(*  Title:      HOL/Unix/Nested_Environment.thy

    Author:     Markus Wenzel, TU Muenchen

*)

header {* Nested environments *}

theory Nested_Environment

imports Main

begin

text {*

  Consider a partial function @{term [source] "e :: 'a => 'b option"};

  this may be understood as an \emph{environment} mapping indexes

  @{typ 'a} to optional entry values @{typ 'b} (cf.\ the basic theory

  @{text Map} of Isabelle/HOL).  This basic idea is easily generalized

  to that of a \emph{nested environment}, where entries may be either

  basic values or again proper environments.  Then each entry is

  accessed by a \emph{path}, i.e.\ a list of indexes leading to its

  position within the structure.

*}

datatype ('a, 'b, 'c) env =

    Val 'a

  | Env 'b  "'c => ('a, 'b, 'c) env option"

text {*

  \medskip In the type @{typ "('a, 'b, 'c) env"} the parameter @{typ

  'a} refers to basic values (occurring in terminal positions), type

  @{typ 'b} to values associated with proper (inner) environments, and

  type @{typ 'c} with the index type for branching.  Note that there

  is no restriction on any of these types.  In particular, arbitrary

  branching may yield rather large (transfinite) tree structures.

*}

subsection {* The lookup operation *}

text {*

  Lookup in nested environments works by following a given path of

  index elements, leading to an optional result (a terminal value or

  nested environment).  A \emph{defined position} within a nested

  environment is one where @{term lookup} at its path does not yield

  @{term None}.

*}

primrec

  lookup :: "('a, 'b, 'c) env => 'c list => ('a, 'b, 'c) env option"

  and lookup_option :: "('a, 'b, 'c) env option => 'c list => ('a, 'b, 'c) env option" where

    "lookup (Val a) xs = (if xs = [] then Some (Val a) else None)"

  | "lookup (Env b es) xs =

      (case xs of

        [] => Some (Env b es)

      | y # ys => lookup_option (es y) ys)"

  | "lookup_option None xs = None"

  | "lookup_option (Some e) xs = lookup e xs"

hide_const lookup_option

text {*

  \medskip The characteristic cases of @{term lookup} are expressed by

  the following equalities.

*}

theorem lookup_nil: "lookup e [] = Some e"

  by (cases e) simp_all

theorem lookup_val_cons: "lookup (Val a) (x # xs) = None"

  by simp

theorem lookup_env_cons:

  "lookup (Env b es) (x # xs) =

    (case es x of

      None => None

    | Some e => lookup e xs)"

  by (cases "es x") simp_all

lemmas lookup_lookup_option.simps [simp del]

  and lookup_simps [simp] = lookup_nil lookup_val_cons lookup_env_cons

theorem lookup_eq:

  "lookup env xs =

    (case xs of

      [] => Some env

    | x # xs =>

      (case env of

        Val a => None

      | Env b es =>

          (case es x of

            None => None

          | Some e => lookup e xs)))"

  by (simp split: list.split env.split)

text {*

  \medskip Displaced @{term lookup} operations, relative to a certain

  base path prefix, may be reduced as follows.  There are two cases,

  depending whether the environment actually extends far enough to

  follow the base path.

*}

theorem lookup_append_none:

  assumes "lookup env xs = None"

  shows "lookup env (xs @ ys) = None"

  using assms

proof (induct xs arbitrary: env)

  case Nil

  then have False by simp

  then show ?case ..

next

  case (Cons x xs)

  show ?case

  proof (cases env)

    case Val

    then show ?thesis by simp

  next

    case (Env b es)

    show ?thesis

    proof (cases "es x")

      case None

      with Env show ?thesis by simp

    next

      case (Some e)

      note es = `es x = Some e`

      show ?thesis

      proof (cases "lookup e xs")

        case None

        then have "lookup e (xs @ ys) = None" by (rule Cons.hyps)

        with Env Some show ?thesis by simp

      next

        case Some

        with Env es have False using Cons.prems by simp

        then show ?thesis ..

