--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/thys/BitCoded.thy Sun Feb 17 22:15:06 2019 +0000
@@ -0,0 +1,326 @@
+
+theory BitCoded
+ imports "Lexer"
+begin
+
+section {* Bit-Encodings *}
+
+datatype bit = Z | S
+
+fun
+ code :: "val \<Rightarrow> bit list"
+where
+ "code Void = []"
+| "code (Char c) = []"
+| "code (Left v) = Z # (code v)"
+| "code (Right v) = S # (code v)"
+| "code (Seq v1 v2) = (code v1) @ (code v2)"
+| "code (Stars []) = [S]"
+| "code (Stars (v # vs)) = (Z # code v) @ code (Stars vs)"
+
+
+fun
+ Stars_add :: "val \<Rightarrow> val \<Rightarrow> val"
+where
+ "Stars_add v (Stars vs) = Stars (v # vs)"
+
+function
+ decode' :: "bit list \<Rightarrow> rexp \<Rightarrow> (val * bit list)"
+where
+ "decode' ds ZERO = (Void, [])"
+| "decode' ds ONE = (Void, ds)"
+| "decode' ds (CHAR d) = (Char d, ds)"
+| "decode' [] (ALT r1 r2) = (Void, [])"
+| "decode' (Z # ds) (ALT r1 r2) = (let (v, ds') = decode' ds r1 in (Left v, ds'))"
+| "decode' (S # ds) (ALT r1 r2) = (let (v, ds') = decode' ds r2 in (Right v, ds'))"
+| "decode' ds (SEQ r1 r2) = (let (v1, ds') = decode' ds r1 in
+ let (v2, ds'') = decode' ds' r2 in (Seq v1 v2, ds''))"
+| "decode' [] (STAR r) = (Void, [])"
+| "decode' (S # ds) (STAR r) = (Stars [], ds)"
+| "decode' (Z # ds) (STAR r) = (let (v, ds') = decode' ds r in
+ let (vs, ds'') = decode' ds' (STAR r)
+ in (Stars_add v vs, ds''))"
+by pat_completeness auto
+
+lemma decode'_smaller:
+ assumes "decode'_dom (ds, r)"
+ shows "length (snd (decode' ds r)) \<le> length ds"
+using assms
+apply(induct ds r)
+apply(auto simp add: decode'.psimps split: prod.split)
+using dual_order.trans apply blast
+by (meson dual_order.trans le_SucI)
+
+termination "decode'"
+apply(relation "inv_image (measure(%cs. size cs) <*lex*> measure(%s. size s)) (%(ds,r). (r,ds))")
+apply(auto dest!: decode'_smaller)
+by (metis less_Suc_eq_le snd_conv)
+
+definition
+ decode :: "bit list \<Rightarrow> rexp \<Rightarrow> val option"
+where
+ "decode ds r \<equiv> (let (v, ds') = decode' ds r
+ in (if ds' = [] then Some v else None))"
+
+lemma decode'_code_Stars:
+ assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> (\<forall>x. decode' (code v @ x) r = (v, x)) \<and> flat v \<noteq> []"
+ shows "decode' (code (Stars vs) @ ds) (STAR r) = (Stars vs, ds)"
+ using assms
+ apply(induct vs)
+ apply(auto)
+ done
+
+lemma decode'_code:
+ assumes "\<Turnstile> v : r"
+ shows "decode' ((code v) @ ds) r = (v, ds)"
+using assms
+ apply(induct v r arbitrary: ds)
+ apply(auto)
+ using decode'_code_Stars by blast
+
+lemma decode_code:
+ assumes "\<Turnstile> v : r"
+ shows "decode (code v) r = Some v"
+ using assms unfolding decode_def
+ by (smt append_Nil2 decode'_code old.prod.case)
+
+
+section {* Annotated Regular Expressions *}
+
+datatype arexp =
+ AZERO
+| AONE "bit list"
+| ACHAR "bit list" char
+| ASEQ "bit list" arexp arexp
+| AALTs "bit list" "arexp list"
+| ASTAR "bit list" arexp
+
+abbreviation
+ "AALT bs r1 r2 \<equiv> AALTs bs [r1, r2]"
+
+
+fun fuse :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp" where
+ "fuse bs AZERO = AZERO"
+| "fuse bs (AONE cs) = AONE (bs @ cs)"
+| "fuse bs (ACHAR cs c) = ACHAR (bs @ cs) c"
+| "fuse bs (AALTs cs rs) = AALTs (bs @ cs) rs"
+| "fuse bs (ASEQ cs r1 r2) = ASEQ (bs @ cs) r1 r2"
+| "fuse bs (ASTAR cs r) = ASTAR (bs @ cs) r"
+
+fun intern :: "rexp \<Rightarrow> arexp" where
+ "intern ZERO = AZERO"
