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theory SizeBound5CT
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428
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imports "Lexer" "PDerivs"
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409
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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' bs ZERO = (undefined, bs)"
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| "decode' bs ONE = (Void, bs)"
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| "decode' bs (CH d) = (Char d, bs)"
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| "decode' [] (ALT r1 r2) = (Void, [])"
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| "decode' (Z # bs) (ALT r1 r2) = (let (v, bs') = decode' bs r1 in (Left v, bs'))"
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| "decode' (S # bs) (ALT r1 r2) = (let (v, bs') = decode' bs r2 in (Right v, bs'))"
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| "decode' bs (SEQ r1 r2) = (let (v1, bs') = decode' bs r1 in
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let (v2, bs'') = decode' bs' r2 in (Seq v1 v2, bs''))"
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| "decode' [] (STAR r) = (Void, [])"
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| "decode' (S # bs) (STAR r) = (Stars [], bs)"
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| "decode' (Z # bs) (STAR r) = (let (v, bs') = decode' bs r in
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let (vs, bs'') = decode' bs' (STAR r)
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in (Stars_add v vs, bs''))"
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by pat_completeness auto
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lemma decode'_smaller:
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assumes "decode'_dom (bs, r)"
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shows "length (snd (decode' bs r)) \<le> length bs"
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using assms
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apply(induct bs 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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apply(auto)
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324 |
using bnullable_correctness apply blast
|
|
|
325 |
apply(case_tac rs)
|
|
|
326 |
apply(auto)
|
|
|
327 |
using bnullable_correctness apply blast
|
|
|
328 |
apply(case_tac rs)
|
|
|
329 |
apply(auto)
|
|
|
330 |
done
|
|
|
331 |
|
|
|
332 |
lemma bmkeps_retrieve_AALTs:
|
|
|
333 |
assumes "\<forall>r \<in> set rs. bnullable r \<longrightarrow> bmkeps r = retrieve r (mkeps (erase r))"
|
|
|
334 |
"bnullables rs"
|
|
|
335 |
shows "bs @ bmkepss rs = retrieve (AALTs bs rs) (mkeps (erase (AALTs bs rs)))"
|
|
|
336 |
using assms
|
|
|
337 |
apply(induct rs arbitrary: bs)
|
|
|
338 |
apply(auto)
|
|
|
339 |
using retrieve_AALTs_bnullable1 apply presburger
|
|
|
340 |
apply (metis retrieve_AALTs_bnullable2)
|
|
|
341 |
apply (simp add: retrieve_AALTs_bnullable1)
|
|
|
342 |
by (metis retrieve_AALTs_bnullable2)
|
|
|
343 |
|
|
|
344 |
|
|
|
345 |
lemma bmkeps_retrieve:
|
|
|
346 |
assumes "bnullable r"
|
|
|
347 |
shows "bmkeps r = retrieve r (mkeps (erase r))"
|
|
|
348 |
using assms
|
|
|
349 |
apply(induct r)
|
|
|
350 |
apply(auto)
|
|
|
351 |
using bmkeps_retrieve_AALTs by auto
|
|
|
352 |
|
|
|
353 |
lemma bder_retrieve:
|
|
|
354 |
assumes "\<Turnstile> v : der c (erase r)"
|
|
|
355 |
shows "retrieve (bder c r) v = retrieve r (injval (erase r) c v)"
|
|
|
356 |
using assms
|
|
|
357 |
apply(induct r arbitrary: v rule: erase.induct)
|
|
|
358 |
using Prf_elims(1) apply auto[1]
|
|
|
359 |
using Prf_elims(1) apply auto[1]
|
|
|
360 |
apply(auto)[1]
|
|
|
361 |
apply (metis Prf_elims(4) injval.simps(1) retrieve.simps(1) retrieve.simps(2))
|
|
|
362 |
using Prf_elims(1) apply blast
|
|
|
363 |
(* AALTs case *)
|
|
|
364 |
apply(simp)
|
|
|
365 |
apply(erule Prf_elims)
|
|
|
366 |
apply(simp)
|
|
|
367 |
apply(simp)
|
|
|
368 |
apply(rename_tac "r\<^sub>1" "r\<^sub>2" rs v)
|
|
|
369 |
apply(erule Prf_elims)
|
|
|
370 |
apply(simp)
|
|
|
371 |
apply(simp)
|
|
|
372 |
apply(case_tac rs)
|
|
|
373 |
apply(simp)
|
|
|
374 |
apply(simp)
|
|
|
375 |
using Prf_elims(3) apply fastforce
|
|
|
376 |
(* ASEQ case *)
|
|
|
377 |
apply(simp)
|
|
|
378 |
apply(case_tac "nullable (erase r1)")
|
|
|
379 |
apply(simp)
|
|
|
380 |
apply(erule Prf_elims)
|
|
|
381 |
using Prf_elims(2) bnullable_correctness apply force
|
|
|
382 |
apply (simp add: bmkeps_retrieve bnullable_correctness retrieve_fuse2)
|
|
|
383 |
apply (simp add: bmkeps_retrieve bnullable_correctness retrieve_fuse2)
|
|
|
384 |
using Prf_elims(2) apply force
|
|
|
385 |
(* ASTAR case *)
|
|
|
386 |
apply(rename_tac bs r v)
|
|
|
387 |
apply(simp)
|
|
|
388 |
apply(erule Prf_elims)
|
|
|
389 |
apply(clarify)
|
|
|
390 |
apply(erule Prf_elims)
|
|
|
391 |
apply(clarify)
|
|
|
392 |
by (simp add: retrieve_fuse2)
|
|
|
393 |
|
|
|
394 |
|
|
|
395 |
lemma MAIN_decode:
|
|
|
396 |
assumes "\<Turnstile> v : ders s r"
|
|
|
397 |
shows "Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r"
|
|
|
398 |
using assms
|
|
|
399 |
proof (induct s arbitrary: v rule: rev_induct)
|
|
|
400 |
case Nil
|
|
|
401 |
have "\<Turnstile> v : ders [] r" by fact
|
|
|
402 |
then have "\<Turnstile> v : r" by simp
|
|
|
403 |
then have "Some v = decode (retrieve (intern r) v) r"
|
|
|
404 |
using decode_code retrieve_code by auto
|
|
|
405 |
then show "Some (flex r id [] v) = decode (retrieve (bders (intern r) []) v) r"
|
|
|
406 |
by simp
|
|
|
407 |
next
|
|
|
408 |
case (snoc c s v)
|
|
|
409 |
have IH: "\<And>v. \<Turnstile> v : ders s r \<Longrightarrow>
|
|
|
410 |
Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r" by fact
|
|
|
411 |
have asm: "\<Turnstile> v : ders (s @ [c]) r" by fact
|
|
|
412 |
then have asm2: "\<Turnstile> injval (ders s r) c v : ders s r"
|
|
|
413 |
by (simp add: Prf_injval ders_append)
|
|
|
414 |
have "Some (flex r id (s @ [c]) v) = Some (flex r id s (injval (ders s r) c v))"
|
|
|
415 |
by (simp add: flex_append)
|
|
|
416 |
also have "... = decode (retrieve (bders (intern r) s) (injval (ders s r) c v)) r"
|
|
|
417 |
using asm2 IH by simp
|
|
|
418 |
also have "... = decode (retrieve (bder c (bders (intern r) s)) v) r"
|
|
|
419 |
using asm by (simp_all add: bder_retrieve ders_append)
|
|
|
420 |
finally show "Some (flex r id (s @ [c]) v) =
|
|
|
421 |
decode (retrieve (bders (intern r) (s @ [c])) v) r" by (simp add: bders_append)
|
|
|
422 |
qed
|
|
|
423 |
|
|
|
424 |
definition blexer where
|
|
|
425 |
"blexer r s \<equiv> if bnullable (bders (intern r) s) then
|
|
|
426 |
decode (bmkeps (bders (intern r) s)) r else None"
|
|
|
427 |
|
|
|
428 |
lemma blexer_correctness:
|
|
|
429 |
shows "blexer r s = lexer r s"
|
|
|
430 |
proof -
|
|
|
431 |
{ define bds where "bds \<equiv> bders (intern r) s"
|
|
|
432 |
define ds where "ds \<equiv> ders s r"
|
|
|
433 |
assume asm: "nullable ds"
|
|
|
434 |
have era: "erase bds = ds"
|
|
|
435 |
unfolding ds_def bds_def by simp
|
|
|
436 |
have mke: "\<Turnstile> mkeps ds : ds"
|
|
|
437 |
using asm by (simp add: mkeps_nullable)
|
|
|
438 |
have "decode (bmkeps bds) r = decode (retrieve bds (mkeps ds)) r"
|
|
|
439 |
using bmkeps_retrieve
|
|
|
440 |
using asm era
|
|
|
441 |
using bnullable_correctness by force
|
|
|
442 |
also have "... = Some (flex r id s (mkeps ds))"
|
|
|
443 |
using mke by (simp_all add: MAIN_decode ds_def bds_def)
|
|
|
444 |
finally have "decode (bmkeps bds) r = Some (flex r id s (mkeps ds))"
|
|
|
445 |
unfolding bds_def ds_def .
|
|
|
446 |
}
|
|
|
447 |
then show "blexer r s = lexer r s"
|
|
|
448 |
unfolding blexer_def lexer_flex
|
|
|
449 |
by (auto simp add: bnullable_correctness[symmetric])
|
|
|
450 |
qed
|
|
|
451 |
|
|
|
452 |
|
|
|
453 |
fun distinctBy :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b set \<Rightarrow> 'a list"
|
|
|
454 |
where
|
|
|
455 |
"distinctBy [] f acc = []"
|
|
|
456 |
| "distinctBy (x#xs) f acc =
|
|
|
457 |
(if (f x) \<in> acc then distinctBy xs f acc
|
|
|
458 |
else x # (distinctBy xs f ({f x} \<union> acc)))"
|
|
|
459 |
|
|
|
460 |
|
|
|
461 |
|
|
|
462 |
fun flts :: "arexp list \<Rightarrow> arexp list"
|
|
|
463 |
where
|
|
|
464 |
"flts [] = []"
|
|
|
465 |
| "flts (AZERO # rs) = flts rs"
|
|
|
466 |
| "flts ((AALTs bs rs1) # rs) = (map (fuse bs) rs1) @ flts rs"
|
|
|
467 |
| "flts (r1 # rs) = r1 # flts rs"
|
|
|
468 |
|
|
|
469 |
|
|
|
470 |
|
|
|
471 |
fun bsimp_ASEQ :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp \<Rightarrow> arexp"
|
|
|
472 |
where
|
|
|
473 |
"bsimp_ASEQ _ AZERO _ = AZERO"
|
|
|
474 |
| "bsimp_ASEQ _ _ AZERO = AZERO"
|
|
|
475 |
| "bsimp_ASEQ bs1 (AONE bs2) r2 = fuse (bs1 @ bs2) r2"
|
|
|
476 |
| "bsimp_ASEQ bs1 r1 r2 = ASEQ bs1 r1 r2"
|
|
|
477 |
|
|
|
478 |
lemma bsimp_ASEQ0[simp]:
|
|
|
479 |
shows "bsimp_ASEQ bs r1 AZERO = AZERO"
|
|
|
480 |
by (case_tac r1)(simp_all)
|
|
|
481 |
|
|
|
482 |
lemma bsimp_ASEQ1:
|
|
|
483 |
assumes "r1 \<noteq> AZERO" "r2 \<noteq> AZERO" "\<nexists>bs. r1 = AONE bs"
|
|
|
484 |
shows "bsimp_ASEQ bs r1 r2 = ASEQ bs r1 r2"
|
|
|
485 |
using assms
|
|
|
486 |
apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
|
|
|
487 |
apply(auto)
|
|
|
488 |
done
|
|
|
489 |
|
|
|
490 |
lemma bsimp_ASEQ2[simp]:
|
|
|
491 |
shows "bsimp_ASEQ bs1 (AONE bs2) r2 = fuse (bs1 @ bs2) r2"
|
|
|
492 |
by (case_tac r2) (simp_all)
|
|
|
493 |
|
|
|
494 |
|
|
|
495 |
fun bsimp_AALTs :: "bit list \<Rightarrow> arexp list \<Rightarrow> arexp"
|
|
|
496 |
where
|
|
|
497 |