      qed

    qed

  qed

qed

theorem lookup_append_some:

  assumes "lookup env xs = Some e"

  shows "lookup env (xs @ ys) = lookup e ys"

  using assms

proof (induct xs arbitrary: env e)

  case Nil

  then have "env = e" by simp

  then show "lookup env ([] @ ys) = lookup e ys" by simp

next

  case (Cons x xs)

  note asm = `lookup env (x # xs) = Some e`

  show "lookup env ((x # xs) @ ys) = lookup e ys"

  proof (cases env)

    case (Val a)

    with asm have False by simp

    then show ?thesis ..

  next

    case (Env b es)

    show ?thesis

    proof (cases "es x")

      case None

      with asm Env have False by simp

      then show ?thesis ..

    next

      case (Some e')

      note es = `es x = Some e'`

      show ?thesis

      proof (cases "lookup e' xs")

        case None

        with asm Env es have False by simp

        then show ?thesis ..

      next

        case Some

        with asm Env es have "lookup e' xs = Some e"

          by simp

        then have "lookup e' (xs @ ys) = lookup e ys" by (rule Cons.hyps)

        with Env es show ?thesis by simp

      qed

    qed

  qed

qed

text {*

  \medskip Successful @{term lookup} deeper down an environment

  structure means we are able to peek further up as well.  Note that

  this is basically just the contrapositive statement of @{thm

  [source] lookup_append_none} above.

*}

theorem lookup_some_append:

  assumes "lookup env (xs @ ys) = Some e"

  shows "\<exists>e. lookup env xs = Some e"

proof -

  from assms have "lookup env (xs @ ys) \<noteq> None" by simp

  then have "lookup env xs \<noteq> None"

    by (rule contrapos_nn) (simp only: lookup_append_none)

  then show ?thesis by (simp)

qed

text {*

  The subsequent statement describes in more detail how a successful

  @{term lookup} with a non-empty path results in a certain situation

  at any upper position.

*}

theorem lookup_some_upper:

  assumes "lookup env (xs @ y # ys) = Some e"

  shows "\<exists>b' es' env'.

    lookup env xs = Some (Env b' es') \<and>

    es' y = Some env' \<and>

    lookup env' ys = Some e"

  using assms

proof (induct xs arbitrary: env e)

  case Nil

  from Nil.prems have "lookup env (y # ys) = Some e"

    by simp

  then obtain b' es' env' where

      env: "env = Env b' es'" and

      es': "es' y = Some env'" and

      look': "lookup env' ys = Some e"

    by (auto simp add: lookup_eq split: option.splits env.splits)

  from env have "lookup env [] = Some (Env b' es')" by simp

  with es' look' show ?case by blast

next

  case (Cons x xs)

  from Cons.prems

  obtain b' es' env' where

      env: "env = Env b' es'" and

      es': "es' x = Some env'" and

      look': "lookup env' (xs @ y # ys) = Some e"

    by (auto simp add: lookup_eq split: option.splits env.splits)

  from Cons.hyps [OF look'] obtain b'' es'' env'' where

      upper': "lookup env' xs = Some (Env b'' es'')" and

      es'': "es'' y = Some env''" and

      look'': "lookup env'' ys = Some e"

    by blast

  from env es' upper' have "lookup env (x # xs) = Some (Env b'' es'')"

    by simp

  with es'' look'' show ?case by blast

qed

subsection {* The update operation *}

text {*

  Update at a certain position in a nested environment may either

  delete an existing entry, or overwrite an existing one.  Note that

  update at undefined positions is simple absorbed, i.e.\ the

  environment is left unchanged.

