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theory SizeBound4CT
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imports
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"Lexer"
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"PDerivs"
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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 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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abbreviation
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bnullables :: "arexp list \<Rightarrow> bool"
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where
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"bnullables rs \<equiv> (\<exists>r \<in> set rs. bnullable r)"
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fun
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bmkeps :: "arexp \<Rightarrow> bit list" and
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bmkepss :: "arexp list \<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 rs) = bs @ (bmkepss rs)"
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| "bmkeps(ASTAR bs r) = bs @ [S]"
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| "bmkepss [] = []"
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| "bmkepss (r # rs) = (if bnullable(r) then (bmkeps r) else (bmkepss rs))"
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lemma bmkepss1:
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assumes "\<not> bnullables rs1"
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shows "bmkepss (rs1 @ rs2) = bmkepss rs2"
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using assms
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by (induct rs1) (auto)
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lemma bmkepss2:
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assumes "bnullables rs1"
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shows "bmkepss (rs1 @ rs2) = bmkepss rs1"
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using assms
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by (induct rs1) (auto)
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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 c (s1 @ s2) = bders (bders c s1) s2"
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apply(induct s1 arbitrary: c 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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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 bnullable_fuse:
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shows "bnullable (fuse bs r) = bnullable r"
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apply(induct r arbitrary: bs)
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apply(auto)
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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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apply(case_tac x2a)
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apply(simp)
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using Prf_elims(1) apply blast
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apply(case_tac x2a)
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apply(simp)
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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(simp)
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apply (smt (verit, best) Prf_elims(3) append_assoc retrieve.simps(4) retrieve.simps(5))
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apply(simp)
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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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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 retrieve_AALTs_bnullable1:
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assumes "bnullable r"
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shows "retrieve (AALTs bs (r # rs)) (mkeps (erase (AALTs bs (r # rs))))
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= bs @ retrieve r (mkeps (erase r))"
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using assms
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apply(case_tac rs)
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apply(auto simp add: bnullable_correctness)
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done
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lemma retrieve_AALTs_bnullable2:
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assumes "\<not>bnullable r" "bnullables rs"
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shows "retrieve (AALTs bs (r # rs)) (mkeps (erase (AALTs bs (r # rs))))
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= retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs)))"
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using assms
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apply(induct rs arbitrary: r bs)
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|
|
326 |
apply(auto)
|
|
|
327 |
using bnullable_correctness apply blast
|
|
|
328 |
apply(case_tac rs)
|
|
|
329 |
apply(auto)
|
|
|
330 |
using bnullable_correctness apply blast
|
|
|
331 |
apply(case_tac rs)
|
|
|
332 |
apply(auto)
|
|
|
333 |
done
|
|
|
334 |
|
|
|
335 |
lemma bmkeps_retrieve_AALTs:
|
|
|
336 |
assumes "\<forall>r \<in> set rs. bnullable r \<longrightarrow> bmkeps r = retrieve r (mkeps (erase r))"
|
|
|
337 |
"bnullables rs"
|
|
|
338 |
shows "bs @ bmkepss rs = retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs)))"
|
|
|
339 |
using assms
|
|
|
340 |
apply(induct rs arbitrary: bs)
|
|
|
341 |
apply(auto)
|
|
|
342 |
using retrieve_AALTs_bnullable1 apply presburger
|
|
|
343 |
apply (metis retrieve_AALTs_bnullable2)
|
|
|
344 |
apply (simp add: retrieve_AALTs_bnullable1)
|
|
|
345 |
by (metis retrieve_AALTs_bnullable2)
|
|
|
346 |
|
|
|
347 |
|
|
|
348 |
lemma bmkeps_retrieve:
|
|
|
349 |
assumes "bnullable r"
|
|
|
350 |
shows "bmkeps r = retrieve r (mkeps (erase r))"
|
|
|
351 |
using assms
|
|
|
352 |
apply(induct r)
|
|
|
353 |
apply(auto)
|
|
|
354 |
using bmkeps_retrieve_AALTs by auto
|
|
|
355 |
|
|
|
356 |
lemma bder_retrieve:
|
|
|
357 |
assumes "\<Turnstile> v : der c (erase r)"
|
|
|
358 |
shows "retrieve (bder c r) v = retrieve r (injval (erase r) c v)"
|
|
|
359 |
using assms
|
|
|
360 |
apply(induct r arbitrary: v rule: erase.induct)
|
|
|
361 |
using Prf_elims(1) apply auto[1]
|
|
|
362 |
using Prf_elims(1) apply auto[1]
|
|
|
363 |
apply(auto)[1]
|
|
|
364 |
apply (metis Prf_elims(4) injval.simps(1) retrieve.simps(1) retrieve.simps(2))
|
|
|
365 |
using Prf_elims(1) apply blast
|
|
|
366 |
(* AALTs case *)
|
|
|
367 |
apply(simp)
|
|
|
368 |
apply(erule Prf_elims)
|
|
|
369 |
apply(simp)
|
|
|
370 |
apply(simp)
|
|
|
371 |
apply(rename_tac "r\<^sub>1" "r\<^sub>2" rs v)
|
|
|
372 |
apply(erule Prf_elims)
|
|
|
373 |
apply(simp)
|
|
|
374 |
apply(simp)
|
|
|
375 |
apply(case_tac rs)
|
|
|
376 |
apply(simp)
|
|
|
377 |
apply(simp)
|
|
|
378 |
using Prf_elims(3) apply fastforce
|
|
|
379 |
(* ASEQ case *)
|
|
|
380 |
apply(simp)
|
|
|
381 |
apply(case_tac "nullable (erase r1)")
|
|
|
382 |
apply(simp)
|
|
|
383 |
apply(erule Prf_elims)
|
|
|
384 |
using Prf_elims(2) bnullable_correctness apply force
|
|
|
385 |
apply (simp add: bmkeps_retrieve bnullable_correctness retrieve_fuse2)
|
|
|
386 |
apply (simp add: bmkeps_retrieve bnullable_correctness retrieve_fuse2)
|
|
|
387 |
using Prf_elims(2) apply force
|
|
|
388 |
(* ASTAR case *)
|
|
|
389 |
apply(rename_tac bs r v)
|
|
|
390 |
apply(simp)
|
|
|
391 |
apply(erule Prf_elims)
|
|
|
392 |
apply(clarify)
|
|
|
393 |
apply(erule Prf_elims)
|
|
|
394 |
apply(clarify)
|
|
|
395 |
by (simp add: retrieve_fuse2)
|
|
|
396 |
|
|
|
397 |
|
|
|
398 |
lemma MAIN_decode:
|
|
|
399 |
assumes "\<Turnstile> v : ders s r"
|
|
|
400 |
shows "Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r"
|
|
|
401 |
using assms
|
|
|
402 |
proof (induct s arbitrary: v rule: rev_induct)
|
|
|
403 |
case Nil
|
|
|
404 |
have "\<Turnstile> v : ders [] r" by fact
|
|
|
405 |
then have "\<Turnstile> v : r" by simp
|
|
|
406 |
then have "Some v = decode (retrieve (intern r) v) r"
|
|
|
407 |
using decode_code retrieve_code by auto
|
|
|
408 |
then show "Some (flex r id [] v) = decode (retrieve (bders (intern r) []) v) r"
|
|
|
409 |
by simp
|
|
|
410 |
next
|
|
|
411 |
case (snoc c s v)
|
|
|
412 |
have IH: "\<And>v. \<Turnstile> v : ders s r \<Longrightarrow>
|
|
|
413 |
Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r" by fact
|
|
|
414 |
have asm: "\<Turnstile> v : ders (s @ [c]) r" by fact
|
|
|
415 |
then have asm2: "\<Turnstile> injval (ders s r) c v : ders s r"
|
|
|
416 |
by (simp add: Prf_injval ders_append)
|
|
|
417 |
have "Some (flex r id (s @ [c]) v) = Some (flex r id s (injval (ders s r) c v))"
|
|
|
418 |
by (simp add: flex_append)
|
|
|
419 |
also have "... = decode (retrieve (bders (intern r) s) (injval (ders s r) c v)) r"
|
|
|
420 |
using asm2 IH by simp
|
|
|
421 |
also have "... = decode (retrieve (bder c (bders (intern r) s)) v) r"
|
|
|
422 |
using asm by (simp_all add: bder_retrieve ders_append)
|
|
|
423 |
finally show "Some (flex r id (s @ [c]) v) =
|
|
|
424 |
decode (retrieve (bders (intern r) (s @ [c])) v) r" by (simp add: bders_append)
|
|
|
425 |
qed
|
|
|
426 |
|
|
|
427 |
definition blexer where
|
|
|
428 |
"blexer r s \<equiv> if bnullable (bders (intern r) s) then
|
|
|
429 |
decode (bmkeps (bders (intern r) s)) r else None"
|
|
|
430 |
|
|
|
431 |
lemma blexer_correctness:
|
|
|
432 |
shows "blexer r s = lexer r s"
|
|
|
433 |
proof -
|
|
|
434 |
{ define bds where "bds \<equiv> bders (intern r) s"
|
|
|
435 |
define ds where "ds \<equiv> ders s r"
|
|
|
436 |
assume asm: "nullable ds"
|
|
|
437 |
have era: "erase bds = ds"
|
|
|
438 |
unfolding ds_def bds_def by simp
|
|
|
439 |
have mke: "\<Turnstile> mkeps ds : ds"
|
|
|
440 |
using asm by (simp add: mkeps_nullable)
|
|
|
441 |
have "decode (bmkeps bds) r = decode (retrieve bds (mkeps ds)) r"
|
|
|
442 |
using bmkeps_retrieve
|
|
|
443 |
using asm era
|
|
|
444 |
using bnullable_correctness by force
|
|
|
445 |
also have "... = Some (flex r id s (mkeps ds))"
|
|
|
446 |
using mke by (simp_all add: MAIN_decode ds_def bds_def)
|
|
|
447 |
finally have "decode (bmkeps bds) r = Some (flex r id s (mkeps ds))"
|
|
|
448 |
unfolding bds_def ds_def .
