2
apply linarith
apply(erule disjE)
apply(simp)
prefer 2
apply(simp)
apply(drule_tac x="AALTs x51 list" in spec)
apply(drule mp)
apply(simp)
using asize0 apply blast
apply(simp)
apply(frule_tac x="a" in spec)
apply(drule mp)
apply(simp)
apply(subgoal_tac "\<exists>r. flts [bsimp a] = [r]")
prefer 2
apply (simp add: length_Suc_conv)
apply(clarify)
apply(simp only: )
apply(case_tac "bsimp a = AZERO")
apply simp
apply(case_tac "\<exists>bs rs. bsimp a = AALTs bs rs")
apply(clarify)
apply(simp)
apply(drule_tac x="AALTs bs rs" in spec)
apply(drule mp)
apply(simp)
apply (metis asize.simps(4) bsimp_size lessI less_le_trans trans_less_add1)
apply(simp)
apply(subst ww)
apply(subst ww)
apply(frule_tac x="fuse x51 r" in spec)
apply(drule mp)
apply(simp)
apply (smt add.commute add_le_cancel_right fuse_size le_add2 le_trans list.map(1) list.simps(9) not_less not_less_eq rt sum_list.Cons)
apply(case_tac "bsimp a = AZERO")
apply simp
apply(case_tac "\<exists>bs rs. bsimp a = AALTs bs rs")
apply(clarify)
defer
apply(
apply(case_tac a)
apply(simp_all)
apply(subgoal_tac "\<exists>r. flts [bsimp a] = [r]")
prefer 2
apply (simp add: length_Suc_conv)
apply auto[1]
apply(case_tac
apply(clarify)
defer
apply(auto)[1]
apply(subst k0)
apply(subst (2) k0)
apply(case_tac "bsimp a = AZERO")
apply(simp)
apply(frule_tac x="AALTs x51 (flts (map bsimp list))" in spec)
apply(drule mp)
apply(simp)
apply (meson add_le_cancel_right asize0 le_trans not_le rt trans_le_add2)
apply(simp)
apply(subst (asm) flts_idem)
apply(auto)[1]
apply(drule_tac x="r" in spec)
apply (metis add.commute add_lessD1 not_add_less1 not_less_eq sum_list_map_remove1)
apply(simp)
apply(subst flts_idem)
apply(auto)[1]
apply(drule_tac x="r" in spec)
apply (metis add.commute add_lessD1 not_add_less1 not_less_eq sum_list_map_remove1)
apply(simp)
apply(case_tac "nonalt (bsimp a)")
apply(subst k0b)
apply(simp)
apply(simp)
apply(subst k0b)
apply(simp)
apply(simp)
apply(auto)[1]
apply(frule_tac x="AALTs x51 (bsimp a # flts (map bsimp list))" in spec)
apply(drule mp)
apply(simp)
prefer 2
apply(simp)
apply(subst (asm) k0)
apply(subst (asm) flts_idem)
apply(auto)[1]
apply (simp add: sum_list_map_remove1)
apply(subst (asm) k0b)
apply(simp)
apply(simp)
apply(simp)
apply(subst k0)
apply(subst flts_idem)
apply(auto)[1]
apply (simp add: sum_list_map_remove1)
apply(subst k0b)
apply(simp)
apply(simp)
apply(simp)
lemma XX_bder:
shows "bsimp (bder c (bsimp r)) = (bsimp \<circ> bder c) r"
apply(induct r)
apply(simp)
apply(simp)
apply(simp)
prefer 3
apply(simp)
prefer 2
apply(simp)
apply(case_tac x2a)
apply(simp)
apply(simp)
apply(auto)[1]
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)
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
fun extr :: "arexp \<Rightarrow> (bit list) set" where
"extr (AONE bs) = {bs}"
| "extr (ACHAR bs c) = {bs}"
| "extr (AALTs bs (r#rs)) =
{bs @ bs' | bs'. bs' \<in> extr r} \<union>
{bs @ bs' | bs'. bs' \<in> extr (AALTs [] rs)}"
| "extr (ASEQ bs r1 r2) =
{bs @ bs1 @ bs2 | bs1 bs2. bs1 \<in> extr r1 \<and> bs2 \<in> extr r2}"
| "extr (ASTAR bs r) = {bs @ [S]} \<union>
