theory BitCoded2
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)"
| "Stars_add v _ = Stars [v]"
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
lemma asize0:
shows "0 < asize r"
apply(induct r)
apply(auto)
done
lemma asize_fuse:
shows "asize (fuse bs r) = asize r"
apply(induct r)
apply(auto)
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 map_bder_fuse:
shows "map (bder c \<circ> fuse bs1) as1 = map (fuse bs1) (map (bder c) as1)"
apply(induct as1)
apply(auto simp add: bder_fuse)
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"
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 spill :: "arexp list \<Rightarrow> arexp list"
where
"spill [] = []"
| "spill ((AALTs bs rs1) # rs) = (map (fuse bs) rs1) @ spill rs"
| "spill (r1 # rs) = r1 # spill rs"
lemma spill_Cons:
shows "spill (r # rs1) = spill [r] @ spill rs1"
apply(induct r arbitrary: rs1)
apply(auto)
done
lemma spill_append:
shows "spill (rs1 @ rs2) = spill rs1 @ spill rs2"
apply(induct rs1 arbitrary: rs2)
apply(auto)
by (metis append.assoc spill_Cons)
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"
inductive contains :: "arexp \<Rightarrow> bit list \<Rightarrow> bool" ("_ >> _" [51, 50] 50)
where
"AONE bs >> bs"
| "ACHAR bs c >> bs"
| "\<lbrakk>a1 >> bs1; a2 >> bs2\<rbrakk> \<Longrightarrow> ASEQ bs a1 a2 >> bs @ bs1 @ bs2"
| "r >> bs1 \<Longrightarrow> AALTs bs (r#rs) >> bs @ bs1"
| "AALTs bs rs >> bs @ bs1 \<Longrightarrow> AALTs bs (r#rs) >> bs @ bs1"
| "ASTAR bs r >> bs @ [S]"
| "\<lbrakk>r >> bs1; ASTAR [] r >> bs2\<rbrakk> \<Longrightarrow> ASTAR bs r >> bs @ Z # bs1 @ bs2"
lemma contains0:
assumes "a >> bs"
shows "(fuse bs1 a) >> bs1 @ bs"
using assms
apply(induct arbitrary: bs1)
apply(auto intro: contains.intros)
apply (metis append.assoc contains.intros(3))
apply (metis append.assoc contains.intros(4))
apply (metis append.assoc contains.intros(5))
apply (metis append.assoc contains.intros(6))
apply (metis append_assoc contains.intros(7))
done
lemma contains1:
assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> intern r >> code v"
shows "ASTAR [] (intern r) >> code (Stars vs)"
using assms
apply(induct vs)
apply(simp)
using contains.simps apply blast
apply(simp)
apply(subst (2) append_Nil[symmetric])
apply(rule contains.intros)
apply(auto)
done
lemma contains2:
assumes "\<Turnstile> v : r"
shows "(intern r) >> code v"
using assms
apply(induct)
prefer 4
apply(simp)
apply(rule contains.intros)
prefer 4
apply(simp)
apply(rule contains.intros)
apply(simp)
apply(subst (3) append_Nil[symmetric])
apply(rule contains.intros)
apply(simp)
apply(simp)
apply(simp)
apply(subst (9) append_Nil[symmetric])
apply(rule contains.intros)
apply (metis append_Cons append_self_conv2 contains0)
apply(simp)
apply(subst (9) append_Nil[symmetric])
apply(rule contains.intros)
back
apply(rule contains.intros)
apply(drule_tac ?bs1.0="[S]" in contains0)
apply(simp)
apply(simp)
apply(case_tac vs)
apply(simp)
apply (metis append_Nil contains.intros(6))
using contains1 by blast
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 qq2a:
assumes "\<not> bnullable r" "\<exists>r \<in> set rs1. bnullable r"
shows "bmkeps (AALTs bs (r # rs1)) = bmkeps (AALTs bs rs1)"
using assms
by (simp add: r1)
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 qq4:
assumes "bnullable (AALTs bs rs)"
shows "bmkeps (AALTs bs rs) = bs @ bmkeps (AALTs [] rs)"
by (metis append_Nil2 assms bmkeps_retrieve bnullable_correctness erase_fuse fuse.simps(4) mkeps_nullable retrieve_fuse2)
lemma contains3a:
assumes "AALTs bs lst >> bs @ bs1"
shows "AALTs bs (a # lst) >> bs @ bs1"
using assms
apply -
by (simp add: contains.intros(5))
lemma contains3b:
assumes "a >> bs1"
shows "AALTs bs (a # lst) >> bs @ bs1"
using assms
apply -
apply(rule contains.intros)
apply(simp)
done
lemma contains3:
assumes "\<And>x. \<lbrakk>x \<in> set rs; bnullable x\<rbrakk> \<Longrightarrow> x >> bmkeps x" "x \<in> set rs" "bnullable x"
shows "AALTs bs rs >> bmkeps (AALTs bs rs)"
using assms
apply(induct rs arbitrary: bs x)
apply simp
by (metis contains.intros(4) contains.intros(5) list.set_intros(1) list.set_intros(2) qq3 qq4 r r0 r1)
lemma cont1:
assumes "\<And>v. \<Turnstile> v : erase r \<Longrightarrow> r >> retrieve r v"
"\<forall>v\<in>set vs. \<Turnstile> v : erase r \<and> flat v \<noteq> []"
shows "ASTAR bs r >> retrieve (ASTAR bs r) (Stars vs)"
using assms
apply(induct vs arbitrary: bs r)
apply(simp)
using contains.intros(6) apply auto[1]
by (simp add: contains.intros(7))
lemma contains4:
assumes "bnullable a"
shows "a >> bmkeps a"
using assms
apply(induct a rule: bnullable.induct)
apply(auto intro: contains.intros)
using contains3 by blast
lemma contains5:
assumes "\<Turnstile> v : r"
shows "(intern r) >> retrieve (intern r) v"
using contains2[OF assms] retrieve_code[OF assms]
by (simp)
lemma contains6:
assumes "\<Turnstile> v : (erase r)"
shows "r >> retrieve r v"
using assms
apply(induct r arbitrary: v rule: erase.induct)
apply(auto)[1]
using Prf_elims(1) apply blast
using Prf_elims(4) contains.intros(1) apply force
using Prf_elims(5) contains.intros(2) apply force
apply(auto)[1]
using Prf_elims(1) apply blast
apply(auto)[1]
using contains3b contains3a apply blast
prefer 2
apply(auto)[1]
apply (metis Prf_elims(2) contains.intros(3) retrieve.simps(6))
prefer 2
apply(auto)[1]
apply (metis Prf_elims(6) cont1)
apply(simp)
apply(erule Prf_elims)
apply(auto)
apply (simp add: contains3b)
using retrieve_fuse2 contains3b contains3a
apply(subst retrieve_fuse2[symmetric])
apply (metis append_Nil2 erase_fuse fuse.simps(4))
apply(simp)
by (metis append_Nil2 erase_fuse fuse.simps(4))
lemma contains7:
assumes "\<Turnstile> v : der c (erase r)"
shows "(bder c r) >> retrieve r (injval (erase r) c v)"
using bder_retrieve[OF assms(1)] retrieve_code[OF assms(1)]
by (metis assms contains6 erase_bder)
lemma contains7a:
assumes "\<Turnstile> v : der c (erase r)"
shows "r >> retrieve r (injval (erase r) c v)"
using assms
apply -
apply(drule Prf_injval)
apply(drule contains6)
apply(simp)
done
fun
bders_simp :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
where
"bders_simp r [] = r"
