365
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theory SizeBound
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imports "Lexer"
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begin
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section \<open>Bit-Encodings\<close>
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datatype bit = Z | S
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fun code :: "val \<Rightarrow> bit list"
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where
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"code Void = []"
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| "code (Char c) = []"
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| "code (Left v) = Z # (code v)"
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| "code (Right v) = S # (code v)"
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| "code (Seq v1 v2) = (code v1) @ (code v2)"
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| "code (Stars []) = [S]"
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| "code (Stars (v # vs)) = (Z # code v) @ code (Stars vs)"
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fun
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Stars_add :: "val \<Rightarrow> val \<Rightarrow> val"
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where
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"Stars_add v (Stars vs) = Stars (v # vs)"
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function
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decode' :: "bit list \<Rightarrow> rexp \<Rightarrow> (val * bit list)"
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where
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"decode' ds ZERO = (Void, [])"
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| "decode' ds ONE = (Void, ds)"
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| "decode' ds (CH d) = (Char d, ds)"
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| "decode' [] (ALT r1 r2) = (Void, [])"
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| "decode' (Z # ds) (ALT r1 r2) = (let (v, ds') = decode' ds r1 in (Left v, ds'))"
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| "decode' (S # ds) (ALT r1 r2) = (let (v, ds') = decode' ds r2 in (Right v, ds'))"
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| "decode' ds (SEQ r1 r2) = (let (v1, ds') = decode' ds r1 in
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let (v2, ds'') = decode' ds' r2 in (Seq v1 v2, ds''))"
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| "decode' [] (STAR r) = (Void, [])"
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| "decode' (S # ds) (STAR r) = (Stars [], ds)"
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| "decode' (Z # ds) (STAR r) = (let (v, ds') = decode' ds r in
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let (vs, ds'') = decode' ds' (STAR r)
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in (Stars_add v vs, ds''))"
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by pat_completeness auto
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lemma decode'_smaller:
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assumes "decode'_dom (ds, r)"
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shows "length (snd (decode' ds r)) \<le> length ds"
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using assms
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apply(induct ds r)
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apply(auto simp add: decode'.psimps split: prod.split)
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using dual_order.trans apply blast
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by (meson dual_order.trans le_SucI)
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termination "decode'"
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apply(relation "inv_image (measure(%cs. size cs) <*lex*> measure(%s. size s)) (%(ds,r). (r,ds))")
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apply(auto dest!: decode'_smaller)
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by (metis less_Suc_eq_le snd_conv)
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definition
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decode :: "bit list \<Rightarrow> rexp \<Rightarrow> val option"
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where
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"decode ds r \<equiv> (let (v, ds') = decode' ds r
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in (if ds' = [] then Some v else None))"
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lemma decode'_code_Stars:
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assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> (\<forall>x. decode' (code v @ x) r = (v, x)) \<and> flat v \<noteq> []"
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shows "decode' (code (Stars vs) @ ds) (STAR r) = (Stars vs, ds)"
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using assms
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apply(induct vs)
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apply(auto)
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done
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lemma decode'_code:
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assumes "\<Turnstile> v : r"
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shows "decode' ((code v) @ ds) r = (v, ds)"
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using assms
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apply(induct v r arbitrary: ds)
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apply(auto)
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using decode'_code_Stars by blast
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lemma decode_code:
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assumes "\<Turnstile> v : r"
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shows "decode (code v) r = Some v"
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using assms unfolding decode_def
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by (smt append_Nil2 decode'_code old.prod.case)
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section {* Annotated Regular Expressions *}
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datatype arexp =
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AZERO
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| AONE "bit list"
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| ACHAR "bit list" char
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| ASEQ "bit list" arexp arexp
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| AALTs "bit list" "arexp list"
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| ASTAR "bit list" arexp
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abbreviation
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"AALT bs r1 r2 \<equiv> AALTs bs [r1, r2]"
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fun asize :: "arexp \<Rightarrow> nat" where
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"asize AZERO = 1"
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| "asize (AONE cs) = 1"
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| "asize (ACHAR cs c) = 1"
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| "asize (AALTs cs rs) = Suc (sum_list (map asize rs))"
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| "asize (ASEQ cs r1 r2) = Suc (asize r1 + asize r2)"
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| "asize (ASTAR cs r) = Suc (asize r)"
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fun
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erase :: "arexp \<Rightarrow> rexp"
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where
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"erase AZERO = ZERO"
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| "erase (AONE _) = ONE"
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| "erase (ACHAR _ c) = CH c"
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| "erase (AALTs _ []) = ZERO"
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| "erase (AALTs _ [r]) = (erase r)"
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| "erase (AALTs bs (r#rs)) = ALT (erase r) (erase (AALTs bs rs))"
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| "erase (ASEQ _ r1 r2) = SEQ (erase r1) (erase r2)"
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| "erase (ASTAR _ r) = STAR (erase r)"
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fun nonalt :: "arexp \<Rightarrow> bool"
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where
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"nonalt (AALTs bs2 rs) = False"
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| "nonalt r = True"
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fun good :: "arexp \<Rightarrow> bool" where
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"good AZERO = False"
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| "good (AONE cs) = True"
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| "good (ACHAR cs c) = True"
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| "good (AALTs cs []) = False"
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| "good (AALTs cs [r]) = False"
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| "good (AALTs cs (r1#r2#rs)) = (\<forall>r' \<in> set (r1#r2#rs). good r' \<and> nonalt r')"
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| "good (ASEQ _ AZERO _) = False"
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| "good (ASEQ _ (AONE _) _) = False"
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| "good (ASEQ _ _ AZERO) = False"
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| "good (ASEQ cs r1 r2) = (good r1 \<and> good r2)"
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| "good (ASTAR cs r) = True"
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fun fuse :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp" where
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"fuse bs AZERO = AZERO"
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| "fuse bs (AONE cs) = AONE (bs @ cs)"
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| "fuse bs (ACHAR cs c) = ACHAR (bs @ cs) c"
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| "fuse bs (AALTs cs rs) = AALTs (bs @ cs) rs"
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| "fuse bs (ASEQ cs r1 r2) = ASEQ (bs @ cs) r1 r2"
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| "fuse bs (ASTAR cs r) = ASTAR (bs @ cs) r"
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lemma fuse_append:
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shows "fuse (bs1 @ bs2) r = fuse bs1 (fuse bs2 r)"
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apply(induct r)
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apply(auto)
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done
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fun intern :: "rexp \<Rightarrow> arexp" where
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"intern ZERO = AZERO"
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| "intern ONE = AONE []"
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| "intern (CH c) = ACHAR [] c"
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| "intern (ALT r1 r2) = AALT [] (fuse [Z] (intern r1))
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(fuse [S] (intern r2))"
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| "intern (SEQ r1 r2) = ASEQ [] (intern r1) (intern r2)"
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| "intern (STAR r) = ASTAR [] (intern r)"
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fun retrieve :: "arexp \<Rightarrow> val \<Rightarrow> bit list" where
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"retrieve (AONE bs) Void = bs"
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| "retrieve (ACHAR bs c) (Char d) = bs"
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| "retrieve (AALTs bs [r]) v = bs @ retrieve r v"
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| "retrieve (AALTs bs (r#rs)) (Left v) = bs @ retrieve r v"
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| "retrieve (AALTs bs (r#rs)) (Right v) = bs @ retrieve (AALTs [] rs) v"
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| "retrieve (ASEQ bs r1 r2) (Seq v1 v2) = bs @ retrieve r1 v1 @ retrieve r2 v2"
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| "retrieve (ASTAR bs r) (Stars []) = bs @ [S]"
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| "retrieve (ASTAR bs r) (Stars (v#vs)) =
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bs @ [Z] @ retrieve r v @ retrieve (ASTAR [] r) (Stars vs)"
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fun
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bnullable :: "arexp \<Rightarrow> bool"
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where
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"bnullable (AZERO) = False"
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| "bnullable (AONE bs) = True"
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| "bnullable (ACHAR bs c) = False"
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| "bnullable (AALTs bs rs) = (\<exists>r \<in> set rs. bnullable r)"
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| "bnullable (ASEQ bs r1 r2) = (bnullable r1 \<and> bnullable r2)"
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| "bnullable (ASTAR bs r) = True"
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fun
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bmkeps :: "arexp \<Rightarrow> bit list"
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where
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"bmkeps(AONE bs) = bs"
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| "bmkeps(ASEQ bs r1 r2) = bs @ (bmkeps r1) @ (bmkeps r2)"
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| "bmkeps(AALTs bs [r]) = bs @ (bmkeps r)"
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| "bmkeps(AALTs bs (r#rs)) = (if bnullable(r) then bs @ (bmkeps r) else (bmkeps (AALTs bs rs)))"
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| "bmkeps(ASTAR bs r) = bs @ [S]"
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fun
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bder :: "char \<Rightarrow> arexp \<Rightarrow> arexp"
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where
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"bder c (AZERO) = AZERO"
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| "bder c (AONE bs) = AZERO"
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| "bder c (ACHAR bs d) = (if c = d then AONE bs else AZERO)"
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| "bder c (AALTs bs rs) = AALTs bs (map (bder c) rs)"
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| "bder c (ASEQ bs r1 r2) =
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(if bnullable r1
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then AALT bs (ASEQ [] (bder c r1) r2) (fuse (bmkeps r1) (bder c r2))
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else ASEQ bs (bder c r1) r2)"
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| "bder c (ASTAR bs r) = ASEQ bs (fuse [Z] (bder c r)) (ASTAR [] r)"
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fun
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bders :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
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where
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"bders r [] = r"
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| "bders r (c#s) = bders (bder c r) s"
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lemma bders_append:
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"bders r (s1 @ s2) = bders (bders r s1) s2"
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apply(induct s1 arbitrary: r s2)
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apply(simp_all)
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done
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lemma bnullable_correctness:
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shows "nullable (erase r) = bnullable r"
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apply(induct r rule: erase.induct)
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apply(simp_all)
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done
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lemma erase_fuse:
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shows "erase (fuse bs r) = erase r"
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apply(induct r rule: erase.induct)
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apply(simp_all)
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done
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thm Posix.induct
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lemma erase_intern [simp]:
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shows "erase (intern r) = r"
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apply(induct r)
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apply(simp_all add: erase_fuse)
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done
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lemma erase_bder [simp]:
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shows "erase (bder a r) = der a (erase r)"
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apply(induct r rule: erase.induct)
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apply(simp_all add: erase_fuse bnullable_correctness)