{bs @ [Z] @ bs1 @ bs2 | bs1 bs2. bs1 \<in> extr r \<and> bs2 \<in> extr (ASTAR [] r)}"
lemma MAIN_decode:
assumes "\<Turnstile> v : ders s r"
shows "Some (flex r id s v) = decode (retrieve (bders_simp (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_simp (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_simp (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_simp (intern r) s) (injval (ders s r) c v)) r"
using asm2 IH by simp
also have "... = decode (retrieve (bder c (bders_simp (intern r) s)) v) r"
using asm bder_retrieve ders_append
apply -
apply(drule_tac x="v" in meta_spec)
apply(drule_tac x="c" in meta_spec)
apply(drule_tac x="bders_simp (intern r) s" in meta_spec)
apply(drule_tac meta_mp)
apply(simp add: ders_append)
defer
apply(simp)
oops
fun vsimp :: "arexp \<Rightarrow> val \<Rightarrow> val"
where
"vsimp (ASEQ _ (AONE _) r) (Seq v1 v2) = vsimp r v1"
| "vsimp _ v = v"
lemma fuse_vsimp:
assumes "\<Turnstile> v : (erase r)"
shows "vsimp (fuse bs r) v = vsimp r v"
using assms
apply(induct r arbitrary: v bs)
apply(simp_all)
apply(case_tac "\<exists>bs. r1 = AONE bs")
apply(auto)
apply (metis Prf_elims(2) vsimp.simps(1))
apply(erule Prf_elims)
apply(auto)
apply(case_tac r1)
apply(auto)
done
lemma retrieve_XXX0:
assumes "\<And>r v. \<lbrakk>r \<in> set rs; \<Turnstile> v : erase r\<rbrakk> \<Longrightarrow>
\<exists>v'. \<Turnstile> v' : erase (bsimp r) \<and> retrieve (bsimp r) v' = retrieve r v"
"\<Turnstile> v : erase (AALTs bs rs)"
shows "\<exists>v'. \<Turnstile> v' : erase (bsimp_AALTs bs (flts (map bsimp rs))) \<and>
retrieve (bsimp_AALTs bs (flts (map bsimp rs))) v' = retrieve (AALTs bs rs) v"
using assms
apply(induct rs arbitrary: bs v taking: size rule: measure_induct)
apply(case_tac x)
apply(simp)
using Prf_elims(1) apply blast
apply(simp)
apply(case_tac list)
apply(simp_all)
apply(case_tac a)
apply(simp_all)
using Prf_elims(1) apply blast
apply (metis erase.simps(2) fuse.simps(2) retrieve_fuse2)
using Prf_elims(5) apply force
apply(erule Prf_elims)
apply(auto)[1]
apply(simp)
apply(erule Prf_elims)
using Prf_elims(1) apply b last
apply(auto)
apply (metis append_Ni l2 erase_fuse fuse.simps(4) retrieve_fuse2)
apply(case_tac rs)
apply(auto)
oops
fun get where
"get (Some v) = v"
lemma retrieve_XXX:
assumes "\<Turnstile> v : erase r"
shows "\<Turnstile> get (decode (code v) (erase (bsimp r))) : erase (bsimp r)"
using assms
apply(induct r arbitrary: v)
apply(simp)
using Prf_elims(1) apply auto[1]
apply(simp)
apply (simp add: decode_code)
apply(simp)
apply (simp add: decode_code)
apply(simp)
apply(erule Prf_elims)
apply(simp)
apply(case_tac "r1 = AZERO")
apply(simp)
apply (meson Prf_elims(1) Prf_elims(2))
apply(case_tac "r2 = AZERO")
apply(simp)
apply (meson Prf_elims(1) Prf_elims(2))
apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
apply(clarify)
apply(simp)
apply(subst bsimp_ASEQ2)
apply(subst bsimp_ASEQ2)
apply(simp add: erase_fuse)
apply(case_tac r1)
apply(simp_all)
using Prf_elims(4) apply fastforce
apply(erule Prf_elims)
apply(simp)
apply(simp)
defer
apply(subst bsimp_ASEQ1)
