theory Sulzmann

begin

section {* Sulzmann's "Ordering" of Values *}

inductive ValOrd :: "val \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<preceq>_ _" [100, 100, 100] 100)

where

  C2: "v1 \<preceq>r1 v1' \<Longrightarrow> (Seq v1 v2) \<preceq>(SEQ r1 r2) (Seq v1' v2')" 

| C1: "v2 \<preceq>r2 v2' \<Longrightarrow> (Seq v1 v2) \<preceq>(SEQ r1 r2) (Seq v1 v2')" 

| A1: "length (flat v2) > length (flat v1) \<Longrightarrow> (Right v2) \<preceq>(ALT r1 r2) (Left v1)"

| A2: "length (flat v1) \<ge> length (flat v2) \<Longrightarrow> (Left v1) \<preceq>(ALT r1 r2) (Right v2)"

| A3: "v2 \<preceq>r2 v2' \<Longrightarrow> (Right v2) \<preceq>(ALT r1 r2) (Right v2')"

| A4: "v1 \<preceq>r1 v1' \<Longrightarrow> (Left v1) \<preceq>(ALT r1 r2) (Left v1')"

| K1: "flat (Stars (v # vs)) = [] \<Longrightarrow> (Stars []) \<preceq>(STAR r) (Stars (v # vs))"

| K2: "flat (Stars (v # vs)) \<noteq> [] \<Longrightarrow> (Stars (v # vs)) \<preceq>(STAR r) (Stars [])"

| K3: "v1 \<preceq>r v2 \<Longrightarrow> (Stars (v1 # vs1)) \<preceq>(STAR r) (Stars (v2 # vs2))"

| K4: "(Stars vs1) \<preceq>(STAR r) (Stars vs2) \<Longrightarrow> (Stars (v # vs1)) \<preceq>(STAR r) (Stars (v # vs2))"

(*| MY1: "Void \<preceq>ONE Void" 

| MY2: "(Char c) \<preceq>(CHAR c) (Char c)" 

| MY3: "(Stars []) \<preceq>(STAR r) (Stars [])" 

*)

(*

lemma ValOrd_refl: 

  assumes "\<turnstile> v : r" 

  shows "v \<preceq>r v"

using assms

apply(induct r rule: Prf.induct)

apply(rule ValOrd.intros)

apply(simp)

apply(rule ValOrd.intros)

apply(simp)

apply(rule ValOrd.intros)

apply(simp)

apply(rule ValOrd.intros)

apply(rule ValOrd.intros)

apply(rule ValOrd.intros)

apply(rule ValOrd.intros)

apply(simp)

done

*)

lemma ValOrd_irrefl: 

  assumes "\<turnstile> v : r"  "v \<preceq>r v" 

  shows "False"

using assms

apply(induct v r rule: Prf.induct)

apply(erule ValOrd.cases)

apply(simp_all)

apply(erule ValOrd.cases)

apply(simp_all)

apply(erule ValOrd.cases)

apply(simp_all)

apply(erule ValOrd.cases)

apply(simp_all)

apply(erule ValOrd.cases)

apply(simp_all)

apply(erule ValOrd.cases)

apply(simp_all)

apply(erule ValOrd.cases)

apply(simp_all)

done

lemma prefix_sprefix:

  shows "xs \<sqsubseteq>pre ys \<longleftrightarrow> (xs = ys \<or> xs \<sqsubset>spre ys)"

apply(auto simp add: sprefix_list_def prefix_list_def)

done

lemma Posix_CPT2:

  shows "v1 :\<sqsubset>val v2" 

using assms

prefer 3

apply(simp)

apply(simp)

apply(simp)

apply(simp add: append_eq_append_conv2)

apply(auto)[1]

apply(simp)

apply(simp)

lemma Posix_CPT:

  assumes "v1 :\<sqsubset>val v2" "v1 \<in> CPT r s" "v2 \<in> CPT r s"

  shows "v1 \<preceq>r v2" 

using assms

apply(induct r arbitrary: v1 v2 s rule: rexp.induct)

apply(simp add: CPT_def)

apply(clarify)

apply(erule CPrf.cases)

apply(simp_all)

apply(simp add: CPT_def)

apply(clarify)

apply(erule CPrf.cases)

apply(simp_all)

apply(erule CPrf.cases)

apply(simp_all)

apply(rule ValOrd.intros)

apply(simp add: CPT_def)

apply(clarify)

apply(erule CPrf.cases)

apply(simp_all)

apply(erule CPrf.cases)

apply(simp_all)

apply(rule ValOrd.intros)