|
|
|
449 |
}
|
|
|
450 |
then show "blexer r s = lexer r s"
|
|
|
451 |
unfolding blexer_def lexer_flex
|
|
|
452 |
by (auto simp add: bnullable_correctness[symmetric])
|
|
|
453 |
qed
|
|
|
454 |
|
|
|
455 |
|
|
|
456 |
fun distinctBy :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b set \<Rightarrow> 'a list"
|
|
|
457 |
where
|
|
|
458 |
"distinctBy [] f acc = []"
|
|
|
459 |
| "distinctBy (x#xs) f acc =
|
|
|
460 |
(if (f x) \<in> acc then distinctBy xs f acc
|
|
|
461 |
else x # (distinctBy xs f ({f x} \<union> acc)))"
|
|
|
462 |
|
|
|
463 |
|
|
|
464 |
|
|
|
465 |
fun flts :: "arexp list \<Rightarrow> arexp list"
|
|
|
466 |
where
|
|
|
467 |
"flts [] = []"
|
|
|
468 |
| "flts (AZERO # rs) = flts rs"
|
|
|
469 |
| "flts ((AALTs bs rs1) # rs) = (map (fuse bs) rs1) @ flts rs"
|
|
|
470 |
| "flts (r1 # rs) = r1 # flts rs"
|
|
|
471 |
|
|
|
472 |
|
|
|
473 |
|
|
|
474 |
fun bsimp_ASEQ :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp \<Rightarrow> arexp"
|
|
|
475 |
where
|
|
|
476 |
"bsimp_ASEQ _ AZERO _ = AZERO"
|
|
|
477 |
| "bsimp_ASEQ _ _ AZERO = AZERO"
|
|
|
478 |
| "bsimp_ASEQ bs1 (AONE bs2) r2 = fuse (bs1 @ bs2) r2"
|
|
|
479 |
| "bsimp_ASEQ bs1 r1 r2 = ASEQ bs1 r1 r2"
|
|
|
480 |
|
|
|
481 |
lemma bsimp_ASEQ0[simp]:
|
|
|
482 |
shows "bsimp_ASEQ bs r1 AZERO = AZERO"
|
|
|
483 |
by (case_tac r1)(simp_all)
|
|
|
484 |
|
|
|
485 |
lemma bsimp_ASEQ1:
|
|
|
486 |
assumes "r1 \<noteq> AZERO" "r2 \<noteq> AZERO" "\<nexists>bs. r1 = AONE bs"
|
|
|
487 |
shows "bsimp_ASEQ bs r1 r2 = ASEQ bs r1 r2"
|
|
|
488 |
using assms
|
|
|
489 |
apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
|
|
|
490 |
apply(auto)
|
|
|
491 |
done
|
|
|
492 |
|
|
|
493 |
lemma bsimp_ASEQ2[simp]:
|
|
|
494 |
shows "bsimp_ASEQ bs1 (AONE bs2) r2 = fuse (bs1 @ bs2) r2"
|
|
|
495 |
by (case_tac r2) (simp_all)
|
|
|
496 |
|
|
|
497 |
|
|
|
498 |
fun bsimp_AALTs :: "bit list \<Rightarrow> arexp list \<Rightarrow> arexp"
|
|
|
499 |
where
|
|
|
500 |
"bsimp_AALTs _ [] = AZERO"
|
|
|
501 |
| "bsimp_AALTs bs1 [r] = fuse bs1 r"
|
|
|
502 |
| "bsimp_AALTs bs1 rs = AALTs bs1 rs"
|
|
|
503 |
|
|
|
504 |
|
|
|
505 |
fun bsimp :: "arexp \<Rightarrow> arexp"
|
|
|
506 |
where
|
|
|
507 |
"bsimp (ASEQ bs1 r1 r2) = bsimp_ASEQ bs1 (bsimp r1) (bsimp r2)"
|
|
|
508 |
| "bsimp (AALTs bs1 rs) = bsimp_AALTs bs1 (distinctBy (flts (map bsimp rs)) erase {}) "
|
|
|
509 |
| "bsimp r = r"
|
|
|
510 |
|
|
|
511 |
|
|
|
512 |
fun
|
|
|
513 |
bders_simp :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
|
|
|
514 |
where
|
|
|
515 |
"bders_simp r [] = r"
|
|
|
516 |
| "bders_simp r (c # s) = bders_simp (bsimp (bder c r)) s"
|
|
|
517 |
|
|
|
518 |
definition blexer_simp where
|
|
|
519 |
"blexer_simp r s \<equiv> if bnullable (bders_simp (intern r) s) then
|
|
|
520 |
decode (bmkeps (bders_simp (intern r) s)) r else None"
|
|
|
521 |
|
|
|
522 |
|
|
|
523 |
|
|
|
524 |
lemma bders_simp_append:
|
|
|
525 |
shows "bders_simp r (s1 @ s2) = bders_simp (bders_simp r s1) s2"
|
|
|
526 |
apply(induct s1 arbitrary: r s2)
|
|
|
527 |
apply(simp_all)
|
|
|
528 |
done
|
|
|
529 |
|
|
|
530 |
|
|
|
531 |
lemma bmkeps_fuse:
|
|
|
532 |
assumes "bnullable r"
|
|
|
533 |
shows "bmkeps (fuse bs r) = bs @ bmkeps r"
|
|
|
534 |
using assms
|
|
|
535 |
by (metis bmkeps_retrieve bnullable_correctness erase_fuse mkeps_nullable retrieve_fuse2)
|
|
|
536 |
|
|
|
537 |
lemma bmkepss_fuse:
|
|
|
538 |
assumes "bnullables rs"
|
|
|
539 |
shows "bmkepss (map (fuse bs) rs) = bs @ bmkepss rs"
|
|
|
540 |
using assms
|
|
|
541 |
apply(induct rs arbitrary: bs)
|
|
|
542 |
apply(auto simp add: bmkeps_fuse bnullable_fuse)
|
|
|
543 |
done
|
|
|
544 |
|
|
|
545 |
lemma bder_fuse:
|
|
|
546 |
shows "bder c (fuse bs a) = fuse bs (bder c a)"
|
|
|
547 |
apply(induct a arbitrary: bs c)
|
|
|
548 |
apply(simp_all)
|
|
|
549 |
done
|
|
|
550 |
|
|
|
551 |
|
|
|
552 |
|
|
|
553 |
|
|
|
554 |
inductive
|
|
|
555 |
rrewrite:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto> _" [99, 99] 99)
|
|
|
556 |
and
|
|
|
557 |
srewrite:: "arexp list \<Rightarrow> arexp list \<Rightarrow> bool" (" _ s\<leadsto> _" [100, 100] 100)
|
|
|
558 |
where
|
|
|
559 |
bs1: "ASEQ bs AZERO r2 \<leadsto> AZERO"
|
|
|
560 |
| bs2: "ASEQ bs r1 AZERO \<leadsto> AZERO"
|
|
|
561 |
| bs3: "ASEQ bs1 (AONE bs2) r \<leadsto> fuse (bs1@bs2) r"
|
|
|
562 |
| bs4: "r1 \<leadsto> r2 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto> ASEQ bs r2 r3"
|
|
|
563 |
| bs5: "r3 \<leadsto> r4 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto> ASEQ bs r1 r4"
|
|
|
564 |
| bs6: "AALTs bs [] \<leadsto> AZERO"
|
|
|
565 |
| bs7: "AALTs bs [r] \<leadsto> fuse bs r"
|
|
|
566 |
| bs8: "rs1 s\<leadsto> rs2 \<Longrightarrow> AALTs bs rs1 \<leadsto> AALTs bs rs2"
|
|
|
567 |
| ss1: "[] s\<leadsto> []"
|
|
|
568 |