using L_bsimp_erase L_flat_Prf1 L_flat_Prf2 apply fast force
using L_bsimp_erase L_
lemma retrieve_XXX:
assumes "\<Turnstile> v : erase r"
shows "\<Turnstile> (vsimp (bsimp r) v : erase (bsimp r) \<and> retrieve (bsimp r) (vsimp (bsimp r) v) = retrieve r v"
using assms
apply(induct r arbitrary: v)
apply(simp)
using Prf_elims(1) apply blast
apply(simp)
using Prf_elims(4) apply fastforce
apply(simp)
apply blast
apply simp
apply(case_tac "r1 = AZERO")
apply(simp)
apply (meson Prf_elims(1) Prf_elims(2))
apply(case_tac "r2 = AZERO")
apply(simp)
apply (meson Prf_elims(1) Prf_elims(2))
apply(erule Prf_elims)
apply(simp)
apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
apply(clarify)
apply(simp)
apply(subst bsimp_ASEQ2)
defer
apply(subst bsimp_ASEQ1)
using L_bsimp_erase L_flat_Prf1 L_flat_Prf2 apply fastforce
using L_bsimp_erase L_flat_Prf1 L_flat_Prf2 apply fastforce
apply(simp)
apply(simp)
apply(drule_tac x="v1" in meta_spec)
apply(drule_tac x="v2" in meta_spec)
apply(simp)
apply(clarify)
apply(rule_tac x="Seq v' v'a" in exI)
apply(simp)
apply (metis Prf.intros(1) Prf_elims(1) bsimp_ASEQ1 erase.simps(1) retrieve.simps(6))
prefer 3
apply(drule_tac x="v1" in meta_spec)
apply(drule_tac x="v2" in meta_spec)
apply(simp)
apply(clarify)
apply(rule_tac x="v'a" in exI)
apply(subst bsimp_ASEQ2)
apply (metis Prf_elims(4) append_assoc erase_fuse retrieve.simps(1) retrieve_fuse2)
prefer 2
apply(auto)
apply(case_tac "x2a")
apply(simp)
using Prf_elims(1) apply blast
apply(simp)
apply(case_tac "list")
apply(simp)
sorry
lemma retrieve_XXX:
assumes "\<Turnstile> v : erase r"
shows "\<exists>v'. \<Turnstile> v' : erase (bsimp r) \<and> retrieve (bsimp r) v' = retrieve r v"
using assms
apply(induct r arbitrary: v)
apply(simp)
using Prf_elims(1) apply blast
apply(simp)
using Prf_elims(4) apply fastforce
apply(simp)
apply blast
apply simp
apply(case_tac "r1 = AZERO")
apply(simp)
apply (meson Prf_elims(1) Prf_elims(2))
apply(case_tac "r2 = AZERO")
apply(simp)
apply (meson Prf_elims(1) Prf_elims(2))
apply(erule Prf_elims)
apply(simp)
apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
apply(clarify)
apply(simp)
apply(subst bsimp_ASEQ2)
defer
apply(subst bsimp_ASEQ1)
using L_bsimp_erase L_flat_Prf1 L_flat_Prf2 apply fastforce
using L_bsimp_erase L_flat_Prf1 L_flat_Prf2 apply fastforce
apply(simp)
apply(simp)
apply(drule_tac x="v1" in meta_spec)
apply(drule_tac x="v2" in meta_spec)
apply(simp)
apply(clarify)
apply(rule_tac x="Seq v' v'a" in exI)
apply(simp)
apply (metis Prf.intros(1) Prf_elims(1) bsimp_ASEQ1 erase.simps(1) retrieve.simps(6))
prefer 3
apply(drule_tac x="v1" in meta_spec)
apply(drule_tac x="v2" in meta_spec)
apply(simp)
apply(clarify)
apply(rule_tac x="v'a" in exI)
apply(subst bsimp_ASEQ2)
apply (metis Prf_elims(4) append_assoc erase_fuse retrieve.simps(1) retrieve_fuse2)
prefer 2
apply(auto)
apply(case_tac "x2a")
apply(simp)
using Prf_elims(1) apply blast
apply(simp)
apply(case_tac "list")
apply(simp)
sorry
lemma TEST:
assumes "\<Turnstile> v : ders s r"
shows "\<exists>v'. retrieve (bders (intern r) s) v' = retrieve (bsimp (bders (intern r) s)) v"
using assms