(*SEQ case *)

apply(simp add: CPT_def)

apply(clarify)

apply(erule CPrf.cases)

apply(simp_all)

apply(clarify)

apply(erule CPrf.cases)

apply(simp_all)

apply(clarify)

thm val_ord_SEQ

apply(drule_tac r="r1a" in  val_ord_SEQ)

apply(simp)

using Prf_CPrf apply blast

using Prf_CPrf apply blast

apply(erule disjE)

apply(rule C2)

prefer 2

apply(simp)

apply(rule C1)

apply blast

apply(simp add: append_eq_append_conv2)

apply(clarify)

apply(auto)[1]

apply(drule_tac x="v1a" in meta_spec)

apply(rotate_tac 8)

apply(drule_tac x="v1b" in meta_spec)

apply(rotate_tac 8)

apply(simp)

(* HERE *)

apply(subst (asm) (3) val_ord_ex_def)

apply(clarify)

apply(subst (asm) val_ord_def)

apply(clarify)

apply(rule ValOrd.intros)

apply(simp add: val_ord_ex_def)

oops

lemma ValOrd_trans:

  assumes "x \<preceq>r y" "y \<preceq>r z"

  and "x \<in> CPT r s" "y \<in> CPT r s" "z \<in> CPT r s"

  shows "x \<preceq>r z"

using assms

apply(induct x r y arbitrary: s z rule: ValOrd.induct)

apply(rotate_tac 2)

apply(erule ValOrd.cases)

apply(simp_all)[13]

apply(rule ValOrd.intros)

apply(drule_tac x="s" in meta_spec)

apply(drule_tac x="v1'a" in meta_spec)

apply(drule_tac meta_mp)

apply(simp)

apply(drule_tac meta_mp)

apply(simp add: CPT_def)

oops

lemma ValOrd_preorder:

  "preorder_on (CPT r s) {(v1, v2). v1 \<preceq>r v2 \<and> v1 \<in> (CPT r s) \<and> v2 \<in> (CPT r s)}"

apply(simp add: preorder_on_def)

apply(rule conjI)

apply(simp add: refl_on_def)

apply(auto)

apply(rule ValOrd_refl)

apply(simp add: CPT_def)

apply(rule Prf_CPrf)

apply(auto)[1]

apply(simp add: trans_def)

apply(auto)

definition ValOrdEq :: "val \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<ge>_ _" [100, 100, 100] 100)

where 

  "v\<^sub>1 \<ge>r v\<^sub>2 \<equiv> v\<^sub>1 = v\<^sub>2 \<or> (v\<^sub>1 >r v\<^sub>2 \<and> flat v\<^sub>1 = flat v\<^sub>2)"

(*

inductive ValOrd :: "val \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<succ>_ _" [100, 100, 100] 100)

where

  "v2 \<succ>r2 v2' \<Longrightarrow> (Seq v1 v2) \<succ>(SEQ r1 r2) (Seq v1 v2')" 

| "\<lbrakk>v1 \<succ>r1 v1'; v1 \<noteq> v1'\<rbrakk> \<Longrightarrow> (Seq v1 v2) \<succ>(SEQ r1 r2) (Seq v1' v2')" 

| "length (flat v1) \<ge> length (flat v2) \<Longrightarrow> (Left v1) \<succ>(ALT r1 r2) (Right v2)"

| "length (flat v2) > length (flat v1) \<Longrightarrow> (Right v2) \<succ>(ALT r1 r2) (Left v1)"