| ss2: "rs1 s\<leadsto> rs2 \<Longrightarrow> (r # rs1) s\<leadsto> (r # rs2)"
|
|
|
569 |
| ss3: "r1 \<leadsto> r2 \<Longrightarrow> (r1 # rs) s\<leadsto> (r2 # rs)"
|
|
|
570 |
| ss4: "(AZERO # rs) s\<leadsto> rs"
|
|
|
571 |
| ss5: "(AALTs bs1 rs1 # rsb) s\<leadsto> ((map (fuse bs1) rs1) @ rsb)"
|
|
|
572 |
| ss6: "erase a1 = erase a2 \<Longrightarrow> (rsa@[a1]@rsb@[a2]@rsc) s\<leadsto> (rsa@[a1]@rsb@rsc)"
|
|
|
573 |
| ss7: "erase a01 = erase a02 \<and> (distinctBy as2 erase (set (map erase as1)) = as2p) \<Longrightarrow>
|
|
|
574 |
(rsa@[ASEQ bs (AALTs bs1 as1) (ASTAR bs01 a01)]@rsb@[ASEQ bs (AALTs bs2 as2) (ASTAR bs02 a02)]@rsc) s\<leadsto>
|
|
|
575 |
(rsa@[ASEQ bs (AALTs bs1 as1) (ASTAR bs01 a01)]@rsb@[ASEQ bs (AALTs bs2 as2p) (ASTAR bs02 a02)]@rsc)"
|
|
|
576 |
| ss8: "erase a01 = erase a02 \<and> (distinctBy [a2] erase (set (map erase as1)) = []) \<Longrightarrow>
|
|
|
577 |
(rsa@[ASEQ bs (AALTs bs1 as1) (ASTAR bs01 a01)]@rsb@[ASEQ bs a2 (ASTAR bs02 a02)]@rsc) s\<leadsto>
|
|
|
578 |
(rsa@[ASEQ bs (AALTs bs1 as1) (ASTAR bs01 a01)]@rsb@rsc)"
|
|
|
579 |
| ss9: "erase a01 = erase a02 \<and> (distinctBy as2 erase {erase a1} = as2p) \<Longrightarrow>
|
|
|
580 |
(rsa@[ASEQ bs a1 (ASTAR bs01 a01)]@rsb@[ASEQ bs (AALTs bs2 as2) (ASTAR bs02 a02)]@rsc) s\<leadsto>
|
|
|
581 |
(rsa@[ASEQ bs a1 (ASTAR bs01 a01)]@rsb@[ASEQ bs (AALTs bs2 as2p) (ASTAR bs02 a02)]@rsc)"
|
|
|
582 |
|
|
|
583 |
|
|
|
584 |
inductive
|
|
|
585 |
rrewrites:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto>* _" [100, 100] 100)
|
|
|
586 |
where
|
|
|
587 |
rs1[intro, simp]:"r \<leadsto>* r"
|
|
|
588 |
| rs2[intro]: "\<lbrakk>r1 \<leadsto>* r2; r2 \<leadsto> r3\<rbrakk> \<Longrightarrow> r1 \<leadsto>* r3"
|
|
|
589 |
|
|
|
590 |
inductive
|
|
|
591 |
srewrites:: "arexp list \<Rightarrow> arexp list \<Rightarrow> bool" ("_ s\<leadsto>* _" [100, 100] 100)
|
|
|
592 |
where
|
|
|
593 |
sss1[intro, simp]:"rs s\<leadsto>* rs"
|
|
|
594 |
| sss2[intro]: "\<lbrakk>rs1 s\<leadsto> rs2; rs2 s\<leadsto>* rs3\<rbrakk> \<Longrightarrow> rs1 s\<leadsto>* rs3"
|
|
|
595 |
|
|
|
596 |
|
|
|
597 |
|
|
|
598 |
|
|
|
599 |
fun fmap :: "arexp \<Rightarrow> rexp set"
|
|
|
600 |
where
|
|
|
601 |
"fmap AALTs bs rs = flatten (map fmap rs)"
|
|
|
602 |
| "fmap ASEQ bs (AALTs bs1 rs1) r2 = (flatten (map fmap rs1)) ` (ASEQ bs _ r2)"
|
|
|
603 |
| "fmap (ASTAR bs r0) = {ASTAR bs r0}"
|
|
|
604 |
|
|
|
605 |
(*r1~r2 --\c> r1c~r2 \<longrightarrow> r1\s~r2 + r2\s1 ... \<longrightarrow> (r1\s ~ r2) +r2\s'+r2\s''+........ *)
|
|
|
606 |
(* r* \<longrightarrow>r\c ~ r* ---> r\s1 ~ r* + r\s1' ~ r* \<longrightarrow> s1, s1' \<in> suffix s *)
|
|
|
607 |
(* {r1, r2', r3} {r2,r3,r4} pders\<longrightarrow> {r1-r*, r2-r*, r3-r*, r4-r*} *)
|
|
|
608 |
|
|
|
609 |
lemma iso_pder:
|
|
|
610 |
"fmap (bders_simp r s) \<subseteq> pderss UNIV r"
|
|
|
611 |
apply(induction r)
|
|
|
612 |
prefer 4
|
|
|
613 |
oops
|
|
|
614 |
|
|
|
615 |
lemma shape_of_star_after_derssimp:
|
|
|
616 |
shows "\<forall>bs r s. \<exists>bs1 r1 r2 rs bs2 bsp. (bders_simp (ASTAR bs r) s = (ASEQ bs1 r1 r2)) \<or>
|
|
|
617 |
(bders_simp (ASTAR bs r) s = (AALTs bs2 rs)) \<or>
|
|
|
618 |
(bders_simp (ASTAR bs r) s = (ASTAR bsp r))"
|
|
|
619 |
|
|
|
620 |
oops
|
|
|
621 |
|
|
|
622 |
|
|
|
623 |
|
|
|
624 |
lemma notManyTerms:
|
|
|
625 |
shows "card {(bders_simp r (Suffix s))| r s. s \<in> L (erase r) } < (size r)"
|
|
|
626 |
|
|
|
627 |
oops
|
|
|
628 |
|
|
|
629 |
lemma rtozero:
|
|
|
630 |
shows "\<lbrakk>s \<notin> L (erase r); s \<noteq> Nil\<rbrakk> \<Longrightarrow> bders_simp r s = AZERO"
|
|
|
631 |
apply(induct r)
|
|
|
632 |
|
|
|
633 |
oops
|
|
|
634 |
|
|
|
635 |
lemma r_in_rstar:
|
|
|
636 |
shows "r1 \<leadsto> r2 \<Longrightarrow> r1 \<leadsto>* r2"
|
|
|
637 |
using rrewrites.intros(1) rrewrites.intros(2) by blast
|
|
|
638 |
|
|
|
639 |
lemma rrewrites_trans[trans]:
|
|
|
640 |
assumes a1: "r1 \<leadsto>* r2" and a2: "r2 \<leadsto>* r3"
|
|
|
641 |
shows "r1 \<leadsto>* r3"
|
|
|
642 |
using a2 a1
|
|
|
643 |
apply(induct r2 r3 arbitrary: r1 rule: rrewrites.induct)
|
|
|
644 |
apply(auto)
|
|
|
645 |
done
|
|
|
646 |
|
|
|
647 |
lemma srewrites_trans[trans]:
|
|
|
648 |
assumes a1: "r1 s\<leadsto>* r2" and a2: "r2 s\<leadsto>* r3"
|
|
|
649 |
shows "r1 s\<leadsto>* r3"
|
|
|
650 |
using a1 a2
|
|
|
651 |
apply(induct r1 r2 arbitrary: r3 rule: srewrites.induct)
|
|
|
652 |
apply(auto)
|
|
|
653 |
done
|
|
|
654 |
|
|
|
655 |
|
|
|
656 |
|
|
|
657 |
lemma contextrewrites0:
|
|
|
658 |
"rs1 s\<leadsto>* rs2 \<Longrightarrow> AALTs bs rs1 \<leadsto>* AALTs bs rs2"
|
|
|
659 |
apply(induct rs1 rs2 rule: srewrites.inducts)
|
|
|
660 |
apply simp
|
|
|
661 |
using bs8 r_in_rstar rrewrites_trans by blast
|
|
|
662 |
|
|
|
663 |
lemma contextrewrites1:
|
|
|
664 |
"r \<leadsto>* r' \<Longrightarrow> AALTs bs (r # rs) \<leadsto>* AALTs bs (r' # rs)"