apply(induct s arbitrary: r v rule: rev_induct)
apply(simp)
defer
apply(simp add: ders_append)
apply(frule Prf_injval)
apply(drule_tac x="r" in meta_spec)
apply(drule_tac x="injval (ders xs r) x v" in meta_spec)
apply(simp)
apply(simp add: bders_append)
apply(subst bder_retrieve)
apply(simp)
apply(simp)
thm bder_retrieve
thm bmkeps_retrieve
lemma bmkeps_simp2:
assumes "bnullable (bder c r)"
shows "bmkeps (bder c (bsimp r)) = bmkeps (bder c r)"
using assms
apply(induct r)
apply(simp)
apply(simp)
apply(simp)
prefer 3
apply(simp)
apply(simp)
apply(auto)[1]
prefer 2
apply(case_tac "r1 = AZERO")
apply(simp)
apply(case_tac "r2 = AZERO")
apply(simp)
apply(case_tac "\<exists>bs. (bsimp r1) = AONE bs")
apply(clarify)
apply(simp)
apply(subst bsimp_ASEQ2)
apply(simp add: bmkeps_simp)
apply(simp add: bders_append)
apply(drule_tac x="bder a r" in meta_spec)
apply(simp)
apply(simp)
apply(simp)
prefer 3
apply(simp)
prefer 2
apply(simp)
apply(case_tac x2a)
apply(simp)
apply(simp add: )
apply(subst k0)
apply(auto)[1]
apply(case_tac list)
apply(simp)
apply(case_tac "r1=AZERO")
apply(simp)
apply(case_tac "r2=AZERO")
apply(simp)
apply(auto)[1]
apply(case_tac "\<exists>bs. r1=AONE bs")
apply(simp)
apply(auto)[1]
apply(subst bsimp_ASEQ2)
prefer 2
apply(simp)
apply(subst bmkeps_bder_AALTs)
apply(case_tac x2a)
apply(simp)
apply(simp)
apply(auto)[1]
apply(subst bmkeps_bder_AALTs)
apply(case_tac a)
apply(simp_all)
apply(auto)[1]
apply(case_tac list)
apply(simp)
apply(simp)
prefer 2
apply(simp)
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 bbs1:
shows "blexer_simp r [c1, c2] = blexer r [c1, c2]"
apply(simp add: blexer_def blexer_simp_def)
apply(auto)
defer
apply (metis L_bsimp_erase bnullable_correctness der_correctness erase_bder lexer.simps(1) lexer_correct_None option.distinct(1))
apply (metis L_bsimp_erase bnullable_correctness der_correctness erase_bder lexer.simps(1) lexer_correct_None option.distinct(1))
apply(subst bmkeps_simp[symmetric])
using b3 apply auto[1]
apply(subst bmkeps_retrieve)
using b3 bnullable_correctness apply blast
apply(subst bder_retrieve)
using b3 bnullable_correctness mkeps_nullable apply fastforce
apply(subst bmkeps_retrieve)
using bnullable_correctness apply blast
apply(subst bder_retrieve)
using bnullable_correctness mkeps_nullable apply force
using bder_retrieve bmkeps_simp bmkeps_retrieve
lemma bsimp_retrieve_bder:
assumes "\<Turnstile> v : der c (erase r)"
shows "retrieve (bder c r) v = retrieve (bsimp (bder c r)) v"
thm bder_retrieve bmkeps_simp
apply(induct r arbitrary: c v)
apply(simp)
apply(simp)
apply(simp)
apply(auto)[1]
apply(case_tac "bsimp (bder c r1) = AZERO")
apply(simp)
prefer 3
apply(simp)
apply(auto elim!: Prf_elims)[1]
apply(case_tac "(bsimp (fuse [Z] (bder c r))) = AZERO")
apply(simp)
apply (met is L_bsimp_erase L_flat_Prf1 L_flat_Prf2 Prf_elims(1) erase.simps(1) erase_bder erase_fuse)
apply(case_tac "\<exists>bs. bsimp (fuse [Z] (bder c r)) = AONE bs")
apply(clarify)
apply(subgoal_tac "L (der c (erase r)) = {[]}")
prefer 2
apply (metis L.simps(2) L_bsimp_erase erase.simps(2) erase_bder erase_fuse)