| "bders_simp r (c # s) = bders_simp (bsimp (bder c r)) s"
definition blexer_simp where
"blexer_simp r s \<equiv> if bnullable (bders_simp (intern r) s) then
decode (bmkeps (bders_simp (intern r) s)) r else None"
lemma bders_simp_append:
shows "bders_simp r (s1 @ s2) = bders_simp (bders_simp r s1) s2"
apply(induct s1 arbitrary: r s2)
apply(simp)
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 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 (simp add: asize_fuse comp_def)
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: asize_fuse)
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: asize_fuse)
done
lemma qq:
shows "map (asize \<circ> fuse bs) rs = map asize rs"
apply(induct rs)
apply(auto simp add: asize_fuse)
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 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
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 contains_ex1:
assumes "a = AALTs bs1 [AZERO, AONE bs2]" "a >> bs"
shows "bsimp a >> bs"
using assms
apply(simp)
apply(erule contains.cases)
apply(auto)
using contains.simps apply blast
apply(erule contains.cases)
apply(auto)
using contains0 apply fastforce
using contains.simps by blast
lemma contains_ex2:
assumes "a = AALTs bs1 [AZERO, AONE bs2, AALTs bs5 [AONE bs3, AZERO, AONE bs4]]" "a >> bs"
shows "bsimp a >> bs"
using assms
apply(simp)
apply(erule contains.cases)
apply(auto)
using contains.simps apply blast
apply(erule contains.cases)
apply(auto)
using contains3b apply blast
apply(erule contains.cases)
apply(auto)
apply(erule contains.cases)
apply(auto)
apply (metis append.left_neutral contains.intros(4) contains.intros(5) contains0 fuse.simps(2))
apply(erule contains.cases)
apply(auto)
using contains.simps apply blast
apply(erule contains.cases)
apply(auto)
apply (metis append.left_neutral contains.intros(4) contains.intros(5) contains0 fuse.simps(2))
apply(erule contains.cases)
apply(auto)
apply(erule contains.cases)
apply(auto)
done
lemma contains48:
assumes "\<And>x2aa bs bs1. \<lbrakk>x2aa \<in> set x2a; fuse bs x2aa >> bs @ bs1\<rbrakk> \<Longrightarrow> x2aa >> bs1"
"AALTs (bs @ x1) x2a >> bs @ bs1"
shows "AALTs x1 x2a >> bs1"
using assms
apply(induct x2a arbitrary: bs x1 bs1)
apply(auto)
apply(erule contains.cases)
apply(auto)
apply(erule contains.cases)
apply(auto)
apply (simp add: contains.intros(4))
using contains.intros(5) by blast
lemma contains49:
assumes "fuse bs a >> bs @ bs1"
shows "a >> bs1"
using assms
apply(induct a arbitrary: bs bs1)
apply(auto)
using contains.simps apply blast
apply(erule contains.cases)
apply(auto)
apply(rule contains.intros)
apply(erule contains.cases)
apply(auto)
apply(rule contains.intros)
apply(erule contains.cases)
apply(auto)
apply(rule contains.intros)
apply(auto)[2]
prefer 2
apply(erule contains.cases)
apply(auto)
apply (simp add: contains.intros(6))
using contains.intros(7) apply blast
using contains48 by blast
lemma contains50:
assumes "bsimp_AALTs bs rs2 >> bs @ bs1"
shows "bsimp_AALTs bs (rs1 @ rs2) >> bs @ bs1"
using assms
apply(induct rs1 arbitrary: bs rs2 bs1)
apply(simp)
apply(auto)
apply(case_tac rs1)
apply(simp)
apply(case_tac rs2)
apply(simp)
using contains.simps apply blast
apply(simp)
apply(case_tac list)
apply(simp)
apply(rule contains.intros)
back
apply(rule contains.intros)
using contains49 apply blast
apply(simp)
using contains.intros(5) apply blast
apply(simp)
by (metis bsimp_AALTs.elims contains.intros(4) contains.intros(5) contains49 list.distinct(1))
lemma contains51:
assumes "bsimp_AALTs bs [r] >> bs @ bs1"
shows "bsimp_AALTs bs ([r] @ rs2) >> bs @ bs1"
using assms
apply(induct rs2 arbitrary: bs r bs1)
apply(simp)
apply(auto)
using contains.intros(4) contains49 by blast
lemma contains51a:
assumes "bsimp_AALTs bs rs2 >> bs @ bs1"
shows "bsimp_AALTs bs (rs2 @ [r]) >> bs @ bs1"
using assms
apply(induct rs2 arbitrary: bs r bs1)
apply(simp)
apply(auto)
using contains.simps apply blast
apply(case_tac rs2)
apply(auto)
using contains3b contains49 apply blast
apply(case_tac list)
apply(auto)
apply(erule contains.cases)
apply(auto)
using contains.intros(4) apply auto[1]
apply(erule contains.cases)
apply(auto)
apply (simp add: contains.intros(4) contains.intros(5))
apply (simp add: contains.intros(5))
apply(erule contains.cases)
apply(auto)
apply (simp add: contains.intros(4))
apply(erule contains.cases)
apply(auto)
using contains.intros(4) contains.intros(5) apply blast
using contains.intros(5) by blast
lemma contains51b:
assumes "bsimp_AALTs bs rs >> bs @ bs1"
shows "bsimp_AALTs bs (rs @ rs2) >> bs @ bs1"
using assms
apply(induct rs2 arbitrary: bs rs bs1)
apply(simp)
using contains51a by fastforce
lemma contains51c:
assumes "AALTs (bs @ bs2) rs >> bs @ bs1"
shows "bsimp_AALTs bs (map (fuse bs2) rs) >> bs @ bs1"
using assms
apply(induct rs arbitrary: bs bs1 bs2)
apply(auto)
apply(erule contains.cases)
apply(auto)
apply(erule contains.cases)
apply(auto)
using contains0 contains51 apply auto[1]
by (metis append.left_neutral append_Cons contains50 list.simps(9))
lemma contains51d:
assumes "fuse bs r >> bs @ bs1"
shows "bsimp_AALTs bs (flts [r]) >> bs @ bs1"
using assms
apply(induct r arbitrary: bs bs1)
apply(auto)
by (simp add: contains51c)
lemma contains52:
assumes "\<exists>r \<in> set rs. (fuse bs r) >> bs @ bs1"
shows "bsimp_AALTs bs (flts rs) >> bs @ bs1"
using assms
apply(induct rs arbitrary: bs bs1)
apply(simp)
apply(auto)
defer
apply (metis contains50 k0)
apply(subst k0)
apply(rule contains51b)
using contains51d by blast
lemma contains55:
assumes "a >> bs"
shows "bsimp a >> bs"
using assms
apply(induct a bs arbitrary:)
apply(auto intro: contains.intros)
apply(case_tac "bsimp a1 = AZERO")
apply(simp)
using contains.simps apply blast
apply(case_tac "bsimp a2 = AZERO")
apply(simp)
using contains.simps apply blast
apply(case_tac "\<exists>bs. bsimp a1 = AONE bs")
apply(auto)[1]
apply(rotate_tac 1)
apply(erule contains.cases)
apply(auto)
apply (simp add: b1 contains0 fuse_append)
apply (simp add: bsimp_ASEQ1 contains.intros(3))
prefer 2
apply(case_tac rs)
apply(simp)
using contains.simps apply blast
apply (metis contains50 k0)
(* AALTS case *)
apply(rule contains52)
apply(rule_tac x="bsimp r" in bexI)
