theory BitCodedCT
imports "Lexer"
begin
section \<open>Bit-Encodings\<close>
datatype bit = Z | S
fun
code :: "val \<Rightarrow> bit list"
where
"code Void = []"
| "code (Char c) = []"
| "code (Left v) = Z # (code v)"
| "code (Right v) = S # (code v)"
| "code (Seq v1 v2) = (code v1) @ (code v2)"
| "code (Stars []) = [S]"
| "code (Stars (v # vs)) = (Z # code v) @ code (Stars vs)"
fun
Stars_add :: "val \<Rightarrow> val \<Rightarrow> val"
where
"Stars_add v (Stars vs) = Stars (v # vs)"
function
decode' :: "bit list \<Rightarrow> rexp \<Rightarrow> (val * bit list)"
where
"decode' ds ZERO = (Void, [])"
| "decode' ds ONE = (Void, ds)"
| "decode' ds (CHAR d) = (Char d, ds)"
| "decode' [] (ALT r1 r2) = (Void, [])"
| "decode' (Z # ds) (ALT r1 r2) = (let (v, ds') = decode' ds r1 in (Left v, ds'))"
| "decode' (S # ds) (ALT r1 r2) = (let (v, ds') = decode' ds r2 in (Right v, ds'))"
| "decode' ds (SEQ r1 r2) = (let (v1, ds') = decode' ds r1 in
let (v2, ds'') = decode' ds' r2 in (Seq v1 v2, ds''))"
| "decode' [] (STAR r) = (Void, [])"
| "decode' (S # ds) (STAR r) = (Stars [], ds)"
| "decode' (Z # ds) (STAR r) = (let (v, ds') = decode' ds r in
let (vs, ds'') = decode' ds' (STAR r)
in (Stars_add v vs, ds''))"
by pat_completeness auto
lemma decode'_smaller:
assumes "decode'_dom (ds, r)"
shows "length (snd (decode' ds r)) \<le> length ds"
using assms
apply(induct ds r)
apply(auto simp add: decode'.psimps split: prod.split)
using dual_order.trans apply blast
by (meson dual_order.trans le_SucI)
termination "decode'"
apply(relation "inv_image (measure(%cs. size cs) <*lex*> measure(%s. size s)) (%(ds,r). (r,ds))")
apply(auto dest!: decode'_smaller)
by (metis less_Suc_eq_le snd_conv)
definition
decode :: "bit list \<Rightarrow> rexp \<Rightarrow> val option"
where
"decode ds r \<equiv> (let (v, ds') = decode' ds r
in (if ds' = [] then Some v else None))"
lemma decode'_code_Stars:
assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> (\<forall>x. decode' (code v @ x) r = (v, x)) \<and> flat v \<noteq> []"
shows "decode' (code (Stars vs) @ ds) (STAR r) = (Stars vs, ds)"
using assms
apply(induct vs)
apply(auto)
done
lemma decode'_code:
assumes "\<Turnstile> v : r"
shows "decode' ((code v) @ ds) r = (v, ds)"
using assms
apply(induct v r arbitrary: ds)
apply(auto)
using decode'_code_Stars by blast
lemma decode_code:
assumes "\<Turnstile> v : r"
shows "decode (code v) r = Some v"
using assms unfolding decode_def
by (smt append_Nil2 decode'_code old.prod.case)
section {* Annotated Regular Expressions *}
datatype arexp =
AZERO
| AONE "bit list"
| ACHAR "bit list" char
| ASEQ "bit list" arexp arexp
| AALTs "bit list" "arexp list"
| ASTAR "bit list" arexp
abbreviation
"AALT bs r1 r2 \<equiv> AALTs bs [r1, r2]"
fun asize :: "arexp \<Rightarrow> nat" where
"asize AZERO = 1"
| "asize (AONE cs) = 1"
| "asize (ACHAR cs c) = 1"
| "asize (AALTs cs rs) = Suc (sum_list (map asize rs))"
| "asize (ASEQ cs r1 r2) = Suc (asize r1 + asize r2)"
| "asize (ASTAR cs r) = Suc (asize r)"
fun
erase :: "arexp \<Rightarrow> rexp"
where
"erase AZERO = ZERO"
| "erase (AONE _) = ONE"
| "erase (ACHAR _ c) = CHAR c"
| "erase (AALTs _ []) = ZERO"
| "erase (AALTs _ [r]) = (erase r)"
| "erase (AALTs bs (r#rs)) = ALT (erase r) (erase (AALTs bs rs))"
| "erase (ASEQ _ r1 r2) = SEQ (erase r1) (erase r2)"
| "erase (ASTAR _ r) = STAR (erase r)"
lemma decode_code_erase:
assumes "\<Turnstile> v : (erase a)"
shows "decode (code v) (erase a) = Some v"
using assms
by (simp add: decode_code)
fun nonalt :: "arexp \<Rightarrow> bool"
where
"nonalt (AALTs bs2 rs) = False"
| "nonalt r = True"
fun good :: "arexp \<Rightarrow> bool" where
"good AZERO = False"
| "good (AONE cs) = True"
| "good (ACHAR cs c) = True"
| "good (AALTs cs []) = False"
| "good (AALTs cs [r]) = False"
| "good (AALTs cs (r1#r2#rs)) = (\<forall>r' \<in> set (r1#r2#rs). good r' \<and> nonalt r')"
| "good (ASEQ _ AZERO _) = False"
| "good (ASEQ _ (AONE _) _) = False"
| "good (ASEQ _ _ AZERO) = False"
| "good (ASEQ cs r1 r2) = (good r1 \<and> good r2)"
| "good (ASTAR cs r) = True"
fun fuse :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp" where
"fuse bs AZERO = AZERO"
| "fuse bs (AONE cs) = AONE (bs @ cs)"
| "fuse bs (ACHAR cs c) = ACHAR (bs @ cs) c"
| "fuse bs (AALTs cs rs) = AALTs (bs @ cs) rs"
| "fuse bs (ASEQ cs r1 r2) = ASEQ (bs @ cs) r1 r2"
| "fuse bs (ASTAR cs r) = ASTAR (bs @ cs) r"
lemma fuse_append:
shows "fuse (bs1 @ bs2) r = fuse bs1 (fuse bs2 r)"
apply(induct r)
apply(auto)
done
fun intern :: "rexp \<Rightarrow> arexp" where
"intern ZERO = AZERO"
| "intern ONE = AONE []"
| "intern (CHAR c) = ACHAR [] c"
| "intern (ALT r1 r2) = AALT [] (fuse [Z] (intern r1))
(fuse [S] (intern r2))"
| "intern (SEQ r1 r2) = ASEQ [] (intern r1) (intern r2)"
| "intern (STAR r) = ASTAR [] (intern r)"
fun retrieve :: "arexp \<Rightarrow> val \<Rightarrow> bit list" where
"retrieve (AONE bs) Void = bs"
| "retrieve (ACHAR bs c) (Char d) = bs"
| "retrieve (AALTs bs [r]) v = bs @ retrieve r v"
| "retrieve (AALTs bs (r#rs)) (Left v) = bs @ retrieve r v"
| "retrieve (AALTs bs (r#rs)) (Right v) = bs @ retrieve (AALTs [] rs) v"
| "retrieve (ASEQ bs r1 r2) (Seq v1 v2) = bs @ retrieve r1 v1 @ retrieve r2 v2"
| "retrieve (ASTAR bs r) (Stars []) = bs @ [S]"
| "retrieve (ASTAR bs r) (Stars (v#vs)) =
bs @ [Z] @ retrieve r v @ retrieve (ASTAR [] r) (Stars vs)"
fun
bnullable :: "arexp \<Rightarrow> bool"
where
"bnullable (AZERO) = False"
| "bnullable (AONE bs) = True"
| "bnullable (ACHAR bs c) = False"
| "bnullable (AALTs bs rs) = (\<exists>r \<in> set rs. bnullable r)"
| "bnullable (ASEQ bs r1 r2) = (bnullable r1 \<and> bnullable r2)"
| "bnullable (ASTAR bs r) = True"
fun
bmkeps :: "arexp \<Rightarrow> bit list"
where
"bmkeps(AONE bs) = bs"
| "bmkeps(ASEQ bs r1 r2) = bs @ (bmkeps r1) @ (bmkeps r2)"
| "bmkeps(AALTs bs [r]) = bs @ (bmkeps r)"
| "bmkeps(AALTs bs (r#rs)) = (if bnullable(r) then bs @ (bmkeps r) else (bmkeps (AALTs bs rs)))"
| "bmkeps(ASTAR bs r) = bs @ [S]"
fun
bder :: "char \<Rightarrow> arexp \<Rightarrow> arexp"
where
"bder c (AZERO) = AZERO"
| "bder c (AONE bs) = AZERO"
| "bder c (ACHAR bs d) = (if c = d then AONE bs else AZERO)"
| "bder c (AALTs bs rs) = AALTs bs (map (bder c) rs)"
| "bder c (ASEQ bs r1 r2) =
(if bnullable r1
then AALT bs (ASEQ [] (bder c r1) r2) (fuse (bmkeps r1) (bder c r2))
else ASEQ bs (bder c r1) r2)"
| "bder c (ASTAR bs r) = ASEQ bs (fuse [Z] (bder c r)) (ASTAR [] r)"
fun
bders :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
where
"bders r [] = r"
| "bders r (c#s) = bders (bder c r) s"
lemma bders_append:
"bders r (s1 @ s2) = bders (bders r s1) s2"
apply(induct s1 arbitrary: r s2)
apply(simp_all)
done
lemma bnullable_correctness:
shows "nullable (erase r) = bnullable r"
apply(induct r rule: erase.induct)
apply(simp_all)
done
lemma erase_fuse:
shows "erase (fuse bs r) = erase r"
apply(induct r rule: erase.induct)
apply(simp_all)
done
lemma erase_intern [simp]:
shows "erase (intern r) = r"
apply(induct r)
apply(simp_all add: erase_fuse)
done
lemma erase_bder [simp]:
shows "erase (bder a r) = der a (erase r)"
apply(induct r rule: erase.induct)
apply(simp_all add: erase_fuse bnullable_correctness)
done
lemma erase_bders [simp]:
shows "erase (bders r s) = ders s (erase r)"
apply(induct s arbitrary: r )
apply(simp_all)
done
lemma retrieve_encode_STARS:
assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> code v = retrieve (intern r) v"
shows "code (Stars vs) = retrieve (ASTAR [] (intern r)) (Stars vs)"
using assms
apply(induct vs)
apply(simp_all)
done
lemma retrieve_fuse2:
assumes "\<Turnstile> v : (erase r)"
shows "retrieve (fuse bs r) v = bs @ retrieve r v"
using assms
apply(induct r arbitrary: v bs)
apply(auto elim: Prf_elims)[4]
defer
using retrieve_encode_STARS
apply(auto elim!: Prf_elims)[1]
apply(case_tac vs)
apply(simp)
apply(simp)
(* AALTs case *)
apply(simp)
apply(case_tac x2a)
apply(simp)
apply(auto elim!: Prf_elims)[1]
apply(simp)
apply(case_tac list)
apply(simp)
apply(auto)
apply(auto elim!: Prf_elims)[1]
done
lemma retrieve_fuse:
assumes "\<Turnstile> v : r"
shows "retrieve (fuse bs (intern r)) v = bs @ retrieve (intern r) v"
using assms
by (simp_all add: retrieve_fuse2)
lemma retrieve_code:
assumes "\<Turnstile> v : r"
shows "code v = retrieve (intern r) v"
using assms
apply(induct v r )
apply(simp_all add: retrieve_fuse retrieve_encode_STARS)
done
lemma r:
assumes "bnullable (AALTs bs (a # rs))"
shows "bnullable a \<or> (\<not> bnullable a \<and> bnullable (AALTs bs rs))"
using assms
apply(induct rs)
apply(auto)
done
lemma r0:
assumes "bnullable a"
shows "bmkeps (AALTs bs (a # rs)) = bs @ (bmkeps a)"
using assms
by (metis bmkeps.simps(3) bmkeps.simps(4) list.exhaust)
lemma r1:
assumes "\<not> bnullable a" "bnullable (AALTs bs rs)"
shows "bmkeps (AALTs bs (a # rs)) = bmkeps (AALTs bs rs)"
using assms
apply(induct rs)
apply(auto)
done
lemma r2:
assumes "x \<in> set rs" "bnullable x"
shows "bnullable (AALTs bs rs)"
using assms
apply(induct rs)
apply(auto)
done
lemma r3:
assumes "\<not> bnullable r"
" \<exists> x \<in> set rs. bnullable x"
shows "retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs))) =
retrieve (AALTs bs (r # rs)) (mkeps (erase (AALTs bs (r # rs))))"
using assms
apply(induct rs arbitrary: r bs)
apply(auto)[1]
apply(auto)
using bnullable_correctness apply blast
apply(auto simp add: bnullable_correctness mkeps_nullable retrieve_fuse2)
apply(subst retrieve_fuse2[symmetric])
apply (smt bnullable.simps(4) bnullable_correctness erase.simps(5) erase.simps(6) insert_iff list.exhaust list.set(2) mkeps.simps(3) mkeps_nullable)
apply(simp)
apply(case_tac "bnullable a")
apply (smt append_Nil2 bnullable.simps(4) bnullable_correctness erase.simps(5) erase.simps(6) fuse.simps(4) insert_iff list.exhaust list.set(2) mkeps.simps(3) mkeps_nullable retrieve_fuse2)
apply(drule_tac x="a" in meta_spec)
apply(drule_tac x="bs" in meta_spec)
apply(drule meta_mp)
apply(simp)
apply(drule meta_mp)
apply(auto)
apply(subst retrieve_fuse2[symmetric])
apply(case_tac rs)
apply(simp)
apply(auto)[1]
apply (simp add: bnullable_correctness)
apply (metis append_Nil2 bnullable_correctness erase_fuse fuse.simps(4) list.set_intros(1) mkeps.simps(3) mkeps_nullable nullable.simps(4) r2)
apply (simp add: bnullable_correctness)
apply (metis append_Nil2 bnullable_correctness erase.simps(6) erase_fuse fuse.simps(4) list.set_intros(2) mkeps.simps(3) mkeps_nullable r2)
apply(simp)
done
lemma t:
assumes "\<forall>r \<in> set rs. nullable (erase r) \<longrightarrow> bmkeps r = retrieve r (mkeps (erase r))"
"nullable (erase (AALTs bs rs))"
shows " bmkeps (AALTs bs rs) = retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs)))"
using assms
apply(induct rs arbitrary: bs)
apply(simp)
apply(auto simp add: bnullable_correctness)
apply(case_tac rs)
apply(auto simp add: bnullable_correctness)[2]
apply(subst r1)
apply(simp)
apply(rule r2)
apply(assumption)
apply(simp)
apply(drule_tac x="bs" in meta_spec)
apply(drule meta_mp)
apply(auto)[1]
prefer 2
apply(case_tac "bnullable a")
apply(subst r0)
apply blast
apply(subgoal_tac "nullable (erase a)")
prefer 2
using bnullable_correctness apply blast
apply (metis (no_types, lifting) erase.simps(5) erase.simps(6) list.exhaust mkeps.simps(3) retrieve.simps(3) retrieve.simps(4))