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done
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lemma erase_bders [simp]:
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shows "erase (bders r s) = ders s (erase r)"
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apply(induct s arbitrary: r )
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apply(simp_all)
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done
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lemma retrieve_encode_STARS:
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assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> code v = retrieve (intern r) v"
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shows "code (Stars vs) = retrieve (ASTAR [] (intern r)) (Stars vs)"
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using assms
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apply(induct vs)
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apply(simp_all)
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done
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lemma retrieve_fuse2:
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assumes "\<Turnstile> v : (erase r)"
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shows "retrieve (fuse bs r) v = bs @ retrieve r v"
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using assms
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apply(induct r arbitrary: v bs)
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apply(auto elim: Prf_elims)[4]
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defer
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using retrieve_encode_STARS
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apply(auto elim!: Prf_elims)[1]
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apply(case_tac vs)
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apply(simp)
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apply(simp)
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(* AALTs case *)
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apply(simp)
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apply(case_tac x2a)
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apply(simp)
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apply(auto elim!: Prf_elims)[1]
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apply(simp)
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apply(case_tac list)
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apply(simp)
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apply(auto)
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apply(auto elim!: Prf_elims)[1]
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done
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lemma retrieve_fuse:
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assumes "\<Turnstile> v : r"
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shows "retrieve (fuse bs (intern r)) v = bs @ retrieve (intern r) v"
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using assms
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by (simp_all add: retrieve_fuse2)
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lemma retrieve_code:
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assumes "\<Turnstile> v : r"
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shows "code v = retrieve (intern r) v"
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using assms
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apply(induct v r )
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apply(simp_all add: retrieve_fuse retrieve_encode_STARS)
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done
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lemma bnullable_Hdbmkeps_Hd:
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assumes "bnullable a"
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shows "bmkeps (AALTs bs (a # rs)) = bs @ (bmkeps a)"
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using assms
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by (metis bmkeps.simps(3) bmkeps.simps(4) list.exhaust)
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lemma r1:
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assumes "\<not> bnullable a" "bnullable (AALTs bs rs)"
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shows "bmkeps (AALTs bs (a # rs)) = bmkeps (AALTs bs rs)"
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using assms
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apply(induct rs)
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apply(auto)
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done
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lemma r2:
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assumes "x \<in> set rs" "bnullable x"
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shows "bnullable (AALTs bs rs)"
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using assms
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apply(induct rs)
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apply(auto)
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done
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lemma r3:
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assumes "\<not> bnullable r"
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" \<exists> x \<in> set rs. bnullable x"
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shows "retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs))) =
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retrieve (AALTs bs (r # rs)) (mkeps (erase (AALTs bs (r # rs))))"
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using assms
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apply(induct rs arbitrary: r bs)
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apply(auto)[1]
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apply(auto)
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using bnullable_correctness apply blast
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apply(auto simp add: bnullable_correctness mkeps_nullable retrieve_fuse2)
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apply(subst retrieve_fuse2[symmetric])
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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)
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apply(simp)
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apply(case_tac "bnullable a")
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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)
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apply(drule_tac x="a" in meta_spec)
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apply(drule_tac x="bs" in meta_spec)
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apply(drule meta_mp)
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apply(simp)
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apply(drule meta_mp)
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apply(auto)
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apply(subst retrieve_fuse2[symmetric])
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apply(case_tac rs)
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apply(simp)
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apply(auto)[1]
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apply (simp add: bnullable_correctness)
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apply (metis append_Nil2 bnullable_correctness erase_fuse fuse.simps(4) list.set_intros(1) mkeps.simps(3) mkeps_nullable nullable.simps(4) r2)
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apply (simp add: bnullable_correctness)
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apply (metis append_Nil2 bnullable_correctness erase.simps(6) erase_fuse fuse.simps(4) list.set_intros(2) mkeps.simps(3) mkeps_nullable r2)
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apply(simp)
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done
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lemma t:
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assumes "\<forall>r \<in> set rs. nullable (erase r) \<longrightarrow> bmkeps r = retrieve r (mkeps (erase r))"
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"nullable (erase (AALTs bs rs))"
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shows " bmkeps (AALTs bs rs) = retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs)))"
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using assms
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apply(induct rs arbitrary: bs)
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apply(simp)
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apply(auto simp add: bnullable_correctness)
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apply(case_tac rs)
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apply(auto simp add: bnullable_correctness)[2]
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apply(subst r1)
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apply(simp)
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apply(rule r2)
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apply(assumption)
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apply(simp)
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apply(drule_tac x="bs" in meta_spec)
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apply(drule meta_mp)
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apply(auto)[1]
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prefer 2
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apply(case_tac "bnullable a")
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apply(subst bnullable_Hdbmkeps_Hd)
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apply blast
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apply(subgoal_tac "nullable (erase a)")
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prefer 2
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using bnullable_correctness apply blast
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apply (metis (no_types, lifting) erase.simps(5) erase.simps(6) list.exhaust mkeps.simps(3) retrieve.simps(3) retrieve.simps(4))
+ − 392
apply(subst r1)
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apply(simp)
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using r2 apply blast
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apply(drule_tac x="bs" in meta_spec)
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apply(drule meta_mp)
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apply(auto)[1]
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apply(simp)
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using r3 apply blast
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apply(auto)
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using r3 by blast
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lemma bmkeps_retrieve:
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assumes "nullable (erase r)"
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shows "bmkeps r = retrieve r (mkeps (erase r))"
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using assms
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apply(induct r)
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apply(simp)
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apply(simp)
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apply(simp)
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apply(simp)
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defer
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apply(simp)
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apply(rule t)
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apply(auto)
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done
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lemma bder_retrieve:
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assumes "\<Turnstile> v : der c (erase r)"
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shows "retrieve (bder c r) v = retrieve r (injval (erase r) c v)"
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using assms
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apply(induct r arbitrary: v rule: erase.induct)
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apply(simp)
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apply(erule Prf_elims)
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apply(simp)
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apply(erule Prf_elims)
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apply(simp)
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apply(case_tac "c = ca")
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apply(simp)
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apply(erule Prf_elims)
+ − 431
apply(simp)
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apply(simp)
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apply(erule Prf_elims)
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apply(simp)
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apply(erule Prf_elims)
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apply(simp)
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apply(simp)
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apply(rename_tac "r\<^sub>1" "r\<^sub>2" rs v)
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apply(erule Prf_elims)
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apply(simp)
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apply(simp)
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apply(case_tac rs)
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apply(simp)
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apply(simp)
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apply (smt Prf_elims(3) injval.simps(2) injval.simps(3) retrieve.simps(4) retrieve.simps(5) same_append_eq)
+ − 446
apply(simp)
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apply(case_tac "nullable (erase r1)")
+ − 448
apply(simp)
+ − 449
apply(erule Prf_elims)
+ − 450
apply(subgoal_tac "bnullable r1")
+ − 451
prefer 2
+ − 452
using bnullable_correctness apply blast
+ − 453
apply(simp)
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apply(erule Prf_elims)
+ − 455
apply(simp)
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apply(subgoal_tac "bnullable r1")
+ − 457
prefer 2
+ − 458
using bnullable_correctness apply blast
+ − 459
apply(simp)
+ − 460
apply(simp add: retrieve_fuse2)
+ − 461
apply(simp add: bmkeps_retrieve)
+ − 462
apply(simp)
+ − 463
apply(erule Prf_elims)
+ − 464
apply(simp)
+ − 465
using bnullable_correctness apply blast
+ − 466
apply(rename_tac bs r v)
+ − 467
apply(simp)
+ − 468
apply(erule Prf_elims)
+ − 469
apply(clarify)
+ − 470
apply(erule Prf_elims)
+ − 471
apply(clarify)
+ − 472
apply(subst injval.simps)
+ − 473
apply(simp del: retrieve.simps)
+ − 474
apply(subst retrieve.simps)
+ − 475
apply(subst retrieve.simps)
+ − 476
apply(simp)
+ − 477
apply(simp add: retrieve_fuse2)
+ − 478
done
+ − 479
+ − 480
+ − 481
+ − 482
lemma MAIN_decode:
+ − 483
assumes "\<Turnstile> v : ders s r"
+ − 484
shows "Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r"
+ − 485
using assms
+ − 486
proof (induct s arbitrary: v rule: rev_induct)
+ − 487
case Nil
+ − 488
have "\<Turnstile> v : ders [] r" by fact
+ − 489
then have "\<Turnstile> v : r" by simp
+ − 490
then have "Some v = decode (retrieve (intern r) v) r"
+ − 491
using decode_code retrieve_code by auto
+ − 492
then show "Some (flex r id [] v) = decode (retrieve (bders (intern r) []) v) r"
+ − 493
by simp
+ − 494
next
+ − 495
case (snoc c s v)
+ − 496
have IH: "\<And>v. \<Turnstile> v : ders s r \<Longrightarrow>
+ − 497
Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r" by fact
+ − 498
have asm: "\<Turnstile> v : ders (s @ [c]) r" by fact
+ − 499
then have asm2: "\<Turnstile> injval (ders s r) c v : ders s r"
+ − 500
by (simp add: Prf_injval ders_append)
+ − 501
have "Some (flex r id (s @ [c]) v) = Some (flex r id s (injval (ders s r) c v))"
+ − 502
by (simp add: flex_append)
+ − 503
also have "... = decode (retrieve (bders (intern r) s) (injval (ders s r) c v)) r"
+ − 504
using asm2 IH by simp
+ − 505
also have "... = decode (retrieve (bder c (bders (intern r) s)) v) r"
+ − 506
using asm by (simp_all add: bder_retrieve ders_append)
+ − 507
finally show "Some (flex r id (s @ [c]) v) =
+ − 508
decode (retrieve (bders (intern r) (s @ [c])) v) r" by (simp add: bders_append)
+ − 509
qed
+ − 510
+ − 511
+ − 512
definition blex where
+ − 513
"blex a s \<equiv> if bnullable (bders a s) then Some (bmkeps (bders a s)) else None"
+ − 514
+ − 515
+ − 516
+ − 517
definition blexer where
+ − 518
"blexer r s \<equiv> if bnullable (bders (intern r) s) then
+ − 519
decode (bmkeps (bders (intern r) s)) r else None"
+ − 520
+ − 521
lemma blexer_correctness:
+ − 522
shows "blexer r s = lexer r s"
+ − 523
proof -
+ − 524
{ define bds where "bds \<equiv> bders (intern r) s"
+ − 525
define ds where "ds \<equiv> ders s r"
+ − 526
assume asm: "nullable ds"
+ − 527
have era: "erase bds = ds"
+ − 528
unfolding ds_def bds_def by simp
+ − 529
have mke: "\<Turnstile> mkeps ds : ds"
+ − 530
using asm by (simp add: mkeps_nullable)
+ − 531
have "decode (bmkeps bds) r = decode (retrieve bds (mkeps ds)) r"
+ − 532
using bmkeps_retrieve
+ − 533
using asm era by (simp add: bmkeps_retrieve)
+ − 534
also have "... = Some (flex r id s (mkeps ds))"
+ − 535
using mke by (simp_all add: MAIN_decode ds_def bds_def)
+ − 536
finally have "decode (bmkeps bds) r = Some (flex r id s (mkeps ds))"
+ − 537
unfolding bds_def ds_def .