apply(auto)
using contains0 by blast
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 qq4a:
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: qq4a)
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 qq4a 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 qq4a 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 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 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 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 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 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 CT1_SEQ:
shows "bsimp (ASEQ bs a1 a2) = bsimp (ASEQ bs (bsimp a1) (bsimp a2))"
apply(simp add: bsimp_idem)
done
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))
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 CT1b:
shows "bsimp (bsimp_AALTs bs as) = bsimp (bsimp_AALTs bs (map bsimp as))"
apply(induct bs as rule: bsimp_AALTs.induct)
apply(auto simp add: bsimp_idem)
apply (simp add: bsimp_fuse bsimp_idem)
by (metis bsimp_idem comp_apply)
(* CT *)
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 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 L_erase_bder_simp:
shows "L (erase (bsimp (bder a r))) = L (der a (erase (bsimp r)))"
using L_bsimp_erase der_correctness by auto
lemma PPP0:
assumes "s \<in> r \<rightarrow> v"
shows "(bders (intern r) s) >> code v"
using assms
by (smt L07 L1 LX0 Posix1(1) Posix_Prf contains6 erase_bders erase_intern lexer_correct_None lexer_flex mkeps_nullable option.inject retrieve_code)
thm L07 L1 LX0 Posix1(1) Posix_Prf contains6 erase_bders erase_intern lexer_correct_None lexer_flex mkeps_nullable option.inject retrieve_code
lemma PPP0_isar:
assumes "s \<in> r \<rightarrow> v"
shows "(bders (intern r) s) >> code v"
proof -
from assms have a1: "\<Turnstile> v : r" using Posix_Prf by simp
from assms have "s \<in> L r" using Posix1(1) by auto
then have "[] \<in> L (ders s r)" by (simp add: ders_correctness Ders_def)
then have a2: "\<Turnstile> mkeps (ders s r) : ders s r"
by (simp add: mkeps_nullable nullable_correctness)
have "retrieve (bders (intern r) s) (mkeps (ders s r)) =
retrieve (intern r) (flex r id s (mkeps (ders s r)))" using a2 LA LB bder_retrieve by simp
also have "... = retrieve (intern r) v"
using LB assms by auto
also have "... = code v" using a1 by (simp add: retrieve_code)
finally have "retrieve (bders (intern r) s) (mkeps (ders s r)) = code v" by simp
moreover
have "\<Turnstile> mkeps (ders s r) : erase (bders (intern r) s)" using a2 by simp
then have "bders (intern r) s >> retrieve (bders (intern r) s) (mkeps (ders s r))"
by (rule contains6)
ultimately
show "(bders (intern r) s) >> code v" by simp
qed
lemma PPP0b:
assumes "s \<in> r \<rightarrow> v"
shows "(intern r) >> code v"
using assms
using Posix_Prf contains2 by auto
lemma PPP0_eq:
assumes "s \<in> r \<rightarrow> v"
shows "(intern r >> code v) = (bders (intern r) s >> code v)"
using assms
using PPP0_isar PPP0b by blast
lemma f_cont1:
assumes "fuse bs1 a >> bs"
shows "\<exists>bs2. bs = bs1 @ bs2"
using assms
apply(induct a arbitrary: bs1 bs)
apply(auto elim: contains.cases)
done
lemma f_cont2:
assumes "bsimp_AALTs bs1 as >> bs"
shows "\<exists>bs2. bs = bs1 @ bs2"
using assms
apply(induct bs1 as arbitrary: bs rule: bsimp_AALTs.induct)
apply(auto elim: contains.cases f_cont1)
done
lemma contains_SEQ1:
assumes "bsimp_ASEQ bs r1 r2 >> bsX"
shows "\<exists>bs1 bs2. r1 >> bs1 \<and> r2 >> bs2 \<and> bsX = bs @ bs1 @ bs2"
using assms
apply(auto)
apply(case_tac "r1 = AZERO")
apply(auto)
using contains.simps apply blast
apply(case_tac "r2 = AZERO")
apply(auto)
apply(simp add: bsimp_ASEQ0)
using contains.simps apply blast
apply(case_tac "\<exists>bsX. r1 = AONE bsX")
apply(auto)
apply(simp add: bsimp_ASEQ2)
apply (metis append_assoc contains.intros(1) contains49 f_cont1)
apply(simp add: bsimp_ASEQ1)
apply(erule contains.cases)
apply(auto)
done
lemma contains59:
assumes "AALTs bs rs >> bs2"
shows "\<exists>r \<in> set rs. (fuse bs r) >> bs2"
using assms
apply(induct rs arbitrary: bs bs2)
apply(auto)
apply(erule contains.cases)
apply(auto)
apply(erule contains.cases)
apply(auto)
using contains0 by blast
lemma contains60:
assumes "\<exists>r \<in> set rs. fuse bs r >> bs2"
shows "AALTs bs rs >> bs2"
using assms
apply(induct rs arbitrary: bs bs2)
apply(auto)
apply (metis contains3b contains49 f_cont1)
using contains.intros(5) f_cont1 by blast
lemma contains61:
assumes "bsimp_AALTs bs rs >> bs2"
shows "\<exists>r \<in> set rs. (fuse bs r) >> bs2"
using assms
apply(induct arbitrary: bs2 rule: bsimp_AALTs.induct)
apply(auto)
using contains.simps apply blast
using contains59 by fastforce
lemma contains61b:
assumes "bsimp_AALTs bs rs >> bs2"
shows "\<exists>r \<in> set (flts rs). (fuse bs r) >> bs2"
using assms
apply(induct bs rs arbitrary: bs2 rule: bsimp_AALTs.induct)
apply(auto)
using contains.simps apply blast
using contains51d contains61 f_cont1 apply blast
by (metis bsimp_AALTs.simps(3) contains52 contains61 f_cont2)
lemma contains61a:
assumes "\<exists>r \<in> set rs. (fuse bs r) >> bs2"
shows "bsimp_AALTs bs rs >> bs2"
using assms
apply(induct rs arbitrary: bs2 bs)
apply(auto)
apply (metis bsimp_AALTs.elims contains60 list.distinct(1) list.inject list.set_intros(1))
by (metis append_Cons append_Nil contains50 f_cont2)
lemma contains62:
assumes "bsimp_AALTs bs (rs1 @ rs2) >> bs2"
shows "bsimp_AALTs bs rs1 >> bs2 \<or> bsimp_AALTs bs rs2 >> bs2"
using assms
apply -
apply(drule contains61)
apply(auto)
apply(case_tac rs1)
apply(auto)
apply(case_tac list)
apply(auto)
apply (simp add: contains60)
apply(case_tac list)
apply(auto)
apply (simp add: contains60)
apply (meson contains60 list.set_intros(2))
apply(case_tac rs2)
apply(auto)
apply(case_tac list)
apply(auto)
apply (simp add: contains60)
apply(case_tac list)
apply(auto)
apply (simp add: contains60)
apply (meson contains60 list.set_intros(2))
done
lemma contains63:
assumes "AALTs bs (map (fuse bs1) rs) >> bs3"
shows "AALTs (bs @ bs1) rs >> bs3"
using assms
apply(induct rs arbitrary: bs bs1 bs3)
apply(auto elim: contains.cases)
apply(erule contains.cases)
apply(auto)