apply(subst r1)
apply(simp)
using r2 apply blast
apply(drule_tac x="bs" in meta_spec)
apply(drule meta_mp)
apply(auto)[1]
apply(simp)
using r3 apply blast
apply(auto)
using r3 by blast
lemma bmkeps_retrieve:
assumes "nullable (erase r)"
shows "bmkeps r = retrieve r (mkeps (erase r))"
using assms
apply(induct r)
apply(simp)
apply(simp)
apply(simp)
apply(simp)
defer
apply(simp)
apply(rule t)
apply(auto)
done
lemma bder_retrieve:
assumes "\<Turnstile> v : der c (erase r)"
shows "retrieve (bder c r) v = retrieve r (injval (erase r) c v)"
using assms
apply(induct r arbitrary: v rule: erase.induct)
apply(simp)
apply(erule Prf_elims)
apply(simp)
apply(erule Prf_elims)
apply(simp)
apply(case_tac "c = ca")
apply(simp)
apply(erule Prf_elims)
apply(simp)
apply(simp)
apply(erule Prf_elims)
apply(simp)
apply(erule Prf_elims)
apply(simp)
apply(simp)
apply(rename_tac "r\<^sub>1" "r\<^sub>2" rs v)
apply(erule Prf_elims)
apply(simp)
apply(simp)
apply(case_tac rs)
apply(simp)
apply(simp)
apply (smt Prf_elims(3) injval.simps(2) injval.simps(3) retrieve.simps(4) retrieve.simps(5) same_append_eq)
apply(simp)
apply(case_tac "nullable (erase r1)")
apply(simp)
apply(erule Prf_elims)
apply(subgoal_tac "bnullable r1")
prefer 2
using bnullable_correctness apply blast
apply(simp)
apply(erule Prf_elims)
apply(simp)
apply(subgoal_tac "bnullable r1")
prefer 2
using bnullable_correctness apply blast
apply(simp)
apply(simp add: retrieve_fuse2)
apply(simp add: bmkeps_retrieve)
apply(simp)
apply(erule Prf_elims)
apply(simp)
using bnullable_correctness apply blast
apply(rename_tac bs r v)
apply(simp)
apply(erule Prf_elims)
apply(clarify)
apply(erule Prf_elims)
apply(clarify)
apply(subst injval.simps)
apply(simp del: retrieve.simps)
apply(subst retrieve.simps)
apply(subst retrieve.simps)
apply(simp)
apply(simp add: retrieve_fuse2)
done
lemma MAIN_decode:
assumes "\<Turnstile> v : ders s r"
shows "Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r"
using assms
proof (induct s arbitrary: v rule: rev_induct)
case Nil
have "\<Turnstile> v : ders [] r" by fact
then have "\<Turnstile> v : r" by simp
then have "Some v = decode (retrieve (intern r) v) r"
using decode_code retrieve_code by auto
then show "Some (flex r id [] v) = decode (retrieve (bders (intern r) []) v) r"
by simp
next
case (snoc c s v)
have IH: "\<And>v. \<Turnstile> v : ders s r \<Longrightarrow>
Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r" by fact
have asm: "\<Turnstile> v : ders (s @ [c]) r" by fact
then have asm2: "\<Turnstile> injval (ders s r) c v : ders s r"
by (simp add: Prf_injval ders_append)
have "Some (flex r id (s @ [c]) v) = Some (flex r id s (injval (ders s r) c v))"
by (simp add: flex_append)
also have "... = decode (retrieve (bders (intern r) s) (injval (ders s r) c v)) r"
using asm2 IH by simp
also have "... = decode (retrieve (bder c (bders (intern r) s)) v) r"
using asm by (simp_all add: bder_retrieve ders_append)
finally show "Some (flex r id (s @ [c]) v) =
decode (retrieve (bders (intern r) (s @ [c])) v) r" by (simp add: bders_append)
qed
definition blex where
"blex a s \<equiv> if bnullable (bders a s) then Some (bmkeps (bders a s)) else None"
definition blexer where
"blexer r s \<equiv> if bnullable (bders (intern r) s) then
decode (bmkeps (bders (intern r) s)) r else None"
lemma blexer_correctness:
shows "blexer r s = lexer r s"
proof -
{ define bds where "bds \<equiv> bders (intern r) s"
define ds where "ds \<equiv> ders s r"
assume asm: "nullable ds"
have era: "erase bds = ds"
unfolding ds_def bds_def by simp
have mke: "\<Turnstile> mkeps ds : ds"
using asm by (simp add: mkeps_nullable)
have "decode (bmkeps bds) r = decode (retrieve bds (mkeps ds)) r"
using bmkeps_retrieve
using asm era by (simp add: bmkeps_retrieve)
also have "... = Some (flex r id s (mkeps ds))"
using mke by (simp_all add: MAIN_decode ds_def bds_def)
finally have "decode (bmkeps bds) r = Some (flex r id s (mkeps ds))"
unfolding bds_def ds_def .
}
then show "blexer r s = lexer r s"
unfolding blexer_def lexer_flex
apply(subst bnullable_correctness[symmetric])
apply(simp)
done
qed
fun distinctBy :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b set \<Rightarrow> 'a list"
where
"distinctBy [] f acc = []"
| "distinctBy (x#xs) f acc =
(if (f x) \<in> acc then distinctBy xs f acc
else x # (distinctBy xs f ({f x} \<union> acc)))"
fun flts :: "arexp list \<Rightarrow> arexp list"
where
"flts [] = []"
| "flts (AZERO # rs) = flts rs"
| "flts ((AALTs bs rs1) # rs) = (map (fuse bs) rs1) @ flts rs"
| "flts (r1 # rs) = r1 # flts rs"
fun li :: "bit list \<Rightarrow> arexp list \<Rightarrow> arexp"
where
"li _ [] = AZERO"
| "li bs [a] = fuse bs a"
| "li bs as = AALTs bs as"
fun bsimp_ASEQ :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp \<Rightarrow> arexp"
where
"bsimp_ASEQ _ AZERO _ = AZERO"
| "bsimp_ASEQ _ _ AZERO = AZERO"
| "bsimp_ASEQ bs1 (AONE bs2) r2 = fuse (bs1 @ bs2) r2"
| "bsimp_ASEQ bs1 r1 r2 = ASEQ bs1 r1 r2"
fun bsimp_AALTs :: "bit list \<Rightarrow> arexp list \<Rightarrow> arexp"
where
"bsimp_AALTs _ [] = AZERO"
| "bsimp_AALTs bs1 [r] = fuse bs1 r"
| "bsimp_AALTs bs1 rs = AALTs bs1 rs"
fun bsimp :: "arexp \<Rightarrow> arexp"
where
"bsimp (ASEQ bs1 r1 r2) = bsimp_ASEQ bs1 (bsimp r1) (bsimp r2)"
| "bsimp (AALTs bs1 rs) = bsimp_AALTs bs1 (flts (map bsimp rs))"
| "bsimp r = r"
fun
bders_simp :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
where
"bders_simp r [] = r"
| "bders_simp r (c # s) = bders_simp (bsimp (bder c r)) s"
definition blexer_simp where
"blexer_simp r s \<equiv> if bnullable (bders_simp (intern r) s) then
decode (bmkeps (bders_simp (intern r) s)) r else None"
lemma asize0:
shows "0 < asize r"
apply(induct r)
apply(auto)
done
lemma bders_simp_append:
shows "bders_simp r (s1 @ s2) = bders_simp (bders_simp r s1) s2"
apply(induct s1 arbitrary: r s2)
apply(simp)
apply(simp)
done
lemma bsimp_ASEQ_size:
shows "asize (bsimp_ASEQ bs r1 r2) \<le> Suc (asize r1 + asize r2)"
apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
apply(auto)
done
lemma fuse_size:
shows "asize (fuse bs r) = asize r"
apply(induct r)
apply(auto)
done
lemma flts_size:
shows "sum_list (map asize (flts rs)) \<le> sum_list (map asize rs)"
apply(induct rs rule: flts.induct)
apply(simp_all)
by (metis (mono_tags, lifting) add_mono comp_apply eq_imp_le fuse_size le_SucI map_eq_conv)
lemma bsimp_AALTs_size:
shows "asize (bsimp_AALTs bs rs) \<le> Suc (sum_list (map asize rs))"
apply(induct rs rule: bsimp_AALTs.induct)
apply(auto simp add: fuse_size)
done
lemma bsimp_size:
shows "asize (bsimp r) \<le> asize r"
apply(induct r)
apply(simp_all)
apply (meson Suc_le_mono add_mono_thms_linordered_semiring(1) bsimp_ASEQ_size le_trans)
apply(rule le_trans)
apply(rule bsimp_AALTs_size)
apply(simp)
apply(rule le_trans)
apply(rule flts_size)
by (simp add: sum_list_mono)
lemma bsimp_asize0:
shows "(\<Sum>x\<leftarrow>rs. asize (bsimp x)) \<le> sum_list (map asize rs)"
apply(induct rs)
apply(auto)
by (simp add: add_mono bsimp_size)
lemma bsimp_AALTs_size2:
assumes "\<forall>r \<in> set rs. nonalt r"
shows "asize (bsimp_AALTs bs rs) \<ge> sum_list (map asize rs)"
using assms
apply(induct rs rule: bsimp_AALTs.induct)
apply(simp_all add: fuse_size)
done
lemma qq:
shows "map (asize \<circ> fuse bs) rs = map asize rs"
apply(induct rs)
apply(auto simp add: fuse_size)
done
lemma flts_size2:
assumes "\<exists>bs rs'. AALTs bs rs' \<in> set rs"
shows "sum_list (map asize (flts rs)) < sum_list (map asize rs)"
using assms
apply(induct rs)
apply(auto simp add: qq)
apply (simp add: flts_size less_Suc_eq_le)
apply(case_tac a)
apply(auto simp add: qq)
prefer 2
apply (simp add: flts_size le_imp_less_Suc)
using less_Suc_eq by auto
lemma bsimp_AALTs_size3:
assumes "\<exists>r \<in> set (map bsimp rs). \<not>nonalt r"
shows "asize (bsimp (AALTs bs rs)) < asize (AALTs bs rs)"
using assms flts_size2
apply -
apply(clarify)
apply(simp)
apply(drule_tac x="map bsimp rs" in meta_spec)
apply(drule meta_mp)
apply (metis list.set_map nonalt.elims(3))
apply(simp)
apply(rule order_class.order.strict_trans1)
apply(rule bsimp_AALTs_size)
apply(simp)
by (smt Suc_leI bsimp_asize0 comp_def le_imp_less_Suc le_trans map_eq_conv not_less_eq)
lemma L_bsimp_ASEQ:
"L (SEQ (erase r1) (erase r2)) = L (erase (bsimp_ASEQ bs r1 r2))"
apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
apply(simp_all)
by (metis erase_fuse fuse.simps(4))
lemma L_bsimp_AALTs:
"L (erase (AALTs bs rs)) = L (erase (bsimp_AALTs bs rs))"
apply(induct bs rs rule: bsimp_AALTs.induct)
apply(simp_all add: erase_fuse)
done
lemma L_erase_AALTs:
shows "L (erase (AALTs bs rs)) = \<Union> (L ` erase ` (set rs))"
apply(induct rs)
apply(simp)
apply(simp)
apply(case_tac rs)
apply(simp)
apply(simp)
done
lemma L_erase_flts:
shows "\<Union> (L ` erase ` (set (flts rs))) = \<Union> (L ` erase ` (set rs))"
apply(induct rs rule: flts.induct)
apply(simp_all)
apply(auto)
using L_erase_AALTs erase_fuse apply auto[1]
by (simp add: L_erase_AALTs erase_fuse)
lemma L_bsimp_erase:
shows "L (erase r) = L (erase (bsimp r))"
apply(induct r)
apply(simp)
apply(simp)
apply(simp)
apply(auto simp add: Sequ_def)[1]
apply(subst L_bsimp_ASEQ[symmetric])
apply(auto simp add: Sequ_def)[1]
apply(subst (asm) L_bsimp_ASEQ[symmetric])
apply(auto simp add: Sequ_def)[1]
apply(simp)
apply(subst L_bsimp_AALTs[symmetric])
defer
apply(simp)
apply(subst (2)L_erase_AALTs)
apply(subst L_erase_flts)
apply(auto)
apply (simp add: L_erase_AALTs)
using L_erase_AALTs by blast
lemma bsimp_ASEQ0:
shows "bsimp_ASEQ bs r1 AZERO = AZERO"
apply(induct r1)
apply(auto)
done
lemma bsimp_ASEQ1:
assumes "r1 \<noteq> AZERO" "r2 \<noteq> AZERO" "\<forall>bs. r1 \<noteq> AONE bs"
shows "bsimp_ASEQ bs r1 r2 = ASEQ bs r1 r2"
using assms
apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
apply(auto)
done
lemma bsimp_ASEQ2:
shows "bsimp_ASEQ bs (AONE bs1) r2 = fuse (bs @ bs1) r2"
apply(induct r2)
apply(auto)
done
lemma L_bders_simp:
shows "L (erase (bders_simp r s)) = L (erase (bders r s))"
apply(induct s arbitrary: r rule: rev_induct)
apply(simp)
apply(simp)
apply(simp add: ders_append)
apply(simp add: bders_simp_append)
apply(simp add: L_bsimp_erase[symmetric])
by (simp add: der_correctness)
lemma b1:
"bsimp_ASEQ bs1 (AONE bs) r = fuse (bs1 @ bs) r"
apply(induct r)
apply(auto)
done
lemma b2:
assumes "bnullable r"
shows "bmkeps (fuse bs r) = bs @ bmkeps r"
by (simp add: assms bmkeps_retrieve bnullable_correctness erase_fuse mkeps_nullable retrieve_fuse2)
lemma b3:
shows "bnullable r = bnullable (bsimp r)"
using L_bsimp_erase bnullable_correctness nullable_correctness by auto
lemma b4:
shows "bnullable (bders_simp r s) = bnullable (bders r s)"
by (metis L_bders_simp bnullable_correctness lexer.simps(1) lexer_correct_None option.distinct(1))
lemma q1:
assumes "\<forall>r \<in> set rs. bmkeps(bsimp r) = bmkeps r"
shows "map (\<lambda>r. bmkeps(bsimp r)) rs = map bmkeps rs"
using assms
apply(induct rs)
apply(simp)
apply(simp)
done
lemma q3:
assumes "\<exists>r \<in> set rs. bnullable r"
shows "bmkeps (AALTs bs rs) = bmkeps (bsimp_AALTs bs rs)"
using assms
apply(induct bs rs rule: bsimp_AALTs.induct)
apply(simp)
apply(simp)
apply (simp add: b2)
apply(simp)
done
lemma qq1:
assumes "\<exists>r \<in> set rs. bnullable r"