"bsimp_AALTs _ [] = AZERO"
|
|
|
498 |
| "bsimp_AALTs bs1 [r] = fuse bs1 r"
|
|
|
499 |
| "bsimp_AALTs bs1 rs = AALTs bs1 rs"
|
|
|
500 |
|
|
|
501 |
|
|
|
502 |
fun bsimp :: "arexp \<Rightarrow> arexp"
|
|
|
503 |
where
|
|
|
504 |
"bsimp (ASEQ bs1 r1 r2) = bsimp_ASEQ bs1 (bsimp r1) (bsimp r2)"
|
|
|
505 |
| "bsimp (AALTs bs1 rs) = bsimp_AALTs bs1 (distinctBy (flts (map bsimp rs)) erase {}) "
|
|
|
506 |
| "bsimp r = r"
|
|
|
507 |
|
|
|
508 |
|
|
|
509 |
fun
|
|
|
510 |
bders_simp :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
|
|
|
511 |
where
|
|
|
512 |
"bders_simp r [] = r"
|
|
|
513 |
| "bders_simp r (c # s) = bders_simp (bsimp (bder c r)) s"
|
|
|
514 |
|
|
|
515 |
definition blexer_simp where
|
|
|
516 |
"blexer_simp r s \<equiv> if bnullable (bders_simp (intern r) s) then
|
|
|
517 |
decode (bmkeps (bders_simp (intern r) s)) r else None"
|
|
|
518 |
|
|
|
519 |
|
|
|
520 |
|
|
|
521 |
lemma bders_simp_append:
|
|
|
522 |
shows "bders_simp r (s1 @ s2) = bders_simp (bders_simp r s1) s2"
|
|
|
523 |
apply(induct s1 arbitrary: r s2)
|
|
|
524 |
apply(simp_all)
|
|
|
525 |
done
|
|
|
526 |
|
|
|
527 |
|
|
|
528 |
lemma bmkeps_fuse:
|
|
|
529 |
assumes "bnullable r"
|
|
|
530 |
shows "bmkeps (fuse bs r) = bs @ bmkeps r"
|
|
|
531 |
using assms
|
|
|
532 |
by (metis bmkeps_retrieve bnullable_correctness erase_fuse mkeps_nullable retrieve_fuse2)
|
|
|
533 |
|
|
|
534 |
lemma bmkepss_fuse:
|
|
|
535 |
assumes "bnullables rs"
|
|
|
536 |
shows "bmkepss (map (fuse bs) rs) = bs @ bmkepss rs"
|
|
|
537 |
using assms
|
|
|
538 |
apply(induct rs arbitrary: bs)
|
|
|
539 |
apply(auto simp add: bmkeps_fuse bnullable_fuse)
|
|
|
540 |
done
|
|
|
541 |
|
|
|
542 |
lemma bder_fuse:
|
|
|
543 |
shows "bder c (fuse bs a) = fuse bs (bder c a)"
|
|
|
544 |
apply(induct a arbitrary: bs c)
|
|
|
545 |
apply(simp_all)
|
|
|
546 |
done
|
|
|
547 |
|
|
|
548 |
|
|
|
549 |
|
|
|
550 |
|
|
|
551 |
inductive
|
|
|
552 |
rrewrite:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto> _" [99, 99] 99)
|
|
|
553 |
and
|
|
|
554 |
srewrite:: "arexp list \<Rightarrow> arexp list \<Rightarrow> bool" (" _ s\<leadsto> _" [100, 100] 100)
|
|
|
555 |
where
|
|
|
556 |
bs1: "ASEQ bs AZERO r2 \<leadsto> AZERO"
|
|
|
557 |
| bs2: "ASEQ bs r1 AZERO \<leadsto> AZERO"
|
|
|
558 |
| bs3: "ASEQ bs1 (AONE bs2) r \<leadsto> fuse (bs1@bs2) r"
|
|
|
559 |
| bs4: "r1 \<leadsto> r2 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto> ASEQ bs r2 r3"
|
|
|
560 |
| bs5: "r3 \<leadsto> r4 \<Longrightarrow> ASEQ bs r1 r3 \<leadsto> ASEQ bs r1 r4"
|
|
|
561 |
| bs6: "AALTs bs [] \<leadsto> AZERO"
|
|
|
562 |
| bs7: "AALTs bs [r] \<leadsto> fuse bs r"
|
|
|
563 |
| bs8: "rs1 s\<leadsto> rs2 \<Longrightarrow> AALTs bs rs1 \<leadsto> AALTs bs rs2"
|
|
|
564 |
(*| ss1: "[] s\<leadsto> []"*)
|
|
|
565 |
| ss2: "rs1 s\<leadsto> rs2 \<Longrightarrow> (r # rs1) s\<leadsto> (r # rs2)"
|
|
|
566 |
| ss3: "r1 \<leadsto> r2 \<Longrightarrow> (r1 # rs) s\<leadsto> (r2 # rs)"
|
|
|
567 |
| ss4: "(AZERO # rs) s\<leadsto> rs"
|
|
|
568 |
| ss5: "(AALTs bs1 rs1 # rsb) s\<leadsto> ((map (fuse bs1) rs1) @ rsb)"
|
|
|
569 |
| ss6: "erase a1 = erase a2 \<Longrightarrow> (rsa@[a1]@rsb@[a2]@rsc) s\<leadsto> (rsa@[a1]@rsb@rsc)"
|
|
|
570 |
|
|
|
571 |
|
|
|
572 |
inductive
|
|
|
573 |
rrewrites:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto>* _" [100, 100] 100)
|
|
|
574 |
where
|
|
|
575 |
rs1[intro, simp]:"r \<leadsto>* r"
|
|
|
576 |
| rs2[intro]: "\<lbrakk>r1 \<leadsto>* r2; r2 \<leadsto> r3\<rbrakk> \<Longrightarrow> r1 \<leadsto>* r3"
|
|
|
577 |
|
|
|
578 |
inductive
|
|
|
579 |
srewrites:: "arexp list \<Rightarrow> arexp list \<Rightarrow> bool" ("_ s\<leadsto>* _" [100, 100] 100)
|
|
|
580 |
where
|
|
|
581 |
sss1[intro, simp]:"rs s\<leadsto>* rs"
|
|
|
582 |
| sss2[intro]: "\<lbrakk>rs1 s\<leadsto> rs2; rs2 s\<leadsto>* rs3\<rbrakk> \<Longrightarrow> rs1 s\<leadsto>* rs3"
|
|
|
583 |
|
|
|
584 |
|
|
|
585 |
lemma r_in_rstar:
|
|
|
586 |
shows "r1 \<leadsto> r2 \<Longrightarrow> r1 \<leadsto>* r2"
|
|
|
587 |
using rrewrites.intros(1) rrewrites.intros(2) by blast
|
|
|
588 |
|
|
|
589 |
lemma rrewrites_trans[trans]:
|
|
|
590 |
assumes a1: "r1 \<leadsto>* r2" and a2: "r2 \<leadsto>* r3"
|
|
|
591 |
shows "r1 \<leadsto>* r3"
|
|
|
592 |
using a2 a1
|
|
|
593 |
apply(induct r2 r3 arbitrary: r1 rule: rrewrites.induct)
|
|
|
594 |
apply(auto)
|
|
|
595 |
done
|
|
|
596 |
|
|
|
597 |
lemma srewrites_trans[trans]:
|
|
|
598 |
assumes a1: "r1 s\<leadsto>* r2" and a2: "r2 s\<leadsto>* r3"
|
|
|
599 |
shows "r1 s\<leadsto>* r3"
|
|
|
600 |
using a1 a2
|
|
|
601 |
apply(induct r1 r2 arbitrary: r3 rule: srewrites.induct)
|
|
|
602 |
apply(auto)
|
|
|
603 |
done
|
|
|
604 |
|
|
|
605 |
|
|
|
606 |
|
|
|
607 |
lemma contextrewrites0:
|
|
|
608 |
"rs1 s\<leadsto>* rs2 \<Longrightarrow> AALTs bs rs1 \<leadsto>* AALTs bs rs2"
|
|
|
609 |
apply(induct rs1 rs2 rule: srewrites.inducts)
|
|
|
610 |
apply simp
|
|
|
611 |
using bs8 r_in_rstar rrewrites_trans by blast
|
|
|
612 |
|
|
|
613 |
lemma contextrewrites1:
|
|
|
614 |
"r \<leadsto>* r' \<Longrightarrow> AALTs bs (r # rs) \<leadsto>* AALTs bs (r' # rs)"
|
|
|
615 |
apply(induct r r' rule: rrewrites.induct)
|
|
|
616 |
apply simp
|
|
|
617 |
using bs8 ss3 by blast
|
|
|
618 |
|
|
|
619 |
lemma srewrite1:
|
|
|
620 |
shows "rs1 s\<leadsto> rs2 \<Longrightarrow> (rs @ rs1) s\<leadsto> (rs @ rs2)"
|
|
|
621 |
apply(induct rs)
|
|
|
622 |
apply(auto)
|
|
|
623 |
using ss2 by auto
|
|
|
624 |
|
|
|
625 |
lemma srewrites1:
|
|
|
626 |
shows "rs1 s\<leadsto>* rs2 \<Longrightarrow> (rs @ rs1) s\<leadsto>* (rs @ rs2)"
|
|
|
627 |
apply(induct rs1 rs2 rule: srewrites.induct)
|
|
|
628 |
apply(auto)
|
|
|
629 |
using srewrite1 by blast
|
|
|
630 |
|
|
|
631 |
lemma srewrite2:
|
|
|
632 |
shows "r1 \<leadsto> r2 \<Longrightarrow> True"
|
|
|
633 |
and "rs1 s\<leadsto> rs2 \<Longrightarrow> (rs1 @ rs) s\<leadsto>* (rs2 @ rs)"
|
|
|
634 |
apply(induct rule: rrewrite_srewrite.inducts)
|
|
|
635 |
apply(auto)
|
|
|
636 |
apply (metis append_Cons append_Nil srewrites1)
|
|
|
637 |
apply(meson srewrites.simps ss3)
|
|
|
638 |
apply (meson srewrites.simps ss4)
|
|
|
639 |
apply (meson srewrites.simps ss5)
|
|
|
640 |
by (metis append_Cons append_Nil srewrites.simps ss6)
|
|
|
641 |
|
|
|
642 |
|
|
|
643 |
lemma srewrites3:
|
|
|
644 |
shows "rs1 s\<leadsto>* rs2 \<Longrightarrow> (rs1 @ rs) s\<leadsto>* (rs2 @ rs)"
|
|
|
645 |
apply(induct rs1 rs2 arbitrary: rs rule: srewrites.induct)
|
|
|
646 |
apply(auto)
|
|
|
647 |
by (meson srewrite2(2) srewrites_trans)
|
|
|
648 |
|
|
|
649 |
(*
|
|
|
650 |
lemma srewrites4:
|
|
|
651 |
assumes "rs3 s\<leadsto>* rs4" "rs1 s\<leadsto>* rs2"
|
|
|
652 |
shows "(rs1 @ rs3) s\<leadsto>* (rs2 @ rs4)"
|
|
|
653 |
using assms
|
|
|
654 |
apply(induct rs3 rs4 arbitrary: rs1 rs2 rule: srewrites.induct)
|
|
|
655 |
apply (simp add: srewrites3)
|
|
|
656 |
using srewrite1 by blast
|
|
|
657 |
*)
|
|
|
658 |
|
|
|
659 |
lemma srewrites6:
|
|
|
660 |
assumes "r1 \<leadsto>* r2"
|
|
|
661 |
shows "[r1] s\<leadsto>* [r2]"
|
|
|
662 |
using assms
|
|
|
663 |
apply(induct r1 r2 rule: rrewrites.induct)
|
|
|
664 |
apply(auto)
|
|
|
665 |
by (meson srewrites.simps srewrites_trans ss3)
|
|
|
666 |
|
|
|
667 |
lemma srewrites7:
|
|
|
668 |
assumes "rs3 s\<leadsto>* rs4" "r1 \<leadsto>* r2"
|
|
|
669 |
shows "(r1 # rs3) s\<leadsto>* (r2 # rs4)"
|
|
|
670 |
using assms
|
|
|
671 |
by (smt (verit, del_insts) append.simps srewrites1 srewrites3 srewrites6 srewrites_trans)
|
|
|
672 |
|
|
|
673 |
lemma ss6_stronger_aux:
|
|
|
674 |
shows "(rs1 @ rs2) s\<leadsto>* (rs1 @ distinctBy rs2 erase (set (map erase rs1)))"
|
|
|
675 |
apply(induct rs2 arbitrary: rs1)
|
|
|
676 |
apply(auto)
|
|
|
677 |
apply (smt (verit, best) append.assoc append.right_neutral append_Cons append_Nil split_list srewrite2(2) srewrites_trans ss6)
|
|
|
678 |
apply(drule_tac x="rs1 @ [a]" in meta_spec)
|
|
|
679 |
apply(simp)
|
|
|
680 |
done
|
|
|
681 |
|
|
|
682 |
lemma ss6_stronger:
|
|
|
683 |
shows "rs1 s\<leadsto>* distinctBy rs1 erase {}"
|
|
|
684 |
using ss6_stronger_aux[of "[]" _] by auto
|
|
|
685 |
|
|
|
686 |
lemma rewrite_preserves_fuse:
|
|
|
687 |
shows "r2 \<leadsto> r3 \<Longrightarrow> fuse bs r2 \<leadsto> fuse bs r3"
|
|
|
688 |
and "rs2 s\<leadsto> rs3 \<Longrightarrow> map (fuse bs) rs2 s\<leadsto> map (fuse bs) rs3"
|
|
|
689 |
proof(induct rule: rrewrite_srewrite.inducts)
|
|
|
690 |
case (bs3 bs1 bs2 r)
|
|
|
691 |
then show "fuse bs (ASEQ bs1 (AONE bs2) r) \<leadsto> fuse bs (fuse (bs1 @ bs2) r)"
|
|
|
692 |
by (metis fuse.simps(5) fuse_append rrewrite_srewrite.bs3)
|
|
|
693 |
next
|
|
|
694 |
case (bs7 bs1 r)
|
|
|
695 |
then show "fuse bs (AALTs bs1 [r]) \<leadsto> fuse bs (fuse bs1 r)"
|
|
|
696 |
by (metis fuse.simps(4) fuse_append rrewrite_srewrite.bs7)
|
|
|
697 |
next
|
|
|
698 |
case (ss2 rs1 rs2 r)
|
|
|
699 |
then show "map (fuse bs) (r # rs1) s\<leadsto> map (fuse bs) (r # rs2)"
|
|
|
700 |
by (simp add: rrewrite_srewrite.ss2)
|
|
|
701 |
next
|
|
|
702 |
case (ss3 r1 r2 rs)
|
|
|
703 |
then show "map (fuse bs) (r1 # rs) s\<leadsto> map (fuse bs) (r2 # rs)"
|
|
|
704 |
by (simp add: rrewrite_srewrite.ss3)
|
|
|
705 |
next
|
|
|
706 |
case (ss5 bs1 rs1 rsb)
|
|
|
707 |
have "map (fuse bs) (AALTs bs1 rs1 # rsb) = AALTs (bs @ bs1) rs1 # (map (fuse bs) rsb)" by simp
|
|
|
708 |
also have "... s\<leadsto> ((map (fuse (bs @ bs1)) rs1) @ (map (fuse bs) rsb))"
|
|
|
709 |
by (simp add: rrewrite_srewrite.ss5)
|
|
|
710 |
finally show "map (fuse bs) (AALTs bs1 rs1 # rsb) s\<leadsto> map (fuse bs) (map (fuse bs1) rs1 @ rsb)"
|
|
|
711 |
by (simp add: comp_def fuse_append)