apply (simp add: contains0 contains60 fuse_append)
by (metis contains.intros(5) contains59 f_cont1)
lemma contains64:
assumes "bsimp_AALTs bs (flts rs1 @ flts rs2) >> bs2" "\<forall>r \<in> set rs2. \<not> fuse bs r >> bs2"
shows "bsimp_AALTs bs (flts rs1) >> bs2"
using assms
apply(induct rs2 arbitrary: rs1 bs bs2)
apply(auto)
apply(drule_tac x="rs1" in meta_spec)
apply(drule_tac x="bs" in meta_spec)
apply(drule_tac x="bs2" in meta_spec)
apply(drule meta_mp)
apply(drule contains61)
apply(auto)
using contains51b contains61a f_cont1 apply blast
apply(subst (asm) k0)
apply(auto)
prefer 2
using contains50 contains61a f_cont1 apply blast
apply(case_tac a)
apply(auto)
by (metis contains60 fuse_append)
lemma contains65:
assumes "bsimp_AALTs bs (flts rs) >> bs2"
shows "\<exists>r \<in> set rs. (fuse bs r) >> bs2"
using assms
apply(induct rs arbitrary: bs bs2 taking: "\<lambda>rs. sum_list (map asize rs)" rule: measure_induct)
apply(case_tac x)
apply(auto elim: contains.cases)
apply(case_tac list)
apply(auto elim: contains.cases)
apply(case_tac a)
apply(auto elim: contains.cases)
apply(drule contains61)
apply(auto)
apply (metis contains60 fuse_append)
apply(case_tac lista)
apply(auto elim: contains.cases)
apply(subst (asm) k0)
apply(drule contains62)
apply(auto)
apply(case_tac a)
apply(auto elim: contains.cases)
apply(case_tac x52)
apply(auto elim: contains.cases)
apply(case_tac list)
apply(auto elim: contains.cases)
apply (simp add: contains60 fuse_append)
apply(erule contains.cases)
apply(auto)
apply (metis append.left_neutral contains0 contains60 fuse.simps(4) in_set_conv_decomp)
apply(erule contains.cases)
apply(auto)
apply (metis contains0 contains60 fuse.simps(4) list.set_intros(1) list.set_intros(2))
apply (simp add: contains.intros(5) contains63)
apply(case_tac aa)
apply(auto)
apply (meson contains60 contains61 contains63)
apply(subst (asm) k0)
apply(drule contains64)
apply(auto)[1]
by (metis append_Nil2 bsimp_AALTs.simps(2) contains50 contains61a contains64 f_cont2 flts.simps(1))
lemma contains55a:
assumes "bsimp r >> bs"
shows "r >> bs"
using assms
apply(induct r arbitrary: bs)
apply(auto)
apply(frule contains_SEQ1)
apply(auto)
apply (simp add: contains.intros(3))
apply(frule f_cont2)
apply(auto)
apply(drule contains65)
apply(auto)
using contains0 contains49 contains60 by blast
lemma PPP1_eq:
shows "bsimp r >> bs \<longleftrightarrow> r >> bs"
using contains55 contains55a by blast
lemma retrieve_code_bder:
assumes "\<Turnstile> v : der c r"
shows "code (injval r c v) = retrieve (bder c (intern r)) v"
using assms
by (simp add: Prf_injval bder_retrieve retrieve_code)
lemma Etrans:
assumes "a >> s" "s = t"
shows "a >> t"
using assms by simp
lemma retrieve_code_bders:
assumes "\<Turnstile> v : ders s r"
shows "code (flex r id s v) = retrieve (bders (intern r) s) v"
using assms
apply(induct s arbitrary: v r rule: rev_induct)
apply(auto simp add: ders_append flex_append bders_append)
apply (simp add: retrieve_code)
apply(frule Prf_injval)
apply(drule_tac meta_spec)+
apply(drule meta_mp)
apply(assumption)
apply(simp)
apply(subst bder_retrieve)
apply(simp)
apply(simp)
done
thm LA LB
lemma contains70:
assumes "s \<in> L(r)"
shows "bders (intern r) s >> code (flex r id s (mkeps (ders s r)))"
apply(subst PPP0_eq[symmetric])
apply (meson assms lexer_correct_None lexer_correctness(1) lexer_flex)
by (metis L07XX PPP0b assms erase_intern)
definition PV where
"PV r s v = flex r id s v"
definition PX where
"PX r s = PV r s (mkeps (ders s r))"
lemma PV_id[simp]:
shows "PV r [] v = v"
by (simp add: PV_def)
lemma PX_id[simp]:
shows "PX r [] = mkeps r"
by (simp add: PX_def)
lemma PV_cons:
shows "PV r (c # s) v = injval r c (PV (der c r) s v)"
apply(simp add: PV_def flex_fun_apply)
done
lemma PX_cons:
shows "PX r (c # s) = injval r c (PX (der c r) s)"
apply(simp add: PX_def PV_cons)
done
lemma PV_append:
shows "PV r (s1 @ s2) v = PV r s1 (PV (ders s1 r) s2 v)"
apply(simp add: PV_def flex_append)
by (simp add: flex_fun_apply2)
lemma PX_append:
shows "PX r (s1 @ s2) = PV r s1 (PX (ders s1 r) s2)"
by (simp add: PV_append PX_def ders_append)
lemma code_PV0:
shows "PV r (c # s) v = injval r c (PV (der c r) s v)"
unfolding PX_def PV_def
apply(simp)
by (simp add: flex_injval)
lemma code_PX0:
shows "PX r (c # s) = injval r c (PX (der c r) s)"
unfolding PX_def
apply(simp add: code_PV0)
done
lemma Prf_PV:
assumes "\<Turnstile> v : ders s r"
shows "\<Turnstile> PV r s v : r"
using assms unfolding PX_def PV_def
apply(induct s arbitrary: v r)
apply(simp)
apply(simp)
by (simp add: Prf_injval flex_injval)
lemma Prf_PX:
assumes "s \<in> L r"
shows "\<Turnstile> PX r s : r"
using assms unfolding PX_def PV_def
using L1 LX0 Posix_Prf lexer_correct_Some by fastforce
lemma PV1:
assumes "\<Turnstile> v : ders s r"
shows "(intern r) >> code (PV r s v)"
using assms
by (simp add: Prf_PV contains2)
lemma PX1:
assumes "s \<in> L r"
shows "(intern r) >> code (PX r s)"
using assms
by (simp add: Prf_PX contains2)
lemma PX2:
assumes "s \<in> L (der c r)"
shows "bder c (intern r) >> code (injval r c (PX (der c r) s))"
using assms
by (simp add: Prf_PX contains6 retrieve_code_bder)
lemma PX2a:
assumes "c # s \<in> L r"
shows "bder c (intern r) >> code (injval r c (PX (der c r) s))"
using assms
using PX2 lexer_correct_None by force
lemma PX2b:
assumes "c # s \<in> L r"
shows "bder c (intern r) >> code (PX r (c # s))"
using assms unfolding PX_def PV_def
by (metis Der_def L07XX PV_def PX2a PX_def Posix_determ Posix_injval der_correctness erase_intern mem_Collect_eq)
lemma PV3:
assumes "\<Turnstile> v : ders s r"
shows "bders (intern r) s >> code (PV r s v)"
using assms
using PX_def PV_def contains70
by (simp add: contains6 retrieve_code_bders)
lemma PX3:
assumes "s \<in> L r"
shows "bders (intern r) s >> code (PX r s)"
using assms
using PX_def PV_def contains70 by auto
lemma PV_bders_iff:
assumes "\<Turnstile> v : ders s r"
shows "bders (intern r) s >> code (PV r s v) \<longleftrightarrow> (intern r) >> code (PV r s v)"
by (simp add: PV1 PV3 assms)
lemma PX_bders_iff:
assumes "s \<in> L r"
shows "bders (intern r) s >> code (PX r s) \<longleftrightarrow> (intern r) >> code (PX r s)"
