
diff -r 9ab8609c66c5 -r 807acaf7f599 thys/Sulzmann.thy
--- a/thys/Sulzmann.thy	Fri Aug 17 12:00:25 2018 +0100
+++ b/thys/Sulzmann.thy	Sat Aug 18 01:54:44 2018 +0100
@@ -5,16 +5,18 @@
 
 section {* Bit-Encodings *}
 
+datatype bit = Z | S
+
 fun 
-  code :: "val \<Rightarrow> bool list"
+  code :: "val \<Rightarrow> bit list"
 where
   "code Void = []"
 | "code (Char c) = []"
-| "code (Left v) = False # (code v)"
-| "code (Right v) = True # (code v)"
+| "code (Left v) = Z # (code v)"
+| "code (Right v) = S # (code v)"
 | "code (Seq v1 v2) = (code v1) @ (code v2)"
-| "code (Stars []) = [True]"
-| "code (Stars (v # vs)) =  False # (code v) @ code (Stars vs)"
+| "code (Stars []) = [S]"
+| "code (Stars (v # vs)) =  (Z # code v) @ code (Stars vs)"
 
 
 fun 
@@ -23,19 +25,19 @@
   "Stars_add v (Stars vs) = Stars (v # vs)"
 
 function
-  decode' :: "bool list \<Rightarrow> rexp \<Rightarrow> (val * bool list)"
+  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' (False # ds) (ALT r1 r2) = (let (v, ds') = decode' ds r1 in (Left v, ds'))"
-| "decode' (True # ds) (ALT r1 r2) = (let (v, ds') = decode' ds r2 in (Right v, ds'))"
+| "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' (True # ds) (STAR r) = (Stars [], ds)"
-| "decode' (False # ds) (STAR r) = (let (v, ds') = decode' ds r in
+| "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
@@ -55,7 +57,7 @@
 by (metis less_Suc_eq_le snd_conv)
 
 definition
-  decode :: "bool list \<Rightarrow> rexp \<Rightarrow> val option"
+  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))"
@@ -76,7 +78,6 @@
   apply(auto)
   using decode'_code_Stars by blast
 
-
 lemma decode_code:
   assumes "\<Turnstile> v : r"
   shows "decode (code v) r = Some v"
@@ -86,13 +87,13 @@
 
 datatype arexp =
   AZERO
-| AONE "bool list"
-| ACHAR "bool list" char
-| ASEQ "bool list" arexp arexp
-| AALT "bool list" arexp arexp
-| ASTAR "bool list" arexp
+| AONE "bit list"
+| ACHAR "bit list" char
+| ASEQ "bit list" arexp arexp
+| AALT "bit list" arexp arexp
+| ASTAR "bit list" arexp
 
-fun fuse :: "bool list \<Rightarrow> arexp \<Rightarrow> arexp" where
+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"
@@ -100,236 +101,219 @@
 | "fuse bs (ASEQ cs r1 r2) = ASEQ (bs @ cs) r1 r2"
 | "fuse bs (ASTAR cs r) = ASTAR (bs @ cs) r"
 
-fun internalise :: "rexp \<Rightarrow> arexp" where
-  "internalise ZERO = AZERO"
-| "internalise ONE = AONE []"
-| "internalise (CHAR c) = ACHAR [] c"
-| "internalise (ALT r1 r2) = AALT [] (fuse [False] (internalise r1)) 
-                                     (fuse [True]  (internalise r2))"
-| "internalise (SEQ r1 r2) = ASEQ [] (internalise r1) (internalise r2)"
-| "internalise (STAR r) = ASTAR [] (internalise 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> bool list" where
+fun retrieve :: "arexp \<Rightarrow> val \<Rightarrow> bit list" where
   "retrieve (AONE bs) Void = bs"
 | "retrieve (ACHAR bs c) (Char d) = bs"
 | "retrieve (AALT bs r1 r2) (Left v) = bs @ retrieve r1 v"
 | "retrieve (AALT bs r1 r2) (Right v) = bs @ retrieve r2 v"
 | "retrieve (ASEQ bs r1 r2) (Seq v1 v2) = bs @ retrieve r1 v1 @ retrieve r2 v2"
-| "retrieve (ASTAR bs r) (Stars []) = bs @ [True]"
+| "retrieve (ASTAR bs r) (Stars []) = bs @ [S]"
 | "retrieve (ASTAR bs r) (Stars (v#vs)) = 
-     bs @ [False] @ retrieve r v @ retrieve (ASTAR [] r) (Stars vs)"
-
+     bs @ [Z] @ retrieve r v @ retrieve (ASTAR [] r) (Stars vs)"
 