|
|
|
665 |
apply(induct r r' rule: rrewrites.induct)
|
|
|
666 |
apply simp
|
|
|
667 |
using bs8 ss3 by blast
|
|
|
668 |
|
|
|
669 |
lemma srewrite1:
|
|
|
670 |
shows "rs1 s\<leadsto> rs2 \<Longrightarrow> (rs @ rs1) s\<leadsto> (rs @ rs2)"
|
|
|
671 |
apply(induct rs)
|
|
|
672 |
apply(auto)
|
|
|
673 |
using ss2 by auto
|
|
|
674 |
|
|
|
675 |
lemma srewrites1:
|
|
|
676 |
shows "rs1 s\<leadsto>* rs2 \<Longrightarrow> (rs @ rs1) s\<leadsto>* (rs @ rs2)"
|
|
|
677 |
apply(induct rs1 rs2 rule: srewrites.induct)
|
|
|
678 |
apply(auto)
|
|
|
679 |
using srewrite1 by blast
|
|
|
680 |
|
|
|
681 |
lemma srewrite2:
|
|
|
682 |
shows "r1 \<leadsto> r2 \<Longrightarrow> True"
|
|
|
683 |
and "rs1 s\<leadsto> rs2 \<Longrightarrow> (rs1 @ rs) s\<leadsto>* (rs2 @ rs)"
|
|
|
684 |
apply(induct rule: rrewrite_srewrite.inducts)
|
|
|
685 |
apply(auto)
|
|
|
686 |
apply (metis append_Cons append_Nil srewrites1)
|
|
|
687 |
apply(meson srewrites.simps ss3)
|
|
|
688 |
apply (meson srewrites.simps ss4)
|
|
|
689 |
apply (meson srewrites.simps ss5)
|
|
|
690 |
by (metis append_Cons append_Nil srewrites.simps ss6)
|
|
|
691 |
|
|
|
692 |
|
|
|
693 |
lemma srewrites3:
|
|
|
694 |
shows "rs1 s\<leadsto>* rs2 \<Longrightarrow> (rs1 @ rs) s\<leadsto>* (rs2 @ rs)"
|
|
|
695 |
apply(induct rs1 rs2 arbitrary: rs rule: srewrites.induct)
|
|
|
696 |
apply(auto)
|
|
|
697 |
by (meson srewrite2(2) srewrites_trans)
|
|
|
698 |
|
|
|
699 |
(*
|
|
|
700 |
lemma srewrites4:
|
|
|
701 |
assumes "rs3 s\<leadsto>* rs4" "rs1 s\<leadsto>* rs2"
|
|
|
702 |
shows "(rs1 @ rs3) s\<leadsto>* (rs2 @ rs4)"
|
|
|
703 |
using assms
|
|
|
704 |
apply(induct rs3 rs4 arbitrary: rs1 rs2 rule: srewrites.induct)
|
|
|
705 |
apply (simp add: srewrites3)
|
|
|
706 |
using srewrite1 by blast
|
|
|
707 |
*)
|
|
|
708 |
|
|
|
709 |
lemma srewrites6:
|
|
|
710 |
assumes "r1 \<leadsto>* r2"
|
|
|
711 |
shows "[r1] s\<leadsto>* [r2]"
|
|
|
712 |
using assms
|
|
|
713 |
apply(induct r1 r2 rule: rrewrites.induct)
|
|
|
714 |
apply(auto)
|
|
|
715 |
by (meson srewrites.simps srewrites_trans ss3)
|
|
|
716 |
|
|
|
717 |
lemma srewrites7:
|
|
|
718 |
assumes "rs3 s\<leadsto>* rs4" "r1 \<leadsto>* r2"
|
|
|
719 |
shows "(r1 # rs3) s\<leadsto>* (r2 # rs4)"
|
|
|
720 |
using assms
|
|
|
721 |
by (smt (verit, del_insts) append.simps srewrites1 srewrites3 srewrites6 srewrites_trans)
|
|
|
722 |
|
|
|
723 |
lemma ss6_stronger_aux:
|
|
|
724 |
shows "(rs1 @ rs2) s\<leadsto>* (rs1 @ distinctBy rs2 erase (set (map erase rs1)))"
|
|
|
725 |
apply(induct rs2 arbitrary: rs1)
|
|
|
726 |
apply(auto)
|
|
|
727 |
apply (smt (verit, best) append.assoc append.right_neutral append_Cons append_Nil split_list srewrite2(2) srewrites_trans ss6)
|
|
|
728 |
apply(drule_tac x="rs1 @ [a]" in meta_spec)
|
|
|
729 |
apply(simp)
|
|
|
730 |
done
|
|
|
731 |
|
|
|
732 |
lemma ss6_stronger:
|
|
|
733 |
shows "rs1 s\<leadsto>* distinctBy rs1 erase {}"
|
|
|
734 |
using ss6_stronger_aux[of "[]" _] by auto
|
|
|
735 |
|
|
|
736 |
lemma rewrite_preserves_fuse:
|
|
|
737 |
shows "r2 \<leadsto> r3 \<Longrightarrow> fuse bs r2 \<leadsto> fuse bs r3"
|
|
|
738 |
and "rs2 s\<leadsto> rs3 \<Longrightarrow> map (fuse bs) rs2 s\<leadsto> map (fuse bs) rs3"
|
|
|
739 |
proof(induct rule: rrewrite_srewrite.inducts)
|
|
|
740 |
case (bs3 bs1 bs2 r)
|
|
|
741 |
then show "fuse bs (ASEQ bs1 (AONE bs2) r) \<leadsto> fuse bs (fuse (bs1 @ bs2) r)"
|
|
|
742 |
by (metis fuse.simps(5) fuse_append rrewrite_srewrite.bs3)
|
|
|
743 |
next
|
|
|
744 |
case (bs7 bs1 r)
|
|
|
745 |
then show "fuse bs (AALTs bs1 [r]) \<leadsto> fuse bs (fuse bs1 r)"
|
|
|
746 |
by (metis fuse.simps(4) fuse_append rrewrite_srewrite.bs7)
|
|
|
747 |
next
|
|
|
748 |
case (ss2 rs1 rs2 r)
|
|
|
749 |
then show "map (fuse bs) (r # rs1) s\<leadsto> map (fuse bs) (r # rs2)"
|
|
|
750 |
by (simp add: rrewrite_srewrite.ss2)
|
|
|
751 |
next
|
|
|
752 |
case (ss3 r1 r2 rs)
|
|
|
753 |
then show "map (fuse bs) (r1 # rs) s\<leadsto> map (fuse bs) (r2 # rs)"
|
|
|
754 |
by (simp add: rrewrite_srewrite.ss3)
|
|
|
755 |
next
|
|
|
756 |
case (ss5 bs1 rs1 rsb)
|
|
|
757 |
have "map (fuse bs) (AALTs bs1 rs1 # rsb) = AALTs (bs @ bs1) rs1 # (map (fuse bs) rsb)" by simp
|
|
|
758 |
also have "... s\<leadsto> ((map (fuse (bs @ bs1)) rs1) @ (map (fuse bs) rsb))"
|
|
|
759 |
by (simp add: rrewrite_srewrite.ss5)
|
|
|
760 |
finally show "map (fuse bs) (AALTs bs1 rs1 # rsb) s\<leadsto> map (fuse bs) (map (fuse bs1) rs1 @ rsb)"
|
|
|
761 |
by (simp add: comp_def fuse_append)
|
|
|
762 |
next
|
|
|
763 |
case (ss6 a1 a2 rsa rsb rsc)
|
|
|
764 |
then show "map (fuse bs) (rsa @ [a1] @ rsb @ [a2] @ rsc) s\<leadsto> map (fuse bs) (rsa @ [a1] @ rsb @ rsc)"
|
|
|
765 |
apply(simp)
|
|
|
766 |
apply(rule rrewrite_srewrite.ss6[simplified])
|
|
|
767 |