apply(simp)
apply(subst bsimp_ASEQ1)
apply(simp)
apply(simp)
apply(auto)[1]
prefer 2
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 oo2a:
assumes "\<forall>r. bmkeps (bders_simp r s) = bmkeps (bders r s)" "c # s \<in> L r"
"bnullable (bders_simp (bsimp (bder c (intern r))) s)"
shows "(case (blexer_simp (der c r) s) of None \<Rightarrow> None | Some v \<Rightarrow> Some (injval r c v)) = blexer_simp r (c # s)"
using assms
apply(simp add: blexer_simp_def)
apply(auto split: option.split)
apply (metis blexer_correctness blexer_def lexer.simps(2) lexer_correct_None option.simps(4))
prefer 2
apply (metis L_bders_simp L_bsimp_erase Posix1(1) Posix_mkeps bnullable_correctness ders_correctness erase_bder erase_bders erase_intern lexer.simps(1) lexer_correct_None)
apply(subst bmkeps_retrieve)
using L_bders_simp bnullable_correctness nullable_correctness apply blast
thm bder_retrieve
apply(subst bder_retrieve[symmetric])
apply(drule_tac x="bsimp (bder c (intern r))" in spec)
apply(drule sym)
apply(simp)
apply(subst blexer_simp_def)
apply(case_tac "bnullable (bders_simp (intern (der c r)) s)")
apply(simp)
apply(auto split: option.split)
apply(subst oo)
apply(simp)
done
lemma oo3:
assumes "\<forall>r. bders r s = bders_simp r s"
shows "blexer_simp r (s @ [c]) = blexer r (s @ [c])"
using assms
apply(simp (no_asm) add: blexer_simp_def)
apply(auto)
prefer 2
apply (metis L_bders_simp blexer_def bnullable_correctness lexer.simps(1) lexer_correct_None option.distinct(1))
apply(simp add: bders_simp_append)
apply(subst bmkeps_simp[symmetric])
using b3 apply auto[1]
apply(simp add: blexer_def)
apply(auto)[1]
prefer 2
apply (metis (mono_tags, lifting) L_bders_simp Posix_mkeps append.right_neutral bders_simp.simps(1) bders_simp.simps(2) bders_simp_append bnullable_correctness lexer.simps(1) lexer_correct_None lexer_correctness(1) option.distinct(1))
apply(simp add: bders_append)
done
lemma oo4:
assumes "\<forall>r. bmkeps (bders r s) = bmkeps (bders_simp r s)" "bnullable (bder c (bders r s))"
shows "bmkeps (bders_simp r (s @ [c])) = bmkeps (bders r (s @ [c]))"
using assms
apply(simp add: bders_simp_append)
apply(subst bmkeps_simp[symmetric])
apply (metis L_bders_simp bnullable_correctness der_correctness erase_bder lexer.simps(1) lexer_correct_None option.distinct(1))
apply(simp add: bders_append)
done
lemma oo4:
shows "blexer_simp r s = blexer r s"
apply(induct s arbitrary: r rule: rev_induct)
apply(simp only: blexer_simp_def blexer_def)
apply(simp)
apply(simp only: blexer_simp_def blexer_def)
apply(subgoal_tac "bnullable (bders_simp (intern r) (xs @ [x])) = bnullable (bders (intern r) (xs @ [x]))")
prefer 2
apply (simp add: b4)
apply(simp)
apply(rule impI)
apply(simp add: bders_simp_append)
apply(subst bmkeps_simp[symmetric])
using b3 apply auto[1]
apply(subst bmkeps_retrieve)
using b3 bnullable_correctness apply blast
apply(subst bder_retrieve)
using b3 bnullable_correctness mkeps_nullable apply fastforce
apply(simp add: bders_append)
apply(subst bmkeps_retrieve)
using bnullable_correctness apply blast
apply(subst bder_retrieve)