|
|
|
712 |
next
|
|
|
713 |
case (ss6 a1 a2 rsa rsb rsc)
|
|
|
714 |
then show "map (fuse bs) (rsa @ [a1] @ rsb @ [a2] @ rsc) s\<leadsto> map (fuse bs) (rsa @ [a1] @ rsb @ rsc)"
|
|
|
715 |
apply(simp)
|
|
|
716 |
apply(rule rrewrite_srewrite.ss6[simplified])
|
|
|
717 |
apply(simp add: erase_fuse)
|
|
|
718 |
done
|
|
|
719 |
qed (auto intro: rrewrite_srewrite.intros)
|
|
|
720 |
|
|
|
721 |
lemma rewrites_fuse:
|
|
|
722 |
assumes "r1 \<leadsto>* r2"
|
|
|
723 |
shows "fuse bs r1 \<leadsto>* fuse bs r2"
|
|
|
724 |
using assms
|
|
|
725 |
apply(induction r1 r2 arbitrary: bs rule: rrewrites.induct)
|
|
|
726 |
apply(auto intro: rewrite_preserves_fuse)
|
|
|
727 |
done
|
|
|
728 |
|
|
|
729 |
|
|
|
730 |
lemma star_seq:
|
|
|
731 |
assumes "r1 \<leadsto>* r2"
|
|
|
732 |
shows "ASEQ bs r1 r3 \<leadsto>* ASEQ bs r2 r3"
|
|
|
733 |
using assms
|
|
|
734 |
apply(induct r1 r2 arbitrary: r3 rule: rrewrites.induct)
|
|
|
735 |
apply(auto intro: rrewrite_srewrite.intros)
|
|
|
736 |
done
|
|
|
737 |
|
|
|
738 |
lemma star_seq2:
|
|
|
739 |
assumes "r3 \<leadsto>* r4"
|
|
|
740 |
shows "ASEQ bs r1 r3 \<leadsto>* ASEQ bs r1 r4"
|
|
|
741 |
using assms
|
|
|
742 |
apply(induct r3 r4 arbitrary: r1 rule: rrewrites.induct)
|
|
|
743 |
apply(auto intro: rrewrite_srewrite.intros)
|
|
|
744 |
done
|
|
|
745 |
|
|
|
746 |
lemma continuous_rewrite:
|
|
|
747 |
assumes "r1 \<leadsto>* AZERO"
|
|
|
748 |
shows "ASEQ bs1 r1 r2 \<leadsto>* AZERO"
|
|
|
749 |
using assms bs1 star_seq by blast
|
|
|
750 |
|
|
|
751 |
(*
|
|
|
752 |
lemma continuous_rewrite2:
|
|
|
753 |
assumes "r1 \<leadsto>* AONE bs"
|
|
|
754 |
shows "ASEQ bs1 r1 r2 \<leadsto>* (fuse (bs1 @ bs) r2)"
|
|
|
755 |
using assms by (meson bs3 rrewrites.simps star_seq)
|
|
|
756 |
*)
|
|
|
757 |
|
|
|
758 |
lemma bsimp_aalts_simpcases:
|
|
|
759 |
shows "AONE bs \<leadsto>* bsimp (AONE bs)"
|
|
|
760 |
and "AZERO \<leadsto>* bsimp AZERO"
|
|
|
761 |
and "ACHAR bs c \<leadsto>* bsimp (ACHAR bs c)"
|
|
|
762 |
by (simp_all)
|
|
|
763 |
|
|
|
764 |
lemma bsimp_AALTs_rewrites:
|
|
|
765 |
shows "AALTs bs1 rs \<leadsto>* bsimp_AALTs bs1 rs"
|
|
|
766 |
by (smt (verit) bs6 bs7 bsimp_AALTs.elims rrewrites.simps)
|
|
|
767 |
|
|
|
768 |
lemma trivialbsimp_srewrites:
|
|
|
769 |
assumes "\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* f x"
|
|
|
770 |
shows "rs s\<leadsto>* (map f rs)"
|
|
|
771 |
using assms
|
|
|
772 |
apply(induction rs)
|
|
|
773 |
apply(simp_all add: srewrites7)
|
|
|
774 |
done
|
|
|
775 |
|
|
|
776 |
lemma fltsfrewrites: "rs s\<leadsto>* flts rs"
|
|
|
777 |
apply(induction rs rule: flts.induct)
|
|
|
778 |
apply(auto intro: rrewrite_srewrite.intros)
|
|
|
779 |
apply (meson srewrites.simps srewrites1 ss5)
|
|
|
780 |
using rs1 srewrites7 apply presburger
|
|
|
781 |
using srewrites7 apply force
|
|
|
782 |
apply (simp add: srewrites7)
|
|
|
783 |
by (simp add: srewrites7)
|
|
|
784 |
|
|
|
785 |
lemma bnullable0:
|
|
|
786 |
shows "r1 \<leadsto> r2 \<Longrightarrow> bnullable r1 = bnullable r2"
|
|
|
787 |
and "rs1 s\<leadsto> rs2 \<Longrightarrow> bnullables rs1 = bnullables rs2"
|
|
|
788 |
apply(induct rule: rrewrite_srewrite.inducts)
|
|
|
789 |
apply(auto simp add: bnullable_fuse)
|
|
|
790 |
apply (meson UnCI bnullable_fuse imageI)
|
|
|
791 |
by (metis bnullable_correctness)
|
|
|
792 |
|
|
|
793 |
|
|
|
794 |
lemma rewrites_bnullable_eq:
|
|
|
795 |
assumes "r1 \<leadsto>* r2"
|
|
|
796 |
shows "bnullable r1 = bnullable r2"
|
|
|
797 |
using assms
|
|
|
798 |
apply(induction r1 r2 rule: rrewrites.induct)
|
|
|
799 |
apply simp
|
|
|
800 |
using bnullable0(1) by auto
|
|
|
801 |
|
|
|
802 |
lemma rewrite_bmkeps_aux:
|
|
|
803 |
shows "r1 \<leadsto> r2 \<Longrightarrow> bnullable r1 \<Longrightarrow> bmkeps r1 = bmkeps r2"
|
|
|
804 |
and "rs1 s\<leadsto> rs2 \<Longrightarrow> bnullables rs1 \<Longrightarrow> bmkepss rs1 = bmkepss rs2"
|
|
|
805 |
proof (induct rule: rrewrite_srewrite.inducts)
|
|
|
806 |
case (bs3 bs1 bs2 r)
|
|
|
807 |
have IH2: "bnullable (ASEQ bs1 (AONE bs2) r)" by fact
|
|
|
808 |
then show "bmkeps (ASEQ bs1 (AONE bs2) r) = bmkeps (fuse (bs1 @ bs2) r)"
|
|
|
809 |
by (simp add: bmkeps_fuse)
|
|
|
810 |
next
|
|
|
811 |
case (bs7 bs r)
|
|
|
812 |
have IH2: "bnullable (AALTs bs [r])" by fact
|
|
|
813 |
then show "bmkeps (AALTs bs [r]) = bmkeps (fuse bs r)"
|
|
|
814 |
by (simp add: bmkeps_fuse)
|
|
|
815 |
next
|
|
|
816 |
case (ss3 r1 r2 rs)
|
|
|
817 |
have IH1: "bnullable r1 \<Longrightarrow> bmkeps r1 = bmkeps r2" by fact
|
|
|
818 |
have as: "r1 \<leadsto> r2" by fact
|
|
|
819 |
from IH1 as show "bmkepss (r1 # rs) = bmkepss (r2 # rs)"
|
|
|
820 |
by (simp add: bnullable0)
|
|
|
821 |
next
|
|
|
822 |
case (ss5 bs1 rs1 rsb)
|
|
|
823 |
have "bnullables (AALTs bs1 rs1 # rsb)" by fact
|
|
|
824 |
then show "bmkepss (AALTs bs1 rs1 # rsb) = bmkepss (map (fuse bs1) rs1 @ rsb)"
|
|
|
825 |
by (simp add: bmkepss1 bmkepss2 bmkepss_fuse bnullable_fuse)
|
|
|
826 |
next
|
|
|
827 |
case (ss6 a1 a2 rsa rsb rsc)
|
|
|
828 |
have as1: "erase a1 = erase a2" by fact
|
|
|
829 |
have as3: "bnullables (rsa @ [a1] @ rsb @ [a2] @ rsc)" by fact
|
|
|
830 |
show "bmkepss (rsa @ [a1] @ rsb @ [a2] @ rsc) = bmkepss (rsa @ [a1] @ rsb @ rsc)" using as1 as3
|
|
|
831 |
by (smt (verit, best) append_Cons bmkeps.simps(3) bmkepss.simps(2) bmkepss1 bmkepss2 bnullable_correctness)
|
|
|
832 |
qed (auto)
|
|
|
833 |
|
|
|
834 |
lemma rewrites_bmkeps:
|
|
|
835 |
assumes "r1 \<leadsto>* r2" "bnullable r1"
|
|
|
836 |
shows "bmkeps r1 = bmkeps r2"
|
|
|
837 |
using assms
|
|
|
838 |
proof(induction r1 r2 rule: rrewrites.induct)
|
|
|
839 |
case (rs1 r)
|
|
|
840 |
then show "bmkeps r = bmkeps r" by simp
|
|
|
841 |
next
|
|
|
842 |
case (rs2 r1 r2 r3)
|
|
|
843 |
then have IH: "bmkeps r1 = bmkeps r2" by simp
|
|
|
844 |
have a1: "bnullable r1" by fact
|
|
|
845 |
have a2: "r1 \<leadsto>* r2" by fact
|
|
|
846 |
have a3: "r2 \<leadsto> r3" by fact
|
|
|
847 |
have a4: "bnullable r2" using a1 a2 by (simp add: rewrites_bnullable_eq)
|
|
|
848 |
then have "bmkeps r2 = bmkeps r3"
|
|
|
849 |
using a3 bnullable0(1) rewrite_bmkeps_aux(1) by blast
|
|
|
850 |
then show "bmkeps r1 = bmkeps r3" using IH by simp
|
|
|
851 |
qed
|
|
|
852 |
|
|
|
853 |
|
|
|
854 |
lemma rewrites_to_bsimp:
|
|
|
855 |
shows "r \<leadsto>* bsimp r"
|
|
|
856 |
proof (induction r rule: bsimp.induct)
|
|
|
857 |
case (1 bs1 r1 r2)
|
|
|
858 |
have IH1: "r1 \<leadsto>* bsimp r1" by fact
|
|
|
859 |
have IH2: "r2 \<leadsto>* bsimp r2" by fact
|
|
|
860 |
{ assume as: "bsimp r1 = AZERO \<or> bsimp r2 = AZERO"
|
|
|
861 |
with IH1 IH2 have "r1 \<leadsto>* AZERO \<or> r2 \<leadsto>* AZERO" by auto
|
|
|
862 |
then have "ASEQ bs1 r1 r2 \<leadsto>* AZERO"
|
|
|
863 |
by (metis bs2 continuous_rewrite rrewrites.simps star_seq2)
|
|
|
864 |
then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" using as by auto
|
|
|
865 |
}
|
|
|
866 |
moreover
|
|
|
867 |
{ assume "\<exists>bs. bsimp r1 = AONE bs"
|
|
|
868 |
then obtain bs where as: "bsimp r1 = AONE bs" by blast
|
|
|
869 |
with IH1 have "r1 \<leadsto>* AONE bs" by simp
|
|
|
870 |
then have "ASEQ bs1 r1 r2 \<leadsto>* fuse (bs1 @ bs) r2" using bs3 star_seq by blast
|
|
|
871 |
with IH2 have "ASEQ bs1 r1 r2 \<leadsto>* fuse (bs1 @ bs) (bsimp r2)"
|
|
|
872 |
using rewrites_fuse by (meson rrewrites_trans)
|
|
|
873 |
then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 (AONE bs) r2)" by simp
|
|
|
874 |
then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by (simp add: as)
|
|
|
875 |
}
|
|
|
876 |
moreover
|
|
|
877 |
{ assume as1: "bsimp r1 \<noteq> AZERO" "bsimp r2 \<noteq> AZERO" and as2: "(\<nexists>bs. bsimp r1 = AONE bs)"
|
|
|
878 |
then have "bsimp_ASEQ bs1 (bsimp r1) (bsimp r2) = ASEQ bs1 (bsimp r1) (bsimp r2)"
|
|
|
879 |
by (simp add: bsimp_ASEQ1)
|
|
|
880 |
then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp_ASEQ bs1 (bsimp r1) (bsimp r2)" using as1 as2 IH1 IH2
|
|
|
881 |
by (metis rrewrites_trans star_seq star_seq2)
|
|
|
882 |
then have "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by simp
|
|
|
883 |
}
|
|
|
884 |
ultimately show "ASEQ bs1 r1 r2 \<leadsto>* bsimp (ASEQ bs1 r1 r2)" by blast
|
|
|
885 |
next
|
|
|
886 |
case (2 bs1 rs)
|
|
|
887 |
have IH: "\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* bsimp x" by fact
|
|
|
888 |
then have "rs s\<leadsto>* (map bsimp rs)" by (simp add: trivialbsimp_srewrites)
|
|
|
889 |
also have "... s\<leadsto>* flts (map bsimp rs)" by (simp add: fltsfrewrites)
|
|
|
890 |
also have "... s\<leadsto>* distinctBy (flts (map bsimp rs)) erase {}" by (simp add: ss6_stronger)
|
|
|
891 |
finally have "AALTs bs1 rs \<leadsto>* AALTs bs1 (distinctBy (flts (map bsimp rs)) erase {})"
|
|
|
892 |
using contextrewrites0 by blast
|
|
|
893 |
also have "... \<leadsto>* bsimp_AALTs bs1 (distinctBy (flts (map bsimp rs)) erase {})"
|
|
|
894 |
by (simp add: bsimp_AALTs_rewrites)
|
|
|
895 |
finally show "AALTs bs1 rs \<leadsto>* bsimp (AALTs bs1 rs)" by simp
|
|
|
896 |
qed (simp_all)
|
|
|
897 |
|
|
|
898 |
|
|
|
899 |
lemma to_zero_in_alt:
|
|
|
900 |
shows "AALT bs (ASEQ [] AZERO r) r2 \<leadsto> AALT bs AZERO r2"
|
|
|
901 |
by (simp add: bs1 bs8 ss3)
|
|
|
902 |
|
|
|
903 |
|
|
|
904 |
|
|
|
905 |
lemma bder_fuse_list:
|
|
|
906 |
shows "map (bder c \<circ> fuse bs1) rs1 = map (fuse bs1 \<circ> bder c) rs1"
|
|
|
907 |
apply(induction rs1)
|
|
|
908 |
apply(simp_all add: bder_fuse)
|
|
|
909 |
done
|
|
|
910 |
|
|
|
911 |
lemma rewrite_preserves_bder:
|
|
|
912 |
shows "r1 \<leadsto> r2 \<Longrightarrow> bder c r1 \<leadsto>* bder c r2"
|
|
|
913 |