by (simp add: PX1 PX3 assms)
lemma PX4:
assumes "(s1 @ s2) \<in> L r"
shows "bders (intern r) (s1 @ s2) >> code (PX r (s1 @ s2))"
using assms
by (simp add: PX3)
lemma PX_bders_iff2:
assumes "(s1 @ s2) \<in> L r"
shows "bders (intern r) (s1 @ s2) >> code (PX r (s1 @ s2)) \<longleftrightarrow>
(intern r) >> code (PX r (s1 @ s2))"
by (simp add: PX1 PX3 assms)
lemma PV_bders_iff3:
assumes "\<Turnstile> v : ders (s1 @ s2) r"
shows "bders (intern r) (s1 @ s2) >> code (PV r (s1 @ s2) v) \<longleftrightarrow>
bders (intern r) s1 >> code (PV r (s1 @ s2) v)"
by (metis PV3 PV_append Prf_PV assms ders_append)
lemma PX_bders_iff3:
assumes "(s1 @ s2) \<in> L r"
shows "bders (intern r) (s1 @ s2) >> code (PX r (s1 @ s2)) \<longleftrightarrow>
bders (intern r) s1 >> code (PX r (s1 @ s2))"
by (metis Ders_def L07XX PV_append PV_def PX4 PX_def Posix_Prf assms contains6 ders_append ders_correctness erase_bders erase_intern mem_Collect_eq retrieve_code_bders)
lemma PV_bder_iff:
assumes "\<Turnstile> v : ders (s1 @ [c]) r"
shows "bder c (bders (intern r) s1) >> code (PV r (s1 @ [c]) v) \<longleftrightarrow>
bders (intern r) s1 >> code (PV r (s1 @ [c]) v)"
by (simp add: PV_bders_iff3 assms bders_snoc)
lemma PV_bder_IFF:
assumes "\<Turnstile> v : ders (s1 @ c # s2) r"
shows "bder c (bders (intern r) s1) >> code (PV r (s1 @ c # s2) v) \<longleftrightarrow>
bders (intern r) s1 >> code (PV r (s1 @ c # s2) v)"
by (metis LA PV3 PV_def Prf_PV assms bders_append code_PV0 contains7 ders.simps(2) erase_bders erase_intern retrieve_code_bders)
lemma PX_bder_iff:
assumes "(s1 @ [c]) \<in> L r"
shows "bder c (bders (intern r) s1) >> code (PX r (s1 @ [c])) \<longleftrightarrow>
bders (intern r) s1 >> code (PX r (s1 @ [c]))"
by (simp add: PX_bders_iff3 assms bders_snoc)
lemma PV_bder_iff2:
assumes "\<Turnstile> v : ders (c # s1) r"
shows "bders (bder c (intern r)) s1 >> code (PV r (c # s1) v) \<longleftrightarrow>
bder c (intern r) >> code (PV r (c # s1) v)"
by (metis PV3 Prf_PV assms bders.simps(2) code_PV0 contains7 ders.simps(2) erase_intern retrieve_code)
lemma PX_bder_iff2:
assumes "(c # s1) \<in> L r"
shows "bders (bder c (intern r)) s1 >> code (PX r (c # s1)) \<longleftrightarrow>
bder c (intern r) >> code (PX r (c # s1))"
using PX2b PX3 assms by force
definition EQ where
"EQ a1 a2 \<equiv> (\<forall>bs. a1 >> bs \<longleftrightarrow> a2 >> bs)"
lemma EQ1:
assumes "EQ (intern r1) (intern r2)"
"bders (intern r1) s >> code (PX r1 s)"
"s \<in> L r1" "s \<in> L r2"
shows "bders (intern r2) s >> code (PX r1 s)"
using assms unfolding EQ_def
thm PX_bders_iff
apply(subst (asm) PX_bders_iff)
apply(assumption)
apply(subgoal_tac "intern r2 >> code (PX r1 s)")
prefer 2
apply(auto)
lemma AA1:
assumes "[c] \<in> L r"
assumes "bder c (intern r) >> code (PX r [c])"
shows "bder c (bsimp (intern r)) >> code (PX r [c])"
using assms
apply(induct a arbitrary: c bs1 bs2 rs)
apply(auto elim: contains.cases)
apply(case_tac "c = x2a")
apply(simp)
apply(case_tac rs)
apply(auto)
using contains0 apply fastforce
apply(case_tac list)
apply(auto)
prefer 2
apply(erule contains.cases)
apply(auto)
lemma TEST:
assumes "bder c a >> bs"
shows "bder c (bsimp a) >> bs"
using assms
apply(induct a arbitrary: c bs)
apply(auto elim: contains.cases)
prefer 2
apply(erule contains.cases)
apply(auto)
lemma PX_bder_simp_iff:
assumes "\<Turnstile> v: ders (s1 @ s2) r"
shows "bders (bsimp (bders (intern r) s1)) s2 >> code (PV r (s1 @ s2) v) \<longleftrightarrow>
bders (intern r) s1 >> code (PV r (s1 @ s2) v)"
using assms
apply(induct s2 arbitrary: r s1 v)
apply(simp)
apply (simp add: PV3 contains55)
apply(drule_tac x="r" in meta_spec)
apply(drule_tac x="s1 @ [a]" in meta_spec)
apply(drule_tac x="v" in meta_spec)
apply(simp)
apply(simp add: bders_append)
apply(subst (asm) PV_bder_IFF)
definition EXs where
"EXs a s \<equiv> \<forall>v \<in> \<lbrace>= v : ders s (erase a).
lemma
assumes "s \<in> L r"
shows "(bders_simp (intern r) s >> code (PX r s)) \<longleftrightarrow> ((intern r) >> code (PX r s))"
using assms
apply(induct s arbitrary: r rule: rev_induct)
apply(simp)
apply(simp add: bders_simp_append)
apply(simp add: PPP1_eq)
find_theorems "retrieve (bders _ _) _"
find_theorems "_ >> retrieve _ _"
find_theorems "bsimp _ >> _"
lemma PX4a:
assumes "(s1 @ s2) \<in> L r"
shows "bders (intern r) (s1 @ s2) >> code (PV r s1 (PX (ders s1 r) s2))"
using PX4[OF assms]
apply(simp add: PX_append)
done
lemma PV5:
assumes "s2 \<in> (ders s1 r) \<rightarrow> v"
shows "bders (intern r) (s1 @ s2) >> code (PV r s1 v)"
by (simp add: PPP0_isar PV_def Posix_flex assms)
lemma PV6:
assumes "s2 \<in> (ders s1 r) \<rightarrow> v"
shows "bders (bders (intern r) s1) s2 >> code (PV r s1 v)"
using PV5 assms bders_append by auto
find_theorems "retrieve (bders _ _) _"
find_theorems "_ >> retrieve _ _"
find_theorems "bder _ _ >> _"
lemma PV6:
assumes "s @[c] \<in> L r"
shows"bder s1 (bders (intern r) s2) >> code (PX r (c # s))"
apply(subst PX_bders_iff)
apply(rule contains7)
apply(simp)
apply(rule assms)
apply(subst retrieve_code)
apply(simp add: PV_def)
apply(simp)
apply(drule_tac x="r" in meta_spec)
apply(drule_tac x="s1 @ [a]" in meta_spec)
apply(simp add: bders_append)
apply(subst PV_cons)
apply(drule_tac x="injval r a v" in meta_spec)
apply(drule meta_mp)
lemma PV8:
assumes "(s1 @ s2) \<in> L r"
shows "bders (bders_simp (intern r) s1) s2 >> code (PX r (s1 @ s2))"
using assms
apply(induct s1 arbitrary: r s2 rule: rev_induct)
apply(simp add: PX3)
apply(simp)
apply(simp add: bders_simp_append)
apply(drule_tac x="r" in meta_spec)
apply(drule_tac x="x # s2" in meta_spec)
apply(simp add: bders_simp_append)
apply(rule iffI)
defer
apply(simp add: PX_append)
apply(simp add: bders_append)
lemma PV6:
assumes "\<Turnstile> v : ders s r"
shows "bders (intern r) s >> code (PV r s v)"
using assms
by (simp add: PV_def contains6 retrieve_code_bders)
lemma OO0_PX:
assumes "s \<in> L r"
shows "bders (intern r) s >> code (PX r s)"
using assms
by (simp add: PX3)
lemma OO1:
assumes "[c] \<in> r \<rightarrow> v"
shows "bder c (intern r) >> code v"
using assms
using PPP0_isar by force
lemma OO1a:
assumes "[c] \<in> L r"