using bnullable_correctness mkeps_nullable apply fastforce
apply(subst erase_bder)
apply(case_tac "xs \<in> L")
apply(subst (asm) (2) bmkeps_retrieve)
thm bmkeps_retrieve bmkeps_retrieve
apply(subst bmkeps_retrieve[symmetric])
apply (simp add: bnullable_correctness)
apply(simp add: bders_simp_append)
apply(induct s arbitrary: r rule: rev_induct)
apply(simp add: blexer_def blexer_simp_def)
apply(rule oo3)
apply(simp (no_asm) add: blexer_simp_def)
apply(auto)
prefer 2
apply (metis L_bders_simp blexer_def bnullable_correctness lexer.simps(1) lexer_correct_None option.distinct(1))
apply(simp add: bders_simp_append)
apply(subst bmkeps_simp[symmetric])
using b3 apply auto[1]
apply(simp add: blexer_def)
apply(auto)[1]
prefer 2
apply (m etis (mono_tags, lifting) L_bders_simp Posix_mkeps append.right_neutral bders_simp.simps(1) bders_simp.simps(2) bders_simp_append bnullable_correctness lexer.simps(1) lexer_correct_None lexer_correctness(1) option.distinct(1))
apply(simp add: bders_append)
oops
lemma bnullable_simp:
assumes "s \<in> L (erase r)"
shows "bmkeps (bders r s) = bmkeps (bders_simp r s)"
using assms
apply(induct s arbitrary: r rule: rev_induct)
apply(simp)
apply(simp add: bders_append bders_simp_append)
apply(subst bmkeps_simp[symmetric])
apply (metis L_bders_simp b3 bders_simp.simps(1) bders_simp.simps(2) bders_simp_append blexer_correctness blexer_def bnullable_correctness erase_bders erase_intern lexer.simps(1) lexer_correct_None lexer_correct_Some lexer_correctness(1))
apply(subst bmkeps_retrieve)
apply (metis bders.simps(1) bders.simps(2) bders_append blexer_correctness blexer_def bnullable_correctness erase_bders erase_intern lexer_correct_Some option.distinct(1))
apply(subst bmkeps_retrieve)
apply (metis L_bders_simp L_bsimp_erase Posix_mkeps bders_simp.simps(1) bders_simp.simps(2) bders_simp_append blexer_correctness blexer_def bnullable_correctness erase_bders erase_intern lexer.simps(1) lexer_correct_None lexer_correctness(2))
apply(subst bder_retrieve)
apply (metis bders.simps(1) bders.simps(2) bders_append blexer_correctness blexer_def bnullable_correctness erase_bder erase_bders erase_intern lexer_correct_Some mkeps_nullable option.distinct(1))
apply(subst bder_retrieve)
apply (metis L_bders_simp L_bsimp_erase Posix_mkeps bders_simp.simps(1) bders_simp.simps(2) bders_simp_append blexer_correctness blexer_def bnullable_correctness erase_bder erase_bders erase_intern lexer.simps(1) lexer_correct_None lexer_correctness(2) mkeps_nullable)
apply(drule_tac x="bder a r" in meta_spec)
apply(drule_tac meta_mp)
apply (me tis erase_bder lexer.simps(2) lexer_correct_None option.simps(4))
apply(simp)
oops
lemma
shows "blexer r s = blexer_simp r s"
apply(induct s arbitrary: r rule: rev_induct)
apply(simp add: blexer_def blexer_simp_def)
apply(case_tac "xs @ [x] \<in> L r")
defer
apply(subgoal_tac "blexer (ders xs r) [x] = None")
prefer 2
apply(subst blexer_correctness)
apply(simp (no_asm) add: lexer_correct_None)
apply(simp add: nullable_correctness)
apply(simp add: der_correctness ders_correctness)
apply(simp add: Der_def Ders_def)