and "rs1 s\<leadsto> rs2 \<Longrightarrow> map (bder c) rs1 s\<leadsto>* map (bder c) rs2"
|
|
|
914 |
proof(induction rule: rrewrite_srewrite.inducts)
|
|
|
915 |
case (bs1 bs r2)
|
|
|
916 |
show "bder c (ASEQ bs AZERO r2) \<leadsto>* bder c AZERO"
|
|
|
917 |
by (simp add: continuous_rewrite)
|
|
|
918 |
next
|
|
|
919 |
case (bs2 bs r1)
|
|
|
920 |
show "bder c (ASEQ bs r1 AZERO) \<leadsto>* bder c AZERO"
|
|
|
921 |
apply(auto)
|
|
|
922 |
apply (meson bs6 contextrewrites0 rrewrite_srewrite.bs2 rs2 ss3 ss4 sss1 sss2)
|
|
|
923 |
by (simp add: r_in_rstar rrewrite_srewrite.bs2)
|
|
|
924 |
next
|
|
|
925 |
case (bs3 bs1 bs2 r)
|
|
|
926 |
show "bder c (ASEQ bs1 (AONE bs2) r) \<leadsto>* bder c (fuse (bs1 @ bs2) r)"
|
|
|
927 |
apply(simp)
|
|
|
928 |
by (metis (no_types, lifting) bder_fuse bs8 bs7 fuse_append rrewrites.simps ss4 to_zero_in_alt)
|
|
|
929 |
next
|
|
|
930 |
case (bs4 r1 r2 bs r3)
|
|
|
931 |
have as: "r1 \<leadsto> r2" by fact
|
|
|
932 |
have IH: "bder c r1 \<leadsto>* bder c r2" by fact
|
|
|
933 |
from as IH show "bder c (ASEQ bs r1 r3) \<leadsto>* bder c (ASEQ bs r2 r3)"
|
|
|
934 |
by (metis bder.simps(5) bnullable0(1) contextrewrites1 rewrite_bmkeps_aux(1) star_seq)
|
|
|
935 |
next
|
|
|
936 |
case (bs5 r3 r4 bs r1)
|
|
|
937 |
have as: "r3 \<leadsto> r4" by fact
|
|
|
938 |
have IH: "bder c r3 \<leadsto>* bder c r4" by fact
|
|
|
939 |
from as IH show "bder c (ASEQ bs r1 r3) \<leadsto>* bder c (ASEQ bs r1 r4)"
|
|
|
940 |
apply(simp)
|
|
|
941 |
apply(auto)
|
|
|
942 |
using contextrewrites0 r_in_rstar rewrites_fuse srewrites6 srewrites7 star_seq2 apply presburger
|
|
|
943 |
using star_seq2 by blast
|
|
|
944 |
next
|
|
|
945 |
case (bs6 bs)
|
|
|
946 |
show "bder c (AALTs bs []) \<leadsto>* bder c AZERO"
|
|
|
947 |
using rrewrite_srewrite.bs6 by force
|
|
|
948 |
next
|
|
|
949 |
case (bs7 bs r)
|
|
|
950 |
show "bder c (AALTs bs [r]) \<leadsto>* bder c (fuse bs r)"
|
|
|
951 |
by (simp add: bder_fuse r_in_rstar rrewrite_srewrite.bs7)
|
|
|
952 |
next
|
|
|
953 |
case (bs8 rs1 rs2 bs)
|
|
|
954 |
have IH1: "map (bder c) rs1 s\<leadsto>* map (bder c) rs2" by fact
|
|
|
955 |
then show "bder c (AALTs bs rs1) \<leadsto>* bder c (AALTs bs rs2)"
|
|
|
956 |
using contextrewrites0 by force
|
|
|
957 |
(*next
|
|
|
958 |
case ss1
|
|
|
959 |
show "map (bder c) [] s\<leadsto>* map (bder c) []" by simp*)
|
|
|
960 |
next
|
|
|
961 |
case (ss2 rs1 rs2 r)
|
|
|
962 |
have IH1: "map (bder c) rs1 s\<leadsto>* map (bder c) rs2" by fact
|
|
|
963 |
then show "map (bder c) (r # rs1) s\<leadsto>* map (bder c) (r # rs2)"
|
|
|
964 |
by (simp add: srewrites7)
|
|
|
965 |
next
|
|
|
966 |
case (ss3 r1 r2 rs)
|
|
|
967 |
have IH: "bder c r1 \<leadsto>* bder c r2" by fact
|
|
|
968 |
then show "map (bder c) (r1 # rs) s\<leadsto>* map (bder c) (r2 # rs)"
|
|
|
969 |
by (simp add: srewrites7)
|
|
|
970 |
next
|
|
|
971 |
case (ss4 rs)
|
|
|
972 |
show "map (bder c) (AZERO # rs) s\<leadsto>* map (bder c) rs"
|
|
|
973 |
using rrewrite_srewrite.ss4 by fastforce
|
|
|
974 |
next
|
|
|
975 |
case (ss5 bs1 rs1 rsb)
|
|
|
976 |
show "map (bder c) (AALTs bs1 rs1 # rsb) s\<leadsto>* map (bder c) (map (fuse bs1) rs1 @ rsb)"
|
|
|
977 |
apply(simp)
|
|
|
978 |
using bder_fuse_list map_map rrewrite_srewrite.ss5 srewrites.simps by blast
|
|
|
979 |
next
|
|
|
980 |
case (ss6 a1 a2 bs rsa rsb)
|
|
|
981 |
have as: "erase a1 = erase a2" by fact
|
|
|
982 |
show "map (bder c) (bs @ [a1] @ rsa @ [a2] @ rsb) s\<leadsto>* map (bder c) (bs @ [a1] @ rsa @ rsb)"
|
|
|
983 |
apply(simp only: map_append)
|
|
|
984 |
by (smt (verit, best) erase_bder list.simps(8) list.simps(9) as rrewrite_srewrite.ss6 srewrites.simps)
|
|
|
985 |
qed
|
|
|
986 |
|
|
|
987 |
lemma rewrites_preserves_bder:
|
|
|
988 |
assumes "r1 \<leadsto>* r2"
|
|
|
989 |
shows "bder c r1 \<leadsto>* bder c r2"
|
|
|
990 |
using assms
|
|
|
991 |
apply(induction r1 r2 rule: rrewrites.induct)
|
|
|
992 |
apply(simp_all add: rewrite_preserves_bder rrewrites_trans)
|
|
|
993 |
done
|
|
|
994 |
|
|
|
995 |
|
|
|
996 |
lemma central:
|
|
|
997 |
shows "bders r s \<leadsto>* bders_simp r s"
|
|
|
998 |
proof(induct s arbitrary: r rule: rev_induct)
|
|
|
999 |
case Nil
|
|
|
1000 |
then show "bders r [] \<leadsto>* bders_simp r []" by simp
|
|
|
1001 |
next
|
|
|
1002 |
case (snoc x xs)
|
|
|
1003 |
have IH: "\<And>r. bders r xs \<leadsto>* bders_simp r xs" by fact
|
|
|
1004 |
have "bders r (xs @ [x]) = bders (bders r xs) [x]" by (simp add: bders_append)
|
|
|
1005 |
also have "... \<leadsto>* bders (bders_simp r xs) [x]" using IH
|
|
|
1006 |
by (simp add: rewrites_preserves_bder)
|
|
|
1007 |
also have "... \<leadsto>* bders_simp (bders_simp r xs) [x]" using IH
|
|
|
1008 |
by (simp add: rewrites_to_bsimp)
|
|
|
1009 |
finally show "bders r (xs @ [x]) \<leadsto>* bders_simp r (xs @ [x])"
|
|
|
1010 |
by (simp add: bders_simp_append)
|
|
|
1011 |
qed
|
|
|
1012 |
|
|
|
1013 |
lemma main_aux:
|
|
|
1014 |
assumes "bnullable (bders r s)"
|
|
|
1015 |
shows "bmkeps (bders r s) = bmkeps (bders_simp r s)"
|
|
|
1016 |
proof -
|
|
|
1017 |
have "bders r s \<leadsto>* bders_simp r s" by (rule central)
|
|
|
1018 |
then
|
|
|
1019 |
show "bmkeps (bders r s) = bmkeps (bders_simp r s)" using assms
|
|
|
1020 |
by (rule rewrites_bmkeps)
|
|
|
1021 |
qed
|
|
|
1022 |
|
|
|
1023 |
|
|
|
1024 |
theorem main_blexer_simp:
|
|
|
1025 |
shows "blexer r s = blexer_simp r s"
|
|
|
1026 |
unfolding blexer_def blexer_simp_def
|
|
|
1027 |
by (metis central main_aux rewrites_bnullable_eq)
|
|
|
1028 |
|
|
|
1029 |
|
|
|
1030 |
theorem blexersimp_correctness:
|
|
|
1031 |
shows "lexer r s = blexer_simp r s"
|
|
|
1032 |
using blexer_correctness main_blexer_simp by simp
|
|
|
1033 |
|
|
|
1034 |
|
|
|
1035 |
(* some tests *)
|
|
|
1036 |
|
|
|
1037 |
lemma asize_fuse:
|
|
|
1038 |
shows "asize (fuse bs r) = asize r"
|
|
|
1039 |
apply(induct r arbitrary: bs)
|
|
|
1040 |
apply(auto)
|
|
|
1041 |
done
|
|
|
1042 |
|
|
|
1043 |
lemma asize_rewrite2:
|
|
|
1044 |
shows "r1 \<leadsto> r2 \<Longrightarrow> asize r1 \<ge> asize r2"
|
|
|
1045 |
and "rs1 s\<leadsto> rs2 \<Longrightarrow> (sum_list (map asize rs1)) \<ge> (sum_list (map asize rs2))"
|
|
|
1046 |
apply(induct rule: rrewrite_srewrite.inducts)
|
|
|
1047 |
apply(auto simp add: asize_fuse comp_def)
|
|
|
1048 |
done
|
|
|
1049 |
|
|
|
1050 |
lemma asize_rrewrites:
|
|
|
1051 |
assumes "r1 \<leadsto>* r2"
|
|
|
1052 |
shows "asize r1 \<ge> asize r2"
|
|
|
1053 |
using assms
|
|
|
1054 |
apply(induct rule: rrewrites.induct)
|
|
|
1055 |
apply(auto)
|
|
|
1056 |
using asize_rewrite2(1) le_trans by blast
|
|
|
1057 |
|
|
|
1058 |
|
|
|
1059 |
|
|
|
1060 |
fun asize2 :: "arexp \<Rightarrow> nat" where
|
|
|
1061 |
"asize2 AZERO = 1"
|
|
|
1062 |
| "asize2 (AONE cs) = 1"
|
|
|
1063 |
| "asize2 (ACHAR cs c) = 1"
|
|
|
1064 |
| "asize2 (AALTs cs rs) = Suc (Suc (sum_list (map asize2 rs)))"
|
|
|
1065 |
| "asize2 (ASEQ cs r1 r2) = Suc (asize2 r1 + asize2 r2)"
|
|
|
1066 |
| "asize2 (ASTAR cs r) = Suc (asize2 r)"
|
|
|
1067 |
|
|
|
1068 |
|
|
|
1069 |
lemma asize2_fuse:
|
|
|
1070 |
shows "asize2 (fuse bs r) = asize2 r"
|
|
|
1071 |
apply(induct r arbitrary: bs)
|
|
|
1072 |
apply(auto)
|
|
|
1073 |
done
|
|
|
1074 |
|
|
|
1075 |
lemma asize2_not_zero:
|
|
|
1076 |
shows "0 < asize2 r"
|
|
|
1077 |
apply(induct r)
|
|
|
1078 |
apply(auto)
|
|
|
1079 |
done
|
|
|
1080 |
|
|
|
1081 |
lemma asize_rewrite:
|
|
|
1082 |
shows "r1 \<leadsto> r2 \<Longrightarrow> asize2 r1 > asize2 r2"
|
|
|
1083 |
and "rs1 s\<leadsto> rs2 \<Longrightarrow> (sum_list (map asize2 rs1)) > (sum_list (map asize2 rs2))"
|
|
|
1084 |
apply(induct rule: rrewrite_srewrite.inducts)
|
|
|
1085 |
apply(auto simp add: asize2_fuse comp_def)
|
|
|
1086 |
apply(simp add: asize2_not_zero)
|
|
|
1087 |
done
|
|
|
1088 |
|
|
|
1089 |
lemma asize2_bsimp_ASEQ:
|
|
|
1090 |
shows "asize2 (bsimp_ASEQ bs r1 r2) \<le> Suc (asize2 r1 + asize2 r2)"
|
|
|
1091 |
apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
|
|
|
1092 |
apply(auto)
|
|
|
1093 |
done
|
|
|
1094 |
|
|
|
1095 |
lemma asize2_bsimp_AALTs:
|
|
|
1096 |
shows "asize2 (bsimp_AALTs bs rs) \<le> Suc (Suc (sum_list (map asize2 rs)))"
|
|
|
1097 |
apply(induct bs rs rule: bsimp_AALTs.induct)
|
|
|
1098 |
apply(auto simp add: asize2_fuse)
|
|
|
1099 |
done
|
|
|
1100 |
|
|
|
1101 |
lemma distinctBy_asize2:
|
|
|
1102 |
shows "sum_list (map asize2 (distinctBy rs f acc)) \<le> sum_list (map asize2 rs)"
|
|
|
1103 |
apply(induct rs f acc rule: distinctBy.induct)
|
|
|
1104 |
apply(auto)
|
|
|
1105 |
done
|
|
|
1106 |
|
|
|
1107 |
lemma flts_asize2:
|
|
|
1108 |
shows "sum_list (map asize2 (flts rs)) \<le> sum_list (map asize2 rs)"
|
|
|
1109 |
apply(induct rs rule: flts.induct)
|
|
|
1110 |
apply(auto simp add: comp_def asize2_fuse)
|
|
|
1111 |
done
|
|
|
1112 |
|
|
|
1113 |
lemma sumlist_asize2:
|
|
|
1114 |
assumes "\<And>x. x \<in> set rs \<Longrightarrow> asize2 (f x) \<le> asize2 x"
|
|
|
1115 |
shows "sum_list (map asize2 (map f rs)) \<le> sum_list (map asize2 rs)"
|
|
|
1116 |
using assms
|
|
|
1117 |
apply(induct rs)
|
|
|
1118 |
apply(auto simp add: comp_def)
|
|
|
1119 |
by (simp add: add_le_mono)
|
|
|
1120 |
|
|
|
1121 |
lemma test0:
|
|
|
1122 |
assumes "r1 \<leadsto>* r2"
|
|
|
1123 |
shows "r1 = r2 \<or> (\<exists>r3. r1 \<leadsto> r3 \<and> r3 \<leadsto>* r2)"
|
|
|
1124 |
using assms
|
|
|
1125 |
apply(induct r1 r2 rule: rrewrites.induct)
|
|
|
1126 |
apply(auto)
|
|
|
1127 |
done
|
|
|
1128 |
|
|
|
1129 |
lemma test2:
|
|
|
1130 |
assumes "r1 \<leadsto>* r2"
|
|
|
1131 |
shows "asize2 r1 \<ge> asize2 r2"
|
|
|
1132 |
using assms
|
|
|
1133 |
apply(induct r1 r2 rule: rrewrites.induct)