shows "bder c (intern r) >> code (PX r [c])"
using assms unfolding PX_def PV_def
using contains70 by fastforce
lemma OO12:
assumes "[c1, c2] \<in> L r"
shows "bders (intern r) [c1, c2] >> code (PX r [c1, c2])"
using assms
using PX_def PV_def contains70 by presburger
lemma OO2:
assumes "[c] \<in> L r"
shows "bders_simp (intern r) [c] >> code (PX r [c])"
using assms
using OO1a Posix1(1) contains55 by auto
lemma OO22:
assumes "[c1, c2] \<in> L r"
shows "bders_simp (intern r) [c1, c2] >> code (PX r [c1, c2])"
using assms
apply(simp)
apply(rule contains55)
apply(rule Etrans)
thm contains7
apply(rule contains7)
lemma contains70:
assumes "s \<in> L(r)"
shows "bders_simp (intern r) s >> code (flex r id s (mkeps (ders s r)))"
using assms
apply(induct s arbitrary: r rule: rev_induct)
apply(simp)
apply (simp add: contains2 mkeps_nullable nullable_correctness)
apply(simp add: bders_simp_append flex_append)
apply(simp add: PPP1_eq)
apply(rule Etrans)
apply(rule_tac v="flex r id xs (mkeps (ders (xs @ [x]) r))" in contains7)
thm L07XX PPP0b erase_intern
find_theorems "retrieve (bders _ _) _"
find_theorems "_ >> retrieve _ _"
find_theorems "bder _ _ >> _"
proof -
from assms have "\<Turnstile> v : erase (bder c r)" by simp
then have "bder c r >> retrieve (bder c r) v"
by (simp add: contains6)
moreover have "retrieve (bder c r) v = retrieve r (injval (erase r) c v)"
using assms bder_retrieve by blast
ultimately have "bder c r >> code (injval (erase r) c v)"
apply -
apply(subst retrieve_code_bder)
apply(simp add: assms)
oops
find_theorems "code _ = retrieve _ _"
find_theorems "_ >> retrieve _ _"
find_theorems "bder _ _ >> _"
lemma
assumes "s \<in> r \<rightarrow> v" "s = [c1, c2]"
shows "bders_simp (intern r) s >> bs \<longleftrightarrow> bders (intern r) s >> bs"
using assms
apply(simp add: PPP1_eq)
lemma PPP10:
assumes "s \<in> r \<rightarrow> v"
shows "bders_simp (intern r) s >> retrieve (intern r) v \<longleftrightarrow> bders (intern r) s >> retrieve (intern r) v"
using assms
apply(induct s arbitrary: r v rule: rev_induct)
apply(auto)
apply(simp_all add: PPP1_eq bders_append bders_simp_append)
find_theorems "bder _ _ >> _"
lemma
shows "bder
find_theorems "bsimp _ >> _"
fun get where
"get (Some v) = v"
lemma decode9:
assumes "decode' bs (STAR r) = (v, bsX)" "bs \<noteq> []"
shows "\<exists>vs. v = Stars vs"
using assms
apply(induct bs\<equiv>"bs" r\<equiv>"STAR r" arbitrary: bs r v bsX rule: decode'.induct)
apply(auto)
apply(case_tac "decode' ds r")
apply(auto)
apply(case_tac "decode' b (STAR r)")
apply(auto)
apply(case_tac aa)
apply(auto)
done
lemma decode10_Stars:
assumes "decode' bs (STAR r) = (Stars vs, bs1)" "\<Turnstile> Stars vs : (STAR r)" "vs \<noteq> []"
shows "decode' (bs @ bsX) (STAR r) = (Stars vs, bs1 @ bsX)"
using assms
apply(induct vs arbitrary: bs r bs1 bsX)
apply(auto elim!: Prf_elims)
apply(case_tac vs)
apply(auto)
apply(case_tac bs)
apply(auto)
apply(case_tac aa)
apply(auto)
apply(case_tac "decode' list r")
apply(auto)
apply(case_tac "decode' b (STAR r)")
apply(auto)
apply(case_tac "decode' (list @ bsX) r")
apply(auto)
apply(case_tac "decode' ba (STAR r)")
apply(auto)
apply(case_tac ba)
apply(auto)
oops
lemma decode10:
assumes "decode' bs r = (v, bs1)" "\<Turnstile> v : r"
shows "decode' (bs @ bsX) r = (v, bs1 @ bsX)"
using assms
apply(induct bs r arbitrary: v bs1 bsX rule: decode'.induct)
apply(auto elim: Prf_elims)[7]
apply(case_tac "decode' ds r1")
apply(auto)[3]
apply(case_tac "decode' (ds @ bsX) r1")
apply(auto)[3]
apply(auto elim: Prf_elims)[4]
apply(case_tac "decode' ds r2")
apply(auto)[1]
apply(case_tac "decode' (ds @ bsX) r2")
apply(auto)[1]
apply(auto elim: Prf_elims)[2]
apply(case_tac "decode' ds r1")
apply(auto)[1]
apply(case_tac "decode' b r2")
apply(auto)[1]
apply(auto elim: Prf_elims)[1]
apply(auto elim: Prf_elims)[1]
apply(auto elim: Prf_elims)[1]
apply(erule Prf_elims)
(* STAR case *)
apply(auto)
apply(case_tac "decode' ds r")
apply(auto)
apply(case_tac "decode' b (STAR r)")
apply(auto)
apply(case_tac aa)
apply(auto)
apply(case_tac "decode' (b @ bsX) (STAR r)")
apply(auto)
oops
lemma contains100:
assumes "(intern r) >> bs"
shows "\<exists>v bsV. decode' bs r = (v, bsV) \<and> \<Turnstile> v : r"
using assms
apply(induct r arbitrary: bs)
apply(auto)
apply(erule contains.cases)
apply(auto)
apply(erule contains.cases)
apply(auto intro: Prf.intros)
apply(erule contains.cases)
apply(auto)
apply(drule_tac x="bs1" in meta_spec)
apply(drule_tac x="bs2" in meta_spec)
apply(auto)[1]
apply(rule_tac x="Seq v va" in exI)
apply(auto)
apply(case_tac "decode' (bs1 @ bs2) r1")
apply(auto)
apply(case_tac "decode' b r2")
apply(auto)
oops
lemma contains101:
assumes "(intern r) >> code v"
shows "\<Turnstile> v : r"
using assms
apply(induct r arbitrary: v)
apply(auto elim: contains.cases)
apply(erule contains.cases)
apply(auto)
apply(case_tac v)
apply(auto intro: Prf.intros)
apply(erule contains.cases)
apply(auto)
apply(case_tac v)
apply(auto intro: Prf.intros)
(*
using contains.simps apply blast
apply(erule contains.cases)
apply(auto)
using L1 Posix_ONE Prf.intros(4) apply force
apply(erule contains.cases)
apply(auto)
apply (metis Prf.intros(5) code.simps(2) decode_code get.simps)
apply(erule contains.cases)
apply(auto)
prefer 2
apply(erule contains.cases)
apply(auto)
apply(frule f_cont1)
apply(auto)
apply(case_tac "decode' bs2 r1")
apply(auto)
apply(rule Prf.intros)
apply (metis Cons_eq_append_conv contains49 f_cont1 fst_conv list.inject self_append_conv2)
apply(erule contains.cases)
apply(auto)
apply(frule f_cont1)
apply(auto)
apply(case_tac "decode' bs2 r2")
apply(auto)
apply(rule Prf.intros)
apply (metis (full_types) append_Cons contains49 append_Nil fst_conv)
apply(erule contains.cases)
apply(auto)
apply(case_tac "decode' (bs1 @ bs2) r1")
apply(auto)
apply(case_tac "decode' b r2")
apply(auto)
apply(rule Prf.intros)
apply (metis fst_conv)
apply(subgoal_tac "b = bs2 @ bsX")
apply(auto)
apply (metis fst_conv)
apply(subgoal_tac "decode' (bs1 @ bs2 @ bsX) r1 = (a, bs2 @ bsX)")
apply simp
*)
apply(case_tac ba)
apply(auto)
apply(drule meta_spec)
apply(drule meta_mp)
apply(assumption)
prefer 2
apply(case_tac v)
apply(auto)
find_theorems "bder _ _ >> _"