apply(subgoal_tac "blexer r (xs @ [x]) = None")
prefer 2
apply (simp add: blexer_correctness)
using lexer_correct_None apply auto[1]
apply(simp)
apply(subgoal_tac "blexer_simp (ders xs r) [x] = None")
prefer 2
apply (metis L_bders_simp Posix_injval Posix_mkeps bders.simps(2) blexer_correctness blexer_simp_def bnullable_correctness ders.simps(1) erase_bder erase_bders erase_intern lexer_correct_None lexer_correctness(2))
apply(subgoal_tac "[] \<notin> L (erase (bders_simp (intern r) (xs @ [x])))")
prefer 2
apply(metis L_bders_simp Posix_injval bders.simps(2) blexer_correctness ders.simps(1) ders_append erase_bder erase_bders erase_intern lexer_correct_None lexer_correctness(2))
using blexer_simp_def bnullable_correctness lexer_correct_None apply auto[1]
(* case xs @ [x] \<in> L r *)
apply(subgoal_tac "\<exists>v. blexer (ders xs r) [x] = Some v \<and> [x] \<in> (ders xs r) \<rightarrow> v")
prefer 2
using blexer_correctness lexer_correct_Some apply auto[1]
apply (simp add: Posix_injval Posix_mkeps)
apply (metis ders.simps(1) ders.simps(2) ders_append lexer_correct_None lexer_flex)
apply(clarify)
apply(subgoal_tac "blexer_simp (ders xs r) [x] = Some v")
prefer 2
apply(simp add: blexer_simp_def)
apply(auto)[1]
apply (metis bders.simps(1) bders.simps(2) blexer_def bmkeps_simp option.simps(3))
using b3 blexer_def apply fastforce
apply(subgoal_tac "blexer_simp (ders xs r) [x] = blexer_simp r (xs @ [x])")
prefer 2
apply(simp add: blexer_simp_def)
apply(simp)
apply(simp)
apply(subst blexer_simp_def)
apply(simp)
apply(auto)
apply(drule_tac x="der a r" in meta_spec)
apply(subst blexer_def)
apply(subgoal_tac "bnullable (bders (intern r) (a # s))")
prefer 2
apply (me tis Posix_injval blexer_correctness blexer_def lexer_correctness(2))
apply(simp)
lemma
shows "blexer r s = blexer_simp r s"
apply(induct s arbitrary: r)
apply(simp add: blexer_def blexer_simp_def)
apply(case_tac "s \<in> L (der a r)")
defer
apply(subgoal_tac "blexer (der a r) s = None")
prefer 2
apply (simp add: blexer_correctness lexer_correct_None)
apply(subgoal_tac "blexer r (a # s) = None")
prefer 2
apply (simp add: blexer_correctness)
apply(simp)
apply(subst blexer_simp_def)
apply(simp)
apply(drule_tac x="der a r" in meta_spec)
apply(subgoal_tac "s \<notin> L(erase (bder a (intern r)))")
prefer 2
apply simp
apply(simp only:)
apply(subst blexer_simp_def)
apply(subgoal_tac "\<not> bnullable (bders_simp (intern r) (a # s))")
apply(simp)
apply(subst bnullable_correctness[symmetric])
apply(simp)
oops
lemma flts_bsimp:
"flts (map bsimp rs) = map bsimp (flts rs)"
apply(induct rs taking: size rule: measure_induct)
apply(case_tac x)
apply(simp)
apply(simp)
apply(induct rs rule: flts.induct)
apply(simp)
apply(simp)
defer
apply(simp)
apply(simp)
defer
apply(simp)
apply(subst List.list.map(2))
apply(simp only: flts.simps)
apply(subst k0)
apply(subst map_append)
apply(simp only:)
apply(simp del: bsimp.simps)
apply(case_tac rs1)
apply(simp)
apply(simp)
apply(case_tac list)
apply(simp_all)
thm map
apply(subst map.simps)
apply(auto)
defer
apply(case_tac "(bsimp va) = AZERO")
apply(simp)
using b3 apply for ce