+ − 538
}
+ − 539
then show "blexer r s = lexer r s"
+ − 540
unfolding blexer_def lexer_flex
+ − 541
apply(subst bnullable_correctness[symmetric])
+ − 542
apply(simp)
+ − 543
done
+ − 544
qed
+ − 545
+ − 546
+ − 547
fun distinctBy :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b set \<Rightarrow> 'a list"
+ − 548
where
+ − 549
"distinctBy [] f acc = []"
+ − 550
| "distinctBy (x#xs) f acc =
+ − 551
(if (f x) \<in> acc then distinctBy xs f acc
+ − 552
else x # (distinctBy xs f ({f x} \<union> acc)))"
+ − 553
+ − 554
+ − 555
+ − 556
+ − 557
fun flts :: "arexp list \<Rightarrow> arexp list"
+ − 558
where
+ − 559
"flts [] = []"
+ − 560
| "flts (AZERO # rs) = flts rs"
+ − 561
| "flts ((AALTs bs rs1) # rs) = (map (fuse bs) rs1) @ flts rs"
+ − 562
| "flts (r1 # rs) = r1 # flts rs"
+ − 563
+ − 564
+ − 565
+ − 566
+ − 567
fun li :: "bit list \<Rightarrow> arexp list \<Rightarrow> arexp"
+ − 568
where
+ − 569
"li _ [] = AZERO"
+ − 570
| "li bs [a] = fuse bs a"
+ − 571
| "li bs as = AALTs bs as"
+ − 572
+ − 573
+ − 574
+ − 575
+ − 576
fun bsimp_ASEQ :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp \<Rightarrow> arexp"
+ − 577
where
+ − 578
"bsimp_ASEQ _ AZERO _ = AZERO"
+ − 579
| "bsimp_ASEQ _ _ AZERO = AZERO"
+ − 580
| "bsimp_ASEQ bs1 (AONE bs2) r2 = fuse (bs1 @ bs2) r2"
+ − 581
| "bsimp_ASEQ bs1 r1 r2 = ASEQ bs1 r1 r2"
+ − 582
+ − 583
+ − 584
fun bsimp_AALTs :: "bit list \<Rightarrow> arexp list \<Rightarrow> arexp"
+ − 585
where
+ − 586
"bsimp_AALTs _ [] = AZERO"
+ − 587
| "bsimp_AALTs bs1 [r] = fuse bs1 r"
+ − 588
| "bsimp_AALTs bs1 rs = AALTs bs1 rs"
+ − 589
+ − 590
+ − 591
fun bsimp :: "arexp \<Rightarrow> arexp"
+ − 592
where
+ − 593
"bsimp (ASEQ bs1 r1 r2) = bsimp_ASEQ bs1 (bsimp r1) (bsimp r2)"
+ − 594
| "bsimp (AALTs bs1 rs) = bsimp_AALTs bs1 (distinctBy (flts (map bsimp rs)) erase {} ) "
+ − 595
| "bsimp r = r"
+ − 596
+ − 597
+ − 598
+ − 599
+ − 600
fun
+ − 601
bders_simp :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
+ − 602
where
+ − 603
"bders_simp r [] = r"
+ − 604
| "bders_simp r (c # s) = bders_simp (bsimp (bder c r)) s"
+ − 605
+ − 606
definition blexer_simp where
+ − 607
"blexer_simp r s \<equiv> if bnullable (bders_simp (intern r) s) then
+ − 608
decode (bmkeps (bders_simp (intern r) s)) r else None"
+ − 609
+ − 610
export_code bders_simp in Scala module_name Example
+ − 611
+ − 612
lemma bders_simp_append:
+ − 613
shows "bders_simp r (s1 @ s2) = bders_simp (bders_simp r s1) s2"
+ − 614
apply(induct s1 arbitrary: r s2)
+ − 615
apply(simp)
+ − 616
apply(simp)
+ − 617
done
+ − 618
+ − 619
+ − 620
+ − 621
+ − 622
+ − 623
+ − 624
+ − 625
lemma L_bsimp_ASEQ:
+ − 626
"L (SEQ (erase r1) (erase r2)) = L (erase (bsimp_ASEQ bs r1 r2))"
+ − 627
apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
+ − 628
apply(simp_all)
+ − 629
by (metis erase_fuse fuse.simps(4))
+ − 630
+ − 631
lemma L_bsimp_AALTs:
+ − 632
"L (erase (AALTs bs rs)) = L (erase (bsimp_AALTs bs rs))"
+ − 633
apply(induct bs rs rule: bsimp_AALTs.induct)
+ − 634
apply(simp_all add: erase_fuse)
+ − 635
done
+ − 636
+ − 637
lemma L_erase_AALTs:
+ − 638
shows "L (erase (AALTs bs rs)) = \<Union> (L ` erase ` (set rs))"
+ − 639
apply(induct rs)
+ − 640
apply(simp)
+ − 641
apply(simp)
+ − 642
apply(case_tac rs)
+ − 643
apply(simp)
+ − 644
apply(simp)
+ − 645
done
+ − 646
+ − 647
lemma L_erase_flts:
+ − 648
shows "\<Union> (L ` erase ` (set (flts rs))) = \<Union> (L ` erase ` (set rs))"
+ − 649
apply(induct rs rule: flts.induct)
+ − 650
apply(simp_all)
+ − 651
apply(auto)
+ − 652
using L_erase_AALTs erase_fuse apply auto[1]
+ − 653
by (simp add: L_erase_AALTs erase_fuse)
+ − 654
+ − 655
lemma L_erase_dB_acc:
+ − 656
shows "( \<Union>(L ` acc) \<union> ( \<Union> (L ` erase ` (set (distinctBy rs erase acc) ) ) )) = \<Union>(L ` acc) \<union> \<Union> (L ` erase ` (set rs))"
+ − 657
apply(induction rs arbitrary: acc)
+ − 658
apply simp
+ − 659
apply simp
+ − 660
by (smt (z3) SUP_absorb UN_insert sup_assoc sup_commute)
+ − 661
+ − 662
lemma L_erase_dB:
+ − 663
shows " ( \<Union> (L ` erase ` (set (distinctBy rs erase {}) ) ) ) = \<Union> (L ` erase ` (set rs))"
+ − 664
by (metis L_erase_dB_acc Un_commute Union_image_empty)
+ − 665
+ − 666
lemma L_bsimp_erase:
+ − 667
shows "L (erase r) = L (erase (bsimp r))"
+ − 668
apply(induct r)
+ − 669
apply(simp)
+ − 670
apply(simp)
+ − 671
apply(simp)
+ − 672
apply(auto simp add: Sequ_def)[1]
+ − 673
apply(subst L_bsimp_ASEQ[symmetric])
+ − 674
apply(auto simp add: Sequ_def)[1]
+ − 675
apply(subst (asm) L_bsimp_ASEQ[symmetric])
+ − 676
apply(auto simp add: Sequ_def)[1]
+ − 677
apply(simp)
+ − 678
apply(subst L_bsimp_AALTs[symmetric])
+ − 679
defer
+ − 680
apply(simp)
+ − 681
apply(subst (2)L_erase_AALTs)
+ − 682
apply(subst L_erase_dB)
+ − 683
apply(subst L_erase_flts)
+ − 684
apply(auto)
+ − 685
apply (simp add: L_erase_AALTs)
+ − 686
using L_erase_AALTs by blast
+ − 687
+ − 688
lemma bsimp_ASEQ0:
+ − 689
shows "bsimp_ASEQ bs r1 AZERO = AZERO"
+ − 690
apply(induct r1)
+ − 691
apply(auto)
+ − 692
done
+ − 693
+ − 694
+ − 695
+ − 696
lemma bsimp_ASEQ1:
+ − 697
assumes "r1 \<noteq> AZERO" "r2 \<noteq> AZERO" "\<forall>bs. r1 \<noteq> AONE bs"
+ − 698
shows "bsimp_ASEQ bs r1 r2 = ASEQ bs r1 r2"
+ − 699
using assms
+ − 700
apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
+ − 701
apply(auto)
+ − 702
done
+ − 703
+ − 704
lemma bsimp_ASEQ2:
+ − 705
shows "bsimp_ASEQ bs (AONE bs1) r2 = fuse (bs @ bs1) r2"
+ − 706
apply(induct r2)
+ − 707
apply(auto)
+ − 708
done
+ − 709
+ − 710
+ − 711
lemma L_bders_simp:
+ − 712
shows "L (erase (bders_simp r s)) = L (erase (bders r s))"
+ − 713
apply(induct s arbitrary: r rule: rev_induct)
+ − 714
apply(simp)
+ − 715
apply(simp)
+ − 716
apply(simp add: ders_append)
+ − 717
apply(simp add: bders_simp_append)
+ − 718
apply(simp add: L_bsimp_erase[symmetric])
+ − 719
by (simp add: der_correctness)
+ − 720
+ − 721
+ − 722
lemma b2:
+ − 723
assumes "bnullable r"
+ − 724
shows "bmkeps (fuse bs r) = bs @ bmkeps r"
+ − 725
by (simp add: assms bmkeps_retrieve bnullable_correctness erase_fuse mkeps_nullable retrieve_fuse2)
+ − 726
+ − 727
+ − 728
lemma b4:
+ − 729
shows "bnullable (bders_simp r s) = bnullable (bders r s)"
+ − 730
by (metis L_bders_simp bnullable_correctness lexer.simps(1) lexer_correct_None option.distinct(1))
+ − 731
+ − 732
+ − 733
lemma qq1:
+ − 734
assumes "\<exists>r \<in> set rs. bnullable r"
+ − 735
shows "bmkeps (AALTs bs (rs @ rs1)) = bmkeps (AALTs bs rs)"
+ − 736
using assms
+ − 737
apply(induct rs arbitrary: rs1 bs)
+ − 738
apply(simp)
+ − 739
apply(simp)
+ − 740
by (metis Nil_is_append_conv bmkeps.simps(4) neq_Nil_conv bnullable_Hdbmkeps_Hd split_list_last)
+ − 741
+ − 742
lemma qq2:
+ − 743
assumes "\<forall>r \<in> set rs. \<not> bnullable r" "\<exists>r \<in> set rs1. bnullable r"
+ − 744
shows "bmkeps (AALTs bs (rs @ rs1)) = bmkeps (AALTs bs rs1)"
+ − 745
using assms
+ − 746
apply(induct rs arbitrary: rs1 bs)
+ − 747
apply(simp)
+ − 748
apply(simp)
+ − 749
by (metis append_assoc in_set_conv_decomp r1 r2)
+ − 750
+ − 751
lemma qq3:
+ − 752
shows "bnullable (AALTs bs rs) = (\<exists>r \<in> set rs. bnullable r)"
+ − 753