|
|
|
1134 |
apply(auto)
|
|
|
1135 |
using asize_rewrite(1) by fastforce
|
|
|
1136 |
|
|
|
1137 |
|
|
|
1138 |
lemma test3:
|
|
|
1139 |
shows "r = bsimp r \<or> (asize2 (bsimp r) < asize2 r)"
|
|
|
1140 |
proof -
|
|
|
1141 |
have "r \<leadsto>* bsimp r"
|
|
|
1142 |
by (simp add: rewrites_to_bsimp)
|
|
|
1143 |
then have "r = bsimp r \<or> (\<exists>r3. r \<leadsto> r3 \<and> r3 \<leadsto>* bsimp r)"
|
|
|
1144 |
using test0 by blast
|
|
|
1145 |
then show ?thesis
|
|
|
1146 |
by (meson asize_rewrite(1) dual_order.strict_trans2 test2)
|
|
|
1147 |
qed
|
|
|
1148 |
|
|
|
1149 |
lemma test3Q:
|
|
|
1150 |
shows "r = bsimp r \<or> (asize (bsimp r) \<le> asize r)"
|
|
|
1151 |
proof -
|
|
|
1152 |
have "r \<leadsto>* bsimp r"
|
|
|
1153 |
by (simp add: rewrites_to_bsimp)
|
|
|
1154 |
then have "r = bsimp r \<or> (\<exists>r3. r \<leadsto> r3 \<and> r3 \<leadsto>* bsimp r)"
|
|
|
1155 |
using test0 by blast
|
|
|
1156 |
then show ?thesis
|
|
|
1157 |
using asize_rewrite2(1) asize_rrewrites le_trans by blast
|
|
|
1158 |
qed
|
|
|
1159 |
|
|
|
1160 |
lemma test4:
|
|
|
1161 |
shows "asize2 (bsimp (bsimp r)) \<le> asize2 (bsimp r)"
|
|
|
1162 |
apply(induct r rule: bsimp.induct)
|
|
|
1163 |
apply(auto)
|
|
|
1164 |
using rewrites_to_bsimp test2 apply fastforce
|
|
|
1165 |
using rewrites_to_bsimp test2 by presburger
|
|
|
1166 |
|
|
|
1167 |
lemma test4Q:
|
|
|
1168 |
shows "asize (bsimp (bsimp r)) \<le> asize (bsimp r)"
|
|
|
1169 |
apply(induct r rule: bsimp.induct)
|
|
|
1170 |
apply(auto)
|
|
|
1171 |
apply (metis order_refl test3Q)
|
|
|
1172 |
by (metis le_refl test3Q)
|
|
|
1173 |
|
|
|
1174 |
|
|
|
1175 |
|
|
|
1176 |
lemma testb0:
|
|
|
1177 |
shows "fuse bs1 (bsimp_ASEQ bs r1 r2) = bsimp_ASEQ (bs1 @ bs) r1 r2"
|
|
|
1178 |
apply(induct bs r1 r2 arbitrary: bs1 rule: bsimp_ASEQ.induct)
|
|
|
1179 |
apply(auto)
|
|
|
1180 |
done
|
|
|
1181 |
|
|
|
1182 |
lemma testb1:
|
|
|
1183 |
shows "fuse bs1 (bsimp_AALTs bs rs) = bsimp_AALTs (bs1 @ bs) rs"
|
|
|
1184 |
apply(induct bs rs arbitrary: bs1 rule: bsimp_AALTs.induct)
|
|
|
1185 |
apply(auto simp add: fuse_append)
|
|
|
1186 |
done
|
|
|
1187 |
|
|
|
1188 |
lemma testb2:
|
|
|
1189 |
shows "bsimp (bsimp_ASEQ bs r1 r2) = bsimp_ASEQ bs (bsimp r1) (bsimp r2)"
|
|
|
1190 |
apply(induct bs r1 r2 rule: bsimp_ASEQ.induct)
|
|
|
1191 |
apply(auto simp add: testb0 testb1)
|
|
|
1192 |
done
|
|
|
1193 |
|
|
|
1194 |
lemma testb3:
|
|
|
1195 |
shows "\<nexists>r'. (bsimp r \<leadsto> r') \<and> asize2 (bsimp r) > asize2 r'"
|
|
|
1196 |
apply(induct r rule: bsimp.induct)
|
|
|
1197 |
apply(auto)
|
|
|
1198 |
defer
|
|
|
1199 |
defer
|
|
|
1200 |
using rrewrite.cases apply blast
|
|
|
1201 |
using rrewrite.cases apply blast
|
|
|
1202 |
using rrewrite.cases apply blast
|
|
|
1203 |
using rrewrite.cases apply blast
|
|
|
1204 |
oops
|
|
|
1205 |
|
|
|
1206 |
lemma testb4:
|
|
|
1207 |
assumes "sum_list (map asize rs1) \<le> sum_list (map asize rs2)"
|
|
|
1208 |
shows "asize (bsimp_AALTs bs1 rs1) \<le> Suc (asize (bsimp_AALTs bs1 rs2))"
|
|
|
1209 |
using assms
|
|
|
1210 |
apply(induct bs1 rs2 arbitrary: rs1 rule: bsimp_AALTs.induct)
|
|
|
1211 |
apply(auto)
|
|
|
1212 |
apply(case_tac rs1)
|
|
|
1213 |
apply(auto)
|
|
|
1214 |
using asize2.elims apply auto[1]
|
|
|
1215 |
apply (metis One_nat_def Zero_not_Suc asize.elims)
|
|
|
1216 |
apply(case_tac rs1)
|
|
|
1217 |
apply(auto)
|
|
|
1218 |
apply(case_tac list)
|
|
|
1219 |
apply(auto)
|
|
|
1220 |
using asize_fuse apply force
|
|
|
1221 |
apply (simp add: asize_fuse)
|
|
|
1222 |
by (smt (verit, ccfv_threshold) One_nat_def add.right_neutral asize.simps(1) asize.simps(4) asize_fuse bsimp_AALTs.elims le_Suc_eq list.map(1) list.map(2) not_less_eq_eq sum_list_simps(1) sum_list_simps(2))
|
|
|
1223 |
|
|
|
1224 |
lemma flts_asize:
|
|
|
1225 |
shows "sum_list (map asize (flts rs)) \<le> sum_list (map asize rs)"
|
|
|
1226 |
apply(induct rs rule: flts.induct)
|
|
|
1227 |
apply(auto simp add: comp_def asize_fuse)
|
|
|
1228 |
done
|
|
|
1229 |
|
|
|
1230 |
|
|
|
1231 |
lemma test5:
|
|
|
1232 |
shows "asize2 r \<ge> asize2 (bsimp r)"
|
|
|
1233 |
apply(induct r rule: bsimp.induct)
|
|
|
1234 |
apply(auto)
|
|
|
1235 |
apply (meson Suc_le_mono add_le_mono asize2_bsimp_ASEQ order_trans)
|
|
|
1236 |
apply(rule order_trans)
|
|
|
1237 |
apply(rule asize2_bsimp_AALTs)
|
|
|
1238 |
apply(simp)
|
|
|
1239 |
apply(rule order_trans)
|
|
|
1240 |
apply(rule distinctBy_asize2)
|
|
|
1241 |
apply(rule order_trans)
|
|
|
1242 |
apply(rule flts_asize2)
|
|
|
1243 |
using sumlist_asize2 by force
|
|
|
1244 |
|
|
|
1245 |
|
|
|
1246 |
fun awidth :: "arexp \<Rightarrow> nat" where
|
|
|
1247 |
"awidth AZERO = 1"
|
|
|
1248 |
| "awidth (AONE cs) = 1"
|
|
|
1249 |
| "awidth (ACHAR cs c) = 1"
|
|
|
1250 |
| "awidth (AALTs cs rs) = (sum_list (map awidth rs))"
|
|
|
1251 |
| "awidth (ASEQ cs r1 r2) = (awidth r1 + awidth r2)"
|
|
|
1252 |
| "awidth (ASTAR cs r) = (awidth r)"
|
|
|
1253 |
|
|
|
1254 |
|
|
|
1255 |
|
|
|
1256 |
lemma
|
|
|
1257 |
shows "s \<notin> L r \<Longrightarrow> blexer_simp r s = None"
|
|
|
1258 |
by (simp add: blexersimp_correctness lexer_correct_None)
|
|
|
1259 |
|
|
|
1260 |
lemma g1:
|
|
|
1261 |
"bders_simp AZERO s = AZERO"
|
|
|
1262 |
apply(induct s)
|
|
|
1263 |
apply(simp)
|
|
|
1264 |
apply(simp)
|
|
|
1265 |
done
|
|
|
1266 |
|
|
|
1267 |
lemma g2:
|
|
|
1268 |
"s \<noteq> Nil \<Longrightarrow> bders_simp (AONE bs) s = AZERO"
|
|
|
1269 |
apply(induct s)
|
|
|
1270 |
apply(simp)
|
|
|
1271 |
apply(simp)
|
|
|
1272 |
apply(case_tac s)
|
|
|
1273 |
apply(simp)
|
|
|
1274 |
apply(simp)
|
|
|
1275 |
done
|
|
|
1276 |
|
|
|
1277 |
lemma finite_pder:
|
|
|
1278 |
shows "finite (pder c r)"
|
|
|
1279 |
apply(induct c r rule: pder.induct)
|
|
|
1280 |
apply(auto)
|
|
|
1281 |
done
|
|
|
1282 |
|
|
|
1283 |
|
|
|
1284 |
|
|
|
1285 |
lemma awidth_fuse:
|
|
|
1286 |
shows "awidth (fuse bs r) = awidth r"
|
|
|
1287 |
apply(induct r arbitrary: bs)
|
|
|
1288 |
apply(auto)
|
|
|
1289 |
done
|
|
|
1290 |
|
|
|
1291 |
lemma pders_SEQs:
|
|
|
1292 |
assumes "finite A"
|
|
|
1293 |
shows "card (SEQs A (STAR r)) \<le> card A"
|
|
|
1294 |
using assms
|
|
|
1295 |
by (simp add: SEQs_eq_image card_image_le)
|
|
|
1296 |
|
|
|
1297 |
lemma binullable_intern:
|
|
|
1298 |
shows "bnullable (intern r) = nullable r"
|
|
|
1299 |
apply(induct r)
|
|
|
1300 |
apply(auto simp add: bnullable_fuse)
|
|
|
1301 |
done
|
|
|
1302 |
|
|
|
1303 |
lemma
|
|
|
1304 |
"card (pder c r) \<le> awidth (bder c (intern r))"
|
|
|
1305 |
apply(induct c r rule: pder.induct)
|
|
|
1306 |
apply(simp)
|
|
|
1307 |
apply(simp)
|
|
|
1308 |
apply(simp)
|
|
|
1309 |
apply(simp)
|
|
|
1310 |
apply(rule order_trans)
|
|
|
1311 |
apply(rule card_Un_le)
|
|
|
1312 |
apply (simp add: awidth_fuse bder_fuse)
|
|
|
1313 |
defer
|
|
|
1314 |
apply(simp)
|
|
|
1315 |
apply(rule order_trans)
|
|
|
1316 |
apply(rule pders_SEQs)
|
|
|
1317 |
using finite_pder apply presburger
|
|
|
1318 |
apply (simp add: awidth_fuse)
|
|
|
1319 |
apply(auto)
|
|
|
1320 |
apply(rule order_trans)
|
|
|
1321 |
apply(rule card_Un_le)
|
|
|
1322 |
apply(simp add: awidth_fuse)
|
|
|
1323 |
defer
|
|
|
1324 |
using binullable_intern apply blast
|
|
|
1325 |
using binullable_intern apply blast
|
|
|
1326 |
apply (smt (verit, best) SEQs_eq_image add.commute add_Suc_right card_image_le dual_order.trans finite_pder trans_le_add2)
|
|
|
1327 |
apply(subgoal_tac "card (SEQs (pder c r1) r2) \<le> card (pder c r1)")
|
|
|
1328 |
apply(linarith)
|
|
|
1329 |
by (simp add: UNION_singleton_eq_range card_image_le finite_pder)
|
|
|
1330 |
|
|
|
1331 |
lemma
|
|
|
1332 |
"card (pder c r) \<le> asize (bder c (intern r))"
|
|
|
1333 |
apply(induct c r rule: pder.induct)
|
|
|
1334 |
apply(simp)
|
|
|
1335 |
apply(simp)
|
|
|
1336 |
apply(simp)
|
|
|
1337 |
apply(simp)
|
|
|
1338 |
apply (metis add_mono_thms_linordered_semiring(1) asize_fuse bder_fuse card_Un_le le_Suc_eq order_trans)
|
|
|
1339 |
defer
|
|
|
1340 |
apply(simp)
|
|
|
1341 |
apply(rule order_trans)
|
|
|
1342 |
apply(rule pders_SEQs)
|
|
|
1343 |
using finite_pder apply presburger
|
|
|
1344 |
apply (simp add: asize_fuse)
|
|
|
1345 |
apply(simp)
|
|
|
1346 |
apply(auto)
|
|
|
1347 |
apply(rule order_trans)
|
|
|
1348 |
apply(rule card_Un_le)
|
|
|
1349 |
apply (smt (z3) SEQs_eq_image add.commute add_Suc_right add_mono_thms_linordered_semiring(1) asize_fuse card_image_le dual_order.trans finite_pder le_add1)
|
|
|
1350 |
apply(rule order_trans)
|
|
|
1351 |
apply(rule card_Un_le)
|
|
|
1352 |
using binullable_intern apply blast
|
|
|
1353 |
using binullable_intern apply blast
|
|
|
1354 |
by (smt (verit, best) SEQs_eq_image add.commute add_Suc_right card_image_le dual_order.trans finite_pder trans_le_add2)
|
|
|
1355 |
|
|
|
1356 |
lemma
|
|
|
1357 |
"card (pder c r) \<le> asize (bsimp (bder c (intern r)))"
|
|
|
1358 |
apply(induct c r rule: pder.induct)
|
|
|
1359 |
apply(simp)
|
|
|
1360 |
apply(simp)
|
|
|
1361 |
apply(simp)
|
|
|
1362 |
apply(simp)
|
|
|
1363 |
apply(rule order_trans)
|
|
|
1364 |
apply(rule card_Un_le)
|
|
|
1365 |
prefer 3
|
|
|
1366 |
apply(simp)
|
|
|
1367 |
apply(rule order_trans)
|
|
|
1368 |
apply(rule pders_SEQs)
|
|
|
1369 |
using finite_pder apply blast
|
|
|
1370 |
oops
|
|
|
1371 |
|
|
|
1372 |
|
|
|
1373 |
(* below is the idempotency of bsimp *)
|
|
|
1374 |
|
|
|
1375 |
lemma bsimp_ASEQ_fuse:
|
|
|
1376 |
shows "fuse bs1 (bsimp_ASEQ bs2 r1 r2) = bsimp_ASEQ (bs1 @ bs2) r1 r2"
|
|
|
1377 |
apply(induct r1 r2 arbitrary: bs1 bs2 rule: bsimp_ASEQ.induct)
|
|
|
1378 |
apply(auto)
|
|
|
1379 |
done
|
|
|
1380 |
|
|
|
1381 |
lemma bsimp_AALTs_fuse:
|
|
|
1382 |
assumes "\<forall>r \<in> set rs. fuse bs1 (fuse bs2 r) = fuse (bs1 @ bs2) r"
|
|
|
1383 |
shows "fuse bs1 (bsimp_AALTs bs2 rs) = bsimp_AALTs (bs1 @ bs2) rs"
|
|
|
1384 |
using assms
|
|
|
1385 |
apply(induct bs2 rs arbitrary: bs1 rule: bsimp_AALTs.induct)
|
|
|
1386 |
apply(auto)
|
|
|
1387 |
done
|
|
|
1388 |
|
|
|
1389 |
lemma bsimp_fuse:
|
|
|
1390 |
shows "fuse bs (bsimp r) = bsimp (fuse bs r)"
|
|
|
1391 |
apply(induct r arbitrary: bs)
|
|
|
1392 |
apply(simp_all add: bsimp_ASEQ_fuse bsimp_AALTs_fuse fuse_append)
|
|
|
1393 |
done
|
|
|
1394 |
|
|
|
1395 |
lemma bsimp_ASEQ_idem:
|
|
|
1396 |
assumes "bsimp (bsimp r1) = bsimp r1" "bsimp (bsimp r2) = bsimp r2"
|
|
|
1397 |
shows "bsimp (bsimp_ASEQ x1 (bsimp r1) (bsimp r2)) = bsimp_ASEQ x1 (bsimp r1) (bsimp r2)"
|
|
|
1398 |
using assms
|
|
|
1399 |
apply(case_tac "bsimp r1 = AZERO")
|
|
|
1400 |
apply(simp)
|
|
|
1401 |
apply(case_tac "bsimp r2 = AZERO")
|
|
|
1402 |
apply(simp)
|
|
|
1403 |
apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
|
|
|
1404 |
apply(auto)[1]
|
|
|
1405 |
apply (metis bsimp_fuse)
|
|
|
1406 |
apply(simp add: bsimp_ASEQ1)
|
|
|
1407 |
done
|
|
|
1408 |
|
|
|
1409 |
lemma bsimp_AALTs_idem:
|
|
|
1410 |
assumes "\<forall>r \<in> set rs. bsimp (bsimp r) = bsimp r"
|
|
|
1411 |
shows "bsimp (bsimp_AALTs bs rs) = bsimp_AALTs bs (map bsimp rs)"
|
|
|
1412 |
using assms
|
|
|
1413 |
apply(induct bs rs rule: bsimp_AALTs.induct)
|
|
|
1414 |
apply(simp)
|
|
|
1415 |
apply(simp)
|
|
|
1416 |
using bsimp_fuse apply presburger
|
|
|
1417 |
oops
|
|
|
1418 |
|
|
|
1419 |
lemma bsimp_idem_rev:
|
|
|
1420 |
shows "\<nexists>r2. bsimp r1 \<leadsto> r2"
|
|
|
1421 |
apply(induct r1 rule: bsimp.induct)
|
|
|
1422 |
apply(auto)
|
|
|
1423 |
defer
|
|
|
1424 |
defer
|
|
|
1425 |
using rrewrite.simps apply blast
|
|
|
1426 |
using rrewrite.cases apply blast
|
|
|
1427 |
using rrewrite.simps apply blast
|
|
|
1428 |
using rrewrite.cases apply blast
|
|
|
1429 |
apply(case_tac "bsimp r1 = AZERO")
|
|
|
1430 |
apply(simp)
|
|
|
1431 |
apply(case_tac "bsimp r2 = AZERO")
|
|
|
1432 |
apply(simp)
|
|
|
1433 |
apply(case_tac "\<exists>bs. bsimp r1 = AONE bs")
|
|
|
1434 |
apply(auto)[1]
|
|
|
1435 |
prefer 2
|
|
|
1436 |
apply (smt (verit, best) arexp.distinct(25) arexp.inject(3) bsimp_ASEQ1 rrewrite.simps)
|
|
|
1437 |
defer
|
|
|
1438 |
oops
|
|
|
1439 |
|
|
|
1440 |
lemma bsimp_idem:
|
|
|
1441 |
shows "bsimp (bsimp r) = bsimp r"
|
|
|
1442 |
apply(induct r rule: bsimp.induct)
|
|
|
1443 |
apply(auto)
|
|
|
1444 |
using bsimp_ASEQ_idem apply presburger
|
|
417
|
1445 |
sorry
|
|
|
1446 |
|
|
409
|
1447 |
|
|
|
1448 |
lemma neg:
|
|
|
1449 |
shows " \<not>(\<exists>r2. r1 \<leadsto> r2 \<and> (r2 \<leadsto>* bsimp r1) )"
|
|
|
1450 |
apply(rule notI)
|
|
|
1451 |
apply(erule exE)
|
|
|
1452 |
apply(erule conjE)
|
|
|
1453 |
oops
|
|
|
1454 |
|
|
|
1455 |
|
|
|
1456 |
|
|
|
1457 |
lemma reduction_always_in_bsimp:
|
|
|
1458 |
shows " \<lbrakk> r1 \<leadsto> r2 ; \<not>(r2 \<leadsto>* bsimp r1)\<rbrakk> \<Longrightarrow> False"
|
|
|
1459 |
apply(erule rrewrite.cases)
|
|
|
1460 |
apply simp
|
|
|
1461 |
apply auto
|
|
|
1462 |
|
|
|
1463 |
oops
|
|
|
1464 |
|
|
417
|
1465 |
|
|
|
1466 |
|
|
409
|
1467 |
(*
|
|
|
1468 |
AALTs [] [AZERO, AALTs(bs1, [a, b]) ]
|
|
|
1469 |
rewrite seq 1: \<leadsto> AALTs [] [ AALTs(bs1, [a, b]) ] \<leadsto>
|
|
|
1470 |
fuse [] (AALTs bs1, [a, b])
|
|
|
1471 |
rewrite seq 2: \<leadsto> AALTs [] [AZERO, (fuse bs1 a), (fuse bs1 b)]) ]
|
|
|
1472 |
|
|
|
1473 |
*)
|
|
|
1474 |
|
|
|
1475 |
lemma normal_bsimp:
|
|
|
1476 |
shows "\<nexists>r'. bsimp r \<leadsto> r'"
|
|
|
1477 |
oops
|
|
|
1478 |
|
|
|
1479 |
(*r' size bsimp r > size r'
|
|
|
1480 |
r' \<leadsto>* bsimp bsimp r
|
|
|
1481 |
size bsimp r > size r' \<ge> size bsimp bsimp r*)
|
|
|
1482 |
|
|
417
|
1483 |
fun orderedSufAux :: "nat \<Rightarrow> char list \<Rightarrow> char list list"
|
|
|
1484 |
where
|
|
421
|
1485 |
"orderedSufAux (Suc i) ss = (drop i ss) # (orderedSufAux i ss)"
|
|
417
|
1486 |
|"orderedSufAux 0 ss = Nil"
|
|
|
1487 |
|
|
|
1488 |
fun
|
|
|
1489 |
orderedSuf :: "char list \<Rightarrow> char list list"
|
|
|
1490 |
where
|
|
|
1491 |
"orderedSuf s = orderedSufAux (length s) s"
|
|
|
1492 |
|
|
421
|
1493 |
fun orderedPrefAux :: "nat \<Rightarrow> char list \<Rightarrow> char list list"
|
|
|
1494 |
where
|
|
|
1495 |
"orderedPrefAux (Suc i) ss = (take i ss) # (orderedPrefAux i ss)"
|
|
|
1496 |
|"orderedPrefAux 0 ss = Nil"
|
|
|
1497 |
|
|
|
1498 |
|
|
|
1499 |
fun orderedPref :: "char list \<Rightarrow> char list list"
|
|
|
1500 |
where
|
|
|
1501 |
"orderedPref s = orderedPrefAux (length s) s"
|
|
|
1502 |
|
|
|
1503 |
lemma shape_of_pref_1list:
|
|
|
1504 |
shows "orderedPref [c] = [[]]"
|
|
417
|
1505 |
apply auto
|
|
|
1506 |
done
|
|
|
1507 |
|
|
421
|
1508 |
lemma shape_of_suf_1list:
|
|
|
1509 |
shows "orderedSuf [c] = [[c]]"
|
|
|
1510 |
by auto
|
|
|
1511 |
|
|
|
1512 |
lemma shape_of_suf_2list:
|
|
|
1513 |
shows "orderedSuf [c2, c3] = [[c3], [c2,c3]]"
|
|
|
1514 |
by auto
|
|
|
1515 |
|
|
|
1516 |
lemma shape_of_prf_2list:
|
|
|
1517 |
shows "orderedPref [c1, c2] = [[c1], []]"
|
|
|
1518 |
by auto
|
|
|
1519 |
|
|
417
|
1520 |
|
|
|
1521 |
lemma shape_of_suf_3list:
|
|
421
|
1522 |
shows "orderedSuf [c1, c2, c3] = [[c3], [c2, c3], [c1, c2, c3]]"
|
|
|
1523 |
by auto
|
|
|
1524 |
|
|
|
1525 |
lemma throwing_elem_around:
|
|
|
1526 |
shows "orderedSuf (s1 @ [a] @ s) = (orderedSuf s) @ (map (\<lambda>s11. s11 @ s) (orderedSuf ( s1 @ [a]) ))"
|
|
|
1527 |
and "orderedSuf (s1 @ [a] @ s) = (orderedSuf ([a] @ s) @ (map (\<lambda>s11. s11 @ ([a] @ s))) (orderedSuf s1) )"
|
|
|
1528 |
sorry
|
|
|
1529 |
|
|
|
1530 |
|
|
|
1531 |
lemma suf_cons:
|
|
|
1532 |
shows "orderedSuf (s1 @ s) = (orderedSuf s) @ (map (\<lambda>s11. s11 @ s) (orderedSuf s1))"
|
|
|
1533 |
apply(induct s arbitrary: s1)
|
|
|
1534 |
apply simp
|
|
|
1535 |
apply(subgoal_tac "s1 @ a # s = (s1 @ [a]) @ s")
|
|
|
1536 |
prefer 2
|
|
|
1537 |
apply simp
|
|
|
1538 |
apply(subgoal_tac "orderedSuf (s1 @ a # s) = orderedSuf ((s1 @ [a]) @ s)")
|
|
|
1539 |
prefer 2
|
|
|
1540 |
apply presburger
|
|
|
1541 |
apply(drule_tac x="s1 @ [a]" in meta_spec)
|
|
|
1542 |
sorry
|
|
|
1543 |
|
|
|
1544 |
|
|
|
1545 |
|
|
|
1546 |
lemma shape_of_prf_3list:
|
|
|
1547 |
shows "orderedPref [c1, c2, c3] = [[c1, c2], [c1], []]"
|
|
|
1548 |
by auto
|
|
|
1549 |
|
|
|
1550 |
fun zip_concat :: "char list list \<Rightarrow> char list list \<Rightarrow> char list list"
|
|
|
1551 |
where
|
|
|
1552 |
"zip_concat (s1#ss1) (s2#ss2) = (s1@s2) # (zip_concat ss1 ss2)"
|
|
|
1553 |
| "zip_concat [] [] = []"
|
|
|
1554 |
| "zip_concat [] (s2#ss2) = s2 # (zip_concat [] ss2)"
|
|
|
1555 |
| "zip_concat (s1#ss1) [] = s1 # (zip_concat ss1 [])"
|
|
|
1556 |
|
|
|
1557 |
|
|
|
1558 |
|
|
|
1559 |
lemma compliment_pref_suf:
|
|
|
1560 |
shows "zip_concat (orderedPref s) (orderedSuf s) = replicate (length s) s "
|
|
|
1561 |
apply(induct s)
|
|
|
1562 |
apply auto[1]
|
|
|
1563 |
sorry
|
|
|
1564 |
|
|
417
|
1565 |
|
|
|
1566 |
|
|
|
1567 |
|
|
|
1568 |
datatype rrexp =
|
|
|
1569 |
RZERO
|
|
|
1570 |
| RONE
|
|
|
1571 |
| RCHAR char
|
|
|
1572 |
| RSEQ rrexp rrexp
|
|
|
1573 |
| RALTS "rrexp list"
|
|
|
1574 |
| RSTAR rrexp
|
|
|
1575 |
|
|
|
1576 |
abbreviation
|
|
|
1577 |
"RALT r1 r2 \<equiv> RALTS [r1, r2]"
|
|
|
1578 |
|
|
|
1579 |
fun
|
|
|
1580 |
rerase :: "arexp \<Rightarrow> rrexp"
|
|
|
1581 |
where
|
|
|
1582 |
"rerase AZERO = RZERO"
|
|
|
1583 |
| "rerase (AONE _) = RONE"
|
|
|
1584 |
| "rerase (ACHAR _ c) = RCHAR c"
|
|
|
1585 |
| "rerase (AALTs bs rs) = RALTS (map rerase rs)"
|
|
|
1586 |
| "rerase (ASEQ _ r1 r2) = RSEQ (rerase r1) (rerase r2)"
|
|
|
1587 |
| "rerase (ASTAR _ r) = RSTAR (rerase r)"
|
|
|
1588 |
|
|
|
1589 |
|
|
|
1590 |
|
|
|
1591 |
|
|
|
1592 |
fun
|
|
|
1593 |
rnullable :: "rrexp \<Rightarrow> bool"
|
|
|
1594 |
where
|
|
|
1595 |
"rnullable (RZERO) = False"
|
|
|
1596 |
| "rnullable (RONE ) = True"
|
|
|
1597 |
| "rnullable (RCHAR c) = False"
|
|
|
1598 |
| "rnullable (RALTS rs) = (\<exists>r \<in> set rs. rnullable r)"
|
|
|
1599 |
| "rnullable (RSEQ r1 r2) = (rnullable r1 \<and> rnullable r2)"
|
|
|
1600 |
| "rnullable (RSTAR r) = True"
|
|
|
1601 |
|
|
|
1602 |
|
|
|
1603 |
fun
|
|
|
1604 |
rder :: "char \<Rightarrow> rrexp \<Rightarrow> rrexp"
|
|
|
1605 |
where
|
|
|
1606 |
"rder c (RZERO) = RZERO"
|
|
|
1607 |
| "rder c (RONE) = RZERO"
|
|
|
1608 |
| "rder c (RCHAR d) = (if c = d then RONE else RZERO)"
|
|
|
1609 |
| "rder c (RALTS rs) = RALTS (map (rder c) rs)"
|
|
|
1610 |
| "rder c (RSEQ r1 r2) =
|
|
|
1611 |
(if rnullable r1
|
|
|
1612 |
then RALT (RSEQ (rder c r1) r2) (rder c r2)
|
|
|
1613 |
else RSEQ (rder c r1) r2)"
|
|
|
1614 |
| "rder c (RSTAR r) = RSEQ (rder c r) (RSTAR r)"
|
|
|
1615 |
|
|
|
1616 |
|
|
|
1617 |
fun
|
|
|
1618 |
rders :: "rrexp \<Rightarrow> string \<Rightarrow> rrexp"
|
|
|
1619 |
where
|
|
|
1620 |
"rders r [] = r"
|
|
|
1621 |
| "rders r (c#s) = rders (rder c r) s"
|
|
|
1622 |
|
|
|
1623 |
fun rdistinct :: "'a list \<Rightarrow>'a set \<Rightarrow> 'a list"
|
|
|
1624 |
where
|
|
|
1625 |
"rdistinct [] acc = []"
|
|
|
1626 |
| "rdistinct (x#xs) acc =
|
|
|
1627 |
(if x \<in> acc then rdistinct xs acc
|
|
|
1628 |
else x # (rdistinct xs ({x} \<union> acc)))"
|
|
|
1629 |
|
|
|
1630 |
|
|
|
1631 |
fun rflts :: "rrexp list \<Rightarrow> rrexp list"
|
|
|
1632 |
where
|
|
|
1633 |
"rflts [] = []"
|
|
|
1634 |
| "rflts (RZERO # rs) = rflts rs"
|
|
|
1635 |
| "rflts ((RALTS rs1) # rs) = rs1 @ rflts rs"
|
|
|
1636 |
| "rflts (r1 # rs) = r1 # rflts rs"
|
|