+| "intern ONE = AONE []"
+| "intern (CHAR c) = ACHAR [] c"
+| "intern (ALT r1 r2) = AALT [] (fuse [Z] (intern r1))
+ (fuse [S] (intern r2))"
+| "intern (SEQ r1 r2) = ASEQ [] (intern r1) (intern r2)"
+| "intern (STAR r) = ASTAR [] (intern r)"
+
+
+fun retrieve :: "arexp \<Rightarrow> val \<Rightarrow> bit list" where
+ "retrieve (AONE bs) Void = bs"
+| "retrieve (ACHAR bs c) (Char d) = bs"
+| "retrieve (AALTs bs (r#rs)) (Left v) = bs @ retrieve r v"
+| "retrieve (AALTs bs (r#rs)) (Right v) = bs @ retrieve (AALTs [] rs) v"
+| "retrieve (ASEQ bs r1 r2) (Seq v1 v2) = bs @ retrieve r1 v1 @ retrieve r2 v2"
+| "retrieve (ASTAR bs r) (Stars []) = bs @ [S]"
+| "retrieve (ASTAR bs r) (Stars (v#vs)) =
+ bs @ [Z] @ retrieve r v @ retrieve (ASTAR [] r) (Stars vs)"
+
+fun
+ erase :: "arexp \<Rightarrow> rexp"
+where
+ "erase AZERO = ZERO"
+| "erase (AONE _) = ONE"
+| "erase (ACHAR _ c) = CHAR c"
+| "erase (AALTs _ []) = ZERO"
+| "erase (AALTs _ [r]) = (erase r)"
+| "erase (AALTs _ (r#rs)) = ALT (erase r) (erase (AALTs [] rs))"
+| "erase (ASEQ _ r1 r2) = SEQ (erase r1) (erase r2)"
+| "erase (ASTAR _ r) = STAR (erase r)"
+
+fun
+ bnullable :: "arexp \<Rightarrow> bool"
+where
+ "bnullable (AZERO) = False"
+| "bnullable (AONE bs) = True"
+| "bnullable (ACHAR bs c) = False"
+| "bnullable (AALTs bs rs) = (\<exists>r \<in> set rs. bnullable r)"
+| "bnullable (ASEQ bs r1 r2) = (bnullable r1 \<and> bnullable r2)"
+| "bnullable (ASTAR bs r) = True"
+
+fun
+ bmkeps :: "arexp \<Rightarrow> bit list"
+where
+ "bmkeps(AONE bs) = bs"
+| "bmkeps(ASEQ bs r1 r2) = bs @ (bmkeps r1) @ (bmkeps r2)"
+| "bmkeps(AALTs bs (r#rs)) = (if bnullable(r) then bs @ (bmkeps r) else (bmkeps (AALTs bs rs)))"
+| "bmkeps(ASTAR bs r) = bs @ [S]"
+
+
+fun
+ bder :: "char \<Rightarrow> arexp \<Rightarrow> arexp"
+where
+ "bder c (AZERO) = AZERO"
+| "bder c (AONE bs) = AZERO"
+| "bder c (ACHAR bs d) = (if c = d then AONE bs else AZERO)"
+| "bder c (AALTs bs rs) = AALTs bs (map (bder c) rs)"
+| "bder c (ASEQ bs r1 r2) =
+ (if bnullable r1
+ then AALT bs (ASEQ [] (bder c r1) r2) (fuse (bmkeps r1) (bder c r2))
+ else ASEQ bs (bder c r1) r2)"
+| "bder c (ASTAR bs r) = ASEQ bs (fuse [Z] (bder c r)) (ASTAR [] r)"
+
+
+fun
+ bders :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
+where
+ "bders r [] = r"
+| "bders r (c#s) = bders (bder c r) s"
+
+lemma bders_append:
+ "bders r (s1 @ s2) = bders (bders r s1) s2"
+ apply(induct s1 arbitrary: r s2)
+ apply(simp_all)
+ done
+
+lemma bnullable_correctness:
+ shows "nullable (erase r) = bnullable r"
+ apply(induct r rule: erase.induct)
+ apply(simp_all)
+ done
+
+lemma erase_fuse:
+ shows "erase (fuse bs r) = erase r"
+ apply(induct r rule: erase.induct)
+ apply(simp_all)
+ done
+
+lemma erase_intern [simp]:
+ shows "erase (intern r) = r"
+ apply(induct r)
+ apply(simp_all add: erase_fuse)
+ done
+
+lemma erase_bder [simp]:
+ shows "erase (bder a r) = der a (erase r)"
+ apply(induct r rule: erase.induct)
+ apply(simp_all add: erase_fuse bnullable_correctness)
+ done
+
+lemma erase_bders [simp]:
+ shows "erase (bders r s) = ders s (erase r)"
+ apply(induct s arbitrary: r )
+ apply(simp_all)
+ done
+
+lemma retrieve_encode_STARS:
+ assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> code v = retrieve (intern r) v"
+ shows "code (Stars vs) = retrieve (ASTAR [] (intern r)) (Stars vs)"
+ using assms
+ apply(induct vs)
+ apply(simp_all)
+ done