apply(induct rs arbitrary: bs)
+ − 754
apply(simp)
+ − 755
apply(simp)
+ − 756
done
+ − 757
+ − 758
+ − 759
+ − 760
+ − 761
+ − 762
fun nonnested :: "arexp \<Rightarrow> bool"
+ − 763
where
+ − 764
"nonnested (AALTs bs2 []) = True"
+ − 765
| "nonnested (AALTs bs2 ((AALTs bs1 rs1) # rs2)) = False"
+ − 766
| "nonnested (AALTs bs2 (r # rs2)) = nonnested (AALTs bs2 rs2)"
+ − 767
| "nonnested r = True"
+ − 768
+ − 769
+ − 770
lemma k0:
+ − 771
shows "flts (r # rs1) = flts [r] @ flts rs1"
+ − 772
apply(induct r arbitrary: rs1)
+ − 773
apply(auto)
+ − 774
done
+ − 775
+ − 776
lemma k00:
+ − 777
shows "flts (rs1 @ rs2) = flts rs1 @ flts rs2"
+ − 778
apply(induct rs1 arbitrary: rs2)
+ − 779
apply(auto)
+ − 780
by (metis append.assoc k0)
+ − 781
+ − 782
lemma k0a:
+ − 783
shows "flts [AALTs bs rs] = map (fuse bs) rs"
+ − 784
apply(simp)
+ − 785
done
+ − 786
+ − 787
+ − 788
+ − 789
+ − 790
+ − 791
+ − 792
+ − 793
+ − 794
lemma bsimp_AALTs_qq:
+ − 795
assumes "1 < length rs"
+ − 796
shows "bsimp_AALTs bs rs = AALTs bs rs"
+ − 797
using assms
+ − 798
apply(case_tac rs)
+ − 799
apply(simp)
+ − 800
apply(case_tac list)
+ − 801
apply(simp_all)
+ − 802
done
+ − 803
+ − 804
+ − 805
+ − 806
lemma bbbbs1:
+ − 807
shows "nonalt r \<or> (\<exists>bs rs. r = AALTs bs rs)"
+ − 808
using nonalt.elims(3) by auto
+ − 809
+ − 810
+ − 811
+ − 812
+ − 813
+ − 814
lemma flts_append:
+ − 815
"flts (xs1 @ xs2) = flts xs1 @ flts xs2"
+ − 816
apply(induct xs1 arbitrary: xs2 rule: rev_induct)
+ − 817
apply(auto)
+ − 818
apply(case_tac xs)
+ − 819
apply(auto)
+ − 820
apply(case_tac x)
+ − 821
apply(auto)
+ − 822
apply(case_tac x)
+ − 823
apply(auto)
+ − 824
done
+ − 825
+ − 826
fun nonazero :: "arexp \<Rightarrow> bool"
+ − 827
where
+ − 828
"nonazero AZERO = False"
+ − 829
| "nonazero r = True"
+ − 830
+ − 831
+ − 832
lemma flts_single1:
+ − 833
assumes "nonalt r" "nonazero r"
+ − 834
shows "flts [r] = [r]"
+ − 835
using assms
+ − 836
apply(induct r)
+ − 837
apply(auto)
+ − 838
done
+ − 839
+ − 840
+ − 841
+ − 842
lemma q3a:
+ − 843
assumes "\<exists>r \<in> set rs. bnullable r"
+ − 844
shows "bmkeps (AALTs bs (map (fuse bs1) rs)) = bmkeps (AALTs (bs@bs1) rs)"
+ − 845
using assms
+ − 846
apply(induct rs arbitrary: bs bs1)
+ − 847
apply(simp)
+ − 848
apply(simp)
+ − 849
apply(auto)
+ − 850
apply (metis append_assoc b2 bnullable_correctness erase_fuse bnullable_Hdbmkeps_Hd)
+ − 851
apply(case_tac "bnullable a")
+ − 852
apply (metis append.assoc b2 bnullable_correctness erase_fuse bnullable_Hdbmkeps_Hd)
+ − 853
apply(case_tac rs)
+ − 854
apply(simp)
+ − 855
apply(simp)
+ − 856
apply(auto)[1]
+ − 857
apply (metis bnullable_correctness erase_fuse)+
+ − 858
done
+ − 859
+ − 860
lemma qq4:
+ − 861
assumes "\<exists>x\<in>set list. bnullable x"
+ − 862
shows "\<exists>x\<in>set (flts list). bnullable x"
+ − 863
using assms
+ − 864
apply(induct list rule: flts.induct)
+ − 865
apply(auto)
+ − 866
by (metis UnCI bnullable_correctness erase_fuse imageI)
+ − 867
+ − 868
+ − 869
lemma qs3:
+ − 870
assumes "\<exists>r \<in> set rs. bnullable r"
+ − 871
shows "bmkeps (AALTs bs rs) = bmkeps (AALTs bs (flts rs))"
+ − 872
using assms
+ − 873
apply(induct rs arbitrary: bs taking: size rule: measure_induct)
+ − 874
apply(case_tac x)
+ − 875
apply(simp)
+ − 876
apply(simp)
+ − 877
apply(case_tac a)
+ − 878
apply(simp)
+ − 879
apply (simp add: r1)
+ − 880
apply(simp)
+ − 881
apply (simp add: bnullable_Hdbmkeps_Hd)
+ − 882
apply(simp)
+ − 883
apply(case_tac "flts list")
+ − 884
apply(simp)
+ − 885
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)
+ − 886
apply(simp)
+ − 887
apply (simp add: r1)
+ − 888
prefer 3
+ − 889
apply(simp)
+ − 890
apply (simp add: bnullable_Hdbmkeps_Hd)
+ − 891
prefer 2
+ − 892
apply(simp)
+ − 893
apply(case_tac "\<exists>x\<in>set x52. bnullable x")
+ − 894
apply(case_tac "list")
+ − 895
apply(simp)
+ − 896
apply (metis b2 fuse.simps(4) q3a r2)
+ − 897
apply(erule disjE)
+ − 898
apply(subst qq1)
+ − 899
apply(auto)[1]
+ − 900
apply (metis bnullable_correctness erase_fuse)
+ − 901
apply(simp)
+ − 902
apply (metis b2 fuse.simps(4) q3a r2)
+ − 903
apply(simp)
+ − 904
apply(auto)[1]
+ − 905
apply(subst qq1)
+ − 906
apply (metis bnullable_correctness erase_fuse image_eqI set_map)
+ − 907
apply (metis b2 fuse.simps(4) q3a r2)
+ − 908
apply(subst qq1)
+ − 909
apply (metis bnullable_correctness erase_fuse image_eqI set_map)
+ − 910
apply (metis b2 fuse.simps(4) q3a r2)
+ − 911
apply(simp)
+ − 912
apply(subst qq2)
+ − 913
apply (metis bnullable_correctness erase_fuse imageE set_map)
+ − 914
prefer 2
+ − 915
apply(case_tac "list")
+ − 916
apply(simp)
+ − 917
apply(simp)
+ − 918
apply (simp add: qq4)
+ − 919
apply(simp)
+ − 920
apply(auto)
+ − 921
apply(case_tac list)
+ − 922
apply(simp)
+ − 923
apply(simp)
+ − 924
apply (simp add: bnullable_Hdbmkeps_Hd)
+ − 925
apply(case_tac "bnullable (ASEQ x41 x42 x43)")
+ − 926
apply(case_tac list)
+ − 927
apply(simp)
+ − 928
apply(simp)
+ − 929
apply (simp add: bnullable_Hdbmkeps_Hd)
+ − 930
apply(simp)
+ − 931
using qq4 r1 r2 by auto
+ − 932
+ − 933
+ − 934
+ − 935
+ − 936
lemma bder_fuse:
+ − 937
shows "bder c (fuse bs a) = fuse bs (bder c a)"
+ − 938
apply(induct a arbitrary: bs c)
+ − 939
apply(simp_all)
+ − 940
done
+ − 941
+ − 942
+ − 943
fun flts2 :: "char \<Rightarrow> arexp list \<Rightarrow> arexp list"
+ − 944
where
+ − 945
"flts2 _ [] = []"
+ − 946
| "flts2 c (AZERO # rs) = flts2 c rs"
+ − 947
| "flts2 c (AONE _ # rs) = flts2 c rs"
+ − 948
| "flts2 c (ACHAR bs d # rs) = (if c = d then (ACHAR bs d # flts2 c rs) else flts2 c rs)"
+ − 949
| "flts2 c ((AALTs bs rs1) # rs) = (map (fuse bs) rs1) @ flts2 c rs"
+ − 950
| "flts2 c (ASEQ bs r1 r2 # rs) = (if (bnullable(r1) \<and> r2 = AZERO) then
+ − 951
flts2 c rs
+ − 952
else ASEQ bs r1 r2 # flts2 c rs)"
+ − 953
| "flts2 c (r1 # rs) = r1 # flts2 c rs"
+ − 954
+ − 955
+ − 956
+ − 957
+ − 958
+ − 959
+ − 960
+ − 961
+ − 962
+ − 963
+ − 964
+ − 965
+ − 966
+ − 967
lemma WQ1:
+ − 968
assumes "s \<in> L (der c r)"
+ − 969
shows "s \<in> der c r \<rightarrow> mkeps (ders s (der c r))"
+ − 970
using assms
+ − 971
oops
+ − 972
+ − 973
+ − 974
+ − 975
lemma bder_bsimp_AALTs:
+ − 976
shows "bder c (bsimp_AALTs bs rs) = bsimp_AALTs bs (map (bder c) rs)"
+ − 977
apply(induct bs rs rule: bsimp_AALTs.induct)
+ − 978
apply(simp)
+ − 979
apply(simp)
+ − 980
apply (simp add: bder_fuse)
+ − 981
apply(simp)
+ − 982
done
+ − 983
+ − 984
+ − 985
+ − 986
lemma
+ − 987
assumes "asize (bsimp a) = asize a" "a = AALTs bs [AALTs bs2 [], AZERO, AONE bs3]"
+ − 988
shows "bsimp a = a"
+ − 989
using assms
+ − 990
apply(simp)
+ − 991
oops
+ − 992
+ − 993
+ − 994
+ − 995
+ − 996
+ − 997
+ − 998
+ − 999
+ − 1000
inductive rrewrite:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto> _" [99, 99] 99)
+ − 1001
where
+ − 1002
"ASEQ bs AZERO r2 \<leadsto> AZERO"
+ − 1003
| "ASEQ bs r1 AZERO \<leadsto> AZERO"
+ − 1004
| "ASEQ bs (AONE bs1) r \<leadsto> fuse (bs@bs1) r"
+ − 1005
| "r1 \<leadsto> r2 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto> ASEQ bs r2 r3"
+ − 1006
| "r3 \<leadsto> r4 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto> ASEQ bs r1 r4"
+ − 1007