lemma PPP0_isar:
assumes "bders r s >> code v"
shows "bders_simp r s >> code v"
using assms
apply(induct s arbitrary: r v)
apply(simp)
apply(auto)
apply(drule_tac x="bsimp (bder a r)" in meta_spec)
apply(drule_tac x="v" in meta_spec)
apply(drule_tac meta_mp)
prefer 2
apply(simp)
using bnullable_correctness nullable_correctness apply fastforce
apply(simp add: bders_append)
lemma PPP0_isar:
assumes "s \<in> r \<rightarrow> v"
shows "(bders (intern r) s) >> code v"
proof -
from assms have a1: "\<Turnstile> v : r" using Posix_Prf by simp
from assms have "s \<in> L r" using Posix1(1) by auto
then have "[] \<in> L (ders s r)" by (simp add: ders_correctness Ders_def)
then have a2: "\<Turnstile> mkeps (ders s r) : ders s r"
by (simp add: mkeps_nullable nullable_correctness)
have "retrieve (bders (intern r) s) (mkeps (ders s r)) =
retrieve (intern r) (flex r id s (mkeps (ders s r)))" using a2 LA by simp
also have "... = retrieve (intern r) v"
using LB assms by auto
also have "... = code v" using a1 by (simp add: retrieve_code)
finally have "retrieve (bders (intern r) s) (mkeps (ders s r)) = code v" by simp
moreover
have "\<Turnstile> mkeps (ders s r) : erase (bders (intern r) s)" using a2 by simp
then have "bders (intern r) s >> retrieve (bders (intern r) s) (mkeps (ders s r))"
by (rule contains6)
ultimately
show "(bders (intern r) s) >> code v" by simp
qed
lemma A0:
assumes "r \<in> set (flts rs)"
shows "r \<in> set rs"
using assms
apply(induct rs arbitrary: r rule: flts.induct)
apply(auto)
oops
lemma A1:
assumes "r \<in> set (flts (map (bder c) (flts rs)))" "\<forall>r \<in> set rs. nonnested r \<and> good r"
shows "r \<in> set (flts (map (bder c) rs))"
using assms
apply(induct rs arbitrary: r c rule: flts.induct)
apply(auto)
apply(subst (asm) map_bder_fuse)
apply(simp add: flts_append)
apply(auto)
apply(auto simp add: comp_def)
apply(subgoal_tac "\<forall>r \<in> set rs1. nonalt r \<and> good r")
prefer 2
apply (metis Nil_is_append_conv good.simps(5) good.simps(6) in_set_conv_decomp neq_Nil_conv)
apply(case_tac rs1)
apply(auto)
apply(subst (asm) k0)
apply(auto)
oops
lemma bsimp_comm2:
assumes "bder c a >> bs"
shows "bder c (bsimp a) >> bs"
using assms
apply(induct a arbitrary: bs c taking: "asize" rule: measure_induct)
apply(case_tac x)
apply(auto)
prefer 2
apply(erule contains.cases)
apply(auto)
apply(subst bder_bsimp_AALTs)
apply(rule contains61a)
apply(rule bexI)
apply(rule contains0)
apply(assumption)
lemma bsimp_comm:
assumes "bder c (bsimp a) >> bs"
shows "bsimp (bder c a) >> bs"
using assms
apply(induct a arbitrary: bs c taking: "asize" rule: measure_induct)
apply(case_tac x)
apply(auto)
prefer 4
apply(erule contains.cases)
apply(auto)
using contains.intros(3) contains55 apply fastforce
prefer 3
apply(subst (asm) bder_bsimp_AALTs)
apply(drule contains61b)
apply(auto)
apply(rule contains61a)
apply(rule bexI)
apply(assumption)
apply(rule_tac t="set (flts (map (bsimp \<circ> bder c) x52))"
and s="set (flts (map (bder c \<circ> bsimp) x52))" in subst)
prefer 2
find_theorems "map (_ \<circ> _) _ = _"
apply(simp add: comp_def)
find_theorems "bder _ (bsimp_AALTs _ _) = _"
apply(drule contains_SEQ1)
apply(auto)[1]
apply(rule contains.intros)
prefer 2
apply(assumption)
apply(case_tac "bnullable x42")
apply(simp)
prefer 2
apply(simp)
apply(case_tac "bsimp x42 = AZERO")
apply (me tis L_erase_bder_simp bder.simps(1) bsimp.simps(3) bsimp_ASEQ.simps(1) good.simps(1) good1a xxx_bder2)
apply(case_tac "bsimp x43 = AZERO")
apply (simp add: bsimp_ASEQ0)
apply(case_tac "\<exists>bs1. bsimp x42 = AONE bs1")
using b3 apply force
apply(subst bsimp_ASEQ1)
apply(auto)[3]
apply(auto)[1]
using b3 apply blast
apply(case_tac "bsimp (bder c x42) = AZERO")
apply(simp)
using contains.simps apply blast
apply(case_tac "\<exists>bs2. bsimp (bder c x42) = AONE bs2")
apply(auto)[1]
apply(subst (asm) bsimp_ASEQ2)
apply(subgoal_tac "\<exists>bsX. bs = x41 @ bs2 @ bsX")
apply(auto)[1]
apply(rule contains.intros)
apply (simp add: contains.intros(1))
apply (metis append_assoc contains49)
using append_assoc f_cont1 apply blast
apply(subst (asm) bsimp_ASEQ1)
apply(auto)[3]
apply(erule contains.cases)
apply(auto)
using contains.intros(3) less_add_Suc1 apply blast
apply(case_tac "bsimp x42 = AZERO")
using b3 apply force
apply(case_tac "bsimp x43 = AZERO")
apply (metis LLLL(1) L_erase_bder_simp bder.simps(1) bsimp_AALTs.simps(1) bsimp_ASEQ0 bsimp_fuse flts.simps(1) flts.simps(2) fuse.simps(1) good.simps(1) good1a xxx_bder2)
apply(case_tac "\<exists>bs1. bsimp x42 = AONE bs1")
apply(auto)[1]
apply(subst bsimp_ASEQ2)
apply(drule_tac x="fuse (x41 @ bs1) x43" in spec)
apply(drule mp)
apply (simp add: asize_fuse)
apply(drule_tac x="bs" in spec)
apply(drule_tac x="c" in spec)
apply(drule mp)
prefer 2
apply (simp add: bsimp_fuse)
apply(subst (asm) k0)
apply(subgoal_tac "\<exists>bsX. bs = x41 @ bsX")
prefer 2
using f_cont2 apply blast
apply(clarify)
apply(drule contains62)
apply(auto)[1]
apply(case_tac "bsimp (bder c x42) = AZERO")
apply (metis append_is_Nil_conv bsimp_ASEQ.simps(1) contains61 flts.simps(1) flts.simps(2) in_set_conv_decomp list.distinct(1))
apply(case_tac "\<exists>bsX. bsimp (bder c x42) = AONE bsX")
apply(clarify)
apply (simp add: L_erase_bder_simp xxx_bder2)
using L_erase_bder_simp xxx_bder2 apply auto[1]
apply(drule contains65)
apply(auto)[1]
apply (simp add: bder_fuse bmkeps_simp bsimp_fuse fuse_append)
apply(subst bsimp_ASEQ1)
apply(auto)[3]
apply(auto)[1]
apply(case_tac "bsimp (bder c x42) = AZERO")
apply(simp add: bsimp_ASEQ0)
apply(drule contains65)
apply(auto)[1]
apply (metis asize_fuse bder_fuse bmkeps_simp bsimp_fuse contains.intros(4) contains.intros(5) contains49 f_cont1 less_add_Suc2)
apply(frule f_cont1)
apply(auto)
apply(case_tac "\<exists>bsX. bsimp (bder c x42) = AONE bsX")
apply(auto)[1]
apply(subst (asm) bsimp_ASEQ2)
apply(auto)
apply(drule contains65)
apply(auto)[1]
apply(frule f_cont1)
apply(auto)
apply(rule contains.intros)
apply (metis (no_types, lifting) append_Nil2 append_eq_append_conv2 contains.intros(1) contains.intros(3) contains49 f_cont1 less_add_Suc1 same_append_eq)
apply(frule f_cont1)
apply(auto)