|
1637 |
|
|
|
1638 |
|
|
|
1639 |
fun rsimp_ALTs :: " rrexp list \<Rightarrow> rrexp"
|
|
|
1640 |
where
|
|
|
1641 |
"rsimp_ALTs [] = RZERO"
|
|
|
1642 |
| "rsimp_ALTs [r] = r"
|
|
|
1643 |
| "rsimp_ALTs rs = RALTS rs"
|
|
|
1644 |
|
|
|
1645 |
fun rsimp_SEQ :: " rrexp \<Rightarrow> rrexp \<Rightarrow> rrexp"
|
|
|
1646 |
where
|
|
|
1647 |
"rsimp_SEQ RZERO _ = RZERO"
|
|
|
1648 |
| "rsimp_SEQ _ RZERO = RZERO"
|
|
|
1649 |
| "rsimp_SEQ RONE r2 = r2"
|
|
|
1650 |
| "rsimp_SEQ r1 r2 = RSEQ r1 r2"
|
|
|
1651 |
|
|
|
1652 |
|
|
|
1653 |
fun rsimp :: "rrexp \<Rightarrow> rrexp"
|
|
|
1654 |
where
|
|
|
1655 |
"rsimp (RSEQ r1 r2) = rsimp_SEQ (rsimp r1) (rsimp r2)"
|
|
|
1656 |
| "rsimp (RALTS rs) = rsimp_ALTs (rdistinct (rflts (map rsimp rs)) {}) "
|
|
|
1657 |
| "rsimp r = r"
|
|
|
1658 |
|
|
|
1659 |
|
|
|
1660 |
fun
|
|
|
1661 |
rders_simp :: "rrexp \<Rightarrow> string \<Rightarrow> rrexp"
|
|
|
1662 |
where
|
|
|
1663 |
"rders_simp r [] = r"
|
|
|
1664 |
| "rders_simp r (c#s) = rders_simp (rsimp (rder c r)) s"
|
|
|
1665 |
|
|
421
|
1666 |
fun rsize :: "rrexp \<Rightarrow> nat" where
|
|
|
1667 |
"rsize RZERO = 1"
|
|
|
1668 |
| "rsize (RONE) = 1"
|
|
|
1669 |
| "rsize (RCHAR c) = 1"
|
|
|
1670 |
| "rsize (RALTS rs) = Suc (sum_list (map rsize rs))"
|
|
|
1671 |
| "rsize (RSEQ r1 r2) = Suc (rsize r1 + rsize r2)"
|
|
|
1672 |
| "rsize (RSTAR r) = Suc (rsize r)"
|
|
|
1673 |
|
|
417
|
1674 |
|
|
427
|
1675 |
|
|
|
1676 |
|
|
|
1677 |
lemma rsimp_aalts_smaller:
|
|
|
1678 |
shows "rsize (rsimp_ALTs rs) \<le> rsize (RALTS rs)"
|
|
|
1679 |
apply(induct rs)
|
|
|
1680 |
apply simp
|
|
|
1681 |
sorry
|
|
|
1682 |
|
|
|
1683 |
lemma finite_list_of_ders:
|
|
|
1684 |
shows"\<exists>dersset. ( (finite dersset) \<and> (\<forall>s. (rders_simp r s) \<in> dersset) )"
|
|
|
1685 |
sorry
|
|
|
1686 |
|
|
|
1687 |
|
|
|
1688 |
|
|
417
|
1689 |
lemma rerase_bsimp:
|
|
|
1690 |
shows "rerase (bsimp r) = rsimp (rerase r)"
|
|
|
1691 |
apply(induct r)
|
|
|
1692 |
apply auto
|
|
|
1693 |
|
|
|
1694 |
|
|
|
1695 |
sorry
|
|
|
1696 |
|
|
421
|
1697 |
|
|
417
|
1698 |
lemma rerase_bder:
|
|
|
1699 |
shows "rerase (bder c r) = rder c (rerase r)"
|
|
|
1700 |
apply(induct r)
|
|
|
1701 |
apply auto
|
|
|
1702 |
sorry
|
|
421
|
1703 |
(*
|
|
417
|
1704 |
lemma rders_shape:
|
|
421
|
1705 |
shows "s \<noteq> [] \<Longrightarrow> rders_simp (RSEQ r1 r2) s =
|
|
417
|
1706 |
rsimp (RALTS ((RSEQ (rders r1 s) r2) #
|
|
|
1707 |
(map (rders r2) (orderedSuf s))) )"
|
|
421
|
1708 |
apply(induct s arbitrary: r1 r2 rule: rev_induct)
|
|
|
1709 |
apply simp
|
|
|
1710 |
apply simp
|
|
|
1711 |
|
|
417
|
1712 |
sorry
|
|
421
|
1713 |
*)
|
|
|
1714 |
|
|
427
|
1715 |
|
|
421
|
1716 |
|
|
|
1717 |
fun rders_cond_list :: "rrexp \<Rightarrow> bool list \<Rightarrow> char list list \<Rightarrow> rrexp list"
|
|
|
1718 |
where
|
|
|
1719 |
"rders_cond_list r2 (True # bs) (s # strs) = (rders r2 s) # (rders_cond_list r2 bs strs)"
|
|
|
1720 |
| "rders_cond_list r2 (False # bs) (s # strs) = rders_cond_list r2 bs strs"
|
|
|
1721 |
| "rders_cond_list r2 [] s = []"
|
|
|
1722 |
| "rders_cond_list r2 bs [] = []"
|
|
|
1723 |
|
|
|
1724 |
fun nullable_bools :: "rrexp \<Rightarrow> char list list \<Rightarrow> bool list"
|
|
|
1725 |
where
|
|
|
1726 |
"nullable_bools r (s#strs) = (rnullable (rders r s)) # (nullable_bools r strs) "
|
|
|
1727 |
|"nullable_bools r [] = []"
|
|
417
|
1728 |
|
|
427
|
1729 |
thm rsimp_SEQ.simps
|
|
|
1730 |
|
|
|
1731 |
lemma idiot:
|
|
|
1732 |
shows "rsimp_SEQ RONE r = r"
|
|
|
1733 |
apply(case_tac r)
|
|
|
1734 |
apply simp_all
|
|
|
1735 |
done
|
|
|
1736 |
|
|
|
1737 |
lemma no_dup_after_simp:
|
|
|
1738 |
shows "RALTS rs = rsimp r \<Longrightarrow> distinct rs"
|
|
|
1739 |
sorry
|
|
|
1740 |
|
|
|
1741 |
lemma no_further_dB_after_simp:
|
|
|
1742 |
shows "RALTS rs = rsimp r \<Longrightarrow> rdistinct rs {} = rs"
|
|
|
1743 |
sorry
|
|
|
1744 |
|
|
|
1745 |
lemma longlist_withstands_rsimp_alts:
|
|
|
1746 |
shows "length rs \<ge> 2 \<Longrightarrow> rsimp_ALTs rs = RALTS rs"
|
|
|
1747 |
sorry
|
|
|
1748 |
|
|
|
1749 |
lemma no_alt_short_list_after_simp:
|
|
|
1750 |
shows "RALTS rs = rsimp r \<Longrightarrow> rsimp_ALTs rs = RALTS rs"
|
|
|
1751 |
sorry
|
|
|
1752 |
|
|
|
1753 |
lemma idiot2:
|
|
|
1754 |
shows " \<lbrakk>r1 \<noteq> RZERO; r1 \<noteq> RONE;r2 \<noteq> RZERO\<rbrakk>
|
|
|
1755 |
\<Longrightarrow> rsimp_SEQ r1 r2 = RSEQ r1 r2"
|
|
|
1756 |
apply(case_tac r1)
|
|
|
1757 |
apply(case_tac r2)
|
|
|
1758 |
apply simp_all
|
|
|
1759 |
apply(case_tac r2)
|
|
|
1760 |
apply simp_all
|
|
|
1761 |
apply(case_tac r2)
|
|
|
1762 |
apply simp_all
|
|
|
1763 |
apply(case_tac r2)
|
|
|
1764 |
apply simp_all
|
|
|
1765 |
apply(case_tac r2)
|
|
|
1766 |
apply simp_all
|
|
|
1767 |
done
|
|
|
1768 |
|
|
|
1769 |
lemma rsimp_aalts_another:
|
|
|
1770 |
shows "\<forall>r \<in> (set (map rsimp ((RSEQ (rders r1 s) r2) #
|
|
|
1771 |
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) )) ). (rsize r) < N "
|
|
|
1772 |
sorry
|
|
|
1773 |
|
|
|
1774 |
|
|
|
1775 |
|
|
|
1776 |
lemma shape_derssimpseq_onechar:
|
|
|
1777 |
shows " rerase (bders_simp r [c]) = rerase (bsimp (bders r [c]))"
|
|
|
1778 |
and "rders_simp (RSEQ r1 r2) [c] =
|
|
|
1779 |
rsimp (RALTS ((RSEQ (rders r1 [c]) r2) #
|
|
|
1780 |
(rders_cond_list r2 (nullable_bools r1 (orderedPref [c])) (orderedSuf [c]))) )"
|
|
|
1781 |
apply simp
|
|
|
1782 |
apply(simp add: rders.simps)
|
|
|
1783 |
apply(case_tac "rsimp (rder c r1) = RZERO")
|
|
|
1784 |
apply auto
|
|
|
1785 |
apply(case_tac "rsimp (rder c r1) = RONE")
|
|
|
1786 |
apply auto
|
|
|
1787 |
apply(subgoal_tac "rsimp_SEQ RONE (rsimp r2) = rsimp r2")
|
|
|
1788 |
prefer 2
|
|
|
1789 |
using idiot
|
|
|
1790 |
apply simp
|
|
|
1791 |
apply(subgoal_tac "rsimp_SEQ RONE (rsimp r2) = rsimp_ALTs (rdistinct (rflts [rsimp r2]) {})")
|
|
|
1792 |
prefer 2
|
|
|
1793 |
apply auto
|
|
|
1794 |
apply(case_tac "rsimp r2")
|
|
|
1795 |
apply auto
|
|
|
1796 |
apply(subgoal_tac "rdistinct x5 {} = x5")
|
|
|
1797 |
prefer 2
|
|
|
1798 |
using no_further_dB_after_simp
|
|
|
1799 |
apply metis
|
|
|
1800 |
apply(subgoal_tac "rsimp_ALTs (rdistinct x5 {}) = rsimp_ALTs x5")
|
|
|
1801 |
prefer 2
|
|
|
1802 |
apply fastforce
|
|
|
1803 |
apply auto
|
|
|
1804 |
apply (metis no_alt_short_list_after_simp)
|
|
|
1805 |
apply (case_tac "rsimp r2 = RZERO")
|
|
|
1806 |
apply(subgoal_tac "rsimp_SEQ (rsimp (rder c r1)) (rsimp r2) = RZERO")
|
|
|
1807 |
prefer 2
|
|
|
1808 |
apply(case_tac "rsimp ( rder c r1)")
|
|
|
1809 |
apply auto
|
|
|
1810 |
apply(subgoal_tac "rsimp_SEQ (rsimp (rder c r1)) (rsimp r2) = RSEQ (rsimp (rder c r1)) (rsimp r2)")
|
|
|
1811 |
prefer 2
|
|
|
1812 |
apply auto
|
|
|
1813 |
apply(metis idiot2)
|
|
|
1814 |
done
|
|
|
1815 |
|
|
|
1816 |
lemma rders__onechar:
|
|
|
1817 |
shows " (rders_simp r [c]) = (rsimp (rders r [c]))"
|
|
|
1818 |
by simp
|
|
|
1819 |
|
|
|
1820 |
lemma rders_append:
|
|
|
1821 |
"rders c (s1 @ s2) = rders (rders c s1) s2"
|
|
|
1822 |
apply(induct s1 arbitrary: c s2)
|
|
|
1823 |
apply(simp_all)
|
|
|
1824 |
done
|
|
|
1825 |
|
|
|
1826 |
lemma rders_simp_append:
|
|
|
1827 |
"rders_simp c (s1 @ s2) = rders_simp (rders_simp c s1) s2"
|
|
|
1828 |
apply(induct s1 arbitrary: c s2)
|
|
|
1829 |
apply(simp_all)
|
|
|
1830 |
done
|
|
|
1831 |
|
|
|
1832 |
lemma inside_simp_removal:
|
|
|
1833 |
shows " rsimp (rder x (rsimp r)) = rsimp (rder x r)"
|
|
417
|
1834 |
|
|
427
|
1835 |
sorry
|
|
|
1836 |
|
|
|
1837 |
lemma set_related_list:
|
|
|
1838 |
shows "distinct rs \<Longrightarrow> length rs = card (set rs)"
|
|
|
1839 |
by (simp add: distinct_card)
|
|
|
1840 |
(*this section deals with the property of distinctBy: creates a list without duplicates*)
|
|
|
1841 |
lemma rdistinct_never_added_twice:
|
|
|
1842 |
shows "rdistinct (a # rs) {a} = rdistinct rs {a}"
|
|
|
1843 |
by force
|
|
|
1844 |
|
|
|
1845 |
|
|
|
1846 |
lemma rdistinct_does_the_job:
|
|
|
1847 |
shows "distinct (rdistinct rs s)"
|
|
|
1848 |
apply(induct rs arbitrary: s)
|
|
|
1849 |
apply simp
|
|
|
1850 |
apply simp
|
|
|
1851 |
sorry
|
|
|
1852 |
|
|
|
1853 |
|
|
|
1854 |
|
|
|
1855 |
|
|
|
1856 |
(*this section deals with the property of distinctBy: creates a list without duplicates*)
|
|
|
1857 |
|
|
|
1858 |
lemma rders_simp_same_simpders:
|
|
|
1859 |
shows "s \<noteq> [] \<Longrightarrow> rders_simp r s = rsimp (rders r s)"
|
|
|
1860 |
apply(induct s rule: rev_induct)
|
|
|
1861 |
apply simp
|
|
|
1862 |
apply(case_tac "xs = []")
|
|
|
1863 |
apply simp
|
|
|
1864 |
apply(simp add: rders_append rders_simp_append)
|
|
|
1865 |
using inside_simp_removal by blast
|
|
|
1866 |
|
|
422
|
1867 |
lemma shape_derssimp_seq:
|
|
427
|
1868 |
shows "\<lbrakk>s \<noteq>[] \<rbrakk> \<Longrightarrow> (rders_simp r s) = (rsimp (rders r s))"
|
|
421
|
1869 |
and "\<lbrakk>s \<noteq> []\<rbrakk> \<Longrightarrow> rders_simp (RSEQ r1 r2) s =
|
|
|
1870 |
rsimp (RALTS ((RSEQ (rders r1 s) r2) #
|
|
|
1871 |
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) )"
|
|
|
1872 |
apply(induct s arbitrary: r r1 r2 rule: rev_induct)
|
|
|
1873 |
apply simp
|
|
|
1874 |
apply simp
|
|
|
1875 |
apply(case_tac "xs = []")
|
|
|
1876 |
|
|
|
1877 |
apply (simp add: bders_simp_append )
|
|
|
1878 |
apply(subgoal_tac "rerase (bsimp (bder x (bders_simp r xs))) = (rsimp (rerase (bder x (bders_simp r xs)))) ")
|
|
|
1879 |
prefer 2
|
|
|
1880 |
apply (simp add: rerase_bsimp)
|
|
|
1881 |
apply(subgoal_tac "(rsimp (rerase (bder x (bders_simp r xs)))) = (rsimp (rder x (rerase (bders_simp r xs))))")
|
|
|
1882 |
|
|
|
1883 |