| "r \<leadsto> r' \<Longrightarrow> (AALTs bs (rs1 @ [r] @ rs2)) \<leadsto> (AALTs bs (rs1 @ [r'] @ rs2))"
+ − 1008
(*context rule for eliminating 0, alts--corresponds to the recursive call flts r::rs = r::(flts rs)*)
+ − 1009
| "AALTs bs (rsa@AZERO # rsb) \<leadsto> AALTs bs (rsa@rsb)"
+ − 1010
| "AALTs bs (rsa@(AALTs bs1 rs1)# rsb) \<leadsto> AALTs bs (rsa@(map (fuse bs1) rs1)@rsb)"
+ − 1011
(*the below rule for extracting common prefixes between a list of rexp's bitcodes*)
+ − 1012
| "AALTs bs (map (fuse bs1) rs) \<leadsto> AALTs (bs@bs1) rs"
+ − 1013
(*opposite direction also allowed, which means bits are free to be moved around
+ − 1014
as long as they are on the right path*)
+ − 1015
| "AALTs (bs@bs1) rs \<leadsto> AALTs bs (map (fuse bs1) rs)"
+ − 1016
| "AALTs bs [] \<leadsto> AZERO"
+ − 1017
| "AALTs bs [r] \<leadsto> fuse bs r"
+ − 1018
| "erase a1 = erase a2 \<Longrightarrow> AALTs bs (rsa@[a1]@rsb@[a2]@rsc) \<leadsto> AALTs bs (rsa@[a1]@rsb@rsc)"
+ − 1019
+ − 1020
+ − 1021
inductive rrewrites:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto>* _" [100, 100] 100)
+ − 1022
where
+ − 1023
rs1[intro, simp]:"r \<leadsto>* r"
+ − 1024
| rs2[intro]: "\<lbrakk>r1 \<leadsto>* r2; r2 \<leadsto> r3\<rbrakk> \<Longrightarrow> r1 \<leadsto>* r3"
+ − 1025
+ − 1026
inductive srewrites:: "arexp list \<Rightarrow> arexp list \<Rightarrow> bool" (" _ s\<leadsto>* _" [100, 100] 100)
+ − 1027
where
+ − 1028
ss1: "[] s\<leadsto>* []"
+ − 1029
|ss2: "\<lbrakk>r \<leadsto>* r'; rs s\<leadsto>* rs'\<rbrakk> \<Longrightarrow> (r#rs) s\<leadsto>* (r'#rs')"
+ − 1030
(*rs1 = [r1, r2, ..., rn] rs2 = [r1', r2', ..., rn']
+ − 1031
[r1, r2, ..., rn] \<leadsto>* [r1', r2, ..., rn] \<leadsto>* [...r2',...] \<leadsto>* [r1', r2',... rn']
+ − 1032
*)
+ − 1033
+ − 1034
+ − 1035
+ − 1036
lemma r_in_rstar : "r1 \<leadsto> r2 \<Longrightarrow> r1 \<leadsto>* r2"
+ − 1037
using rrewrites.intros(1) rrewrites.intros(2) by blast
+ − 1038
+ − 1039
lemma real_trans:
+ − 1040
assumes a1: "r1 \<leadsto>* r2" and a2: "r2 \<leadsto>* r3"
+ − 1041
shows "r1 \<leadsto>* r3"
+ − 1042
using a2 a1
+ − 1043
apply(induct r2 r3 arbitrary: r1 rule: rrewrites.induct)
+ − 1044
apply(auto)
+ − 1045
done
+ − 1046
+ − 1047
+ − 1048
lemma many_steps_later: "\<lbrakk>r1 \<leadsto> r2; r2 \<leadsto>* r3 \<rbrakk> \<Longrightarrow> r1 \<leadsto>* r3"
+ − 1049
by (meson r_in_rstar real_trans)
+ − 1050
+ − 1051
+ − 1052
lemma contextrewrites1: "r \<leadsto>* r' \<Longrightarrow> (AALTs bs (r#rs)) \<leadsto>* (AALTs bs (r'#rs))"
+ − 1053
apply(induct r r' rule: rrewrites.induct)
+ − 1054
apply simp
+ − 1055
by (metis append_Cons append_Nil rrewrite.intros(6) rs2)
+ − 1056
+ − 1057
+ − 1058
lemma contextrewrites2: "r \<leadsto>* r' \<Longrightarrow> (AALTs bs (rs1@[r]@rs)) \<leadsto>* (AALTs bs (rs1@[r']@rs))"
+ − 1059
apply(induct r r' rule: rrewrites.induct)
+ − 1060
apply simp
+ − 1061
using rrewrite.intros(6) by blast
+ − 1062
+ − 1063
+ − 1064
+ − 1065
lemma srewrites_alt: "rs1 s\<leadsto>* rs2 \<Longrightarrow> (AALTs bs (rs@rs1)) \<leadsto>* (AALTs bs (rs@rs2))"
+ − 1066
+ − 1067
apply(induct rs1 rs2 arbitrary: bs rs rule: srewrites.induct)
+ − 1068
apply(rule rs1)
+ − 1069
apply(drule_tac x = "bs" in meta_spec)
+ − 1070
apply(drule_tac x = "rsa@[r']" in meta_spec)
+ − 1071
apply simp
+ − 1072
apply(rule real_trans)
+ − 1073
prefer 2
+ − 1074
apply(assumption)
+ − 1075
apply(drule contextrewrites2)
+ − 1076
apply auto
+ − 1077
done
+ − 1078
+ − 1079
+ − 1080
corollary srewrites_alt1: "rs1 s\<leadsto>* rs2 \<Longrightarrow> AALTs bs rs1 \<leadsto>* AALTs bs rs2"
+ − 1081
by (metis append.left_neutral srewrites_alt)
+ − 1082
+ − 1083
+ − 1084
lemma star_seq: "r1 \<leadsto>* r2 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto>* ASEQ bs r2 r3"
+ − 1085
apply(induct r1 r2 arbitrary: r3 rule: rrewrites.induct)
+ − 1086
apply(rule rs1)
+ − 1087
apply(erule rrewrites.cases)
+ − 1088
apply(simp)
+ − 1089
apply(rule r_in_rstar)
+ − 1090
apply(rule rrewrite.intros(4))
+ − 1091
apply simp
+ − 1092
apply(rule rs2)
+ − 1093
apply(assumption)
+ − 1094
apply(rule rrewrite.intros(4))
+ − 1095
by assumption
+ − 1096
+ − 1097
lemma star_seq2: "r3 \<leadsto>* r4 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto>* ASEQ bs r1 r4"
+ − 1098
apply(induct r3 r4 arbitrary: r1 rule: rrewrites.induct)
+ − 1099
apply auto
+ − 1100
using rrewrite.intros(5) by blast
+ − 1101
+ − 1102
+ − 1103
lemma continuous_rewrite: "\<lbrakk>r1 \<leadsto>* AZERO\<rbrakk> \<Longrightarrow> ASEQ bs1 r1 r2 \<leadsto>* AZERO"
+ − 1104
apply(induction ra\<equiv>"r1" rb\<equiv>"AZERO" arbitrary: bs1 r1 r2 rule: rrewrites.induct)
+ − 1105
apply (simp add: r_in_rstar rrewrite.intros(1))
+ − 1106
+ − 1107
by (meson rrewrite.intros(1) rrewrites.intros(2) star_seq)
+ − 1108
+ − 1109
+ − 1110
+ − 1111
lemma bsimp_aalts_simpcases: "AONE bs \<leadsto>* (bsimp (AONE bs))" "AZERO \<leadsto>* bsimp AZERO" "ACHAR bs c \<leadsto>* (bsimp (ACHAR bs c))"
+ − 1112
apply (simp add: rrewrites.intros(1))
+ − 1113
apply (simp add: rrewrites.intros(1))
+ − 1114
by (simp add: rrewrites.intros(1))
+ − 1115
+ − 1116
lemma trivialbsimpsrewrites: "\<lbrakk>\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* f x \<rbrakk> \<Longrightarrow> rs s\<leadsto>* (map f rs)"
+ − 1117
+ − 1118
apply(induction rs)
+ − 1119
apply simp
+ − 1120
apply(rule ss1)
+ − 1121
by (metis insert_iff list.simps(15) list.simps(9) srewrites.simps)
+ − 1122
+ − 1123
+ − 1124
lemma bsimp_AALTsrewrites: "AALTs bs1 rs \<leadsto>* bsimp_AALTs bs1 rs"
+ − 1125
apply(induction rs)
+ − 1126
apply simp
+ − 1127
apply(rule r_in_rstar)
+ − 1128
apply(simp add: rrewrite.intros(11))
+ − 1129
apply(case_tac "rs = Nil")
+ − 1130
apply(simp)
+ − 1131
using rrewrite.intros(12) apply auto[1]
+ − 1132
apply(subgoal_tac "length (a#rs) > 1")
+ − 1133
apply(simp add: bsimp_AALTs_qq)
+ − 1134
apply(simp)
+ − 1135
done
+ − 1136
+ − 1137
inductive frewrites:: "arexp list \<Rightarrow> arexp list \<Rightarrow> bool" (" _ f\<leadsto>* _" [100, 100] 100)
+ − 1138
where
+ − 1139
fs1: "[] f\<leadsto>* []"
+ − 1140
|fs2: "\<lbrakk>rs f\<leadsto>* rs'\<rbrakk> \<Longrightarrow> (AZERO#rs) f\<leadsto>* rs'"
+ − 1141
|fs3: "\<lbrakk>rs f\<leadsto>* rs'\<rbrakk> \<Longrightarrow> ((AALTs bs rs1) # rs) f\<leadsto>* ((map (fuse bs) rs1) @ rs')"
+ − 1142
|fs4: "\<lbrakk>rs f\<leadsto>* rs';nonalt r; nonazero r\<rbrakk> \<Longrightarrow> (r#rs) f\<leadsto>* (r#rs')"
+ − 1143
+ − 1144
+ − 1145
+ − 1146
+ − 1147
+ − 1148
lemma flts_prepend: "\<lbrakk>nonalt a; nonazero a\<rbrakk> \<Longrightarrow> flts (a#rs) = a # (flts rs)"
+ − 1149
by (metis append_Cons append_Nil flts_single1 k00)
+ − 1150
+ − 1151
lemma fltsfrewrites: "rs f\<leadsto>* (flts rs)"
+ − 1152
apply(induction rs)
+ − 1153
apply simp
+ − 1154
apply(rule fs1)
+ − 1155
+ − 1156
apply(case_tac "a = AZERO")
+ − 1157
+ − 1158
+ − 1159
using fs2 apply auto[1]
+ − 1160
apply(case_tac "\<exists>bs rs. a = AALTs bs rs")
+ − 1161
apply(erule exE)+
+ − 1162
+ − 1163
apply (simp add: fs3)
+ − 1164
apply(subst flts_prepend)
+ − 1165
apply(rule nonalt.elims(2))
+ − 1166
prefer 2
+ − 1167
thm nonalt.elims
+ − 1168