apply(case_tac "(bsimp a2) = AZERO")
apply(simp)
apply (me tis bder.simps(1) bsimp.simps(3) bsimp_AALTs.simps(1) bsimp_ASEQ0 bsimp_fuse flts.simps(1) flts.simps(2) fuse.simps(1))
apply(case_tac "\<exists>bs. (bsimp a1) = AONE bs")
apply(clarify)
apply(simp)
lemma XXX:
shows "bsimp (bsimp a) = bsimp a"
sorry
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:
shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
apply(induct a arbitrary: c)
apply(simp)
apply(simp)
apply(simp)
prefer 3
apply(simp)
apply(auto)[1]
apply(case_tac "(bsimp a1) = AZERO")
apply(simp)
using b3 apply force
apply(case_tac "(bsimp a2) = AZERO")
apply(simp)
apply (metis bder.simps(1) bsimp.simps(3) bsimp_AALTs.simps(1) bsimp_ASEQ0 bsimp_fuse flts.simps(1) flts.simps(2) fuse.simps(1))
apply(case_tac "\<exists>bs. (bsimp a1) = AONE bs")
apply(clarify)
apply(simp)
apply(subst bsimp_ASEQ2)
apply(subgoal_tac "bmkeps a1 = bs")
prefer 2
apply (simp add: bmkeps_simp)
apply(simp)
apply(subst (1) bsimp_fuse[symmetric])
defer
apply(subst bsimp_ASEQ1)
apply(simp)
apply(simp)
apply(simp)
apply(auto)[1]
apply (metis XXX bmkeps_simp bsimp_fuse)
using b3 apply blast
apply (smt XXX b3 bder.simps(1) bder.simps(5) bnullable.simps(2) bsimp.simps(1) bsimp_ASEQ.simps(1) bsimp_ASEQ0 bsimp_ASEQ1)
apply(simp)
prefer 2
apply(subst bder_fuse)
apply(subst bsimp_fuse[symmetric])
apply(simp)
sorry
thm bsimp_AALTs.simps
thm bsimp.simps
thm flts.simps
lemma XXX3:
"bsimp (bders (bsimp r) s) = bsimp (bders r s)"
apply(induct s arbitrary: r rule: rev_induct)
apply(simp)
apply (simp add: XXX)
apply(simp add: bders_append)
apply(subst (2) XXX2[symmetric])
apply(subst XXX2[symmetric])
apply(drule_tac x="r" in meta_spec)
apply(simp)
done
lemma XXX4:
"bders_simp (bsimp r) s = bsimp (bders r s)"
apply(induct s arbitrary: r)
apply(simp)
apply(simp)
by (metis XXX2)
lemma
assumes "bnullable (bder c r)" "bnullable (bder c (bsimp r))"
shows "bmkeps (bder c r) = bmkeps (bder c (bsimp r))"
using assms
apply(induct r arbitrary: c)
apply(simp)
apply(simp)
apply(simp)
prefer 3
apply(simp)
apply(auto)[1]
apply(case_tac "(bsimp r1) = AZERO")
apply(simp)
apply(case_tac "(bsimp r2) = AZERO")
apply(simp)
apply (simp add: bsimp_ASEQ0)
apply(case_tac "\<exists>bs. (bsimp r1) = AONE bs")
apply(clarify)
apply(simp)
apply(subgoal_tac "bnullable r1")
prefer 2
using b3 apply force
apply(simp)
apply(simp add: bsimp_ASEQ2)
prefer 2
apply(subst bsimp_ASEQ2)
lemma
assumes "bnullable (bders a (s1 @ s2))" "bnullable (bders (bsimp (bders a s1)) s2)"
shows "bmkeps (bders a (s1 @ s2)) = bmkeps (bders (bsimp (bders a s1)) s2)"
using assms
apply(induct s2 arbitrary: a s1)
apply(simp)
using bmkeps_simp apply blast
apply(simp add: bders_append)
apply(drule_tac x="aa" in meta_spec)
apply(drule_tac x="s1 @ [a]" in meta_spec)
apply(drule meta_mp)
apply(simp add: bders_append)
apply(simp add: bders_append)
apply(drule meta_mp)
apply (metis b4 bders.simps(2) bders_simp.simps(2))
apply(simp)
apply (met is b4 bders.simps(2) bders_simp.simps(2))
using b3 apply blast
using b3 apply auto[1]
apply(auto simp add: blex_def)
prefer 3
end