shows "bmkeps (AALTs bs (rs @ rs1)) = bmkeps (AALTs bs rs)"
using assms
apply(induct rs arbitrary: rs1 bs)
apply(simp)
apply(simp)
by (metis Nil_is_append_conv bmkeps.simps(4) neq_Nil_conv r0 split_list_last)
lemma qq2:
assumes "\<forall>r \<in> set rs. \<not> bnullable r" "\<exists>r \<in> set rs1. bnullable r"
shows "bmkeps (AALTs bs (rs @ rs1)) = bmkeps (AALTs bs rs1)"
using assms
apply(induct rs arbitrary: rs1 bs)
apply(simp)
apply(simp)
by (metis append_assoc in_set_conv_decomp r1 r2)
lemma qq3:
shows "bnullable (AALTs bs rs) = (\<exists>r \<in> set rs. bnullable r)"
apply(induct rs arbitrary: bs)
apply(simp)
apply(simp)
done
lemma fuse_empty:
shows "fuse [] r = r"
apply(induct r)
apply(auto)
done
lemma flts_fuse:
shows "map (fuse bs) (flts rs) = flts (map (fuse bs) rs)"
apply(induct rs arbitrary: bs rule: flts.induct)
apply(auto simp add: fuse_append)
done
lemma bsimp_ASEQ_fuse:
shows "fuse bs1 (bsimp_ASEQ bs2 r1 r2) = bsimp_ASEQ (bs1 @ bs2) r1 r2"
apply(induct r1 r2 arbitrary: bs1 bs2 rule: bsimp_ASEQ.induct)
apply(auto)
done
lemma bsimp_AALTs_fuse:
assumes "\<forall>r \<in> set rs. fuse bs1 (fuse bs2 r) = fuse (bs1 @ bs2) r"
shows "fuse bs1 (bsimp_AALTs bs2 rs) = bsimp_AALTs (bs1 @ bs2) rs"
using assms
apply(induct bs2 rs arbitrary: bs1 rule: bsimp_AALTs.induct)
apply(auto)
done
lemma bsimp_fuse:
shows "fuse bs (bsimp r) = bsimp (fuse bs r)"
apply(induct r arbitrary: bs)
apply(simp)
apply(simp)
apply(simp)
prefer 3
apply(simp)
apply(simp)
apply (simp add: bsimp_ASEQ_fuse)
apply(simp)
by (simp add: bsimp_AALTs_fuse fuse_append)
lemma bsimp_fuse_AALTs:
shows "fuse bs (bsimp (AALTs [] rs)) = bsimp (AALTs bs rs)"
apply(subst bsimp_fuse)
apply(simp)
done
lemma bsimp_fuse_AALTs2:
shows "fuse bs (bsimp_AALTs [] rs) = bsimp_AALTs bs rs"
using bsimp_AALTs_fuse fuse_append by auto
lemma bsimp_ASEQ_idem:
assumes "bsimp (bsimp r1) = bsimp r1" "bsimp (bsimp r2) = bsimp r2"
shows "bsimp (bsimp_ASEQ x1 (bsimp r1) (bsimp r2)) = bsimp_ASEQ x1 (bsimp r1) (bsimp r2)"
using assms
apply(case_tac "bsimp r1 = AZERO")
apply(simp)
apply(case_tac "bsimp r2 = AZERO")
apply(simp)
apply (metis bnullable.elims(2) bnullable.elims(3) bsimp.simps(3) bsimp_ASEQ.simps(2) bsimp_ASEQ.simps(3) bsimp_ASEQ.simps(4) bsimp_ASEQ.simps(5) bsimp_ASEQ.simps(6))
apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
apply(auto)[1]
apply(subst bsimp_ASEQ2)
apply(subst bsimp_ASEQ2)
apply (metis assms(2) bsimp_fuse)
apply(subst bsimp_ASEQ1)
apply(auto)
done
fun nonnested :: "arexp \<Rightarrow> bool"
where
"nonnested (AALTs bs2 []) = True"
| "nonnested (AALTs bs2 ((AALTs bs1 rs1) # rs2)) = False"
| "nonnested (AALTs bs2 (r # rs2)) = nonnested (AALTs bs2 rs2)"
| "nonnested r = True"
lemma k0:
shows "flts (r # rs1) = flts [r] @ flts rs1"
apply(induct r arbitrary: rs1)
apply(auto)
done
lemma k00:
shows "flts (rs1 @ rs2) = flts rs1 @ flts rs2"
apply(induct rs1 arbitrary: rs2)
apply(auto)
by (metis append.assoc k0)
lemma k0a:
shows "flts [AALTs bs rs] = map (fuse bs) rs"
apply(simp)
done
lemma k0b:
assumes "nonalt r" "r \<noteq> AZERO"
shows "flts [r] = [r]"
using assms
apply(case_tac r)
apply(simp_all)
done
lemma nn1:
assumes "nonnested (AALTs bs rs)"
shows "\<nexists>bs1 rs1. flts rs = [AALTs bs1 rs1]"
using assms
apply(induct rs rule: flts.induct)
apply(auto)
done
lemma nn1q:
assumes "nonnested (AALTs bs rs)"
shows "\<nexists>bs1 rs1. AALTs bs1 rs1 \<in> set (flts rs)"
using assms
apply(induct rs rule: flts.induct)
apply(auto)
done
lemma nn1qq:
assumes "nonnested (AALTs bs rs)"
shows "\<nexists>bs1 rs1. AALTs bs1 rs1 \<in> set rs"
using assms
apply(induct rs rule: flts.induct)
apply(auto)
done
lemma nn10:
assumes "nonnested (AALTs cs rs)"
shows "nonnested (AALTs (bs @ cs) rs)"
using assms
apply(induct rs arbitrary: cs bs)
apply(simp_all)
apply(case_tac a)
apply(simp_all)
done
lemma nn11a:
assumes "nonalt r"
shows "nonalt (fuse bs r)"
using assms
apply(induct r)
apply(auto)
done
lemma nn1a:
assumes "nonnested r"
shows "nonnested (fuse bs r)"
using assms
apply(induct bs r arbitrary: rule: fuse.induct)
apply(simp_all add: nn10)
done
lemma n0:
shows "nonnested (AALTs bs rs) \<longleftrightarrow> (\<forall>r \<in> set rs. nonalt r)"
apply(induct rs arbitrary: bs)
apply(auto)
apply (metis list.set_intros(1) nn1qq nonalt.elims(3))
apply (metis list.set_intros(2) nn1qq nonalt.elims(3))
by (metis nonalt.elims(2) nonnested.simps(3) nonnested.simps(4) nonnested.simps(5) nonnested.simps(6) nonnested.simps(7))
lemma nn1c:
assumes "\<forall>r \<in> set rs. nonnested r"
shows "\<forall>r \<in> set (flts rs). nonalt r"
using assms
apply(induct rs rule: flts.induct)
apply(auto)
apply(rule nn11a)
by (metis nn1qq nonalt.elims(3))
lemma nn1bb:
assumes "\<forall>r \<in> set rs. nonalt r"
shows "nonnested (bsimp_AALTs bs rs)"
using assms
apply(induct bs rs rule: bsimp_AALTs.induct)
apply(auto)
apply (metis nn11a nonalt.simps(1) nonnested.elims(3))
using n0 by auto
lemma nn1b:
shows "nonnested (bsimp r)"
apply(induct r)
apply(simp_all)
apply(case_tac "bsimp r1 = AZERO")
apply(simp)
apply(case_tac "bsimp r2 = AZERO")
apply(simp)
apply(subst bsimp_ASEQ0)
apply(simp)
apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
apply(auto)[1]
apply(subst bsimp_ASEQ2)
apply (simp add: nn1a)
apply(subst bsimp_ASEQ1)
apply(auto)
apply(rule nn1bb)
apply(auto)
by (metis (mono_tags, hide_lams) imageE nn1c set_map)
lemma nn1d:
assumes "bsimp r = AALTs bs rs"
shows "\<forall>r1 \<in> set rs. \<forall> bs. r1 \<noteq> AALTs bs rs2"
using nn1b assms
by (metis nn1qq)
lemma nn_flts:
assumes "nonnested (AALTs bs rs)"
shows "\<forall>r \<in> set (flts rs). nonalt r"
using assms
apply(induct rs arbitrary: bs rule: flts.induct)
apply(auto)
done
lemma rt:
shows "sum_list (map asize (flts (map bsimp rs))) \<le> sum_list (map asize rs)"
apply(induct rs)
apply(simp)
apply(simp)
apply(subst k0)
apply(simp)
by (smt add_le_cancel_right add_mono bsimp_size flts.simps(1) flts_size k0 le_iff_add list.simps(9) map_append sum_list.Cons sum_list.append trans_le_add1)
lemma bsimp_AALTs_qq:
assumes "1 < length rs"
shows "bsimp_AALTs bs rs = AALTs bs rs"
using assms
apply(case_tac rs)
apply(simp)
apply(case_tac list)
apply(simp_all)
done
lemma bsimp_AALTs1:
assumes "nonalt r"
shows "bsimp_AALTs bs (flts [r]) = fuse bs r"
using assms
apply(case_tac r)
apply(simp_all)
done
lemma bbbbs:
assumes "good r" "r = AALTs bs1 rs"
shows "bsimp_AALTs bs (flts [r]) = AALTs bs (map (fuse bs1) rs)"
using assms
by (metis (no_types, lifting) Nil_is_map_conv append.left_neutral append_butlast_last_id bsimp_AALTs.elims butlast.simps(2) good.simps(4) good.simps(5) k0a map_butlast)
lemma bbbbs1:
shows "nonalt r \<or> (\<exists>bs rs. r = AALTs bs rs)"
using nonalt.elims(3) by auto
lemma good_fuse:
shows "good (fuse bs r) = good r"
apply(induct r arbitrary: bs)
apply(auto)
apply(case_tac r1)
apply(simp_all)
apply(case_tac r2)
apply(simp_all)
apply(case_tac r2)
apply(simp_all)
apply(case_tac r2)
apply(simp_all)
apply(case_tac r2)
apply(simp_all)
apply(case_tac r1)
apply(simp_all)
apply(case_tac r2)
apply(simp_all)
apply(case_tac r2)
apply(simp_all)
apply(case_tac r2)
apply(simp_all)
apply(case_tac r2)
apply(simp_all)
apply(case_tac x2a)
apply(simp_all)
apply(case_tac list)
apply(simp_all)
apply(case_tac x2a)
apply(simp_all)
apply(case_tac list)
apply(simp_all)
done
lemma good0:
assumes "rs \<noteq> Nil" "\<forall>r \<in> set rs. nonalt r"
shows "good (bsimp_AALTs bs rs) \<longleftrightarrow> (\<forall>r \<in> set rs. good r)"
using assms
apply(induct bs rs rule: bsimp_AALTs.induct)
apply(auto simp add: good_fuse)
done
lemma good0a:
assumes "flts (map bsimp rs) \<noteq> Nil" "\<forall>r \<in> set (flts (map bsimp rs)). nonalt r"
shows "good (bsimp (AALTs bs rs)) \<longleftrightarrow> (\<forall>r \<in> set (flts (map bsimp rs)). good r)"
using assms
apply(simp)
apply(auto)
apply(subst (asm) good0)
apply(simp)
apply(auto)
apply(subst good0)
apply(simp)
apply(auto)
done
lemma flts0:
assumes "r \<noteq> AZERO" "nonalt r"
shows "flts [r] \<noteq> []"
using assms
apply(induct r)
apply(simp_all)
done
lemma flts1:
assumes "good r"
shows "flts [r] \<noteq> []"
using assms
apply(induct r)
apply(simp_all)
apply(case_tac x2a)
apply(simp)
apply(simp)
done
lemma flts2:
assumes "good r"
shows "\<forall>r' \<in> set (flts [r]). good r' \<and> nonalt r'"
using assms
apply(induct r)
apply(simp)
apply(simp)
apply(simp)
prefer 2
apply(simp)
apply(auto)[1]
apply (metis bsimp_AALTs.elims good.simps(4) good.simps(5) good.simps(6) good_fuse)
apply (metis bsimp_AALTs.elims good.simps(4) good.simps(5) good.simps(6) nn11a)
apply fastforce
apply(simp)
done
lemma flts3:
assumes "\<forall>r \<in> set rs. good r \<or> r = AZERO"
shows "\<forall>r \<in> set (flts rs). good r"
using assms
apply(induct rs arbitrary: rule: flts.induct)
apply(simp_all)
by (metis UnE flts2 k0a set_map)
lemma flts3b:
assumes "\<exists>r\<in>set rs. good r"
shows "flts rs \<noteq> []"
using assms
apply(induct rs arbitrary: rule: flts.induct)
apply(simp)
apply(simp)
apply(simp)
apply(auto)
done
lemma flts4:
assumes "bsimp_AALTs bs (flts rs) = AZERO"
shows "\<forall>r \<in> set rs. \<not> good r"
using assms
apply(induct rs arbitrary: bs rule: flts.induct)
apply(auto)
defer
apply (metis (no_types, lifting) Nil_is_append_conv append_self_conv2 bsimp_AALTs.elims butlast.simps(2) butlast_append flts3b nonalt.simps(1) nonalt.simps(2))
apply (metis arexp.distinct(7) bsimp_AALTs.elims flts2 good.simps(1) good.simps(2) good0 k0b list.distinct(1) list.inject nonalt.simps(3))
apply (metis arexp.distinct(3) arexp.distinct(7) bsimp_AALTs.elims fuse.simps(3) list.distinct(1) list.inject)
apply (metis arexp.distinct(7) bsimp_AALTs.elims good.simps(1) good_fuse list.distinct(1) list.inject)
apply (metis arexp.distinct(7) bsimp_AALTs.elims list.distinct(1) list.inject)
apply (metis arexp.distinct(7) bsimp_AALTs.elims flts2 good.simps(1) good.simps(33) good0 k0b list.distinct(1) list.inject nonalt.simps(6))
by (metis (no_types, lifting) Nil_is_append_conv append_Nil2 arexp.distinct(7) bsimp_AALTs.elims butlast.simps(2) butlast_append flts1 flts2 good.simps(1) good0 k0a)
lemma flts_nil:
assumes "\<forall>y. asize y < Suc (sum_list (map asize rs)) \<longrightarrow>
good (bsimp y) \<or> bsimp y = AZERO"
and "\<forall>r\<in>set rs. \<not> good (bsimp r)"
shows "flts (map bsimp rs) = []"
using assms
apply(induct rs)
apply(simp)
apply(simp)
apply(subst k0)
apply(simp)
by force
lemma flts_nil2:
assumes "\<forall>y. asize y < Suc (sum_list (map asize rs)) \<longrightarrow>
good (bsimp y) \<or> bsimp y = AZERO"
and "bsimp_AALTs bs (flts (map bsimp rs)) = AZERO"
shows "flts (map bsimp rs) = []"
using assms
apply(induct rs arbitrary: bs)
apply(simp)
apply(simp)
apply(subst k0)
apply(simp)
apply(subst (asm) k0)
apply(auto)
apply (metis flts.simps(1) flts.simps(2) flts4 k0 less_add_Suc1 list.set_intros(1))
by (metis flts.simps(2) flts4 k0 less_add_Suc1 list.set_intros(1))
lemma good_SEQ:
assumes "r1 \<noteq> AZERO" "r2 \<noteq> AZERO" "\<forall>bs. r1 \<noteq> AONE bs"
shows "good (ASEQ bs r1 r2) = (good r1 \<and> good r2)"
using assms
apply(case_tac r1)
apply(simp_all)
apply(case_tac r2)
apply(simp_all)
apply(case_tac r2)
apply(simp_all)
apply(case_tac r2)
apply(simp_all)
apply(case_tac r2)
apply(simp_all)
done
lemma good1:
shows "good (bsimp a) \<or> bsimp a = AZERO"
apply(induct a taking: asize rule: measure_induct)
apply(case_tac x)
apply(simp)
apply(simp)
apply(simp)
prefer 3
apply(simp)
prefer 2
(* AALTs case *)
apply(simp only:)
apply(case_tac "x52")
apply(simp)
thm good0a
(* AALTs list at least one - case *)
apply(simp only: )
apply(frule_tac x="a" in spec)
apply(drule mp)
apply(simp)
(* either first element is good, or AZERO *)
apply(erule disjE)
prefer 2
apply(simp)
(* in the AZERO case, the size is smaller *)
apply(drule_tac x="AALTs x51 list" in spec)
apply(drule mp)
apply(simp add: asize0)
apply(subst (asm) bsimp.simps)
apply(subst (asm) bsimp.simps)
apply(assumption)
(* in the good case *)
apply(frule_tac x="AALTs x51 list" in spec)
apply(drule mp)
apply(simp add: asize0)
apply(erule disjE)
apply(rule disjI1)
apply(simp add: good0)
apply(subst good0)
apply (metis Nil_is_append_conv flts1 k0)
apply (metis ex_map_conv list.simps(9) nn1b nn1c)
apply(simp)
apply(subst k0)
apply(simp)
apply(auto)[1]
using flts2 apply blast
apply(subst (asm) good0)
prefer 3
apply(auto)[1]
apply auto[1]
apply (metis ex_map_conv nn1b nn1c)
(* in the AZERO case *)
apply(simp)
apply(frule_tac x="a" in spec)
apply(drule mp)
apply(simp)
apply(erule disjE)
apply(rule disjI1)
apply(subst good0)
apply(subst k0)
using flts1 apply blast
apply(auto)[1]
apply (metis (no_types, hide_lams) ex_map_conv list.simps(9) nn1b nn1c)
apply(auto)[1]
apply(subst (asm) k0)
apply(auto)[1]
using flts2 apply blast
apply(frule_tac x="AALTs x51 list" in spec)
apply(drule mp)
apply(simp add: asize0)
apply(erule disjE)
apply(simp)
apply(simp)
apply (metis add.left_commute flts_nil2 less_add_Suc1 less_imp_Suc_add list.distinct(1) list.set_cases nat.inject)
apply(subst (2) k0)
apply(simp)
(* SEQ case *)
apply(simp)
apply(case_tac "bsimp x42 = AZERO")
apply(simp)
apply(case_tac "bsimp x43 = AZERO")
apply(simp)
apply(subst (2) bsimp_ASEQ0)
apply(simp)
apply(case_tac "\<exists>bs. bsimp x42 = AONE bs")
apply(auto)[1]
apply(subst bsimp_ASEQ2)
using good_fuse apply force
apply(subst bsimp_ASEQ1)
apply(auto)
apply(subst good_SEQ)
apply(simp)
apply(simp)
apply(simp)
using less_add_Suc1 less_add_Suc2 by blast
lemma good1a:
assumes "L(erase a) \<noteq> {}"
shows "good (bsimp a)"
using good1 assms
using L_bsimp_erase by force
lemma flts_append:
"flts (xs1 @ xs2) = flts xs1 @ flts xs2"
apply(induct xs1 arbitrary: xs2 rule: rev_induct)
apply(auto)
apply(case_tac xs)
apply(auto)
apply(case_tac x)
apply(auto)
apply(case_tac x)
apply(auto)
done
lemma g1:
assumes "good (bsimp_AALTs bs rs)"
shows "bsimp_AALTs bs rs = AALTs bs rs \<or> (\<exists>r. rs = [r] \<and> bsimp_AALTs bs [r] = fuse bs r)"
using assms
apply(induct rs arbitrary: bs)
apply(simp)
apply(case_tac rs)
apply(simp only:)
apply(simp)
apply(case_tac list)
apply(simp)
by simp
lemma flts_0:
assumes "nonnested (AALTs bs rs)"
shows "\<forall>r \<in> set (flts rs). r \<noteq> AZERO"
using assms
apply(induct rs arbitrary: bs rule: flts.induct)
apply(simp)
apply(simp)
defer
apply(simp)
apply(simp)
apply(simp)
apply(simp)
apply(rule ballI)
apply(simp)
done
lemma flts_0a:
assumes "nonnested (AALTs bs rs)"
shows "AZERO \<notin> set (flts rs)"
using assms
using flts_0 by blast
lemma qqq1:
shows "AZERO \<notin> set (flts (map bsimp rs))"
by (metis ex_map_conv flts3 good.simps(1) good1)
fun nonazero :: "arexp \<Rightarrow> bool"
where
"nonazero AZERO = False"
| "nonazero r = True"
lemma flts_concat:
shows "flts rs = concat (map (\<lambda>r. flts [r]) rs)"
apply(induct rs)
apply(auto)
apply(subst k0)
apply(simp)
done
lemma flts_single1:
assumes "nonalt r" "nonazero r"
shows "flts [r] = [r]"
using assms
apply(induct r)
apply(auto)
done
lemma flts_qq:
assumes "\<forall>y. asize y < Suc (sum_list (map asize rs)) \<longrightarrow> good y \<longrightarrow> bsimp y = y"
"\<forall>r'\<in>set rs. good r' \<and> nonalt r'"
shows "flts (map bsimp rs) = rs"
using assms
apply(induct rs)
apply(simp)
apply(simp)
apply(subst k0)
apply(subgoal_tac "flts [bsimp a] = [a]")
prefer 2
apply(drule_tac x="a" in spec)
apply(drule mp)
apply(simp)
apply(auto)[1]
using good.simps(1) k0b apply blast
apply(auto)[1]
done
lemma test:
assumes "good r"
shows "bsimp r = r"
using assms
apply(induct r taking: "asize" rule: measure_induct)
apply(erule good.elims)
apply(simp_all)
apply(subst k0)
apply(subst (2) k0)
apply(subst flts_qq)
apply(auto)[1]
apply(auto)[1]
apply (metis append_Cons append_Nil bsimp_AALTs.simps(3) good.simps(1) k0b)
apply force+
apply (metis (no_types, lifting) add_Suc add_Suc_right asize.simps(5) bsimp.simps(1) bsimp_ASEQ.simps(19) less_add_Suc1 less_add_Suc2)
apply (metis add_Suc add_Suc_right arexp.distinct(5) arexp.distinct(7) asize.simps(4) asize.simps(5) bsimp.simps(1) bsimp.simps(2) bsimp_ASEQ1 good.simps(21) good.simps(8) less_add_Suc1 less_add_Suc2)
apply force+
apply (metis (no_types, lifting) add_Suc add_Suc_right arexp.distinct(5) arexp.distinct(7) asize.simps(4) asize.simps(5) bsimp.simps(1) bsimp.simps(2) bsimp_ASEQ1 good.simps(25) good.simps(8) less_add_Suc1 less_add_Suc2)
apply (metis add_Suc add_Suc_right arexp.distinct(7) asize.simps(4) bsimp.simps(2) bsimp_ASEQ1 good.simps(26) good.simps(8) less_add_Suc1 less_add_Suc2)
apply force+
done
lemma test2:
assumes "good r"
shows "bsimp r = r"
using assms
apply(induct r taking: "asize" rule: measure_induct)
apply(case_tac x)
apply(simp_all)
defer
(* AALT case *)
apply(subgoal_tac "1 < length x52")
prefer 2
apply(case_tac x52)
apply(simp)
apply(simp)
apply(case_tac list)
apply(simp)
apply(simp)
apply(subst bsimp_AALTs_qq)
prefer 2
apply(subst flts_qq)
apply(auto)[1]
apply(auto)[1]
apply(case_tac x52)
apply(simp)
apply(simp)
apply(case_tac list)
apply(simp)
apply(simp)
apply(auto)[1]
apply (metis (no_types, lifting) bsimp_AALTs.elims good.simps(6) length_Cons length_pos_if_in_set list.size(3) nat_neq_iff)
apply(simp)
apply(case_tac x52)
apply(simp)
apply(simp)
apply(case_tac list)
apply(simp)
apply(simp)
apply(subst k0)
apply(simp)
apply(subst (2) k0)
apply(simp)
apply (simp add: Suc_lessI flts1 one_is_add)
(* SEQ case *)
apply(case_tac "bsimp x42 = AZERO")
apply simp
apply (metis asize.elims good.simps(10) good.simps(11) good.simps(12) good.simps(2) good.simps(7) good.simps(9) good_SEQ less_add_Suc1)
apply(case_tac "\<exists>bs'. bsimp x42 = AONE bs'")
apply(auto)[1]
defer
apply(case_tac "bsimp x43 = AZERO")
apply(simp)
apply (metis bsimp.elims bsimp.simps(3) good.simps(10) good.simps(11) good.simps(12) good.simps(8) good.simps(9) good_SEQ less_add_Suc2)
apply(auto)
apply (subst bsimp_ASEQ1)
apply(auto)[3]
apply(auto)[1]
apply (metis bsimp.simps(3) good.simps(2) good_SEQ less_add_Suc1)
apply (metis bsimp.simps(3) good.simps(2) good_SEQ less_add_Suc1 less_add_Suc2)
apply (subst bsimp_ASEQ2)
apply(drule_tac x="x42" in spec)
apply(drule mp)
apply(simp)
apply(drule mp)
apply (metis bsimp.elims bsimp.simps(3) good.simps(10) good.simps(11) good.simps(2) good_SEQ)
apply(simp)
done
lemma bsimp_idem:
shows "bsimp (bsimp r) = bsimp r"
using test good1
by force
lemma q3a:
assumes "\<exists>r \<in> set rs. bnullable r"
shows "bmkeps (AALTs bs (map (fuse bs1) rs)) = bmkeps (AALTs (bs@bs1) rs)"
using assms
apply(induct rs arbitrary: bs bs1)
apply(simp)
apply(simp)
apply(auto)
apply (metis append_assoc b2 bnullable_correctness erase_fuse r0)
apply(case_tac "bnullable a")
apply (metis append.assoc b2 bnullable_correctness erase_fuse r0)
apply(case_tac rs)
apply(simp)
apply(simp)
apply(auto)[1]
apply (metis bnullable_correctness erase_fuse)+
done
lemma qq4:
assumes "\<exists>x\<in>set list. bnullable x"
shows "\<exists>x\<in>set (flts list). bnullable x"
using assms
apply(induct list rule: flts.induct)
apply(auto)
by (metis UnCI bnullable_correctness erase_fuse imageI)
lemma qs3:
assumes "\<exists>r \<in> set rs. bnullable r"
shows "bmkeps (AALTs bs rs) = bmkeps (AALTs bs (flts rs))"
using assms
apply(induct rs arbitrary: bs taking: size rule: measure_induct)
apply(case_tac x)
apply(simp)
apply(simp)
apply(case_tac a)
apply(simp)
apply (simp add: r1)
apply(simp)
apply (simp add: r0)
apply(simp)
apply(case_tac "flts list")
apply(simp)
apply (metis L_erase_AALTs L_erase_flts L_flat_Prf1 L_flat_Prf2 Prf_elims(1) bnullable_correctness erase.simps(4) mkeps_nullable r2)
apply(simp)
apply (simp add: r1)
prefer 3
apply(simp)
apply (simp add: r0)
prefer 2
apply(simp)
apply(case_tac "\<exists>x\<in>set x52. bnullable x")
apply(case_tac "list")
apply(simp)
apply (metis b2 fuse.simps(4) q3a r2)
apply(erule disjE)
apply(subst qq1)
apply(auto)[1]
apply (metis bnullable_correctness erase_fuse)
apply(simp)
apply (metis b2 fuse.simps(4) q3a r2)
apply(simp)
apply(auto)[1]
apply(subst qq1)
apply (metis bnullable_correctness erase_fuse image_eqI set_map)
apply (metis b2 fuse.simps(4) q3a r2)
apply(subst qq1)
apply (metis bnullable_correctness erase_fuse image_eqI set_map)
apply (metis b2 fuse.simps(4) q3a r2)
apply(simp)
apply(subst qq2)
apply (metis bnullable_correctness erase_fuse imageE set_map)
prefer 2
apply(case_tac "list")
apply(simp)
apply(simp)
apply (simp add: qq4)
apply(simp)
apply(auto)
apply(case_tac list)
apply(simp)
apply(simp)
apply (simp add: r0)
apply(case_tac "bnullable (ASEQ x41 x42 x43)")
apply(case_tac list)
apply(simp)
apply(simp)
apply (simp add: r0)
apply(simp)
using qq4 r1 r2 by auto
lemma k1:
assumes "\<And>x2aa. \<lbrakk>x2aa \<in> set x2a; bnullable x2aa\<rbrakk> \<Longrightarrow> bmkeps x2aa = bmkeps (bsimp x2aa)"
"\<exists>x\<in>set x2a. bnullable x"
shows "bmkeps (AALTs x1 (flts x2a)) = bmkeps (AALTs x1 (flts (map bsimp x2a)))"
using assms
apply(induct x2a)
apply fastforce
apply(simp)
apply(subst k0)
apply(subst (2) k0)
apply(auto)[1]
apply (metis b3 k0 list.set_intros(1) qs3 r0)
by (smt b3 imageI insert_iff k0 list.set(2) qq3 qs3 r0 r1 set_map)
lemma bmkeps_simp:
assumes "bnullable r"
shows "bmkeps r = bmkeps (bsimp r)"
using assms
apply(induct r)
apply(simp)
apply(simp)
apply(simp)
apply(simp)
prefer 3
apply(simp)
apply(case_tac "bsimp r1 = AZERO")
apply(simp)
apply(auto)[1]
apply (metis L_bsimp_erase L_flat_Prf1 L_flat_Prf2 Prf_elims(1) bnullable_correctness erase.simps(1) mkeps_nullable)
apply(case_tac "bsimp r2 = AZERO")
apply(simp)
apply(auto)[1]
apply (metis L_bsimp_erase L_flat_Prf1 L_flat_Prf2 Prf_elims(1) bnullable_correctness erase.simps(1) mkeps_nullable)
apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
apply(auto)[1]
apply(subst b1)
apply(subst b2)
apply(simp add: b3[symmetric])
apply(simp)
apply(subgoal_tac "bsimp_ASEQ x1 (bsimp r1) (bsimp r2) = ASEQ x1 (bsimp r1) (bsimp r2)")
prefer 2
apply (smt b3 bnullable.elims(2) bsimp_ASEQ.simps(17) bsimp_ASEQ.simps(19) bsimp_ASEQ.simps(20) bsimp_ASEQ.simps(21) bsimp_ASEQ.simps(22) bsimp_ASEQ.simps(24) bsimp_ASEQ.simps(25) bsimp_ASEQ.simps(26) bsimp_ASEQ.simps(27) bsimp_ASEQ.simps(29) bsimp_ASEQ.simps(30) bsimp_ASEQ.simps(31))
apply(simp)
apply(simp)
thm q3
apply(subst q3[symmetric])
apply simp
using b3 qq4 apply auto[1]
apply(subst qs3)
apply simp
using k1 by blast
thm bmkeps_retrieve bmkeps_simp bder_retrieve
lemma bmkeps_bder_AALTs:
assumes "\<exists>r \<in> set rs. bnullable (bder c r)"
shows "bmkeps (bder c (bsimp_AALTs bs rs)) = bmkeps (bsimp_AALTs bs (map (bder c) rs))"
using assms
apply(induct rs)
apply(simp)
apply(simp)
apply(auto)