+ − 1169
apply blast
+ − 1170
+ − 1171
using bbbbs1 apply blast
+ − 1172
apply(simp add: nonalt.simps)+
+ − 1173
+ − 1174
apply (meson nonazero.elims(3))
+ − 1175
+ − 1176
by (meson fs4 nonalt.elims(3) nonazero.elims(3))
+ − 1177
+ − 1178
+ − 1179
lemma rrewrite0away: "AALTs bs ( AZERO # rsb) \<leadsto> AALTs bs rsb"
+ − 1180
by (metis append_Nil rrewrite.intros(7))
+ − 1181
+ − 1182
+ − 1183
lemma frewritesaalts:"rs f\<leadsto>* rs' \<Longrightarrow> (AALTs bs (rs1@rs)) \<leadsto>* (AALTs bs (rs1@rs'))"
+ − 1184
apply(induct rs rs' arbitrary: bs rs1 rule:frewrites.induct)
+ − 1185
apply(rule rs1)
+ − 1186
apply(drule_tac x = "bs" in meta_spec)
+ − 1187
apply(drule_tac x = "rs1 @ [AZERO]" in meta_spec)
+ − 1188
apply(rule real_trans)
+ − 1189
apply simp
+ − 1190
using r_in_rstar rrewrite.intros(7) apply presburger
+ − 1191
apply(drule_tac x = "bsa" in meta_spec)
+ − 1192
apply(drule_tac x = "rs1a @ [AALTs bs rs1]" in meta_spec)
+ − 1193
apply(rule real_trans)
+ − 1194
apply simp
+ − 1195
using r_in_rstar rrewrite.intros(8) apply presburger
+ − 1196
apply(drule_tac x = "bs" in meta_spec)
+ − 1197
apply(drule_tac x = "rs1@[r]" in meta_spec)
+ − 1198
apply(rule real_trans)
+ − 1199
apply simp
+ − 1200
apply auto
+ − 1201
done
+ − 1202
+ − 1203
lemma fltsrewrites: " AALTs bs1 rs \<leadsto>* AALTs bs1 (flts rs)"
+ − 1204
apply(induction rs)
+ − 1205
apply simp
+ − 1206
apply(case_tac "a = AZERO")
+ − 1207
apply (metis append_Nil flts.simps(2) many_steps_later rrewrite.intros(7))
+ − 1208
+ − 1209
+ − 1210
+ − 1211
apply(case_tac "\<exists>bs2 rs2. a = AALTs bs2 rs2")
+ − 1212
apply(erule exE)+
+ − 1213
apply(simp add: flts.simps)
+ − 1214
prefer 2
+ − 1215
+ − 1216
apply(subst flts_prepend)
+ − 1217
+ − 1218
apply (meson nonalt.elims(3))
+ − 1219
+ − 1220
apply (meson nonazero.elims(3))
+ − 1221
apply(subgoal_tac "(a#rs) f\<leadsto>* (a#flts rs)")
+ − 1222
apply (metis append_Nil frewritesaalts)
+ − 1223
apply (meson fltsfrewrites fs4 nonalt.elims(3) nonazero.elims(3))
+ − 1224
by (metis append_Cons append_Nil fltsfrewrites frewritesaalts k00 k0a)
+ − 1225
+ − 1226
lemma alts_simpalts: "\<And>bs1 rs. (\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* bsimp x) \<Longrightarrow>
+ − 1227
AALTs bs1 rs \<leadsto>* AALTs bs1 (map bsimp rs)"
+ − 1228
apply(subgoal_tac " rs s\<leadsto>* (map bsimp rs)")
+ − 1229
prefer 2
+ − 1230
using trivialbsimpsrewrites apply auto[1]
+ − 1231
using srewrites_alt1 by auto
+ − 1232
+ − 1233
+ − 1234
lemma threelistsappend: "rsa@a#rsb = (rsa@[a])@rsb"
+ − 1235
apply auto
+ − 1236
done
+ − 1237
+ − 1238
fun distinctByAcc :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b set \<Rightarrow> 'b set"
+ − 1239
where
+ − 1240
"distinctByAcc [] f acc = acc"
+ − 1241
| "distinctByAcc (x#xs) f acc =
+ − 1242
(if (f x) \<in> acc then distinctByAcc xs f acc
+ − 1243
else (distinctByAcc xs f ({f x} \<union> acc)))"
+ − 1244
+ − 1245
lemma dB_single_step: "distinctBy (a#rs) f {} = a # distinctBy rs f {f a}"
+ − 1246
apply simp
+ − 1247
done
+ − 1248
+ − 1249
lemma somewhereInside: "r \<in> set rs \<Longrightarrow> \<exists>rs1 rs2. rs = rs1@[r]@rs2"
+ − 1250
using split_list by fastforce
+ − 1251
+ − 1252
lemma somewhereMapInside: "f r \<in> f ` set rs \<Longrightarrow> \<exists>rs1 rs2 a. rs = rs1@[a]@rs2 \<and> f a = f r"
+ − 1253
apply auto
+ − 1254
by (metis split_list)
+ − 1255
+ − 1256
lemma alts_dBrewrites_withFront: " AALTs bs (rsa @ rs) \<leadsto>* AALTs bs (rsa @ distinctBy rs erase (erase ` set rsa))"
+ − 1257
apply(induction rs arbitrary: rsa)
+ − 1258
apply simp
+ − 1259
apply(drule_tac x = "rsa@[a]" in meta_spec)
+ − 1260
apply(subst threelistsappend)
+ − 1261
apply(rule real_trans)
+ − 1262
apply simp
+ − 1263
apply(case_tac "a \<in> set rsa")
+ − 1264
apply simp
+ − 1265
apply(drule somewhereInside)
+ − 1266
apply(erule exE)+
+ − 1267
apply simp
+ − 1268
apply(subgoal_tac " AALTs bs
+ − 1269
(rs1 @
+ − 1270
a #
+ − 1271
rs2 @
+ − 1272
a #
+ − 1273
distinctBy rs erase
+ − 1274
(insert (erase a)
+ − 1275
(erase `
+ − 1276
(set rs1 \<union> set rs2)))) \<leadsto> AALTs bs (rs1@ a # rs2 @ distinctBy rs erase
+ − 1277
(insert (erase a)
+ − 1278
(erase `
+ − 1279
(set rs1 \<union> set rs2)))) ")
+ − 1280
prefer 2
+ − 1281
using rrewrite.intros(13) apply force
+ − 1282
using r_in_rstar apply force
+ − 1283
apply(subgoal_tac "erase ` set (rsa @ [a]) = insert (erase a) (erase ` set rsa)")
+ − 1284
prefer 2
+ − 1285
+ − 1286
apply auto[1]
+ − 1287
apply(case_tac "erase a \<in> erase `set rsa")
+ − 1288
+ − 1289
apply simp
+ − 1290
apply(subgoal_tac "AALTs bs (rsa @ a # distinctBy rs erase (insert (erase a) (erase ` set rsa))) \<leadsto>
+ − 1291
AALTs bs (rsa @ distinctBy rs erase (insert (erase a) (erase ` set rsa)))")
+ − 1292
apply force
+ − 1293
apply (smt (verit, ccfv_threshold) append_Cons append_assoc append_self_conv2 r_in_rstar rrewrite.intros(13) same_append_eq somewhereMapInside)
+ − 1294
by force
+ − 1295
+ − 1296
+ − 1297
+ − 1298
lemma alts_dBrewrites: "AALTs bs rs \<leadsto>* AALTs bs (distinctBy rs erase {})"
+ − 1299
apply(induction rs)
+ − 1300
apply simp
+ − 1301
apply simp
+ − 1302
using alts_dBrewrites_withFront
+ − 1303
by (metis append_Nil dB_single_step empty_set image_empty)
+ − 1304
+ − 1305
+ − 1306
+ − 1307
+ − 1308
+ − 1309
+ − 1310
lemma bsimp_rewrite: " (rrewrites r ( bsimp r))"
+ − 1311
apply(induction r rule: bsimp.induct)
+ − 1312
apply simp
+ − 1313
apply(case_tac "bsimp r1 = AZERO")
+ − 1314
apply simp
+ − 1315
using continuous_rewrite apply blast
+ − 1316
apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
+ − 1317
apply(erule exE)
+ − 1318
apply simp
+ − 1319
apply(subst bsimp_ASEQ2)
+ − 1320
apply (meson real_trans rrewrite.intros(3) rrewrites.intros(2) star_seq star_seq2)
+ − 1321
apply (smt (verit, best) bsimp_ASEQ0 bsimp_ASEQ1 real_trans rrewrite.intros(2) rs2 star_seq star_seq2)
+ − 1322
defer
+ − 1323
using bsimp_aalts_simpcases(2) apply blast
+ − 1324
apply simp
+ − 1325
apply simp
+ − 1326
apply simp
+ − 1327
+ − 1328
apply auto
+ − 1329
+ − 1330
+ − 1331
apply(subgoal_tac "AALTs bs1 rs \<leadsto>* AALTs bs1 (map bsimp rs)")
+ − 1332
apply(subgoal_tac "AALTs bs1 (map bsimp rs) \<leadsto>* AALTs bs1 (flts (map bsimp rs))")
+ − 1333
apply(subgoal_tac "AALTs bs1 (flts (map bsimp rs)) \<leadsto>* AALTs bs1 (distinctBy (flts (map bsimp rs)) erase {})")
+ − 1334
apply(subgoal_tac "AALTs bs1 (distinctBy (flts (map bsimp rs)) erase {}) \<leadsto>* bsimp_AALTs bs1 (distinctBy (flts (map bsimp rs)) erase {} )")
+ − 1335
+ − 1336
+ − 1337
apply (meson real_trans)
+ − 1338
+ − 1339
apply (meson bsimp_AALTsrewrites)
+ − 1340
+ − 1341
apply (meson alts_dBrewrites)
+ − 1342
+ − 1343
using fltsrewrites apply auto[1]
+ − 1344
+ − 1345
using alts_simpalts by force
+ − 1346
+ − 1347
+ − 1348
lemma rewritenullable: "\<lbrakk>r1 \<leadsto> r2; bnullable r1 \<rbrakk> \<Longrightarrow> bnullable r2"
+ − 1349
apply(induction r1 r2 rule: rrewrite.induct)
+ − 1350
apply(simp)+
+ − 1351
apply (metis bnullable_correctness erase_fuse)
+ − 1352
apply simp
+ − 1353
apply simp
+ − 1354
apply auto[1]
+ − 1355
apply auto[1]
+ − 1356
apply auto[4]
+ − 1357