 fun 
-  aerase :: "arexp \<Rightarrow> rexp"
+  erase :: "arexp \<Rightarrow> rexp"
+where
+  "erase AZERO = ZERO"
+| "erase (AONE _) = ONE"
+| "erase (ACHAR _ c) = CHAR c"
+| "erase (AALT _ r1 r2) = ALT (erase r1) (erase r2)"
+| "erase (ASEQ _ r1 r2) = SEQ (erase r1) (erase r2)"
+| "erase (ASTAR _ r) = STAR (erase r)"
+
+fun
+ bnullable :: "arexp \<Rightarrow> bool"
 where
-  "aerase AZERO = ZERO"
-| "aerase (AONE _) = ONE"
-| "aerase (ACHAR _ c) = CHAR c"
-| "aerase (AALT _ r1 r2) = ALT (aerase r1) (aerase r2)"
-| "aerase (ASEQ _ r1 r2) = SEQ (aerase r1) (aerase r2)"
-| "aerase (ASTAR _ r) = STAR (aerase r)"
+  "bnullable (AZERO) = False"
+| "bnullable (AONE bs) = True"
+| "bnullable (ACHAR bs c) = False"
+| "bnullable (AALT bs r1 r2) = (bnullable r1 \<or> bnullable r2)"
+| "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(AALT bs r1 r2) = (if bnullable(r1) then bs @ (bmkeps r1) else bs @ (bmkeps r2))"
+| "bmkeps(ASTAR bs r) = bs @ [S]"
 
 
 fun
- anullable :: "arexp \<Rightarrow> bool"
-where
-  "anullable (AZERO) = False"
-| "anullable (AONE bs) = True"
-| "anullable (ACHAR bs c) = False"
-| "anullable (AALT bs r1 r2) = (anullable r1 \<or> anullable r2)"
-| "anullable (ASEQ bs r1 r2) = (anullable r1 \<and> anullable r2)"
-| "anullable (ASTAR bs r) = True"
-
-fun 
-  amkeps :: "arexp \<Rightarrow> bool list"
+ bder :: "char \<Rightarrow> arexp \<Rightarrow> arexp"
 where
-  "amkeps(AONE bs) = bs"
-| "amkeps(ASEQ bs r1 r2) = bs @ (amkeps r1) @ (amkeps r2)"
-| "amkeps(AALT bs r1 r2) = (if anullable(r1) then bs @ (amkeps r1) else bs @ (amkeps r2))"
-| "amkeps(ASTAR bs r) = bs @ [True]"
-
-
-fun
- ader :: "char \<Rightarrow> arexp \<Rightarrow> arexp"
-where
-  "ader c (AZERO) = AZERO"
-| "ader c (AONE bs) = AZERO"
-| "ader c (ACHAR bs d) = (if c = d then AONE bs else AZERO)"
-| "ader c (AALT bs r1 r2) = AALT bs (ader c r1) (ader c r2)"
-| "ader c (ASEQ bs r1 r2) = 
-     (if anullable r1
-      then AALT bs (ASEQ [] (ader c r1) r2) (fuse (amkeps r1) (ader c r2))
-      else ASEQ bs (ader c r1) r2)"
-| "ader c (ASTAR bs r) = ASEQ bs (fuse [False] (ader c r)) (ASTAR [] r)"
+  "bder c (AZERO) = AZERO"
+| "bder c (AONE bs) = AZERO"
+| "bder c (ACHAR bs d) = (if c = d then AONE bs else AZERO)"
+| "bder c (AALT bs r1 r2) = AALT bs (bder c r1) (bder c r2)"
+| "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 
-  aders :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
+  bders :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
 where
-  "aders r [] = r"
-| "aders r (c#s) = aders (ader c r) s"
+  "bders r [] = r"
+| "bders r (c#s) = bders (bder c r) s"
 