apply(simp add: erase_fuse)
|
|
|
768 |
done
|
|
|
769 |
qed (auto intro: rrewrite_srewrite.intros)
|
|
|
770 |
|
|
|
771 |
lemma rewrites_fuse:
|
|
|
772 |
assumes "r1 \<leadsto>* r2"
|
|
|
773 |
shows "fuse bs r1 \<leadsto>* fuse bs r2"
|
|
|
774 |
using assms
|
|
|
775 |
apply(induction r1 r2 arbitrary: bs rule: rrewrites.induct)
|
|
|
776 |
apply(auto intro: rewrite_preserves_fuse)
|
|
|
777 |
done
|
|
|
778 |
|
|
|
779 |
|
|
|
780 |
lemma star_seq:
|
|
|
781 |
assumes "r1 \<leadsto>* r2"
|
|
|
782 |
shows "ASEQ bs r1 r3 \<leadsto>* ASEQ bs r2 r3"
|
|
|
783 |
using assms
|
|
|
784 |
apply(induct r1 r2 arbitrary: r3 rule: rrewrites.induct)
|
|
|
785 |
apply(auto intro: rrewrite_srewrite.intros)
|
|
|
786 |
done
|
|
|
787 |
|
|
|
788 |
lemma star_seq2:
|
|
|
789 |
assumes "r3 \<leadsto>* r4"
|
|
|
790 |
shows "ASEQ bs r1 r3 \<leadsto>* ASEQ bs r1 r4"
|
|
|
791 |
using assms
|
|
|
792 |
apply(induct r3 r4 arbitrary: r1 rule: rrewrites.induct)
|
|
|
793 |
apply(auto intro: rrewrite_srewrite.intros)
|
|
|
794 |
done
|
|
|
795 |
|
|
|
796 |
lemma continuous_rewrite:
|
|
|
797 |
assumes "r1 \<leadsto>* AZERO"
|
|
|
798 |
shows "ASEQ bs1 r1 r2 \<leadsto>* AZERO"
|
|
|
799 |
using assms bs1 star_seq by blast
|
|
|
800 |
|
|
|
801 |
(*
|
|
|
802 |
lemma continuous_rewrite2:
|
|
|
803 |
assumes "r1 \<leadsto>* AONE bs"
|
|
|
804 |
shows "ASEQ bs1 r1 r2 \<leadsto>* (fuse (bs1 @ bs) r2)"
|
|
|
805 |
using assms by (meson bs3 rrewrites.simps star_seq)
|
|
|
806 |
*)
|
|
|
807 |
|
|
|
808 |
lemma bsimp_aalts_simpcases:
|
|
|
809 |
shows "AONE bs \<leadsto>* bsimp (AONE bs)"
|
|
|
810 |
and "AZERO \<leadsto>* bsimp AZERO"
|
|
|
811 |
and "ACHAR bs c \<leadsto>* bsimp (ACHAR bs c)"
|
|
|
812 |
by (simp_all)
|
|
|
813 |
|
|
|
814 |
lemma bsimp_AALTs_rewrites:
|
|
|
815 |
shows "AALTs bs1 rs \<leadsto>* bsimp_AALTs bs1 rs"
|
|
|
816 |
by (smt (verit) bs6 bs7 bsimp_AALTs.elims rrewrites.simps)
|
|
|
817 |
|
|
|
818 |
lemma trivialbsimp_srewrites:
|
|
|
819 |
assumes "\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* f x"
|
|
|
820 |
shows "rs s\<leadsto>* (map f rs)"
|
|
|
821 |
using assms
|
|
|
822 |
apply(induction rs)
|
|
|
823 |
apply(simp_all add: srewrites7)
|
|
|
824 |
done
|
|
|
825 |
|
|
|
826 |
lemma fltsfrewrites: "rs s\<leadsto>* flts rs"
|
|
|
827 |
apply(induction rs rule: flts.induct)
|
|
|
828 |
apply(auto intro: rrewrite_srewrite.intros)
|
|
|
829 |
apply (meson srewrites.simps srewrites1 ss5)
|
|
|
830 |
using rs1 srewrites7 apply presburger
|
|
|
831 |
using srewrites7 apply force
|
|
|
832 |
apply (simp add: srewrites7)
|
|
|
833 |
by (simp add: srewrites7)
|
|
|
834 |
|
|
|
835 |
lemma bnullable0:
|
|
|
836 |
shows "r1 \<leadsto> r2 \<Longrightarrow> bnullable r1 = bnullable r2"
|
|
|
837 |
and "rs1 s\<leadsto> rs2 \<Longrightarrow> bnullables rs1 = bnullables rs2"
|
|
|
838 |
apply(induct rule: rrewrite_srewrite.inducts)
|
|
|
839 |
apply(auto simp add: bnullable_fuse)
|
|
|
840 |
apply (meson UnCI bnullable_fuse imageI)
|
|
|
841 |
by (metis bnullable_correctness)
|
|
|
842 |
|
|
|
843 |
|
|
|
844 |
lemma rewrites_bnullable_eq:
|
|
|
845 |
assumes "r1 \<leadsto>* r2"
|
|
|
846 |
shows "bnullable r1 = bnullable r2"
|
|
|
847 |
using assms
|
|
|
848 |
apply(induction r1 r2 rule: rrewrites.induct)
|
|
|
849 |
apply simp
|
|
|
850 |
using bnullable0(1) by auto
|
|
|
851 |
|
|
|
852 |
lemma rewrite_bmkeps_aux:
|
|
|
853 |
shows "r1 \<leadsto> r2 \<Longrightarrow> bnullable r1 \<Longrightarrow> bmkeps r1 = bmkeps r2"
|
|
|
854 |
and "rs1 s\<leadsto> rs2 \<Longrightarrow> bnullables rs1 \<Longrightarrow> bmkepss rs1 = bmkepss rs2"
|
|
|
855 |
proof (induct rule: rrewrite_srewrite.inducts)
|
|
|
856 |
case (bs3 bs1 bs2 r)
|
|
|
857 |
have IH2: "bnullable (ASEQ bs1 (AONE bs2) r)" by fact
|
|
|
858 |
then show "bmkeps (ASEQ bs1 (AONE bs2) r) = bmkeps (fuse (bs1 @ bs2) r)"
|
|
|
859 |
by (simp add: bmkeps_fuse)
|
|
|
860 |
next
|
|
|
861 |
case (bs7 bs r)
|
|
|
862 |
have IH2: "bnullable (AALTs bs [r])" by fact
|
|
|
863 |
then show "bmkeps (AALTs bs [r]) = bmkeps (fuse bs r)"
|
|
|
864 |
by (simp add: bmkeps_fuse)
|
|
|
865 |
next
|
|
|
866 |
case (ss3 r1 r2 rs)
|
|
|
867 |
have IH1: "bnullable r1 \<Longrightarrow> bmkeps r1 = bmkeps r2" by fact
|
|
|
868 |
have as: "r1 \<leadsto> r2" by fact
|
|
|
869 |
from IH1 as show "bmkepss (r1 # rs) = bmkepss (r2 # rs)"
|
|
|
870 |
by (simp add: bnullable0)
|
|
|
871 |
next
|
|
|
872 |
case (ss5 bs1 rs1 rsb)
|
|
|
873 |
have "bnullables (AALTs bs1 rs1 # rsb)" by fact
|
|
|
874 |
then show "bmkepss (AALTs bs1 rs1 # rsb) = bmkepss (map (fuse bs1) rs1 @ rsb)"
|
|
|
875 |
by (simp add: bmkepss1 bmkepss2 bmkepss_fuse bnullable_fuse)
|
|
|
876 |
next
|
|
|
877 |
case (ss6 a1 a2 rsa rsb rsc)
|
|
|
878 |
have as1: "erase a1 = erase a2" by fact
|
|
|
879 |
have as3: "bnullables (rsa @ [a1] @ rsb @ [a2] @ rsc)" by fact