apply(subgoal_tac "xs \<noteq> [] \<Longrightarrow> rsimp (rder x (rerase (bders_simp r xs))) = rsimp (rder x (rerase (bsimp (bders r xs))))")
|
|
|
1884 |
prefer 2
|
|
427
|
1885 |
apply presburger
|
|
|
1886 |
apply(case_tac "xs = []")
|
|
421
|
1887 |
sorry
|
|
417
|
1888 |
|
|
427
|
1889 |
|
|
422
|
1890 |
lemma shape_derssimp_alts:
|
|
427
|
1891 |
shows "s \<noteq> [] \<Longrightarrow> rders_simp (RALTS rs) s = rsimp (RALTS (map (\<lambda>r. rders r s) rs))"
|
|
|
1892 |
apply(case_tac "s")
|
|
|
1893 |
apply simp
|
|
|
1894 |
apply simp
|
|
|
1895 |
sorry
|
|
428
|
1896 |
(*
|
|
|
1897 |
fun rexp_encode :: "rrexp \<Rightarrow> nat"
|
|
|
1898 |
where
|
|
|
1899 |
"rexp_encode RZERO = 0"
|
|
|
1900 |
|"rexp_encode RONE = 1"
|
|
|
1901 |
|"rexp_encode (RCHAR c) = 2"
|
|
|
1902 |
|"rexp_encode (RSEQ r1 r2) = ( 2 ^ (rexp_encode r1)) "
|
|
|
1903 |
*)
|
|
|
1904 |
lemma finite_chars:
|
|
|
1905 |
shows " \<exists>N. ( (\<forall>r \<in> (set rs). \<exists>c. r = RCHAR c) \<and> (distinct rs) \<longrightarrow> length rs < N)"
|
|
|
1906 |
apply(rule_tac x = "Suc 256" in exI)
|
|
|
1907 |
sorry
|
|
|
1908 |
|
|
430
|
1909 |
definition all_chars :: "int \<Rightarrow> char list"
|
|
|
1910 |
where "all_chars n = map char_of [0..n]"
|
|
428
|
1911 |
|
|
|
1912 |
fun rexp_enum :: "nat \<Rightarrow> rrexp list"
|
|
|
1913 |
where
|
|
|
1914 |
"rexp_enum 0 = []"
|
|
430
|
1915 |
|"rexp_enum (Suc 0) = RALTS [] # RZERO # (RONE # (map RCHAR (all_chars 255)))"
|
|
|
1916 |
|"rexp_enum (Suc n) = [(RSEQ r1 r2). r1 \<in> set (rexp_enum i) \<and> r2 \<in> set (rexp_enum j) \<and> i + j = n]"
|
|
|
1917 |
|
|
428
|
1918 |
|
|
|
1919 |
lemma finite_sized_rexp_forms_finite_set:
|
|
|
1920 |
shows " \<exists>SN. ( \<forall>r.( rsize r < N \<longrightarrow> r \<in> SN)) \<and> (finite SN)"
|
|
|
1921 |
apply(induct N)
|
|
|
1922 |
apply simp
|
|
|
1923 |
apply auto
|
|
|
1924 |
(*\<lbrakk>\<forall>r. rsize r < N \<longrightarrow> r \<in> SN; finite SN\<rbrakk> \<Longrightarrow> \<exists>SN. (\<forall>r. rsize r < Suc N \<longrightarrow> r \<in> SN) \<and> finite SN*)
|
|
|
1925 |
(* \<And>N. \<exists>SN. (\<forall>r. rsize r < N \<longrightarrow> r \<in> SN) \<and> finite SN \<Longrightarrow> \<exists>SN. (\<forall>r. rsize r < Suc N \<longrightarrow> r \<in> SN) \<and> finite SN*)
|
|
|
1926 |
sorry
|
|
422
|
1927 |
|
|
|
1928 |
|
|
421
|
1929 |
lemma finite_size_finite_regx:
|
|
428
|
1930 |
shows " \<exists>l. \<forall>rs. ((\<forall>r \<in> (set rs). rsize r < N) \<and> (distinct rs) \<longrightarrow> (length rs) < l) "
|
|
430
|
1931 |
sorry
|
|
421
|
1932 |
|
|
|
1933 |
(*below probably needs proved concurrently*)
|
|
|
1934 |
|
|
|
1935 |
lemma finite_r1r2_ders_list:
|
|
427
|
1936 |
shows "(\<forall>s. rsize (rders_simp r1 s) < N1 \<and> rsize (rders_simp r2 s) < N2)
|
|
|
1937 |
\<Longrightarrow> \<exists>l. \<forall>s.
|
|
|
1938 |
(length (rdistinct (map rsimp (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) {}) ) < l "
|
|
421
|
1939 |
sorry
|
|
|
1940 |
|
|
427
|
1941 |
(*
|
|
|
1942 |
\<lbrakk>s \<noteq> []\<rbrakk> \<Longrightarrow> rders_simp (RSEQ r1 r2) s =
|
|
|
1943 |
rsimp (RALTS ((RSEQ (rders r1 s) r2) #
|
|
|
1944 |
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) )
|
|
|
1945 |
*)
|
|
|
1946 |
|
|
|
1947 |
lemma finite_width_alt:
|
|
|
1948 |
shows "(\<forall>s. rsize (rders_simp r1 s) < N1 \<and> rsize (rders_simp r2 s) < N2)
|
|
|
1949 |
\<Longrightarrow> \<exists>N3. \<forall>s. rsize (rsimp (RALTS ((RSEQ (rders r1 s) r2) #
|
|
|
1950 |
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) )) < N3"
|
|
|
1951 |
|
|
|
1952 |
sorry
|
|
|
1953 |
|
|
|
1954 |
|
|
|
1955 |
lemma empty_diff:
|
|
|
1956 |
shows "s = [] \<Longrightarrow>
|
|
|
1957 |
(rsize (rders_simp (RSEQ r1 r2) s)) \<le>
|
|
|
1958 |
(max
|
|
|
1959 |
(rsize (rsimp (RALTS (RSEQ (rders r1 s) r2 # rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)))))
|
|
|
1960 |
(Suc (rsize r1 + rsize r2)) ) "
|
|
|
1961 |
apply simp
|
|
|
1962 |
done
|
|
|
1963 |
|
|
421
|
1964 |
lemma finite_seq:
|
|
|
1965 |
shows "(\<forall>s. rsize (rders_simp r1 s) < N1 \<and> rsize (rders_simp r2 s) < N2)
|
|
|
1966 |
\<Longrightarrow> \<exists>N3.\<forall>s.(rsize (rders_simp (RSEQ r1 r2) s)) < N3"
|
|
427
|
1967 |
apply(frule finite_width_alt)
|
|
|
1968 |
apply(erule exE)
|
|
|
1969 |
apply(rule_tac x = "max (N3+2) (Suc (Suc (rsize r1) + (rsize r2)))" in exI)
|
|
|
1970 |
apply(rule allI)
|
|
|
1971 |
apply(case_tac "s = []")
|
|
|
1972 |
prefer 2
|
|
|
1973 |
apply (simp add: less_SucI shape_derssimp_seq(2))
|
|
|
1974 |
apply (meson less_SucI less_max_iff_disj)
|
|
|
1975 |
apply simp
|
|
|
1976 |
done
|
|
|
1977 |
|
|
|
1978 |
|
|
|
1979 |
(*For star related error bound*)
|
|
|
1980 |
|
|
|
1981 |
lemma star_is_a_singleton_list_derc:
|
|
|
1982 |
shows " \<exists>Ss. rders_simp (RSTAR r) [c] = rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)"
|
|
|
1983 |
apply simp
|
|
|
1984 |
apply(rule_tac x = "[[c]]" in exI)
|
|
|
1985 |
apply auto
|
|
|
1986 |
done
|
|
|
1987 |
|
|
|
1988 |
lemma rder_rsimp_ALTs_commute:
|
|
|
1989 |
shows " (rder x (rsimp_ALTs rs)) = rsimp_ALTs (map (rder x) rs)"
|
|
|
1990 |
apply(induct rs)
|
|
|
1991 |
apply simp
|
|
|
1992 |
apply(case_tac rs)
|
|
|
1993 |
apply simp
|
|
|
1994 |
apply auto
|
|
|
1995 |
done
|
|
|
1996 |
|
|
|
1997 |
lemma double_nested_ALTs_under_rsimp:
|
|
|
1998 |
shows "rsimp (rsimp_ALTs ((RALTS rs1) # rs)) = rsimp (RALTS (rs1 @ rs))"
|
|
|
1999 |
apply(case_tac rs1)
|
|
|
2000 |
apply simp
|
|
|
2001 |
|
|
|
2002 |
apply (metis list.simps(8) list.simps(9) neq_Nil_conv rdistinct.simps(1) rflts.simps(1) rflts.simps(2) rsimp.simps(2) rsimp_ALTs.simps(1) rsimp_ALTs.simps(2) rsimp_ALTs.simps(3))
|
|
|
2003 |
apply(case_tac rs)
|
|
|
2004 |
apply simp
|
|
|
2005 |
apply auto
|
|
421
|
2006 |
sorry
|
|
|
2007 |
|
|
427
|
2008 |
lemma star_seqs_produce_star_seqs:
|
|
|
2009 |
shows "rsimp (rsimp_ALTs (map (rder x \<circ> (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss))
|
|
|
2010 |
= rsimp (rsimp_ALTs (map ( (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r)))) Ss))"
|
|
|
2011 |
by (meson comp_apply)
|
|
|
2012 |
|
|
|
2013 |
lemma der_seqstar_res:
|
|
428
|
2014 |
shows "rder x (RSEQ r1 r2) = RSEQ r3 r4"
|
|
427
|
2015 |
|
|
|
2016 |
|
|
|
2017 |
lemma linearity_of_list_of_star_or_starseqs:
|
|
|
2018 |
shows "\<exists>Ssa. rsimp (rder x (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))) =
|
|
|
2019 |
rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ssa)"
|
|
|
2020 |
apply(simp add: rder_rsimp_ALTs_commute)
|
|
|
2021 |
apply(induct Ss)
|
|
|
2022 |
apply simp
|
|
|
2023 |
apply (metis list.simps(8) rsimp_ALTs.simps(1))
|
|
|
2024 |
|
|
428
|
2025 |
|
|
427
|
2026 |
sorry
|
|
|
2027 |
|
|
|
2028 |
lemma starder_is_a_list_of_stars_or_starseqs:
|
|
|
2029 |
shows "s \<noteq> [] \<Longrightarrow> \<exists>Ss. rders_simp (RSTAR r) s = rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)"
|
|
|
2030 |
apply(induct s rule: rev_induct)
|
|
|
2031 |
apply simp
|
|
|
2032 |
apply(case_tac "xs = []")
|
|
|
2033 |
using star_is_a_singleton_list_derc
|
|
|
2034 |
apply(simp)
|
|
|
2035 |
apply auto
|
|
|
2036 |
apply(simp add: rders_simp_append)
|
|
|
2037 |
using linearity_of_list_of_star_or_starseqs by blast
|
|
|
2038 |
|
|
421
|
2039 |
|
|
422
|
2040 |
lemma finite_star:
|
|
|
2041 |
shows "(\<forall>s. rsize (rders_simp r0 s) < N0 )
|
|
|
2042 |
\<Longrightarrow> \<exists>N3. \<forall>s.(rsize (rders_simp (RSTAR r0) s)) < N3"
|
|
427
|
2043 |
|
|
422
|
2044 |
sorry
|
|
|
2045 |
|
|
|
2046 |
|
|
|
2047 |
lemma rderssimp_zero:
|
|
|
2048 |
shows"rders_simp RZERO s = RZERO"
|
|
|
2049 |
apply(induction s)
|
|
|
2050 |
apply simp
|
|
|
2051 |
by simp
|
|
|
2052 |
|
|
|
2053 |
lemma rderssimp_one:
|
|
|
2054 |
shows"rders_simp RONE (a # s) = RZERO"
|
|
|
2055 |
apply(induction s)
|
|
|
2056 |
apply simp
|
|
|
2057 |
by simp
|
|
|
2058 |
|
|
|
2059 |
lemma rderssimp_char:
|
|
|
2060 |
shows "rders_simp (RCHAR c) s = RONE \<or> rders_simp (RCHAR c) s = RZERO \<or> rders_simp (RCHAR c) s = (RCHAR c)"
|
|
|
2061 |
apply auto
|
|
|
2062 |
by (metis rder.simps(2) rder.simps(3) rders_simp.elims rders_simp.simps(2) rderssimp_one rsimp.simps(4))
|
|
|
2063 |
|
|
421
|
2064 |
lemma finite_size_ders:
|
|
422
|
2065 |
shows " \<exists>Nr. \<forall>s. rsize (rders_simp r s) < Nr"
|
|
|
2066 |
apply(induct r rule: rrexp.induct)
|
|
|
2067 |
apply auto
|
|
|
2068 |
apply(rule_tac x = "2" in exI)
|
|
|
2069 |
using rderssimp_zero rsize.simps(1) apply presburger
|
|
|
2070 |
apply(rule_tac x = "2" in exI)
|
|
|
2071 |
apply (metis Suc_1 lessI rders_simp.elims rderssimp_one rsize.simps(1) rsize.simps(2))
|
|
|
2072 |
apply(rule_tac x = "2" in meta_spec)
|
|
|
2073 |
apply (metis lessI rderssimp_char rsize.simps(1) rsize.simps(2) rsize.simps(3))
|
|
|
2074 |
|
|
|
2075 |
using finite_seq apply blast
|
|
|
2076 |
prefer 2
|
|
|
2077 |
|
|
|
2078 |
apply (simp add: finite_star)
|
|
421
|
2079 |
sorry
|
|
409
|
2080 |
|
|
|
2081 |
|
|
|
2082 |
unused_thms
|
|
417
|
2083 |
lemma seq_ders_shape:
|
|
|
2084 |
shows "E"
|
|
|
2085 |
|
|
|
2086 |
oops
|
|
|
2087 |
|
|
|
2088 |
(*rsimp (rders (RSEQ r1 r2) s) =
|
|
|
2089 |
rsimp RALT [RSEQ (rders r1 s) r2, rders r2 si, ...]
|
|
|
2090 |
where si is the i-th shortest suffix of s such that si \<in> L r2"
|
|
|
2091 |
*)
|
|
409
|
2092 |
|
|
|
2093 |
|
|
|
2094 |
inductive aggressive:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto>? _" [99, 99] 99)
|
|
|
2095 |
where
|
|
|
2096 |
"ASEQ bs (AALTs bs1 rs) r \<leadsto>? AALTs (bs@bs1) (map (\<lambda>r'. ASEQ [] r' r) rs) "
|
|
|
2097 |
|
|
|
2098 |
|
|
|
2099 |
|
|
|
2100 |
end
|