apply(case_tac rs)
apply(simp)
apply (metis (full_types) Prf_injval bder_retrieve bmkeps_retrieve bnullable_correctness erase_bder erase_fuse mkeps_nullable retrieve_fuse2)
apply(simp)
apply(case_tac rs)
apply(simp_all)
done
lemma bbs0:
shows "blexer_simp r [] = blexer r []"
apply(simp add: blexer_def blexer_simp_def)
done
lemma bbs1:
shows "blexer_simp r [c] = blexer r [c]"
apply(simp add: blexer_def blexer_simp_def)
apply(auto)
defer
using b3 apply auto[1]
using b3 apply auto[1]
apply(subst bmkeps_simp[symmetric])
apply(simp)
apply(simp)
done
lemma oo:
shows "(case (blexer (der c r) s) of None \<Rightarrow> None | Some v \<Rightarrow> Some (injval r c v)) = blexer r (c # s)"
apply(simp add: blexer_correctness)
done
lemma bder_fuse:
shows "bder c (fuse bs a) = fuse bs (bder c a)"
apply(induct a arbitrary: bs c)
apply(simp_all)
done
lemma XXX2_helper:
assumes "\<forall>y. asize y < Suc (sum_list (map asize rs)) \<longrightarrow> good y \<longrightarrow> bsimp y = y"
"\<forall>r'\<in>set rs. good r' \<and> nonalt r'"
shows "flts (map (bsimp \<circ> bder c) (flts (map bsimp rs))) = flts (map (bsimp \<circ> bder c) rs)"
using assms
apply(induct rs arbitrary: c)
apply(simp)
apply(simp)
apply(subst k0)
apply(simp add: flts_append)
apply(subst (2) k0)
apply(simp add: flts_append)
apply(subgoal_tac "flts [a] = [a]")
prefer 2
using good.simps(1) k0b apply blast
apply(simp)
done
lemma bmkeps_good:
assumes "good a"
shows "bmkeps (bsimp a) = bmkeps a"
using assms
using test2 by auto
lemma xxx_bder:
assumes "good r"
shows "L (erase r) \<noteq> {}"
using assms
apply(induct r rule: good.induct)
apply(auto simp add: Sequ_def)
done
lemma xxx_bder2:
assumes "L (erase (bsimp r)) = {}"
shows "bsimp r = AZERO"
using assms xxx_bder test2 good1
by blast
lemma XXX2aa:
assumes "good a"
shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
using assms
by (simp add: test2)
lemma XXX2aa_ders:
assumes "good a"
shows "bsimp (bders (bsimp a) s) = bsimp (bders a s)"
using assms
by (simp add: test2)
lemma XXX4a:
shows "good (bders_simp (bsimp r) s) \<or> bders_simp (bsimp r) s = AZERO"
apply(induct s arbitrary: r rule: rev_induct)
apply(simp)
apply (simp add: good1)
apply(simp add: bders_simp_append)
apply (simp add: good1)
done
lemma XXX4a_good:
assumes "good a"
shows "good (bders_simp a s) \<or> bders_simp a s = AZERO"
using assms
apply(induct s arbitrary: a rule: rev_induct)
apply(simp)
apply(simp add: bders_simp_append)
apply (simp add: good1)
done
lemma XXX4a_good_cons:
assumes "s \<noteq> []"
shows "good (bders_simp a s) \<or> bders_simp a s = AZERO"
using assms
apply(case_tac s)
apply(auto)
using XXX4a by blast
lemma XXX4b:
assumes "good a" "L (erase (bders_simp a s)) \<noteq> {}"
shows "good (bders_simp a s)"
using assms
apply(induct s arbitrary: a)
apply(simp)
apply(simp)
apply(subgoal_tac "L (erase (bder a aa)) = {} \<or> L (erase (bder a aa)) \<noteq> {}")
prefer 2
apply(auto)[1]
apply(erule disjE)
apply(subgoal_tac "bsimp (bder a aa) = AZERO")
prefer 2
using L_bsimp_erase xxx_bder2 apply auto[1]
apply(simp)
apply (metis L.simps(1) XXX4a erase.simps(1))
apply(drule_tac x="bsimp (bder a aa)" in meta_spec)
apply(drule meta_mp)
apply simp
apply(rule good1a)
apply(auto)
done
lemma bders_AZERO:
shows "bders AZERO s = AZERO"
and "bders_simp AZERO s = AZERO"
apply (induct s)
apply(auto)
done
lemma LA:
assumes "\<Turnstile> v : ders s (erase r)"
shows "retrieve (bders r s) v = retrieve r (flex (erase r) id s v)"
using assms
apply(induct s arbitrary: r v rule: rev_induct)
apply(simp)
apply(simp add: bders_append ders_append)
apply(subst bder_retrieve)
apply(simp)
apply(drule Prf_injval)
by (simp add: flex_append)
lemma LB:
assumes "s \<in> (erase r) \<rightarrow> v"
shows "retrieve r v = retrieve r (flex (erase r) id s (mkeps (ders s (erase r))))"
using assms
apply(induct s arbitrary: r v rule: rev_induct)
apply(simp)
apply(subgoal_tac "v = mkeps (erase r)")
prefer 2
apply (simp add: Posix1(1) Posix_determ Posix_mkeps nullable_correctness)
apply(simp)
apply(simp add: flex_append ders_append)
by (metis Posix_determ Posix_flex Posix_injval Posix_mkeps ders_snoc lexer_correctness(2) lexer_flex)
lemma LB_sym:
assumes "s \<in> (erase r) \<rightarrow> v"
shows "retrieve r v = retrieve r (flex (erase r) id s (mkeps (erase (bders r s))))"
using assms
by (simp add: LB)
lemma LC:
assumes "s \<in> (erase r) \<rightarrow> v"
shows "retrieve r v = retrieve (bders r s) (mkeps (erase (bders r s)))"
apply(simp)
by (metis LA LB Posix1(1) assms lexer_correct_None lexer_flex mkeps_nullable)
lemma L0:
assumes "bnullable a"
shows "retrieve (bsimp a) (mkeps (erase (bsimp a))) = retrieve a (mkeps (erase a))"
using assms
by (metis b3 bmkeps_retrieve bmkeps_simp bnullable_correctness)
thm bmkeps_retrieve
lemma L0a:
assumes "s \<in> L(erase a)"
shows "retrieve (bsimp (bders a s)) (mkeps (erase (bsimp (bders a s)))) =
retrieve (bders a s) (mkeps (erase (bders a s)))"
using assms
by (metis L0 bnullable_correctness erase_bders lexer_correct_None lexer_flex)
lemma L0aa:
assumes "s \<in> L (erase a)"
shows "[] \<in> erase (bsimp (bders a s)) \<rightarrow> mkeps (erase (bsimp (bders a s)))"
using assms
by (metis Posix_mkeps b3 bnullable_correctness erase_bders lexer_correct_None lexer_flex)
lemma L0aaa:
assumes "[c] \<in> L (erase a)"
shows "[c] \<in> (erase a) \<rightarrow> flex (erase a) id [c] (mkeps (erase (bder c a)))"
using assms
by (metis bders.simps(1) bders.simps(2) erase_bders lexer_correct_None lexer_correct_Some lexer_flex option.inject)
lemma L0aaaa:
assumes "[c] \<in> L (erase a)"
shows "[c] \<in> (erase a) \<rightarrow> flex (erase a) id [c] (mkeps (erase (bders a [c])))"
using assms
using L0aaa by auto
lemma L02:
assumes "bnullable (bder c a)"
shows "retrieve (bsimp a) (flex (erase (bsimp a)) id [c] (mkeps (erase (bder c (bsimp a))))) =
retrieve (bder c (bsimp a)) (mkeps (erase (bder c (bsimp a))))"
using assms
apply(simp)
using bder_retrieve L0 bmkeps_simp bmkeps_retrieve L0 LA LB
apply(subst bder_retrieve[symmetric])
apply (metis L_bsimp_erase bnullable_correctness der_correctness erase_bder mkeps_nullable nullable_correctness)
apply(simp)
done
lemma L02_bders:
assumes "bnullable (bders a s)"
shows "retrieve (bsimp a) (flex (erase (bsimp a)) id s (mkeps (erase (bders (bsimp a) s)))) =
retrieve (bders (bsimp a) s) (mkeps (erase (bders (bsimp a) s)))"
using assms
by (metis LA L_bsimp_erase bnullable_correctness ders_correctness erase_bders mkeps_nullable nullable_correctness)
lemma L03:
assumes "bnullable (bder c a)"
shows "retrieve (bder c (bsimp a)) (mkeps (erase (bder c (bsimp a)))) =
bmkeps (bsimp (bder c (bsimp a)))"
using assms
by (metis L0 L_bsimp_erase bmkeps_retrieve bnullable_correctness der_correctness erase_bder nullable_correctness)
lemma L04:
assumes "bnullable (bder c a)"
shows "retrieve (bder c (bsimp a)) (mkeps (erase (bder c (bsimp a)))) =
retrieve (bsimp (bder c (bsimp a))) (mkeps (erase (bsimp (bder c (bsimp a)))))"
using assms
by (metis L0 L_bsimp_erase bnullable_correctness der_correctness erase_bder nullable_correctness)
lemma L05:
assumes "bnullable (bder c a)"
shows "retrieve (bder c (bsimp a)) (mkeps (erase (bder c (bsimp a)))) =
retrieve (bsimp (bder c (bsimp a))) (mkeps (erase (bsimp (bder c (bsimp a)))))"
using assms
using L04 by auto
lemma L06:
assumes "bnullable (bder c a)"
shows "bmkeps (bder c (bsimp a)) = bmkeps (bsimp (bder c (bsimp a)))"
using assms
by (metis L03 L_bsimp_erase bmkeps_retrieve bnullable_correctness der_correctness erase_bder nullable_correctness)
lemma L07:
assumes "s \<in> L (erase r)"
shows "retrieve r (flex (erase r) id s (mkeps (ders s (erase r))))
= retrieve (bders r s) (mkeps (erase (bders r s)))"
using assms
using LB LC lexer_correct_Some by auto
lemma LXXX:
assumes "s \<in> (erase r) \<rightarrow> v" "s \<in> (erase (bsimp r)) \<rightarrow> v'"
shows "retrieve r v = retrieve (bsimp r) v'"
using assms
apply -
thm LC
apply(subst LC)
apply(assumption)
apply(subst L0[symmetric])
using bnullable_correctness lexer_correctness(2) lexer_flex apply fastforce
apply(subst (2) LC)
apply(assumption)
apply(subst (2) L0[symmetric])
using bnullable_correctness lexer_correctness(2) lexer_flex apply fastforce
oops
lemma L07a:
assumes "s \<in> L (erase r)"
shows "retrieve (bsimp r) (flex (erase (bsimp r)) id s (mkeps (ders s (erase (bsimp r)))))
= retrieve r (flex (erase r) id s (mkeps (ders s (erase r))))"
using assms
apply(induct s arbitrary: r)
apply(simp)
using L0a apply force
apply(drule_tac x="(bder a r)" in meta_spec)
apply(drule meta_mp)
apply (metis L_bsimp_erase erase_bder lexer.simps(2) lexer_correct_None option.case(1))
apply(drule sym)
apply(simp)
apply(subst (asm) bder_retrieve)
apply (metis Posix_Prf Posix_flex Posix_mkeps ders.simps(2) lexer_correct_None lexer_flex)
apply(simp only: flex_fun_apply)
apply(simp)
using L0[no_vars] bder_retrieve[no_vars] LA[no_vars] LC[no_vars] L07[no_vars]
oops
lemma L08:
assumes "s \<in> L (erase r)"
shows "retrieve (bders (bsimp r) s) (mkeps (erase (bders (bsimp r) s)))
= retrieve (bders r s) (mkeps (erase (bders r s)))"
using assms
apply(induct s arbitrary: r)
apply(simp)
using L0 bnullable_correctness nullable_correctness apply blast
apply(simp add: bders_append)
apply(drule_tac x="(bder a (bsimp r))" in meta_spec)
apply(drule meta_mp)
apply (metis L_bsimp_erase erase_bder lexer.simps(2) lexer_correct_None option.case(1))
apply(drule sym)
apply(simp)
apply(subst LA)
apply (metis L0aa L_bsimp_erase Posix1(1) ders.simps(2) ders_correctness erase_bder erase_bders mkeps_nullable nullable_correctness)
apply(subst LA)
using lexer_correct_None lexer_flex mkeps_nullable apply force
using L0[no_vars] bder_retrieve[no_vars] LA[no_vars] LC[no_vars] L07[no_vars]
thm L0[no_vars] bder_retrieve[no_vars] LA[no_vars] LC[no_vars] L07[no_vars]
oops
lemma test:
assumes "s = [c]"
shows "retrieve (bders r s) v = XXX" and "YYY = retrieve r (flex (erase r) id s v)"
using assms
apply(simp only: bders.simps)
defer
using assms
apply(simp only: flex.simps id_simps)
using L0[no_vars] bder_retrieve[no_vars] LA[no_vars] LC[no_vars]
find_theorems "retrieve (bders _ _) _"
find_theorems "retrieve _ (mkeps _)"
oops
lemma L06X:
assumes "bnullable (bder c a)"
shows "bmkeps (bder c (bsimp a)) = bmkeps (bder c a)"
using assms
apply(induct a arbitrary: c)
apply(simp)
apply(simp)
apply(simp)
prefer 3
apply(simp)
prefer 2
apply(simp)
defer
oops
lemma L06_2:
assumes "bnullable (bders a [c,d])"
shows "bmkeps (bders (bsimp a) [c,d]) = bmkeps (bsimp (bders (bsimp a) [c,d]))"
using assms
apply(simp)
by (metis L_bsimp_erase bmkeps_simp bnullable_correctness der_correctness erase_bder nullable_correctness)
lemma L06_bders:
assumes "bnullable (bders a s)"
shows "bmkeps (bders (bsimp a) s) = bmkeps (bsimp (bders (bsimp a) s))"
using assms
by (metis L_bsimp_erase bmkeps_simp bnullable_correctness ders_correctness erase_bders nullable_correctness)
lemma LLLL:
shows "L (erase a) = L (erase (bsimp a))"
and "L (erase a) = {flat v | v. \<Turnstile> v: (erase a)}"
and "L (erase a) = {flat v | v. \<Turnstile> v: (erase (bsimp a))}"
using L_bsimp_erase apply(blast)
apply (simp add: L_flat_Prf)
using L_bsimp_erase L_flat_Prf apply(auto)[1]
done
lemma L07XX:
assumes "s \<in> L (erase a)"