+
+lemma retrieve_fuse2:
+ assumes "\<Turnstile> v : (erase r)"
+ shows "retrieve (fuse bs r) v = bs @ retrieve r v"
+ using assms
+ apply(induct r arbitrary: v bs rule: erase.induct)
+ apply(auto elim: Prf_elims)[1]
+ sorry
+
+lemma retrieve_fuse:
+ assumes "\<Turnstile> v : r"
+ shows "retrieve (fuse bs (intern r)) v = bs @ retrieve (intern r) v"
+ using assms
+ by (simp_all add: retrieve_fuse2)
+
+
+lemma retrieve_code:
+ assumes "\<Turnstile> v : r"
+ shows "code v = retrieve (intern r) v"
+ using assms
+ apply(induct v r )
+ apply(simp_all add: retrieve_fuse retrieve_encode_STARS)
+ sorry
+
+
+lemma bmkeps_retrieve:
+ assumes "nullable (erase r)"
+ shows "bmkeps r = retrieve r (mkeps (erase r))"
+ using assms
+ apply(induct r)
+ apply(auto simp add: bnullable_correctness)
+ sorry
+
+lemma bder_retrieve:
+ assumes "\<Turnstile> v : der c (erase r)"
+ shows "retrieve (bder c r) v = retrieve r (injval (erase r) c v)"
+ using assms
+ apply(induct r arbitrary: v)
+ apply(auto elim!: Prf_elims simp add: retrieve_fuse2 bnullable_correctness bmkeps_retrieve)
+ sorry
+
+lemma MAIN_decode:
+ assumes "\<Turnstile> v : ders s r"
+ shows "Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r"
+ using assms
+proof (induct s arbitrary: v rule: rev_induct)
+ case Nil
+ have "\<Turnstile> v : ders [] r" by fact
+ then have "\<Turnstile> v : r" by simp
+ then have "Some v = decode (retrieve (intern r) v) r"
+ using decode_code retrieve_code by auto
+ then show "Some (flex r id [] v) = decode (retrieve (bders (intern r) []) v) r"
+ by simp
+next
+ case (snoc c s v)
+ have IH: "\<And>v. \<Turnstile> v : ders s r \<Longrightarrow>
+ Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r" by fact
+ have asm: "\<Turnstile> v : ders (s @ [c]) r" by fact
+ then have asm2: "\<Turnstile> injval (ders s r) c v : ders s r"
+ by(simp add: Prf_injval ders_append)
+ have "Some (flex r id (s @ [c]) v) = Some (flex r id s (injval (ders s r) c v))"
+ by (simp add: flex_append)
+ also have "... = decode (retrieve (bders (intern r) s) (injval (ders s r) c v)) r"
+ using asm2 IH by simp
+ also have "... = decode (retrieve (bder c (bders (intern r) s)) v) r"
+ using asm by(simp_all add: bder_retrieve ders_append)
+ finally show "Some (flex r id (s @ [c]) v) =
+ decode (retrieve (bders (intern r) (s @ [c])) v) r" by (simp add: bders_append)
+qed
+
+
+definition blexer where
+ "blexer r s \<equiv> if bnullable (bders (intern r) s) then
+ decode (bmkeps (bders (intern r) s)) r else None"
+
+lemma blexer_correctness:
+ shows "blexer r s = lexer r s"
+proof -
+ { define bds where "bds \<equiv> bders (intern r) s"
+ define ds where "ds \<equiv> ders s r"
+ assume asm: "nullable ds"
+ have era: "erase bds = ds"
+ unfolding ds_def bds_def by simp
+ have mke: "\<Turnstile> mkeps ds : ds"
+ using asm by (simp add: mkeps_nullable)
+ have "decode (bmkeps bds) r = decode (retrieve bds (mkeps ds)) r"
+ using bmkeps_retrieve
+ using asm era by (simp add: bmkeps_retrieve)
+ also have "... = Some (flex r id s (mkeps ds))"
+ using mke by (simp_all add: MAIN_decode ds_def bds_def)
+ finally have "decode (bmkeps bds) r = Some (flex r id s (mkeps ds))"
+ unfolding bds_def ds_def .
+ }
+ then show "blexer r s = lexer r s"
+ unfolding blexer_def lexer_flex
+ apply(subst bnullable_correctness[symmetric])
+ apply(simp)
+ done
+qed
+
+
+
+end
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