-lemma aders_append:
-  "aders r (s1 @ s2) = aders (aders r s1) s2"
+lemma bders_append:
+  "bders r (s1 @ s2) = bders (bders r s1) s2"
   apply(induct s1 arbitrary: r s2)
   apply(simp_all)
   done
 
-lemma anullable_correctness:
-  shows "nullable (aerase r) = anullable r"
+lemma bnullable_correctness:
+  shows "nullable (erase r) = bnullable r"
   apply(induct r)
   apply(simp_all)
   done
 
-lemma aerase_fuse:
-  shows "aerase (fuse bs r) = aerase r"
+lemma erase_fuse:
+  shows "erase (fuse bs r) = erase r"
   apply(induct r)
   apply(simp_all)
   done
 
-lemma aerase_internalise:
-  shows "aerase (internalise r) = r"
+lemma erase_intern[simp]:
+  shows "erase (intern r) = r"
   apply(induct r)
-  apply(simp_all add: aerase_fuse)
+  apply(simp_all add: erase_fuse)
   done
 
-lemma aerase_ader:
-  shows "aerase (ader a r) = der a (aerase r)"
+lemma erase_bder[simp]:
+  shows "erase (bder a r) = der a (erase r)"
   apply(induct r)
-  apply(simp_all add: aerase_fuse anullable_correctness)
+  apply(simp_all add: erase_fuse bnullable_correctness)
   done
 
-lemma aerase_aders:
-  shows "aerase (aders r s) = ders s (aerase r)"
+lemma erase_bders[simp]:
+  shows "erase (bders r s) = ders s (erase r)"
   apply(induct s arbitrary: r )
-  apply(simp_all add: aerase_ader)
+  apply(simp_all)
   done
 
 lemma retrieve_encode_STARS:
-  assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> code v = retrieve (internalise r) v"
-  shows "code (Stars vs) = retrieve (ASTAR [] (internalise r)) (Stars vs)"
+  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)
+  using retrieve_encode_STARS
+  apply(auto elim!: Prf_elims)
+  apply(case_tac vs)
   apply(simp)
   apply(simp)
   done
 
-lemma retrieve_afuse2:
-  assumes "\<Turnstile> v : (aerase r)"
-  shows "retrieve (fuse bs r) v = bs @ retrieve r v"
+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 r arbitrary: v bs)
-  apply(auto)
-  using Prf_elims(1) apply blast
-  using Prf_elims(4) apply fastforce
-  using Prf_elims(5) apply fastforce
-  apply (smt Prf_elims(2) append_assoc retrieve.simps(5))
-   apply(erule Prf_elims)
-    apply(simp)
-   apply(simp)
-  apply(erule Prf_elims)
-  apply(simp)
-  apply(case_tac vs)
-   apply(simp)
-  apply(simp)
+  apply(induct v r)
+  apply(simp_all add: retrieve_fuse retrieve_encode_STARS)
   done
 
-lemma retrieve_afuse:
-  assumes "\<Turnstile> v : r"
-  shows "retrieve (fuse bs (internalise r)) v = bs @ retrieve (internalise r) v"
-  using assms 
-  by (simp_all add: retrieve_afuse2 aerase_internalise)
 