shows "s \<in> erase a \<rightarrow> flex (erase a) id s (mkeps (ders s (erase a)))"
using assms
by (meson lexer_correct_None lexer_correctness(1) lexer_flex)
lemma LX0:
assumes "s \<in> L r"
shows "decode (bmkeps (bders (intern r) s)) r = Some(flex r id s (mkeps (ders s r)))"
by (metis assms blexer_correctness blexer_def lexer_correct_None lexer_flex)
lemma L02_bders2:
assumes "bnullable (bders a s)" "s = [c]"
shows "retrieve (bders (bsimp a) s) (mkeps (erase (bders (bsimp a) s))) =
retrieve (bders a s) (mkeps (erase (bders a s)))"
using assms
apply(simp)
apply(induct s arbitrary: a)
apply(simp)
using L0 apply auto[1]
oops
thm bmkeps_retrieve bmkeps_simp Posix_mkeps
lemma WQ1:
assumes "s \<in> L (der c r)"
shows "s \<in> der c r \<rightarrow> mkeps (ders s (der c r))"
using assms
oops
lemma L02_bsimp:
assumes "bnullable (bders a s)"
shows "retrieve (bsimp a) (flex (erase (bsimp a)) id s (mkeps (erase (bders (bsimp a) s)))) =
retrieve a (flex (erase a) id s (mkeps (erase (bders a s))))"
using assms
apply(induct s arbitrary: a)
apply(simp)
apply (simp add: L0)
apply(simp)
apply(drule_tac x="bder a aa" in meta_spec)
apply(simp)
apply(subst (asm) bder_retrieve)
using Posix_Prf Posix_flex Posix_mkeps bnullable_correctness apply fastforce
apply(simp add: flex_fun_apply)
apply(drule sym)
apply(simp)
apply(subst flex_injval)
apply(subst bder_retrieve[symmetric])
apply (metis L_bsimp_erase Posix_Prf Posix_flex Posix_mkeps bders.simps(2) bnullable_correctness ders.simps(2) erase_bders lexer_correct_None lexer_flex option.distinct(1))
apply(simp only: erase_bder[symmetric] erase_bders[symmetric])
apply(subst LB_sym[symmetric])
apply(simp)
oops
lemma L1:
assumes "s \<in> r \<rightarrow> v"
shows "decode (bmkeps (bders (intern r) s)) r = Some v"
using assms
by (metis blexer_correctness blexer_def lexer_correctness(1) option.distinct(1))
lemma L2:
assumes "s \<in> (der c r) \<rightarrow> v"
shows "decode (bmkeps (bders (intern r) (c # s))) r = Some (injval r c v)"
using assms
apply(subst bmkeps_retrieve)
using Posix1(1) lexer_correct_None lexer_flex apply fastforce
using MAIN_decode
apply(subst MAIN_decode[symmetric])
apply(simp)
apply (meson Posix1(1) lexer_correct_None lexer_flex mkeps_nullable)
apply(simp)
apply(subgoal_tac "v = flex (der c r) id s (mkeps (ders s (der c r)))")
prefer 2
apply (metis Posix_determ lexer_correctness(1) lexer_flex option.distinct(1))
apply(simp)
apply(subgoal_tac "injval r c (flex (der c r) id s (mkeps (ders s (der c r)))) =
(flex (der c r) ((\<lambda>v. injval r c v) o id) s (mkeps (ders s (der c r))))")
apply(simp)
using flex_fun_apply by blast
lemma L3:
assumes "s2 \<in> (ders s1 r) \<rightarrow> v"
shows "decode (bmkeps (bders (intern r) (s1 @ s2))) r = Some (flex r id s1 v)"
using assms
apply(induct s1 arbitrary: r s2 v rule: rev_induct)
apply(simp)
using L1 apply blast
apply(simp add: ders_append)
apply(drule_tac x="r" in meta_spec)
apply(drule_tac x="x # s2" in meta_spec)
apply(drule_tac x="injval (ders xs r) x v" in meta_spec)
apply(drule meta_mp)
defer
apply(simp)
apply(simp add: flex_append)
by (simp add: Posix_injval)
lemma bders_snoc:
"bder c (bders a s) = bders a (s @ [c])"
apply(simp add: bders_append)
done
lemma QQ1:
shows "bsimp (bders (bsimp a) []) = bders_simp (bsimp a) []"
apply(simp)
apply(simp add: bsimp_idem)
done
lemma QQ2:
shows "bsimp (bders (bsimp a) [c]) = bders_simp (bsimp a) [c]"
apply(simp)
done
lemma XXX2a_long:
assumes "good a"
shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
using assms
apply(induct a arbitrary: c taking: asize rule: measure_induct)
apply(case_tac x)
apply(simp)
apply(simp)
apply(simp)
prefer 3
apply(simp)
apply(simp)
apply(auto)[1]
apply(case_tac "x42 = AZERO")
apply(simp)
apply(case_tac "x43 = AZERO")
apply(simp)
using test2 apply force
apply(case_tac "\<exists>bs. x42 = AONE bs")
apply(clarify)
apply(simp)
apply(subst bsimp_ASEQ1)
apply(simp)
using b3 apply force
using bsimp_ASEQ0 test2 apply force
thm good_SEQ test2
apply (simp add: good_SEQ test2)
apply (simp add: good_SEQ test2)
apply(case_tac "x42 = AZERO")
apply(simp)
apply(case_tac "x43 = AZERO")
apply(simp)
apply (simp add: bsimp_ASEQ0)
apply(case_tac "\<exists>bs. x42 = AONE bs")
apply(clarify)
apply(simp)
apply(subst bsimp_ASEQ1)
apply(simp)
using bsimp_ASEQ0 test2 apply force
apply (simp add: good_SEQ test2)
apply (simp add: good_SEQ test2)
apply (simp add: good_SEQ test2)
(* AALTs case *)
apply(simp)
using test2 by fastforce
lemma XXX2a_long_without_good:
assumes "a = AALTs bs0 [AALTs bs1 [AALTs bs2 [ASTAR [] (AONE bs7), AONE bs6, ASEQ bs3 (ACHAR bs4 d) (AONE bs5)]]]"
shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
"bsimp (bder c (bsimp a)) = XXX"
"bsimp (bder c a) = YYY"
using assms
apply(simp)
using assms
apply(simp)
prefer 2
using assms
apply(simp)
oops
lemma bder_bsimp_AALTs:
shows "bder c (bsimp_AALTs bs rs) = bsimp_AALTs bs (map (bder c) rs)"
apply(induct bs rs rule: bsimp_AALTs.induct)
apply(simp)
apply(simp)
apply (simp add: bder_fuse)
apply(simp)
done
lemma flts_nothing:
assumes "\<forall>r \<in> set rs. r \<noteq> AZERO" "\<forall>r \<in> set rs. nonalt r"
shows "flts rs = rs"
using assms
apply(induct rs rule: flts.induct)
apply(auto)
done
lemma flts_flts:
assumes "\<forall>r \<in> set rs. good r"
shows "flts (flts rs) = flts rs"
using assms
apply(induct rs taking: "\<lambda>rs. sum_list (map asize rs)" rule: measure_induct)
apply(case_tac x)
apply(simp)
apply(simp)
apply(case_tac a)
apply(simp_all add: bder_fuse flts_append)
apply(subgoal_tac "\<forall>r \<in> set x52. r \<noteq> AZERO")
prefer 2
apply (metis Nil_is_append_conv bsimp_AALTs.elims good.simps(1) good.simps(5) good0 list.distinct(1) n0 nn1b split_list_last test2)
apply(subgoal_tac "\<forall>r \<in> set x52. nonalt r")
prefer 2
apply (metis n0 nn1b test2)
by (metis flts_fuse flts_nothing)
lemma PP:
assumes "bnullable (bders r s)"
shows "bmkeps (bders (bsimp r) s) = bmkeps (bders r s)"
using assms
apply(induct s arbitrary: r)
apply(simp)
using bmkeps_simp apply auto[1]
apply(simp add: bders_append bders_simp_append)
oops
lemma PP:
assumes "bnullable (bders r s)"
shows "bmkeps (bders_simp (bsimp r) s) = bmkeps (bders r s)"
using assms
apply(induct s arbitrary: r rule: rev_induct)
apply(simp)
using bmkeps_simp apply auto[1]
apply(simp add: bders_append bders_simp_append)
apply(drule_tac x="bder a (bsimp r)" in meta_spec)
apply(drule_tac meta_mp)
defer
oops
lemma
assumes "asize (bsimp a) = asize a" "a = AALTs bs [AALTs bs2 [], AZERO, AONE bs3]"
shows "bsimp a = a"
using assms
apply(simp)
oops
lemma iii:
assumes "bsimp_AALTs bs rs \<noteq> AZERO"
shows "rs \<noteq> []"
using assms
apply(induct bs rs rule: bsimp_AALTs.induct)
apply(auto)
done
lemma
assumes "\<forall>y. asize y < Suc (sum_list (map asize x52)) \<longrightarrow> asize (bsimp y) = asize y \<longrightarrow> bsimp y \<noteq> AZERO \<longrightarrow> bsimp y = y"
"asize (bsimp_AALTs x51 (flts (map bsimp x52))) = Suc (sum_list (map asize x52))"
"bsimp_AALTs x51 (flts (map bsimp x52)) \<noteq> AZERO"
shows "bsimp_AALTs x51 (flts (map bsimp x52)) = AALTs x51 x52"
using assms
apply(induct x52 arbitrary: x51)
apply(simp)
oops
lemma
assumes "asize (bsimp a) = asize a" "bsimp a \<noteq> AZERO"
shows "bsimp a = a"
using assms
apply(induct a taking: asize rule: measure_induct)
apply(case_tac x)
apply(simp_all)
apply(case_tac "(bsimp x42) = AZERO")
apply(simp add: asize0)
apply(case_tac "(bsimp x43) = AZERO")
apply(simp add: asize0)
apply (metis bsimp_ASEQ0)
apply(case_tac "\<exists>bs. (bsimp x42) = AONE bs")
apply(auto)[1]
apply (metis b1 bsimp_size fuse_size less_add_Suc2 not_less)
apply (metis Suc_inject add.commute asize.simps(5) bsimp_ASEQ1 bsimp_size leD le_neq_implies_less less_add_Suc2 less_add_eq_less)
(* ALT case *)
apply(frule iii)
apply(case_tac x52)
apply(simp)
apply(simp)
apply(subst k0)
apply(subst (asm) k0)
apply(subst (asm) (2) k0)
apply(subst (asm) (3) k0)
apply(case_tac "(bsimp a) = AZERO")
apply(simp)
apply (metis (no_types, lifting) Suc_le_lessD asize0 bsimp_AALTs_size le_less_trans less_add_same_cancel2 not_less_eq rt)
apply(simp)
apply(case_tac "nonalt (bsimp a)")
prefer 2
apply(drule_tac x="AALTs x51 (bsimp a # list)" in spec)
apply(drule mp)
apply (metis asize.simps(4) bsimp.simps(2) bsimp_AALTs_size3 k0 less_not_refl list.set_intros(1) list.simps(9) sum_list.Cons)
apply(drule mp)
apply(simp)
apply (metis asize.simps(4) bsimp.simps(2) bsimp_AALTs_size3 k0 lessI list.set_intros(1) list.simps(9) not_less_eq sum_list.Cons)
apply(drule mp)
apply(simp)
using bsimp_idem apply auto[1]
apply(simp add: bsimp_idem)
apply (metis append.left_neutral append_Cons asize.simps(4) bsimp.simps(2) bsimp_AALTs_size3 k00 less_not_refl list.set_intros(1) list.simps(9) sum_list.Cons)
apply (metis bsimp.simps(2) bsimp_idem k0 list.simps(9) nn1b nonalt.elims(3) nonnested.simps(2))
apply(subgoal_tac "flts [bsimp a] = [bsimp a]")
prefer 2
using k0b apply blast
apply(clarify)
apply(simp only:)
apply(simp)
apply(case_tac "flts (map bsimp list) = Nil")
apply (metis bsimp_AALTs1 bsimp_size fuse_size less_add_Suc1 not_less)
apply (subgoal_tac "bsimp_AALTs x51 (bsimp a # flts (map bsimp list)) = AALTs x51 (bsimp a # flts (map bsimp list))")
prefer 2
apply (metis bsimp_AALTs.simps(3) neq_Nil_conv)
apply(auto)
apply (metis add.commute bsimp_size leD le_neq_implies_less less_add_Suc1 less_add_eq_less rt)
oops
lemma OOO:
shows "bsimp (bsimp_AALTs bs rs) = bsimp_AALTs bs (flts (map bsimp rs))"
apply(induct rs arbitrary: bs taking: "\<lambda>rs. sum_list (map asize rs)" rule: measure_induct)
apply(case_tac x)
apply(simp)
apply(simp)
apply(case_tac "a = AZERO")
apply(simp)
apply(case_tac "list")
apply(simp)
apply(simp)
apply(case_tac "bsimp a = AZERO")
apply(simp)
apply(case_tac "list")
apply(simp)
apply(simp add: bsimp_fuse[symmetric])
apply(simp)
apply(case_tac "nonalt (bsimp a)")
apply(case_tac list)
apply(simp)
apply(subst k0b)
apply(simp)
apply(simp)
apply(simp add: bsimp_fuse)
apply(simp)
apply(subgoal_tac "asize (bsimp a) < asize a \<or> asize (bsimp a) = asize a")
prefer 2
using bsimp_size le_neq_implies_less apply blast
apply(erule disjE)
apply(drule_tac x="(bsimp a) # list" in spec)
apply(drule mp)
apply(simp)
apply(simp)
apply (metis bsimp.simps(2) bsimp_AALTs.elims bsimp_AALTs.simps(2) bsimp_fuse bsimp_idem list.distinct(1) list.inject list.simps(9))
apply(subgoal_tac "\<exists>bs rs. bsimp a = AALTs bs rs \<and> rs \<noteq> Nil \<and> length rs > 1")
prefer 2
apply (metis bbbbs1 bsimp.simps(2) bsimp_AALTs.simps(1) bsimp_idem flts.simps(1) good.simps(5) good1 length_0_conv length_Suc_conv less_one list.simps(8) nat_neq_iff not_less_eq)
apply(auto)
oops
lemma
assumes "rs = [AALTs bsa [AONE bsb, AONE bsb]]"
shows "bsimp (bsimp_AALTs bs rs) = bsimp_AALTs bs (flts (map bsimp rs))"
using assms
apply(simp)
oops
lemma CT1:
shows "bsimp (AALTs bs as) = bsimp(AALTs bs (map bsimp as))"
apply(induct as arbitrary: bs)
apply(simp)
apply(simp)
by (simp add: bsimp_idem comp_def)
lemma CT1a:
shows "bsimp (AALT bs a1 a2) = bsimp(AALT bs (bsimp a1) (bsimp a2))"
by (metis CT1 list.simps(8) list.simps(9))
(* CT *)
lemma CTU:
shows "bsimp_AALTs bs as = li bs as"
apply(induct bs as rule: li.induct)
apply(auto)
done
lemma CTa:
assumes "\<forall>r \<in> set as. nonalt r \<and> r \<noteq> AZERO"