| "v2 \<succ>r2 v2' \<Longrightarrow> (Right v2) \<succ>(ALT r1 r2) (Right v2')"

| "v1 \<succ>r1 v1' \<Longrightarrow> (Left v1) \<succ>(ALT r1 r2) (Left v1')"

| "Void \<succ>EMPTY Void"

| "(Char c) \<succ>(CHAR c) (Char c)"

| "flat (Stars (v # vs)) = [] \<Longrightarrow> (Stars []) \<succ>(STAR r) (Stars (v # vs))"

| "flat (Stars (v # vs)) \<noteq> [] \<Longrightarrow> (Stars (v # vs)) \<succ>(STAR r) (Stars [])"

| "\<lbrakk>v1 \<succ>r v2; v1 \<noteq> v2\<rbrakk> \<Longrightarrow> (Stars (v1 # vs1)) \<succ>(STAR r) (Stars (v2 # vs2))"

| "(Stars vs1) \<succ>(STAR r) (Stars vs2) \<Longrightarrow> (Stars (v # vs1)) \<succ>(STAR r) (Stars (v # vs2))"

| "(Stars []) \<succ>(STAR r) (Stars [])"

*)

section {* Bit-Encodings *}

fun 

  code :: "val \<Rightarrow> rexp \<Rightarrow> bool list"

where

  "code Void ONE = []"

| "code (Char c) (CHAR d) = []"

| "code (Left v) (ALT r1 r2) = False # (code v r1)"

| "code (Right v) (ALT r1 r2) = True # (code v r2)"

| "code (Seq v1 v2) (SEQ r1 r2) = (code v1 r1) @ (code v2 r2)"

| "code (Stars []) (STAR r) = [True]"

| "code (Stars (v # vs)) (STAR r) =  False # (code v r) @ code (Stars vs) (STAR r)"

fun 

  Stars_add :: "val \<Rightarrow> val \<Rightarrow> val"

where

  "Stars_add v (Stars vs) = Stars (v # vs)"

function

  decode' :: "bool list \<Rightarrow> rexp \<Rightarrow> (val * bool 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' 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

                                    let (vs, ds'') = decode' ds' (STAR r) 

                                    in (Stars_add v vs, ds''))"

by pat_completeness auto

termination

apply(size_change)

oops

term "inv_image (measure(%cs. size cs) <*lex*> measure(%s. size s)) (%(ds,r). (r,ds))"

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)

fun 

  decode :: "bool list \<Rightarrow> rexp \<Rightarrow> val option"

where

  "decode ds r = (let (v, ds') = decode' ds r 

                  in (if ds' = [] then Some v else None))"

lemma decode'_code:

  assumes "\<turnstile> v : r"

  shows "decode' ((code v r) @ ds) r = (v, ds)"

using assms

by (induct v r arbitrary: ds) (auto)

lemma decode_code:

  assumes "\<turnstile> v : r"

  shows "decode (code v r) r = Some v"

using assms decode'_code[of _ _ "[]"]

by auto

datatype arexp =

  AZERO

| AONE "bool list"

| ACHAR "bool list" char

| ASEQ "bool list" arexp arexp

| AALT "bool list" arexp arexp

| ASTAR "bool list" arexp

fun fuse :: "bool 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 (AALT cs r1 r2) = AALT (bs @ cs) r1 r2"

| "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 retrieve :: "arexp \<Rightarrow> val \<Rightarrow> bool 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 (v#vs)) = 

     bs @ [False] @ retrieve r v @ retrieve (ASTAR [] r) (Stars vs)"

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"

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)"

lemma

  assumes "\<turnstile> v : der c r"

  shows "Some (injval r c v) = decode (retrieve (ader c (internalise r)) v) r"

using assms

apply(induct c r arbitrary: v rule: der.induct)

apply(simp_all)

apply(erule Prf_elims)

apply(erule Prf_elims)

apply(case_tac "c = d")

apply(simp)

apply(erule Prf_elims)

apply(simp)

apply(simp)

apply(erule Prf_elims)

apply(auto split: prod.splits)[1]

oops

end