apply (metis UnCI bnullable_correctness erase_fuse imageI)
+ − 1358
apply (metis bnullable_correctness erase_fuse)
+ − 1359
apply (metis bnullable_correctness erase_fuse)
+ − 1360
+ − 1361
apply (metis bnullable_correctness erase.simps(5) erase_fuse)
+ − 1362
+ − 1363
+ − 1364
by (smt (z3) Un_iff bnullable_correctness insert_iff list.set(2) qq3 set_append)
+ − 1365
+ − 1366
lemma rewrite_non_nullable: "\<lbrakk>r1 \<leadsto> r2; \<not>bnullable r1 \<rbrakk> \<Longrightarrow> \<not>bnullable r2"
+ − 1367
apply(induction r1 r2 rule: rrewrite.induct)
+ − 1368
apply auto
+ − 1369
apply (metis bnullable_correctness erase_fuse)+
+ − 1370
done
+ − 1371
+ − 1372
+ − 1373
lemma rewritesnullable: "\<lbrakk> r1 \<leadsto>* r2; bnullable r1 \<rbrakk> \<Longrightarrow> bnullable r2"
+ − 1374
apply(induction r1 r2 rule: rrewrites.induct)
+ − 1375
apply simp
+ − 1376
apply(rule rewritenullable)
+ − 1377
apply simp
+ − 1378
apply simp
+ − 1379
done
+ − 1380
+ − 1381
lemma nonbnullable_lists_concat: " \<lbrakk> \<not> (\<exists>r0\<in>set rs1. bnullable r0); \<not> bnullable r; \<not> (\<exists>r0\<in>set rs2. bnullable r0)\<rbrakk> \<Longrightarrow>
+ − 1382
\<not>(\<exists>r0 \<in> (set (rs1@[r]@rs2)). bnullable r0 ) "
+ − 1383
apply simp
+ − 1384
apply blast
+ − 1385
done
+ − 1386
+ − 1387
+ − 1388
+ − 1389
lemma nomember_bnullable: "\<lbrakk> \<not> (\<exists>r0\<in>set rs1. bnullable r0); \<not> bnullable r; \<not> (\<exists>r0\<in>set rs2. bnullable r0)\<rbrakk>
+ − 1390
\<Longrightarrow> \<not>bnullable (AALTs bs (rs1 @ [r] @ rs2))"
+ − 1391
using nonbnullable_lists_concat qq3 by presburger
+ − 1392
+ − 1393
lemma bnullable_segment: " bnullable (AALTs bs (rs1@[r]@rs2)) \<Longrightarrow> bnullable (AALTs bs rs1) \<or> bnullable (AALTs bs rs2) \<or> bnullable r"
+ − 1394
apply(case_tac "\<exists>r0\<in>set rs1. bnullable r0")
+ − 1395
+ − 1396
using qq3 apply blast
+ − 1397
apply(case_tac "bnullable r")
+ − 1398
+ − 1399
apply blast
+ − 1400
apply(case_tac "\<exists>r0\<in>set rs2. bnullable r0")
+ − 1401
+ − 1402
using bnullable.simps(4) apply presburger
+ − 1403
apply(subgoal_tac "False")
+ − 1404
+ − 1405
apply blast
+ − 1406
+ − 1407
using nomember_bnullable by blast
+ − 1408
+ − 1409
+ − 1410
+ − 1411
lemma bnullablewhichbmkeps: "\<lbrakk>bnullable (AALTs bs (rs1@[r]@rs2)); \<not> bnullable (AALTs bs rs1); bnullable r \<rbrakk>
+ − 1412
\<Longrightarrow> bmkeps (AALTs bs (rs1@[r]@rs2)) = bs @ (bmkeps r)"
+ − 1413
using qq2 bnullable_Hdbmkeps_Hd by force
+ − 1414
+ − 1415
lemma rrewrite_nbnullable: "\<lbrakk> r1 \<leadsto> r2 ; \<not> bnullable r1 \<rbrakk> \<Longrightarrow> \<not>bnullable r2"
+ − 1416
apply(induction rule: rrewrite.induct)
+ − 1417
apply auto[1]
+ − 1418
apply auto[1]
+ − 1419
apply auto[1]
+ − 1420
apply (metis bnullable_correctness erase_fuse)
+ − 1421
apply auto[1]
+ − 1422
apply auto[1]
+ − 1423
apply auto[1]
+ − 1424
apply auto[1]
+ − 1425
apply auto[1]
+ − 1426
apply (metis bnullable_correctness erase_fuse)
+ − 1427
apply auto[1]
+ − 1428
apply (metis bnullable_correctness erase_fuse)
+ − 1429
apply auto[1]
+ − 1430
apply (metis bnullable_correctness erase_fuse)
+ − 1431
apply auto[1]
+ − 1432
apply auto[1]
+ − 1433
+ − 1434
apply (metis bnullable_correctness erase_fuse)
+ − 1435
+ − 1436
by (meson rewrite_non_nullable rrewrite.intros(13))
+ − 1437
+ − 1438
+ − 1439
+ − 1440
+ − 1441
lemma spillbmkepslistr: "bnullable (AALTs bs1 rs1)
+ − 1442
\<Longrightarrow> bmkeps (AALTs bs (AALTs bs1 rs1 # rsb)) = bmkeps (AALTs bs ( map (fuse bs1) rs1 @ rsb))"
+ − 1443
apply(subst bnullable_Hdbmkeps_Hd)
+ − 1444
+ − 1445
apply simp
+ − 1446
by (metis bmkeps.simps(3) k0a list.set_intros(1) qq1 qq4 qs3)
+ − 1447
+ − 1448
lemma third_segment_bnullable: "\<lbrakk>bnullable (AALTs bs (rs1@rs2@rs3)); \<not>bnullable (AALTs bs rs1); \<not>bnullable (AALTs bs rs2)\<rbrakk> \<Longrightarrow>
+ − 1449
bnullable (AALTs bs rs3)"
+ − 1450
+ − 1451
by (metis append.left_neutral append_Cons bnullable.simps(1) bnullable_segment rrewrite.intros(7) rrewrite_nbnullable)
+ − 1452
+ − 1453
+ − 1454
lemma third_segment_bmkeps: "\<lbrakk>bnullable (AALTs bs (rs1@rs2@rs3)); \<not>bnullable (AALTs bs rs1); \<not>bnullable (AALTs bs rs2)\<rbrakk> \<Longrightarrow>
+ − 1455
bmkeps (AALTs bs (rs1@rs2@rs3) ) = bmkeps (AALTs bs rs3)"
+ − 1456
apply(subgoal_tac "bnullable (AALTs bs rs3)")
+ − 1457
apply(subgoal_tac "\<forall>r \<in> set (rs1@rs2). \<not>bnullable r")
+ − 1458
apply(subgoal_tac "bmkeps (AALTs bs (rs1@rs2@rs3)) = bmkeps (AALTs bs ((rs1@rs2)@rs3) )")
+ − 1459
apply (metis qq2 qq3)
+ − 1460
+ − 1461
apply (metis append.assoc)
+ − 1462
+ − 1463
apply (metis append.assoc in_set_conv_decomp r2 third_segment_bnullable)
+ − 1464
+ − 1465
using third_segment_bnullable by blast
+ − 1466
+ − 1467
+ − 1468
lemma rewrite_bmkepsalt: " \<lbrakk>bnullable (AALTs bs (rsa @ AALTs bs1 rs1 # rsb)); bnullable (AALTs bs (rsa @ map (fuse bs1) rs1 @ rsb))\<rbrakk>
+ − 1469
\<Longrightarrow> bmkeps (AALTs bs (rsa @ AALTs bs1 rs1 # rsb)) = bmkeps (AALTs bs (rsa @ map (fuse bs1) rs1 @ rsb))"
+ − 1470
apply(case_tac "bnullable (AALTs bs rsa)")
+ − 1471
+ − 1472
using qq1 apply force
+ − 1473
apply(case_tac "bnullable (AALTs bs1 rs1)")
+ − 1474
apply(subst qq2)
+ − 1475
+ − 1476
+ − 1477
using r2 apply blast
+ − 1478
+ − 1479
apply (metis list.set_intros(1))
+ − 1480
apply (smt (verit, ccfv_threshold) append_eq_append_conv2 list.set_intros(1) qq2 qq3 rewritenullable rrewrite.intros(8) self_append_conv2 spillbmkepslistr)
+ − 1481
+ − 1482
+ − 1483
thm qq1
+ − 1484
apply(subgoal_tac "bmkeps (AALTs bs (rsa @ AALTs bs1 rs1 # rsb)) = bmkeps (AALTs bs rsb) ")
+ − 1485
prefer 2
+ − 1486
+ − 1487
apply (metis append_Cons append_Nil bnullable.simps(1) bnullable_segment rewritenullable rrewrite.intros(11) third_segment_bmkeps)
+ − 1488
+ − 1489
by (metis bnullable.simps(4) rewrite_non_nullable rrewrite.intros(10) third_segment_bmkeps)
+ − 1490
+ − 1491
+ − 1492
+ − 1493
lemma rewrite_bmkeps: "\<lbrakk> r1 \<leadsto> r2; (bnullable r1)\<rbrakk> \<Longrightarrow> bmkeps r1 = bmkeps r2"
+ − 1494
+ − 1495
apply(frule rewritenullable)
+ − 1496
apply simp
+ − 1497
apply(induction r1 r2 rule: rrewrite.induct)
+ − 1498
apply simp
+ − 1499
using bnullable.simps(1) bnullable.simps(5) apply blast
+ − 1500
apply (simp add: b2)
+ − 1501
apply simp
+ − 1502
apply simp
+ − 1503
apply(frule bnullable_segment)
+ − 1504
apply(case_tac "bnullable (AALTs bs rs1)")
+ − 1505
using qq1 apply force
+ − 1506
apply(case_tac "bnullable r")
+ − 1507
using bnullablewhichbmkeps rewritenullable apply presburger
+ − 1508
apply(subgoal_tac "bnullable (AALTs bs rs2)")
+ − 1509
apply(subgoal_tac "\<not> bnullable r'")
+ − 1510
apply (simp add: qq2 r1)
+ − 1511
+ − 1512
using rrewrite_nbnullable apply blast
+ − 1513
+ − 1514
apply blast
+ − 1515
apply (simp add: flts_append qs3)
+ − 1516
+ − 1517
apply (meson rewrite_bmkepsalt)
+ − 1518
+ − 1519
using bnullable.simps(4) q3a apply blast
+ − 1520
+ − 1521
apply (simp add: q3a)
+ − 1522
+ − 1523
using bnullable.simps(1) apply blast
+ − 1524
+ − 1525
apply (simp add: b2)
+ − 1526
+ − 1527
by (smt (z3) Un_iff bnullable_correctness erase.simps(5) qq1 qq2 qq3 set_append)