shows "flts as = as"
using assms
apply(induct as)
apply(simp)
apply(case_tac as)
apply(simp)
apply (simp add: k0b)
using flts_nothing by auto
lemma CT0:
assumes "\<forall>r \<in> set as1. nonalt r \<and> r \<noteq> AZERO"
shows "flts [bsimp_AALTs bs1 as1] = flts (map (fuse bs1) as1)"
using assms CTa
apply(induct as1 arbitrary: bs1)
apply(simp)
apply(simp)
apply(case_tac as1)
apply(simp)
apply(simp)
proof -
fix a :: arexp and as1a :: "arexp list" and bs1a :: "bit list" and aa :: arexp and list :: "arexp list"
assume a1: "nonalt a \<and> a \<noteq> AZERO \<and> nonalt aa \<and> aa \<noteq> AZERO \<and> (\<forall>r\<in>set list. nonalt r \<and> r \<noteq> AZERO)"
assume a2: "\<And>as. \<forall>r\<in>set as. nonalt r \<and> r \<noteq> AZERO \<Longrightarrow> flts as = as"
assume a3: "as1a = aa # list"
have "flts [a] = [a]"
using a1 k0b by blast
then show "fuse bs1a a # fuse bs1a aa # map (fuse bs1a) list = flts (fuse bs1a a # fuse bs1a aa # map (fuse bs1a) list)"
using a3 a2 a1 by (metis (no_types) append.left_neutral append_Cons flts_fuse k00 k0b list.simps(9))
qed
lemma CT01:
assumes "\<forall>r \<in> set as1. nonalt r \<and> r \<noteq> AZERO" "\<forall>r \<in> set as2. nonalt r \<and> r \<noteq> AZERO"
shows "flts [bsimp_AALTs bs1 as1, bsimp_AALTs bs2 as2] = flts ((map (fuse bs1) as1) @ (map (fuse bs2) as2))"
using assms CT0
by (metis k0 k00)
lemma
shows "bsimp (AALT bs (AALTs bs1 (map (bder c) as1)) (AALTs bs2 (map (bder c) as2)))
= bsimp (AALTs bs ((map (fuse bs1) (map (bder c) as1)) @
(map (fuse bs2) (map (bder c) as2))))"
apply(subst bsimp_idem[symmetric])
apply(simp)
oops
lemma CT_exp:
assumes "\<forall>a \<in> set as. bsimp (bder c (bsimp a)) = bsimp (bder c a)"
shows "map bsimp (map (bder c) as) = map bsimp (map (bder c) (map bsimp as))"
using assms
apply(induct as)
apply(auto)
done
lemma asize_set:
assumes "a \<in> set as"
shows "asize a < Suc (sum_list (map asize as))"
using assms
apply(induct as arbitrary: a)
apply(auto)
using le_add2 le_less_trans not_less_eq by blast
lemma XXX2a_long_without_good:
shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
apply(induct a arbitrary: c taking: "\<lambda>a. asize a" rule: measure_induct)
apply(case_tac x)
apply(simp)
apply(simp)
apply(simp)
prefer 3
apply(simp)
(* AALT case *)
prefer 2
apply(simp del: bsimp.simps)
apply(subst (2) CT1)
apply(subst CT_exp)
apply(auto)[1]
using asize_set apply blast
apply(subst CT1[symmetric])
apply(simp)
oops
lemma YY:
assumes "flts (map bsimp as1) = xs"
shows "flts (map bsimp (map (fuse bs1) as1)) = map (fuse bs1) xs"
using assms
apply(induct as1 arbitrary: bs1 xs)
apply(simp)
apply(auto)
by (metis bsimp_fuse flts_fuse k0 list.simps(9))
lemma flts_nonalt:
assumes "flts (map bsimp xs) = ys"
shows "\<forall>y \<in> set ys. nonalt y"
using assms
apply(induct xs arbitrary: ys)
apply(auto)
apply(case_tac xs)
apply(auto)
using flts2 good1 apply fastforce
by (smt ex_map_conv list.simps(9) nn1b nn1c)
lemma WWW2:
shows "bsimp (bsimp_AALTs bs1 (flts (map bsimp as1))) =
bsimp_AALTs bs1 (flts (map bsimp as1))"
by (metis bsimp.simps(2) bsimp_idem)
lemma WWW3:
shows "flts [bsimp_AALTs bs1 (flts (map bsimp as1))] =
flts (map bsimp (map (fuse bs1) as1))"
by (metis CT0 YY flts_nonalt flts_nothing qqq1)
lemma WWW4:
shows "map (bder c \<circ> fuse bs1) as1 = map (fuse bs1) (map (bder c) as1)"
apply(induct as1)
apply(auto)
using bder_fuse by blast
lemma WWW5:
shows "map (bsimp \<circ> bder c) as1 = map bsimp (map (bder c) as1)"
apply(induct as1)
apply(auto)
done
lemma WWW6:
shows "bsimp (bder c (bsimp_AALTs x51 (flts [bsimp a1, bsimp a2]) ) ) =
bsimp(bsimp_AALTs x51 (map (bder c) (flts [bsimp a1, bsimp a2]))) "
using bder_bsimp_AALTs by auto
lemma WWW7:
shows "bsimp (bsimp_AALTs x51 (map (bder c) (flts [bsimp a1, bsimp a2]))) =
bsimp(bsimp_AALTs x51 (flts (map (bder c) [bsimp a1, bsimp a2])))"
sorry
lemma stupid:
assumes "a = b"
shows "bsimp(a) = bsimp(b)"
using assms
apply(auto)
done
(*
proving idea:
bsimp_AALTs x51 (map (bder c) (flts [a1, a2])) = bsimp_AALTs x51 (map (bder c) (flts [a1]++[a2]))
= bsimp_AALTs x51 (map (bder c) ((flts [a1])++(flts [a2]))) =
bsimp_AALTs x51 (map (bder c) (flts [a1]))++(map (bder c) (flts [a2])) = A
and then want to prove that
map (bder c) (flts [a]) = flts [bder c a] under the condition
that a is either a seq with the first elem being not nullable, or a character equal to c,
or an AALTs, or a star
Then, A = bsimp_AALTs x51 (flts [bder c a]) ++ (map (bder c) (flts [a2])) = A1
Using the same condition for a2, we get
A1 = bsimp_AALTs x51 (flts [bder c a1]) ++ (flts [bder c a2])
=bsimp_AALTs x51 flts ([bder c a1] ++ [bder c a2])
=bsimp_AALTs x51 flts ([bder c a1, bder c a2])
*)
lemma manipulate_flts:
shows "bsimp_AALTs x51 (map (bder c) (flts [a1, a2])) =
bsimp_AALTs x51 ((map (bder c) (flts [a1])) @ (map (bder c) (flts [a2])))"
by (metis k0 map_append)
lemma go_inside_flts:
assumes " (bder c a1 \<noteq> AZERO) "
"\<not>(\<exists> a01 a02 x02. ( (a1 = ASEQ x02 a01 a02) \<and> bnullable(a01) ) )"
shows "map (bder c) (flts [a1]) = flts [bder c a1]"
using assms
apply -
apply(case_tac a1)
apply(simp)
apply(simp)
apply(case_tac "x32 = c")
prefer 2
apply(simp)
apply(simp)
apply(simp)
apply (simp add: WWW4)
apply(simp add: bder_fuse)
done
lemma medium010:
assumes " (bder c a1 = AZERO) "
shows "map (bder c) (flts [a1]) = [AZERO] \<or> map (bder c) (flts [a1]) = []"
using assms
apply -
apply(case_tac a1)
apply(simp)
apply(simp)
apply(simp)
apply(simp)
apply(simp)
apply(simp)
done
lemma medium011:
assumes " (bder c a1 = AZERO) "
shows "flts (map (bder c) [a1, a2]) = flts [bder c a2]"
using assms
apply -
apply(simp)
done
lemma medium01central:
shows "bsimp(bsimp_AALTs x51 (map (bder c) (flts [a2])) ) = bsimp(bsimp_AALTs x51 (flts [bder c a2]))"
sorry
lemma plus_bsimp:
assumes "bsimp( bsimp a) = bsimp (bsimp b)"
shows "bsimp a = bsimp b"
using assms
apply -
by (simp add: bsimp_idem)
lemma patience_good5:
assumes "bsimp r = AALTs x y"
shows " \<exists> a aa list. y = a#aa#list"
by (metis Nil_is_map_conv arexp.simps(13) assms bsimp_AALTs.elims flts1 good.simps(5) good1 k0a)
(*SAD*)
(*this does not hold actually
lemma bsimp_equiv0:
shows "bsimp(bsimp r) = bsimp(bsimp (AALTs [] [r]))"
apply(simp)
apply(case_tac "bsimp r")
apply(simp)
apply(simp)
apply(simp)
apply(simp)
thm good1
using good1
apply -
apply(drule_tac x="r" in meta_spec)
apply(erule disjE)
apply(simp only: bsimp_AALTs.simps)
apply(simp only:flts.simps)
apply(drule patience_good5)
apply(clarify)
apply(subst bsimp_AALTs_qq)
apply simp
prefer 2
sorry*)
(*exercise: try multiple ways of proving this*)
(*this lemma does not hold.........
lemma bsimp_equiv1:
shows "bsimp r = bsimp (AALTs [] [r])"
using plus_bsimp
apply -
using bsimp_equiv0 by blast
(*apply(simp)
apply(case_tac "bsimp r")
apply(simp)
apply(simp)
apply(simp)
apply(simp)
(*use lemma good1*)
thm good1
using good1
apply -
apply(drule_tac x="r" in meta_spec)
apply(erule disjE)
apply(subst flts_single1)
apply(simp only: bsimp_AALTs.simps)
prefer 2
thm flts_single1
find_theorems "flts _ = _"*)
*)
lemma bsimp_equiv2:
shows "bsimp (AALTs x51 [r]) = bsimp (AALT x51 AZERO r)"
sorry
lemma medium_stupid_isabelle:
assumes "rs = a # list"
shows "bsimp_AALTs x51 (AZERO # rs) = AALTs x51 (AZERO#rs)"
using assms
apply -
apply(simp)
done
(*
lemma mediumlittle:
shows "bsimp(bsimp_AALTs x51 rs) = bsimp(bsimp_AALTs x51 (AZERO # rs))"
apply(case_tac rs)
apply(simp)
apply(case_tac list)
apply(subst medium_stupid_isabelle)
apply(simp)
prefer 2
apply simp
apply(rule_tac s="a#list" and t="rs" in subst)
apply(simp)
apply(rule_tac t="list" and s= "[]" in subst)
apply(simp)
(*dunno what is the rule for x#nil = x*)
apply(case_tac a)
apply(simp)
apply(simp)
apply(simp)
prefer 3
apply simp
apply(simp only:bsimp_AALTs.simps)
apply simp
apply(case_tac "bsimp x42")
apply(simp)
apply simp
apply(case_tac "bsimp x43")
apply simp
apply simp
apply simp
apply simp
apply(simp only:bsimp_ASEQ.simps)
using good1
apply -
apply(drule_tac x="x43" in meta_spec)
apply(erule disjE)
apply(subst bsimp_AALTs_qq)
using patience_good5 apply force
apply(simp only:bsimp_AALTs.simps)
apply(simp only:fuse.simps)
apply(simp only:flts.simps)
(*OK from here you actually realize this lemma doesnt hold*)
apply(simp)
apply(simp)
apply(rule_tac t="rs" and s="a#list" in subst)
apply(simp)
apply(rule_tac t="list" and s="[]" in subst)
apply(simp)
(*apply(simp only:bsimp_AALTs.simps)*)
(*apply(simp only:fuse.simps)*)
sorry
*)
lemma singleton_list_map:
shows"map f [a] = [f a]"
apply simp
done
lemma map_application2:
shows"map f [a,b] = [f a, f b]"
apply simp
done
(*SAD*)
(* bsimp (bder c (bsimp_AALTs x51 (flts [bsimp a1, bsimp a2]))) =
bsimp (AALT x51 (bder c (bsimp a1)) (bder c (bsimp a2)))*)
(*This equality does not hold*)
lemma medium01:
assumes " (bder c a1 = AZERO) "
shows "bsimp(bsimp_AALTs x51 (map (bder c) (flts [ a1, a2]))) =
bsimp(bsimp_AALTs x51 (flts (map (bder c) [ a1, a2])))"
apply(subst manipulate_flts)
using assms
apply -
apply(subst medium011)
apply(simp)
apply(case_tac "map (bder c) (flts [a1]) = []")
apply(simp)
using medium01central apply blast
apply(frule medium010)
apply(erule disjE)
prefer 2
apply(simp)
apply(simp)
apply(case_tac a2)
apply simp
apply simp
apply simp
apply(simp only:flts.simps)
(*HOW do i say here to replace ASEQ ..... back into a2*)
(*how do i say here to use the definition of map function
without lemma, of course*)
(*how do i say here that AZERO#map (bder c) [ASEQ x41 x42 x43]'s list.len >1
without a lemma, of course*)
apply(subst singleton_list_map)
apply(simp only: bsimp_AALTs.simps)
apply(case_tac "bder c (ASEQ x41 x42 x43)")
apply simp
apply simp
apply simp
prefer 3
apply simp
apply(rule_tac t="bder c (ASEQ x41 x42 x43)"
and s="ASEQ x41a x42a x43a" in subst)
apply simp
apply(simp only: flts.simps)
apply(simp only: bsimp_AALTs.simps)
apply(simp only: fuse.simps)
apply(subst (2) bsimp_idem[symmetric])
apply(subst (1) bsimp_idem[symmetric])
apply(simp only:bsimp.simps)
apply(subst map_application2)
apply(simp only: bsimp.simps)
apply(simp only:flts.simps)
(*want to happily change between a2 and ASEQ x41 42 43, and eliminate now
redundant conditions such as map (bder c) (flts [a1]) = [AZERO] *)
apply(case_tac "bsimp x42a")
apply(simp)
apply(case_tac "bsimp x43a")
apply(simp)
apply(simp)
apply(simp)
apply(simp)
prefer 2
apply(simp)
apply(rule_tac t="bsimp x43a"
and s="AALTs x51a x52" in subst)
apply simp
apply(simp only:bsimp_ASEQ.simps)
apply(simp only:fuse.simps)
apply(simp only:flts.simps)
using medium01central mediumlittle by auto
lemma medium1:
assumes " (bder c a1 \<noteq> AZERO) "
"\<not>(\<exists> a01 a02 x02. ( (a1 = ASEQ x02 a01 a02) \<and> bnullable(a01) ) )"
" (bder c a2 \<noteq> AZERO)"
"\<not>(\<exists> a11 a12 x12. ( (a2 = ASEQ x12 a11 a12) \<and> bnullable(a11) ) )"
shows "bsimp_AALTs x51 (map (bder c) (flts [ a1, a2])) =
bsimp_AALTs x51 (flts (map (bder c) [ a1, a2]))"
using assms
apply -
apply(subst manipulate_flts)
apply(case_tac "a1")
apply(simp)
apply(simp)
apply(case_tac "x32 = c")
prefer 2
apply(simp)
prefer 2
apply(case_tac "bnullable x42")
apply(simp)
apply(simp)
apply(case_tac "a2")
apply(simp)
apply(simp)
apply(case_tac "x32 = c")
prefer 2
apply(simp)
apply(simp)
apply(case_tac "bnullable x42a")
apply(simp)