|
|
|
880 |
show "bmkepss (rsa @ [a1] @ rsb @ [a2] @ rsc) = bmkepss (rsa @ [a1] @ rsb @ rsc)" using as1 as3
|
|
|
881 |
by (smt (verit, best) append_Cons bmkeps.simps(3) bmkepss.simps(2) bmkepss1 bmkepss2 bnullable_correctness)
|
|
|
882 |
qed (auto)
|
|
|
883 |
|
|
|
884 |
lemma rewrites_bmkeps:
|
|
|
885 |
assumes "r1 \<leadsto>* r2" "bnullable r1"
|
|
|
886 |
shows "bmkeps r1 = bmkeps r2"
|
|
|
887 |
using assms
|
|
|
888 |
proof(induction r1 r2 rule: rrewrites.induct)
|
|
|
889 |
case (rs1 r)
|
|
|
890 |
then show "bmkeps r = bmkeps r" by simp
|
|
|
891 |
next
|
|
|
892 |
case (rs2 r1 r2 r3)
|
|
|
893 |
then have IH: "bmkeps r1 = bmkeps r2" by simp
|
|
|
894 |
have a1: "bnullable r1" by fact
|
|
|
895 |
have a2: "r1 \<leadsto>* r2" by fact
|
|
|
896 |
have a3: "r2 \<leadsto> r3" by fact
|
|
|
897 |
have a4: "bnullable r2" using a1 a2 by (simp add: rewrites_bnullable_eq)
|
|
|
898 |
then have "bmkeps r2 = bmkeps r3"
|
|
|
899 |
using a3 bnullable0(1) rewrite_bmkeps_aux(1) by blast
|
|
|
900 |
then show "bmkeps r1 = bmkeps r3" using IH by simp
|
|
|
901 |
qed
|
|
|
902 |
|
|
|
903 |
|
|
|
904 |
lemma rewrites_to_bsimp:
|
|
|
905 |
shows "r \<leadsto>* bsimp r"
|
|
|
906 |
proof (induction r rule: bsimp.induct)
|
|
|
907 |
case (1 bs1 r1 r2)
|
|
|
908 |
have IH1: "r1 \<leadsto>* bsimp r1" by fact
|
|
|
909 |
have IH2: "r2 \<leadsto>* bsimp r2" by fact
|
|
|
910 |
{ assume as: "bsimp r1 = AZERO \<or> bsimp r2 = AZERO"
|
|
|
911 |
with IH1 IH2 have "r1 \<leadsto>* AZERO \<or> r2 \<leadsto>* AZERO" by auto
|
|
|
912 |
then have "ASEQ bs1 r1 r2 \<leadsto>* AZERO"
|
|
|
913 |
by (metis bs2 continuous_rewrite rrewrites.simps star_seq2)
|
|
|
914 |
then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" using as by auto
|
|
|
915 |
}
|
|
|
916 |
moreover
|
|
|
917 |
{ assume "\<exists>bs. bsimp r1 = AONE bs"
|
|
|
918 |
then obtain bs where as: "bsimp r1 = AONE bs" by blast
|
|
|
919 |
with IH1 have "r1 \<leadsto>* AONE bs" by simp
|
|
|
920 |
then have "ASEQ bs1 r1 r2 \<leadsto>* fuse (bs1 @ bs) r2" using bs3 star_seq by blast
|
|
|
921 |
with IH2 have "ASEQ bs1 r1 r2 \<leadsto>* fuse (bs1 @ bs) (bsimp r2)"
|
|
|
922 |
using rewrites_fuse by (meson rrewrites_trans)
|
|
|
923 |
then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 (AONE bs) r2)" by simp
|
|
|
924 |
then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by (simp add: as)
|
|
|
925 |
}
|
|
|
926 |
moreover
|
|
|
927 |
{ assume as1: "bsimp r1 \<noteq> AZERO" "bsimp r2 \<noteq> AZERO" and as2: "(\<nexists>bs. bsimp r1 = AONE bs)"
|
|
|
928 |
then have "bsimp_ASEQ bs1 (bsimp r1) (bsimp r2) = ASEQ bs1 (bsimp r1) (bsimp r2)"
|
|
|
929 |
by (simp add: bsimp_ASEQ1)
|
|
|
930 |
then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp_ASEQ bs1 (bsimp r1) (bsimp r2)" using as1 as2 IH1 IH2
|
|
|
931 |
by (metis rrewrites_trans star_seq star_seq2)
|
|
|
932 |
then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by simp
|
|
|
933 |
}
|
|
|
934 |
ultimately show "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by blast
|
|
|
935 |
next
|
|
|
936 |
case (2 bs1 rs)
|
|
|
937 |
have IH: "\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* bsimp x" by fact
|
|
|
938 |
then have "rs s\<leadsto>* (map bsimp rs)" by (simp add: trivialbsimp_srewrites)
|
|
|
939 |
also have "... s\<leadsto>* flts (map bsimp rs)" by (simp add: fltsfrewrites)
|
|
|
940 |
also have "... s\<leadsto>* distinctBy (flts (map bsimp rs)) erase {}" by (simp add: ss6_stronger)
|
|
|
941 |
finally have "AALTs bs1 rs \<leadsto>* AALTs bs1 (distinctBy (flts (map bsimp rs)) erase {})"
|
|
|
942 |
using contextrewrites0 by blast
|
|
|
943 |
also have "... \<leadsto>* bsimp_AALTs bs1 (distinctBy (flts (map bsimp rs)) erase {})"
|
|
|
944 |
by (simp add: bsimp_AALTs_rewrites)
|
|
|
945 |
finally show "AALTs bs1 rs \<leadsto>* bsimp (AALTs bs1 rs)" by simp
|
|
|
946 |
qed (simp_all)
|
|
|
947 |
|
|
|
948 |
|
|
|
949 |
lemma to_zero_in_alt:
|
|
|
950 |
shows "AALT bs (ASEQ [] AZERO r) r2 \<leadsto> AALT bs AZERO r2"
|
|
|
951 |
by (simp add: bs1 bs8 ss3)
|
|
|
952 |
|
|
|
953 |
|
|
|
954 |
|
|
|
955 |
lemma bder_fuse_list:
|
|
|
956 |
shows "map (bder c \<circ> fuse bs1) rs1 = map (fuse bs1 \<circ> bder c) rs1"
|
|
|
957 |
apply(induction rs1)
|
|
|
958 |
apply(simp_all add: bder_fuse)
|
|
|
959 |
done
|
|
|
960 |
|
|
|
961 |
lemma rewrite_preserves_bder:
|
|
|
962 |
shows "r1 \<leadsto> r2 \<Longrightarrow> bder c r1 \<leadsto>* bder c r2"
|
|
|
963 |
and "rs1 s\<leadsto> rs2 \<Longrightarrow> map (bder c) rs1 s\<leadsto>* map (bder c) rs2"
|
|
|
964 |
proof(induction rule: rrewrite_srewrite.inducts)
|
|
|
965 |
case (bs1 bs r2)
|
|
|
966 |
show "bder c (ASEQ bs AZERO r2) \<leadsto>* bder c AZERO"
|
|
|
967 |
by (simp add: continuous_rewrite)
|
|
|
968 |
next
|
|
|