apply(rule contains.intros)
apply(drule contains49)
apply(subst (asm) bsimp_fuse[symmetric])
apply(frule f_cont1)
apply(auto)
apply(subst (3) append_Nil[symmetric])
apply(rule contains.intros)
apply(drule contains49)
prefer 2
apply(simp)
find_theorems "fuse _ _ >> _"
apply(erule contains.cases)
apply(auto)
thm bder_retrieve
find_theorems "_ >> retrieve _ _"
lemma TEST:
assumes "\<Turnstile> v : ders s (erase r)"
shows "bders r s >> retrieve r (flex (erase r) id s v)"
using assms
apply(induct s arbitrary: v r rule: rev_induct)
apply(simp)
apply (simp add: contains6)
apply(simp add: bders_append ders_append)
apply(rule Etrans)
apply(rule contains7)
apply(simp)
by (metis LA bder_retrieve bders_snoc ders_snoc erase_bders)
lemma TEST1:
assumes "bder c r >> retrieve r (injval (erase r) c v)"
shows "r >> retrieve r v"
oops
lemma TEST2:
assumes "bders (intern r) s >> retrieve (intern r) (flex r id s (mkeps (ders s r)))" "s = [c1, c2]"
shows "bders_simp (intern r) s >> retrieve (intern r) (flex r id s (mkeps (ders s r)))"
using assms
apply(simp)
apply(induct s arbitrary: r rule: rev_induct)
apply(simp)
apply(simp add: bders_simp_append ders_append flex_append bders_append)
apply(rule contains55)
apply(drule_tac x="bsimp (bder a r)" in meta_spec)
thm L02_bders
apply(subst L02_bders)
find_theorems "retrieve (bsimp _) _ = _"
apply(drule_tac "" in Etrans)
lemma TEST2:
assumes "bders r s >> retrieve r (flex (erase r) id s (mkeps (ders s (erase r))))"
shows "bders_simp r s >> retrieve r (flex (erase r) id s (mkeps (ders s (erase r))))"
using assms
apply(induct s arbitrary: r rule: rev_induct)
apply(simp)
apply(simp add: bders_simp_append ders_append flex_append bders_append)
apply(subgoal_tac "bder x (bders r xs) >> retrieve r (flex (erase r) id xs (injval (ders xs (erase r)) x (mkeps (ders xs (erase r)))))")
find_theorems "bders _ _ >> _"
apply(drule_tac x="bsimp (bder a r)" in meta_spec)
thm L02_bders
apply(subst L02_bders)
find_theorems "retrieve (bsimp _) _ = _"
apply(drule_tac "" in Etrans)
apply(rule contains55)
apply(rule Etrans)
apply(rule contains7)
apply(subgoal_tac "\<Turnstile> v : der x (erase (bders_simp r xs))")
apply(assumption)
prefer 2
apply(simp)
by (m etis LA bder_retrieve bders_snoc ders_snoc erase_bders)
lemma PPP0A:
assumes "s \<in> L (r)"
shows "(bders (intern r) s) >> code (flex r id s (mkeps (ders s r)))"
using assms
by (metis L07XX PPP0 erase_intern)
lemma PPP1:
assumes "bder c (intern r) >> code v" "\<Turnstile> v : der c r"
shows "(intern r) >> code (injval r c v)"
using assms
by (simp add: Prf_injval contains2)
(*
lemma PPP1:
assumes "bder c r >> code v" "\<Turnstile> v : der c (erase r)"
shows "r >> code (injval (erase r) c v)"
using assms contains7[OF assms(2)] retrieve_code[OF assms(2)]
find_theorems "bder _ _ >> _"
by (simp add: Prf_injval contains2)
*)
lemma PPP3:
assumes "\<Turnstile> v : ders s (erase a)"
shows "bders a s >> retrieve a (flex (erase a) id s v)"
using LA[OF assms] contains6 erase_bders assms by metis
find_theorems "bder _ _ >> _"
lemma QQQ0:
assumes "bder c a >> code v"
shows "a >> code (injval (erase a) c v)"
using assms
apply(induct a arbitrary: c v)
apply(auto)
using contains.simps apply blast
using contains.simps apply blast
apply(case_tac "c = x2a")
apply(simp)
apply(erule contains.cases)
apply(auto)
lemma PPP4:
assumes "bders (intern a) [c1, c2] >> bs"
shows "bders_simp (intern a) [c1, c2] >> bs"
using assms
apply(simp)
apply(rule contains55)
find_theorems "bder _ _ >> _"
apply(induct s arbitrary: a v rule: rev_induct)
apply(simp)
apply (simp add: contains6)
apply(simp add: bders_append bders_simp_append ders_append flex_append)
(*apply(rule contains55)*)
apply(drule Prf_injval)
apply(drule_tac x="a" in meta_spec)
apply(drule_tac x="injval (ders xs (erase a)) x v" in meta_spec)
apply(drule meta_mp)
apply(assumption)
apply(thin_tac "\<Turnstile> injval (ders xs (erase a)) x v : ders xs (erase a)")
apply(thin_tac "bders a xs >> retrieve a (flex (erase a) id xs (injval (ders xs (erase a)) x v))")
apply(rule Etrans)
apply(rule contains7)
lemma PPP4:
assumes "bders a s >> code v" "\<Turnstile> v : ders s (erase a)"
shows "bders_simp a s >> code v"
using assms
apply(induct s arbitrary: a v rule: rev_induct)
apply(simp)
apply(simp add: bders_append bders_simp_append ders_append)
apply(rule contains55)
find_theorems "bder _ _ >> _"
lemma PPP0:
assumes "s \<in> L (r)"
shows "(bders (intern r) s) >> code (flex r id s (mkeps (ders s r)))"
using assms
apply(induct s arbitrary: r rule: rev_induct)
apply(simp)
apply (simp add: contains2 mkeps_nullable nullable_correctness)
apply(simp add: bders_simp_append flex_append)
apply(rule contains55)
apply(rule Etrans)
apply(rule contains7)
defer
find_theorems "_ >> _"
apply(drule_tac x="der a r" in meta_spec)
apply(drule meta_mp)
find_theorems "bder _ _ >> _"
apply(subgoal_tac "s \<in> L(der a r)")
prefer 2
apply (simp add: Posix_Prf contains2)
apply(simp add: bders_simp_append)
apply(rule contains55)
apply(frule PPP0)
apply(simp add: bders_append)
using Posix_injval contains7
apply(subgoal_tac "retrieve r (injval (erase r) x v)")
find_theorems "bders _ _ >> _"
lemma PPP1:
assumes "\<Turnstile> v : ders s r"
shows "bders (intern r) s >> code v"
using assms
apply(induct s arbitrary: r v rule: rev_induct)
apply(simp)
apply (simp add: Posix_Prf contains2)
apply(simp add: bders_append ders_append flex_append)
apply(frule Prf_injval)
apply(drule meta_spec)
apply(drule meta_spec)
apply(drule meta_mp)
apply(assumption)
apply(subst retrieve_code)
apply(assumption)
apply(subst (asm) retrieve_code)
apply(assumption)
using contains7 contains7a contains6 retrieve_code
apply(rule contains7)
find_theorems "bder _ _ >> _"
find_theorems "code _ = _"
find_theorems "\<Turnstile> _ : der _ _"
find_theorems "_ >> (code _)"
apply(induct s arbitrary: a bs rule: rev_induct)
apply(simp)
apply(simp add: bders_simp_append bders_append)
apply(rule contains55)
find_theorems "bder _ _ >> _"
apply(drule_tac x="bder a aa" in meta_spec)
apply(drule_tac x="bs" in meta_spec)
apply(simp)
apply(rule contains55)
find_theorems "bsimp _ >> _"
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