-
-lemma retrieve_encode:
-  assumes "\<Turnstile> v : r"
-  shows "code v = retrieve (internalise r) v"
-  using assms
-  apply(induct v r)
-  apply(simp_all add: retrieve_afuse retrieve_encode_STARS)
-  done
-
-lemma Q00:
-  assumes "s \<in> r \<rightarrow> v"
-  shows "\<Turnstile> v : r"
-  using assms 
-  apply(induct) 
-        apply(auto intro: Prf.intros)
-  by (metis Prf.intros(6) Prf_elims(6) insert_iff list.simps(15) val.inject(5))
-
-lemma Qa:
-  assumes "anullable r"
-  shows "retrieve r (mkeps (aerase r)) = amkeps r"
+lemma bmkeps_retrieve:
+  assumes "nullable (erase r)"
+  shows "bmkeps r = retrieve r (mkeps (erase r))"
   using assms
   apply(induct r)
-       apply(auto)
-  using anullable_correctness apply auto[1]
-  apply (simp add: anullable_correctness)
-  by (simp add: anullable_correctness)
-
+  apply(auto simp add: bnullable_correctness)
+  done
   
-lemma Qb:
-  assumes "\<Turnstile> v : der c (aerase r)"
-  shows "retrieve (ader c r) v = retrieve r (injval (aerase r) c v)"
+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 c)
-       apply(simp_all)
-  using Prf_elims(1) apply blast
-  using Prf_elims(1) apply blast
-     apply(auto)[1]
-  using Prf_elims(4) apply fastforce
-  using Prf_elims(1) apply blast
-    apply(auto split: if_splits)[1]
-  apply(auto elim!: Prf_elims)[1]
-       apply(rotate_tac 1)
-       apply(drule_tac x="v2" in meta_spec)
-       apply(drule_tac x="c" in meta_spec)
-       apply(drule meta_mp)
-        apply(simp)
-       apply(drule sym)
-       apply(simp)
-       apply(subst retrieve_afuse2)
-        apply (simp add: aerase_ader)
-       apply (simp add: Qa)
-  using anullable_correctness apply auto[1]
-     apply(auto elim!: Prf_elims)[1]
-  using anullable_correctness apply auto[1]
-    apply(auto elim!: Prf_elims)[1]
-   apply(auto elim!: Prf_elims)[1]
-  apply(auto elim!: Prf_elims)[1]
-  by (simp add: retrieve_afuse2 aerase_ader)
-
-lemma MAIN:
-  assumes "\<Turnstile> v : ders s r"
-  shows "code (flex r id s v) = retrieve (aders (internalise r) s) v"
-  using assms
-  apply(induct s arbitrary: v rule: rev_induct)
-  apply(simp)
-  apply(simp add: retrieve_encode)
-  apply(simp add: flex_append aders_append)
-  apply(subst Qb)
-  apply(simp add: aerase_internalise ders_append aerase_aders)
-  apply(simp add: aerase_aders aerase_internalise)
-  apply(drule_tac x="injval (ders xs r) x v" in meta_spec)
-  apply(drule meta_mp)
-  apply(simp add: Prf_injval ders_append)
-  apply(simp)
+  apply(induct r arbitrary: v)
+  apply(auto elim!: Prf_elims simp add: retrieve_fuse2 bnullable_correctness bmkeps_retrieve)
   done
 
-fun alexer where
- "alexer r s = (if anullable (aders (internalise r) s) then 
-                decode (amkeps (aders (internalise r) s)) r else None)"
+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
 
 
-lemma FIN:
-  "alexer r s = lexer r s"
-  apply(auto simp add: lexer_flex)
-  apply (smt MAIN Q00 Qa aerase_aders aerase_internalise decode_code lexer_correctness(1) lexer_flex mkeps_nullable)
-  apply (metis aerase_aders aerase_internalise anullable_correctness)
-  using aerase_aders aerase_internalise anullable_correctness by force
+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
+    by(auto simp add: bnullable_correctness[symmetric])
+qed
 
 unused_thms
   