969 |
case (bs2 bs r1)
|
|
|
970 |
show "bder c (ASEQ bs r1 AZERO) \<leadsto>* bder c AZERO"
|
|
|
971 |
apply(auto)
|
|
|
972 |
apply (meson bs6 contextrewrites0 rrewrite_srewrite.bs2 rs2 ss3 ss4 sss1 sss2)
|
|
|
973 |
by (simp add: r_in_rstar rrewrite_srewrite.bs2)
|
|
|
974 |
next
|
|
|
975 |
case (bs3 bs1 bs2 r)
|
|
|
976 |
show "bder c (ASEQ bs1 (AONE bs2) r) \<leadsto>* bder c (fuse (bs1 @ bs2) r)"
|
|
|
977 |
apply(simp)
|
|
|
978 |
by (metis (no_types, lifting) bder_fuse bs8 bs7 fuse_append rrewrites.simps ss4 to_zero_in_alt)
|
|
|
979 |
next
|
|
|
980 |
case (bs4 r1 r2 bs r3)
|
|
|
981 |
have as: "r1 \<leadsto> r2" by fact
|
|
|
982 |
have IH: "bder c r1 \<leadsto>* bder c r2" by fact
|
|
|
983 |
from as IH show "bder c (ASEQ bs r1 r3) \<leadsto>* bder c (ASEQ bs r2 r3)"
|
|
|
984 |
by (metis bder.simps(5) bnullable0(1) contextrewrites1 rewrite_bmkeps_aux(1) star_seq)
|
|
|
985 |
next
|
|
|
986 |
case (bs5 r3 r4 bs r1)
|
|
|
987 |
have as: "r3 \<leadsto> r4" by fact
|
|
|
988 |
have IH: "bder c r3 \<leadsto>* bder c r4" by fact
|
|
|
989 |
from as IH show "bder c (ASEQ bs r1 r3) \<leadsto>* bder c (ASEQ bs r1 r4)"
|
|
|
990 |
apply(simp)
|
|
|
991 |
apply(auto)
|
|
|
992 |
using contextrewrites0 r_in_rstar rewrites_fuse srewrites6 srewrites7 star_seq2 apply presburger
|
|
|
993 |
using star_seq2 by blast
|
|
|
994 |
next
|
|
|
995 |
case (bs6 bs)
|
|
|
996 |
show "bder c (AALTs bs []) \<leadsto>* bder c AZERO"
|
|
|
997 |
using rrewrite_srewrite.bs6 by force
|
|
|
998 |
next
|
|
|
999 |
case (bs7 bs r)
|
|
|
1000 |
show "bder c (AALTs bs [r]) \<leadsto>* bder c (fuse bs r)"
|
|
|
1001 |
by (simp add: bder_fuse r_in_rstar rrewrite_srewrite.bs7)
|
|
|
1002 |
next
|
|
|
1003 |
case (bs8 rs1 rs2 bs)
|
|
|
1004 |
have IH1: "map (bder c) rs1 s\<leadsto>* map (bder c) rs2" by fact
|
|
|
1005 |
then show "bder c (AALTs bs rs1) \<leadsto>* bder c (AALTs bs rs2)"
|
|
|
1006 |
using contextrewrites0 by force
|
|
|
1007 |
next
|
|
|
1008 |
case ss1
|
|
|
1009 |
show "map (bder c) [] s\<leadsto>* map (bder c) []" by simp
|
|
|
1010 |
next
|
|
|
1011 |
case (ss2 rs1 rs2 r)
|
|
|
1012 |
have IH1: "map (bder c) rs1 s\<leadsto>* map (bder c) rs2" by fact
|
|
|
1013 |
then show "map (bder c) (r # rs1) s\<leadsto>* map (bder c) (r # rs2)"
|
|
|
1014 |
by (simp add: srewrites7)
|
|
|
1015 |
next
|
|
|
1016 |
case (ss3 r1 r2 rs)
|
|
|
1017 |
have IH: "bder c r1 \<leadsto>* bder c r2" by fact
|
|
|
1018 |
then show "map (bder c) (r1 # rs) s\<leadsto>* map (bder c) (r2 # rs)"
|
|
|
1019 |
by (simp add: srewrites7)
|
|
|
1020 |
next
|
|
|
1021 |
case (ss4 rs)
|
|
|
1022 |
show "map (bder c) (AZERO # rs) s\<leadsto>* map (bder c) rs"
|
|
|
1023 |
using rrewrite_srewrite.ss4 by fastforce
|
|
|
1024 |
next
|
|
|
1025 |
case (ss5 bs1 rs1 rsb)
|
|
|
1026 |
show "map (bder c) (AALTs bs1 rs1 # rsb) s\<leadsto>* map (bder c) (map (fuse bs1) rs1 @ rsb)"
|
|
|
1027 |
apply(simp)
|
|
|
1028 |
using bder_fuse_list map_map rrewrite_srewrite.ss5 srewrites.simps by blast
|
|
|
1029 |
next
|
|
|
1030 |
case (ss6 a1 a2 bs rsa rsb)
|
|
|
1031 |
have as: "erase a1 = erase a2" by fact
|
|
|
1032 |
show "map (bder c) (bs @ [a1] @ rsa @ [a2] @ rsb) s\<leadsto>* map (bder c) (bs @ [a1] @ rsa @ rsb)"
|
|
|
1033 |
apply(simp only: map_append)
|
|
|
1034 |
by (smt (verit, best) erase_bder list.simps(8) list.simps(9) as rrewrite_srewrite.ss6 srewrites.simps)
|
|
|
1035 |
qed
|
|
|
1036 |
|
|
|
1037 |
lemma rewrites_preserves_bder:
|
|
|
1038 |
assumes "r1 \<leadsto>* r2"
|
|
|
1039 |
shows "bder c r1 \<leadsto>* bder c r2"
|
|
|
1040 |
using assms
|
|
|
1041 |
apply(induction r1 r2 rule: rrewrites.induct)
|
|
|
1042 |
apply(simp_all add: rewrite_preserves_bder rrewrites_trans)
|
|
|
1043 |
done
|
|
|
1044 |
|
|
|
1045 |
|
|
|
1046 |
lemma central:
|
|
|
1047 |
shows "bders r s \<leadsto>* bders_simp r s"
|
|
|
1048 |
proof(induct s arbitrary: r rule: rev_induct)
|
|
|
1049 |
case Nil
|
|
|
1050 |
then show "bders r [] \<leadsto>* bders_simp r []" by simp
|
|
|
1051 |
next
|
|
|
1052 |
case (snoc x xs)
|
|
|
1053 |
have IH: "\<And>r. bders r xs \<leadsto>* bders_simp r xs" by fact
|
|
|
1054 |
have "bders r (xs @ [x]) = bders (bders r xs) [x]" by (simp add: bders_append)
|
|
|
1055 |
also have "... \<leadsto>* bders (bders_simp r xs) [x]" using IH
|
|
|
1056 |
by (simp add: rewrites_preserves_bder)
|
|
|
1057 |
also have "... \<leadsto>* bders_simp (bders_simp r xs) [x]" using IH
|
|
|
1058 |
by (simp add: rewrites_to_bsimp)
|
|
|
1059 |
finally show "bders r (xs @ [x]) \<leadsto>* bders_simp r (xs @ [x])"
|
|
|
1060 |
by (simp add: bders_simp_append)
|
|
|
1061 |
qed
|
|
|
1062 |
|
|
|
1063 |
lemma main_aux:
|
|
|
1064 |
assumes "bnullable (bders r s)"
|
|
|
1065 |
shows "bmkeps (bders r s) = bmkeps (bders_simp r s)"
|
|
|
1066 |
proof -
|
|
|
1067 |