apply(subst go_inside_flts)
apply(simp)
apply(simp)
apply(simp)
apply(simp)
apply (simp add: WWW4)
apply(simp)
apply (simp add: WWW4)
apply (simp add: go_inside_flts)
apply (metis (no_types, lifting) go_inside_flts k0 list.simps(8) list.simps(9))
by (smt bder.simps(6) flts.simps(1) flts.simps(6) flts.simps(7) go_inside_flts k0 list.inject list.simps(9))
lemma big0:
shows "bsimp (AALT x51 (AALTs bs1 as1) (AALTs bs2 as2)) =
bsimp (AALTs x51 ((map (fuse bs1) as1) @ (map (fuse bs2) as2)))"
by (smt WWW3 bsimp.simps(2) k0 k00 list.simps(8) list.simps(9) map_append)
lemma bignA:
shows "bsimp (AALTs x51 (AALTs bs1 as1 # as2)) =
bsimp (AALTs x51 ((map (fuse bs1) as1) @ as2))"
apply(simp)
apply(subst k0)
apply(subst WWW3)
apply(simp add: flts_append)
done
lemma hardest:
shows "bsimp (bder c (bsimp_AALTs x51 (flts [bsimp a1, bsimp a2]))) =
bsimp (AALT x51 (bder c (bsimp a1)) (bder c (bsimp a2)))"
apply(case_tac "bsimp a1")
apply(case_tac "bsimp a2")
apply simp
apply simp
apply(rule_tac t="bsimp a1"
and s="AZERO" in subst)
apply simp
apply(rule_tac t="bsimp a2"
and s="ACHAR x31 x32" in subst)
apply simp
apply simp
apply(rule_tac t="bsimp a1"
and s="AZERO" in subst)
apply simp
apply(rule_tac t="bsimp a2"
and s="ASEQ x41 x42 x43" in subst)
apply simp
apply(case_tac "bnullable x42")
apply(simp only: bder.simps)
apply(simp)
apply(case_tac "flts
[bsimp_ASEQ [] (bsimp (bder c x42)) (bsimp x43),
bsimp (fuse (bmkeps x42) (bder c x43))]")
apply(simp)
apply simp
(*counterexample finder*)
lemma XXX2a_long_without_good:
shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
apply(induct a arbitrary: c taking: "\<lambda>a. asize a" rule: measure_induct)
apply(case_tac x)
apply(simp)
apply(simp)
apply(simp)
prefer 3
apply(simp)
(* AALT case *)
prefer 2
apply(simp only:)
apply(case_tac "\<exists>a1 a2. x52 = [a1, a2]")
apply(clarify)
apply(simp del: bsimp.simps)
apply(subst (2) CT1)
apply(simp del: bsimp.simps)
apply(rule_tac t="bsimp (bder c a1)" and s="bsimp (bder c (bsimp a1))" in subst)
apply(simp del: bsimp.simps)
apply(rule_tac t="bsimp (bder c a2)" and s="bsimp (bder c (bsimp a2))" in subst)
apply(simp del: bsimp.simps)
apply(subst CT1a[symmetric])
apply(subst bsimp.simps)
apply(simp del: bsimp.simps)
(*bsimp_AALTs x51 (map (bder c) (flts [a1, a2])) =
bsimp_AALTs x51 (flts (map (bder c) [a1, a2]))*)
apply(case_tac "\<exists>bs1 as1. bsimp a1 = AALTs bs1 as1")
apply(case_tac "\<exists>bs2 as2. bsimp a2 = AALTs bs2 as2")
apply(clarify)
apply(simp only:)
apply(simp del: bsimp.simps bder.simps)
apply(subst bsimp_AALTs_qq)
prefer 2
apply(simp del: bsimp.simps)
apply(subst big0)
apply(simp add: WWW4)
apply (metis One_nat_def Suc_eq_plus1 Suc_lessI arexp.distinct(7) bsimp.simps(2) bsimp_AALTs.simps(1) bsimp_idem flts.simps(1) length_append length_greater_0_conv length_map not_add_less2 not_less_eq)
oops
lemma XXX2a_long_without_good:
shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
apply(induct a arbitrary: c taking: "\<lambda>a. asize a" rule: measure_induct)
apply(case_tac x)
apply(simp)
apply(simp)
apply(simp)
prefer 3
apply(simp)
(* AALT case *)
prefer 2
apply(subgoal_tac "nonnested (bsimp x)")
prefer 2
using nn1b apply blast
apply(simp only:)
apply(drule_tac x="AALTs x51 (flts x52)" in spec)
apply(drule mp)
defer
apply(drule_tac x="c" in spec)
apply(simp)
apply(rotate_tac 2)
apply(drule sym)
apply(simp)
apply(simp only: bder.simps)
apply(simp only: bsimp.simps)
apply(subst bder_bsimp_AALTs)
apply(case_tac x52)
apply(simp)
apply(simp)
apply(case_tac list)
apply(simp)
apply(case_tac a)
apply(simp)
apply(simp)
apply(simp)
defer
apply(simp)
(* case AALTs list is not empty *)
apply(simp)
apply(subst k0)
apply(subst (2) k0)
apply(simp)
apply(case_tac "bsimp a = AZERO")
apply(subgoal_tac "bsimp (bder c a) = AZERO")
prefer 2
using less_iff_Suc_add apply auto[1]
apply(simp)
apply(drule_tac x="AALTs x51 list" in spec)
apply(drule mp)
apply(simp add: asize0)
apply(drule_tac x="c" in spec)
apply(simp add: bder_bsimp_AALTs)
apply(case_tac "nonalt (bsimp a)")
prefer 2
apply(drule_tac x="bsimp (AALTs x51 (a#list))" in spec)
apply(drule mp)
apply(rule order_class.order.strict_trans2)
apply(rule bsimp_AALTs_size3)
apply(auto)[1]
apply(simp)
apply(subst (asm) bsimp_idem)
apply(drule_tac x="c" in spec)
apply(simp)
find_theorems "_ < _ \<Longrightarrow> _ \<le> _ \<Longrightarrow>_ < _"
apply(rule le_trans)
apply(subgoal_tac "flts [bsimp a] = [bsimp a]")
prefer 2
using k0b apply blast
apply(simp)
find_theorems "asize _ < asize _"
using bder_bsimp_AALTs
apply(case_tac list)
apply(simp)
sledgeha mmer [timeout=6000]
apply(case_tac "\<exists>r \<in> set (map bsimp x52). \<not>nonalt r")
apply(drule_tac x="bsimp (AALTs x51 x52)" in spec)
apply(drule mp)
using bsimp_AALTs_size3 apply blast
apply(drule_tac x="c" in spec)
apply(subst (asm) (2) test)
apply(case_tac x52)
apply(simp)
apply(simp)
apply(case_tac "bsimp a = AZERO")
apply(simp)
apply(subgoal_tac "bsimp (bder c a) = AZERO")
prefer 2
apply auto[1]
apply (metis L.simps(1) L_bsimp_erase der.simps(1) der_correctness erase.simps(1) erase_bder xxx_bder2)
apply(simp)
apply(drule_tac x="AALTs x51 list" in spec)
apply(drule mp)
apply(simp add: asize0)
apply(simp)
apply(case_tac list)
prefer 2
apply(simp)
apply(case_tac "bsimp aa = AZERO")
apply(simp)
apply(subgoal_tac "bsimp (bder c aa) = AZERO")
prefer 2
apply auto[1]
apply (metis add.left_commute bder.simps(1) bsimp.simps(3) less_add_Suc1)
apply(simp)
apply(drule_tac x="AALTs x51 (a#lista)" in spec)
apply(drule mp)
apply(simp add: asize0)
apply(simp)
apply (metis flts.simps(2) k0)
apply(subst k0)
apply(subst (2) k0)
using less_add_Suc1 apply fastforce
apply(subst k0)
apply(simp)
apply(case_tac "bsimp a = AZERO")
apply(simp)
apply(subgoal_tac "bsimp (bder c a) = AZERO")
prefer 2
apply auto[1]
apply(simp)
apply(case_tac "nonalt (bsimp a)")
apply(subst bsimp_AALTs1)
apply(simp)
using less_add_Suc1 apply fastforce
apply(subst bsimp_AALTs1)
using nn11a apply b last
(* SEQ case *)
apply(clarify)
apply(subst bsimp.simps)
apply(simp del: bsimp.simps)
apply(auto simp del: bsimp.simps)[1]
apply(subgoal_tac "bsimp x42 \<noteq> AZERO")
prefer 2
using b3 apply force
apply(case_tac "bsimp x43 = AZERO")
apply(simp)
apply (simp add: bsimp_ASEQ0)
apply (metis bder.simps(1) bsimp.simps(3) bsimp_AALTs.simps(1) bsimp_fuse flts.simps(1) flts.simps(2) fuse.simps(1) less_add_Suc2)
apply(case_tac "\<exists>bs. bsimp x42 = AONE bs")
apply(clarify)
apply(simp)
apply(subst bsimp_ASEQ2)
apply(subgoal_tac "bsimp (bder c x42) = AZERO")
prefer 2
using less_add_Suc1 apply fastforce
apply(simp)
apply(frule_tac x="x43" in spec)
apply(drule mp)
apply(simp)
apply(drule_tac x="c" in spec)
apply(subst bder_fuse)
apply(subst bsimp_fuse[symmetric])
apply(simp)
apply(subgoal_tac "bmkeps x42 = bs")
prefer 2
apply (simp add: bmkeps_simp)
apply(simp)
apply(subst bsimp_fuse[symmetric])
apply(case_tac "nonalt (bsimp (bder c x43))")
apply(subst bsimp_AALTs1)
using nn11a apply blast
using fuse_append apply blast
apply(subgoal_tac "\<exists>bs rs. bsimp (bder c x43) = AALTs bs rs")
prefer 2
using bbbbs1 apply blast
apply(clarify)
apply(simp)
apply(case_tac rs)
apply(simp)
apply (metis arexp.distinct(7) good.simps(4) good1)
apply(simp)
apply(case_tac list)
apply(simp)
apply (metis arexp.distinct(7) good.simps(5) good1)
apply(simp del: bsimp_AALTs.simps)
apply(simp only: bsimp_AALTs.simps)
apply(simp)
(* HERE *)
apply(case_tac "x42 = AZERO")
apply(simp)
apply(case_tac "bsimp x43 = AZERO")
apply(simp)
apply (simp add: bsimp_ASEQ0)
apply(subgoal_tac "bsimp (fuse (bmkeps x42) (bder c x43)) = AZERO")
apply(simp)
apply (met is bder.simps(1) bsimp.simps(3) bsimp_fuse fuse.simps(1) less_add_Suc2)
apply(case_tac "\<exists>bs. bsimp x42 = AONE bs")
apply(clarify)
apply(simp)
apply(subst bsimp_ASEQ2)
apply(subgoal_tac "bsimp (bder c x42) = AZERO")
apply(simp)
prefer 2
using less_add_Suc1 apply fastforce
apply(subgoal_tac "bmkeps x42 = bs")
prefer 2
apply (simp add: bmkeps_simp)
apply(simp)
apply(case_tac "nonalt (bsimp (bder c x43))")
apply (metis bder_fuse bsimp_AALTs.simps(1) bsimp_AALTs.simps(2) bsimp_fuse flts.simps(1) flts.simps(2) fuse.simps(1) fuse_append k0b less_add_Suc2 nn11a)
apply(subgoal_tac "nonnested (bsimp (bder c x43))")
prefer 2
using nn1b apply blast
apply(case_tac x43)
apply(simp)
apply(simp)
apply(simp)
prefer 3
apply(simp)
apply (metis arexp.distinct(25) arexp.distinct(7) arexp.distinct(9) bsimp_ASEQ.simps(1) bsimp_ASEQ.simps(11) bsimp_ASEQ1 nn11a nonalt.elims(3) nonalt.simps(6))
apply(simp)
apply(auto)[1]
apply(case_tac "(bsimp (bder c x42a)) = AZERO")
apply(simp)
apply(simp)
apply(subgoal_tac "(\<exists>bs1 rs1. 1 < length rs1 \<and> bsimp (bder c x43) = AALTs bs1 rs1 ) \<or>
(\<exists>bs1 r. bsimp (bder c x43) = fuse bs1 r)")
prefer 2
apply (metis fuse_empty)
apply(erule disjE)
prefer 2
apply(clarify)
apply(simp only:)
apply(simp)
apply(simp add: fuse_append)
apply(subst bder_fuse)
apply(subst bsimp_fuse[symmetric])
apply(subst bder_fuse)
apply(subst bsimp_fuse[symmetric])
apply(subgoal_tac "bsimp (bder c (bsimp x43)) = bsimp (bder c x43)")
prefer 2
using less_add_Suc2 apply bl ast
apply(simp only: )
apply(subst bsimp_fuse[symmetric])
apply(simp only: )
apply(simp only: fuse.simps)
apply(simp)
apply(case_tac rs1)
apply(simp)
apply (me tis arexp.distinct(7) fuse.simps(1) good.simps(4) good1 good_fuse)
apply(simp)
apply(case_tac list)
apply(simp)
apply (me tis arexp.distinct(7) fuse.simps(1) good.simps(5) good1 good_fuse)
apply(simp only: bsimp_AALTs.simps map_cons.simps)
apply(auto)[1]
apply(subst bsimp_fuse[symmetric])
apply(subgoal_tac "bmkeps x42 = bs")
prefer 2
apply (simp add: bmkeps_simp)
apply(simp)
using b3 apply force
using bsimp_ASEQ0 test2 apply fo rce
thm good_SEQ test2
apply (simp add: good_SEQ test2)
apply (simp add: good_SEQ test2)
apply(case_tac "x42 = AZERO")
apply(simp)
apply(case_tac "x43 = AZERO")
apply(simp)
apply (simp add: bsimp_ASEQ0)
apply(case_tac "\<exists>bs. x42 = AONE bs")
apply(clarify)
apply(simp)
apply(subst bsimp_ASEQ1)
apply(simp)
using bsimp_ASEQ0 test2 apply fo rce
apply (simp add: good_SEQ test2)
apply (simp add: good_SEQ test2)
apply (simp add: good_SEQ test2)
(* AALTs case *)
apply(simp)
using test2 by fa st force
lemma XXX4ab:
shows "good (bders_simp (bsimp r) s) \<or> bders_simp (bsimp r) s = AZERO"
apply(induct s arbitrary: r rule: rev_induct)
apply(simp)
apply (simp add: good1)
apply(simp add: bders_simp_append)
apply (simp add: good1)
done
lemma XXX4:
assumes "good a"
shows "bders_simp a s = bsimp (bders a s)"
using assms
apply(induct s arbitrary: a rule: rev_induct)
apply(simp)
apply (simp add: test2)
apply(simp add: bders_append bders_simp_append)
oops
lemma MAINMAIN:
"blexer r s = blexer_simp r s"
apply(induct s arbitrary: r)
apply(simp add: blexer_def blexer_simp_def)
apply(simp add: blexer_def blexer_simp_def del: bders.simps bders_simp.simps)
apply(auto simp del: bders.simps bders_simp.simps)
prefer 2
apply (metis b4 bders.simps(2) bders_simp.simps(2))
prefer 2
apply (metis b4 bders.simps(2))
apply(subst bmkeps_simp)
apply(simp)
apply(case_tac s)
apply(simp only: bders.simps)
apply(subst bders_simp.simps)
apply(simp)
oops
lemma
fixes n :: nat
shows "(\<Sum>i \<in> {0..n}. i) = n * (n + 1) div 2"
apply(induct n)
apply(simp)
apply(simp)
done
end