+ − 1528
+ − 1529
+ − 1530
+ − 1531
lemma rewrites_bmkeps: "\<lbrakk> (r1 \<leadsto>* r2); (bnullable r1)\<rbrakk> \<Longrightarrow> bmkeps r1 = bmkeps r2"
+ − 1532
apply(induction r1 r2 rule: rrewrites.induct)
+ − 1533
apply simp
+ − 1534
apply(subgoal_tac "bnullable r2")
+ − 1535
prefer 2
+ − 1536
apply(metis rewritesnullable)
+ − 1537
apply(subgoal_tac "bmkeps r1 = bmkeps r2")
+ − 1538
prefer 2
+ − 1539
apply fastforce
+ − 1540
using rewrite_bmkeps by presburger
+ − 1541
+ − 1542
+ − 1543
thm rrewrite.intros(12)
+ − 1544
lemma alts_rewrite_front: "r \<leadsto> r' \<Longrightarrow> AALTs bs (r # rs) \<leadsto> AALTs bs (r' # rs)"
+ − 1545
by (metis append_Cons append_Nil rrewrite.intros(6))
+ − 1546
+ − 1547
lemma alt_rewrite_front: "r \<leadsto> r' \<Longrightarrow> AALT bs r r2 \<leadsto> AALT bs r' r2"
+ − 1548
using alts_rewrite_front by blast
+ − 1549
+ − 1550
lemma to_zero_in_alt: " AALT bs (ASEQ [] AZERO r) r2 \<leadsto> AALT bs AZERO r2"
+ − 1551
by (simp add: alts_rewrite_front rrewrite.intros(1))
+ − 1552
+ − 1553
lemma alt_remove0_front: " AALT bs AZERO r \<leadsto> AALTs bs [r]"
+ − 1554
by (simp add: rrewrite0away)
+ − 1555
+ − 1556
lemma alt_rewrites_back: "r2 \<leadsto>* r2' \<Longrightarrow>AALT bs r1 r2 \<leadsto>* AALT bs r1 r2'"
+ − 1557
apply(induction r2 r2' arbitrary: bs rule: rrewrites.induct)
+ − 1558
apply simp
+ − 1559
by (meson rs1 rs2 srewrites_alt1 ss1 ss2)
+ − 1560
+ − 1561
lemma rewrite_fuse: " r2 \<leadsto> r3 \<Longrightarrow> fuse bs r2 \<leadsto>* fuse bs r3"
+ − 1562
apply(induction r2 r3 arbitrary: bs rule: rrewrite.induct)
+ − 1563
apply auto
+ − 1564
+ − 1565
apply (simp add: continuous_rewrite)
+ − 1566
+ − 1567
apply (simp add: r_in_rstar rrewrite.intros(2))
+ − 1568
+ − 1569
apply (metis fuse_append r_in_rstar rrewrite.intros(3))
+ − 1570
+ − 1571
using r_in_rstar star_seq apply blast
+ − 1572
+ − 1573
using r_in_rstar star_seq2 apply blast
+ − 1574
+ − 1575
using contextrewrites2 r_in_rstar apply auto[1]
+ − 1576
+ − 1577
apply (simp add: r_in_rstar rrewrite.intros(7))
+ − 1578
+ − 1579
using rrewrite.intros(8) apply auto[1]
+ − 1580
+ − 1581
apply (metis append_assoc r_in_rstar rrewrite.intros(9))
+ − 1582
+ − 1583
apply (metis append_assoc r_in_rstar rrewrite.intros(10))
+ − 1584
+ − 1585
apply (simp add: r_in_rstar rrewrite.intros(11))
+ − 1586
+ − 1587
apply (metis fuse_append r_in_rstar rrewrite.intros(12))
+ − 1588
+ − 1589
using rrewrite.intros(13) by auto
+ − 1590
+ − 1591
+ − 1592
+ − 1593
lemma rewrites_fuse: "r2 \<leadsto>* r2' \<Longrightarrow> (fuse bs1 r2) \<leadsto>* (fuse bs1 r2')"
+ − 1594
apply(induction r2 r2' arbitrary: bs1 rule: rrewrites.induct)
+ − 1595
apply simp
+ − 1596
by (meson real_trans rewrite_fuse)
+ − 1597
+ − 1598
lemma bder_fuse_list: " map (bder c \<circ> fuse bs1) rs1 = map (fuse bs1 \<circ> bder c) rs1"
+ − 1599
apply(induction rs1)
+ − 1600
apply simp
+ − 1601
by (simp add: bder_fuse)
+ − 1602
+ − 1603
+ − 1604
+ − 1605
lemma rewrite_der_altmiddle: "bder c (AALTs bs (rsa @ AALTs bs1 rs1 # rsb)) \<leadsto>* bder c (AALTs bs (rsa @ map (fuse bs1) rs1 @ rsb))"
+ − 1606
apply simp
+ − 1607
apply(simp add: bder_fuse_list)
+ − 1608
apply(rule many_steps_later)
+ − 1609
apply(subst rrewrite.intros(8))
+ − 1610
apply simp
+ − 1611
+ − 1612
by fastforce
+ − 1613
+ − 1614
lemma lock_step_der_removal:
+ − 1615
shows " erase a1 = erase a2 \<Longrightarrow>
+ − 1616
bder c (AALTs bs (rsa @ [a1] @ rsb @ [a2] @ rsc)) \<leadsto>*
+ − 1617
bder c (AALTs bs (rsa @ [a1] @ rsb @ rsc))"
+ − 1618
apply(simp)
+ − 1619
+ − 1620
using rrewrite.intros(13) by auto
+ − 1621
+ − 1622
lemma rewrite_after_der: "r1 \<leadsto> r2 \<Longrightarrow> (bder c r1) \<leadsto>* (bder c r2)"
+ − 1623
apply(induction r1 r2 arbitrary: c rule: rrewrite.induct)
+ − 1624
+ − 1625
apply (simp add: r_in_rstar rrewrite.intros(1))
+ − 1626
apply simp
+ − 1627
+ − 1628
apply (meson contextrewrites1 r_in_rstar rrewrite.intros(11) rrewrite.intros(2) rrewrite0away rs2)
+ − 1629
apply(simp)
+ − 1630
apply(rule many_steps_later)
+ − 1631
apply(rule to_zero_in_alt)
+ − 1632
apply(rule many_steps_later)
+ − 1633
apply(rule alt_remove0_front)
+ − 1634
apply(rule many_steps_later)
+ − 1635
apply(rule rrewrite.intros(12))
+ − 1636
using bder_fuse fuse_append rs1 apply presburger
+ − 1637
apply(case_tac "bnullable r1")
+ − 1638
prefer 2
+ − 1639
apply(subgoal_tac "\<not>bnullable r2")
+ − 1640
prefer 2
+ − 1641
using rewrite_non_nullable apply presburger
+ − 1642
apply simp+
+ − 1643
+ − 1644
using star_seq apply auto[1]
+ − 1645
apply(subgoal_tac "bnullable r2")
+ − 1646
apply simp+
+ − 1647
apply(subgoal_tac "bmkeps r1 = bmkeps r2")
+ − 1648
prefer 2
+ − 1649
using rewrite_bmkeps apply auto[1]
+ − 1650
using contextrewrites1 star_seq apply auto[1]
+ − 1651
using rewritenullable apply auto[1]
+ − 1652
apply(case_tac "bnullable r1")
+ − 1653
apply simp
+ − 1654
apply(subgoal_tac "ASEQ [] (bder c r1) r3 \<leadsto> ASEQ [] (bder c r1) r4")
+ − 1655
prefer 2
+ − 1656
using rrewrite.intros(5) apply blast
+ − 1657
apply(rule many_steps_later)
+ − 1658
apply(rule alt_rewrite_front)
+ − 1659
apply assumption
+ − 1660
apply (meson alt_rewrites_back rewrites_fuse)
+ − 1661
+ − 1662
apply (simp add: r_in_rstar rrewrite.intros(5))
+ − 1663
+ − 1664
using contextrewrites2 apply force
+ − 1665
+ − 1666
using rrewrite.intros(7) apply force
+ − 1667
+ − 1668
using rewrite_der_altmiddle apply auto[1]
+ − 1669
+ − 1670
apply (metis bder.simps(4) bder_fuse_list map_map r_in_rstar rrewrite.intros(9))
+ − 1671
+ − 1672
apply (metis List.map.compositionality bder.simps(4) bder_fuse_list r_in_rstar rrewrite.intros(10))
+ − 1673
+ − 1674
apply (simp add: r_in_rstar rrewrite.intros(11))
+ − 1675
+ − 1676
apply (metis bder.simps(4) bder_bsimp_AALTs bsimp_AALTs.simps(2) bsimp_AALTsrewrites)
+ − 1677
+ − 1678
+ − 1679
using lock_step_der_removal by auto
+ − 1680
+ − 1681
+ − 1682
+ − 1683
lemma rewrites_after_der: " r1 \<leadsto>* r2 \<Longrightarrow> (bder c r1) \<leadsto>* (bder c r2)"
+ − 1684
apply(induction r1 r2 rule: rrewrites.induct)
+ − 1685
apply(rule rs1)
+ − 1686
by (meson real_trans rewrite_after_der)
+ − 1687
+ − 1688
+ − 1689
+ − 1690
+ − 1691
lemma central: " (bders r s) \<leadsto>* (bders_simp r s)"
+ − 1692
apply(induct s arbitrary: r rule: rev_induct)
+ − 1693
+ − 1694
apply simp
+ − 1695
apply(subst bders_append)
+ − 1696
apply(subst bders_simp_append)
+ − 1697
by (metis bders.simps(1) bders.simps(2) bders_simp.simps(1) bders_simp.simps(2) bsimp_rewrite real_trans rewrites_after_der)
+ − 1698
+ − 1699
+ − 1700
+ − 1701
thm arexp.induct
+ − 1702
+ − 1703
lemma quasi_main: "bnullable (bders r s) \<Longrightarrow> bmkeps (bders r s) = bmkeps (bders_simp r s)"
+ − 1704
using central rewrites_bmkeps by blast
+ − 1705
+ − 1706
theorem main_main: "blexer r s = blexer_simp r s"
+ − 1707
by (simp add: b4 blexer_def blexer_simp_def quasi_main)
+ − 1708
+ − 1709
+ − 1710
theorem blexersimp_correctness: "blexer_simp r s= lexer r s"
+ − 1711
using blexer_correctness main_main by auto
+ − 1712
+ − 1713
+ − 1714
unused_thms
+ − 1715
+ − 1716
+ − 1717
end