have "bders r s \<leadsto>* bders_simp r s" by (rule central)
|
|
|
1068 |
then
|
|
|
1069 |
show "bmkeps (bders r s) = bmkeps (bders_simp r s)" using assms
|
|
|
1070 |
by (rule rewrites_bmkeps)
|
|
|
1071 |
qed
|
|
|
1072 |
|
|
|
1073 |
|
|
|
1074 |
theorem main_blexer_simp:
|
|
|
1075 |
shows "blexer r s = blexer_simp r s"
|
|
|
1076 |
unfolding blexer_def blexer_simp_def
|
|
|
1077 |
by (metis central main_aux rewrites_bnullable_eq)
|
|
|
1078 |
|
|
|
1079 |
|
|
|
1080 |
theorem blexersimp_correctness:
|
|
|
1081 |
shows "lexer r s = blexer_simp r s"
|
|
|
1082 |
using blexer_correctness main_blexer_simp by simp
|
|
|
1083 |
|
|
|
1084 |
|
|
|
1085 |
(* below is the idempotency of bsimp *)
|
|
|
1086 |
|
|
|
1087 |
lemma bsimp_ASEQ_fuse:
|
|
|
1088 |
shows "fuse bs1 (bsimp_ASEQ bs2 r1 r2) = bsimp_ASEQ (bs1 @ bs2) r1 r2"
|
|
|
1089 |
apply(induct r1 r2 arbitrary: bs1 bs2 rule: bsimp_ASEQ.induct)
|
|
|
1090 |
apply(auto)
|
|
|
1091 |
done
|
|
|
1092 |
|
|
|
1093 |
lemma bsimp_AALTs_fuse:
|
|
|
1094 |
assumes "\<forall>r \<in> set rs. fuse bs1 (fuse bs2 r) = fuse (bs1 @ bs2) r"
|
|
|
1095 |
shows "fuse bs1 (bsimp_AALTs bs2 rs) = bsimp_AALTs (bs1 @ bs2) rs"
|
|
|
1096 |
using assms
|
|
|
1097 |
apply(induct bs2 rs arbitrary: bs1 rule: bsimp_AALTs.induct)
|
|
|
1098 |
apply(auto)
|
|
|
1099 |
done
|
|
|
1100 |
|
|
|
1101 |
lemma bsimp_fuse:
|
|
|
1102 |
shows "fuse bs (bsimp r) = bsimp (fuse bs r)"
|
|
|
1103 |
apply(induct r arbitrary: bs)
|
|
|
1104 |
apply(simp_all add: bsimp_ASEQ_fuse bsimp_AALTs_fuse fuse_append)
|
|
|
1105 |
done
|
|
|
1106 |
|
|
|
1107 |
lemma bsimp_ASEQ_idem:
|
|
|
1108 |
assumes "bsimp (bsimp r1) = bsimp r1" "bsimp (bsimp r2) = bsimp r2"
|
|
|
1109 |
shows "bsimp (bsimp_ASEQ x1 (bsimp r1) (bsimp r2)) = bsimp_ASEQ x1 (bsimp r1) (bsimp r2)"
|
|
|
1110 |
using assms
|
|
|
1111 |
apply(case_tac "bsimp r1 = AZERO")
|
|
|
1112 |
apply(simp)
|
|
|
1113 |
apply(case_tac "bsimp r2 = AZERO")
|
|
|
1114 |
apply(simp)
|
|
|
1115 |
apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
|
|
|
1116 |
apply(auto)[1]
|
|
|
1117 |
apply (metis bsimp_fuse)
|
|
|
1118 |
apply(simp add: bsimp_ASEQ1)
|
|
|
1119 |
done
|
|
|
1120 |
|
|
|
1121 |
lemma bsimp_AALTs_idem:
|
|
|
1122 |
assumes "\<forall>r \<in> set rs. bsimp (bsimp r) = bsimp r"
|
|
|
1123 |
shows "bsimp (bsimp_AALTs bs rs) = bsimp_AALTs bs (map bsimp rs)"
|
|
|
1124 |
using assms
|
|
|
1125 |
apply(induct bs rs rule: bsimp_AALTs.induct)
|
|
|
1126 |
apply(simp)
|
|
|
1127 |
apply(simp)
|
|
|
1128 |
using bsimp_fuse apply presburger
|
|
|
1129 |
oops
|
|
|
1130 |
|
|
|
1131 |
lemma bsimp_idem_rev:
|
|
|
1132 |
shows "\<nexists>r2. bsimp r1 \<leadsto> r2"
|
|
|
1133 |
apply(induct r1 rule: bsimp.induct)
|
|
|
1134 |
apply(auto)
|
|
|
1135 |
defer
|
|
|
1136 |
defer
|
|
|
1137 |
using rrewrite.simps apply blast
|
|
|
1138 |
using rrewrite.cases apply blast
|
|
|
1139 |
using rrewrite.simps apply blast
|
|
|
1140 |
using rrewrite.cases apply blast
|
|
|
1141 |
apply(case_tac "bsimp r1 = AZERO")
|
|
|
1142 |
apply(simp)
|
|
|
1143 |
apply(case_tac "bsimp r2 = AZERO")
|
|
|
1144 |
apply(simp)
|
|
|
1145 |
apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
|
|
|
1146 |
apply(auto)[1]
|
|
|
1147 |
prefer 2
|
|
|
1148 |
apply (smt (verit, best) arexp.distinct(25) arexp.inject(3) bsimp_ASEQ1 rrewrite.simps)
|
|
|
1149 |
defer
|
|
|
1150 |
oops
|
|
|
1151 |
|
|
|
1152 |
lemma bsimp_idem:
|
|
|
1153 |
shows "bsimp (bsimp r) = bsimp r"
|
|
|
1154 |
apply(induct r rule: bsimp.induct)
|
|
|
1155 |
apply(auto)
|
|
|
1156 |
using bsimp_ASEQ_idem apply presburger
|
|
|
1157 |
oops
|
|
|
1158 |
|
|
409
|
1159 |
|
|
|
1160 |
lemma normal_form:
|
|
|
1161 |
shows "\<forall>r. \<nexists> r'. bsimp r \<leadsto> r'"
|
|
|
1162 |
|
|
|
1163 |
oops
|
|
|
1164 |
|
|
|
1165 |
lemma another_normal:
|
|
|
1166 |
shows "\<nexists>r'. bsimp r \<leadsto> r'"
|
|
|
1167 |
oops
|
|
|
1168 |
|
|
406
|
1169 |
export_code blexer_simp blexer lexer bders bders_simp in Scala module_name VerifiedLexers
|
|
|
1170 |
|
|
|
1171 |
|
|
|
1172 |
unused_thms
|
|
|
1173 |
|
|
|
1174 |
|
|
|
1175 |
fun concatLen:: "arexp \<Rightarrow> int"
|
|
|
1176 |
where
|
|
|
1177 |
"concatLen AZERO = 0"
|
|
|
1178 |
| "concatLen (AONE bs) = 0"
|
|
|
1179 |
| "concatLen (ACHAR bs a) = 0"
|
|
|
1180 |
| "concatLen (ASEQ bs a1 a2) = 1 + (max (concatLen a1) ( concatLen a2))"
|
|
|
1181 |
| "concatLen (ASTAR bs r0) = 1 + (concatLen r0)"
|
|
|
1182 |
| "concatLen (AALTS bs as) = foldl max (map concatLen as)"
|
|
|
1183 |
|
|
|
1184 |
|
|
|
1185 |
|
|
|
1186 |
inductive aggressive:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto>? _" [99, 99] 99)
|
|
|
1187 |
where
|
|
|
1188 |
"ASEQ bs (AALTs bs1 rs) r \<leadsto>? AALTs (bs@bs1) (map (\<lambda>r'. ASEQ [] r' r) rs) "
|
|
|
1189 |
|
|
|
1190 |
|
|
|
1191 |
|
|
|
1192 |
end
|