*}

primrec

  update :: "'c list => ('a, 'b, 'c) env option

    => ('a, 'b, 'c) env => ('a, 'b, 'c) env"

  and update_option :: "'c list => ('a, 'b, 'c) env option

    => ('a, 'b, 'c) env option => ('a, 'b, 'c) env option" where

    "update xs opt (Val a) =

      (if xs = [] then (case opt of None => Val a | Some e => e)

      else Val a)"

  | "update xs opt (Env b es) =

      (case xs of

        [] => (case opt of None => Env b es | Some e => e)

      | y # ys => Env b (es (y := update_option ys opt (es y))))"

  | "update_option xs opt None =

      (if xs = [] then opt else None)"

  | "update_option xs opt (Some e) =

      (if xs = [] then opt else Some (update xs opt e))"

hide_const update_option

text {*

  \medskip The characteristic cases of @{term update} are expressed by

  the following equalities.

*}

theorem update_nil_none: "update [] None env = env"

  by (cases env) simp_all

theorem update_nil_some: "update [] (Some e) env = e"

  by (cases env) simp_all

theorem update_cons_val: "update (x # xs) opt (Val a) = Val a"

  by simp

theorem update_cons_nil_env:

    "update [x] opt (Env b es) = Env b (es (x := opt))"

  by (cases "es x") simp_all

theorem update_cons_cons_env:

  "update (x # y # ys) opt (Env b es) =

    Env b (es (x :=

      (case es x of

        None => None

      | Some e => Some (update (y # ys) opt e))))"

  by (cases "es x") simp_all

lemmas update_update_option.simps [simp del]

  and update_simps [simp] = update_nil_none update_nil_some

    update_cons_val update_cons_nil_env update_cons_cons_env

lemma update_eq:

  "update xs opt env =

    (case xs of

      [] =>

        (case opt of

          None => env

        | Some e => e)

    | x # xs =>

        (case env of

          Val a => Val a

        | Env b es =>

            (case xs of

              [] => Env b (es (x := opt))

            | y # ys =>

                Env b (es (x :=

                  (case es x of

                    None => None

                  | Some e => Some (update (y # ys) opt e)))))))"

  by (simp split: list.split env.split option.split)

text {*

  \medskip The most basic correspondence of @{term lookup} and @{term

  update} states that after @{term update} at a defined position,

  subsequent @{term lookup} operations would yield the new value.

*}

theorem lookup_update_some:

  assumes "lookup env xs = Some e"

  shows "lookup (update xs (Some env') env) xs = Some env'"

  using assms

proof (induct xs arbitrary: env e)

  case Nil

  then have "env = e" by simp

  then show ?case by simp

next

  case (Cons x xs)

  note hyp = Cons.hyps

    and asm = `lookup env (x # xs) = Some e`

  show ?case

  proof (cases env)

    case (Val a)

    with asm have False by simp

    then show ?thesis ..

  next

    case (Env b es)

    show ?thesis

    proof (cases "es x")

      case None

      with asm Env have False by simp

      then show ?thesis ..

    next

      case (Some e')

      note es = `es x = Some e'`

      show ?thesis

      proof (cases xs)

        case Nil

        with Env show ?thesis by simp

      next

        case (Cons x' xs')

        from asm Env es have "lookup e' xs = Some e" by simp

        then have "lookup (update xs (Some env') e') xs = Some env'" by (rule hyp)

        with Env es Cons show ?thesis by simp

      qed

    qed

  qed

qed

text {*

  \medskip The properties of displaced @{term update} operations are

  analogous to those of @{term lookup} above.  There are two cases:

  below an undefined position @{term update} is absorbed altogether,

  and below a defined positions @{term update} affects subsequent

  @{term lookup} operations in the obvious way.

*}

theorem update_append_none:

  assumes "lookup env xs = None"

  shows "update (xs @ y # ys) opt env = env"

  using assms

proof (induct xs arbitrary: env)

  case Nil

  then have False by simp

  then show ?case ..

next

  case (Cons x xs)

  note hyp = Cons.hyps

    and asm = `lookup env (x # xs) = None`

  show "update ((x # xs) @ y # ys) opt env = env"

  proof (cases env)

    case (Val a)

    then show ?thesis by simp

  next

    case (Env b es)

    show ?thesis

    proof (cases "es x")

      case None

      note es = `es x = None`

      show ?thesis

        by (cases xs) (simp_all add: es Env fun_upd_idem_iff)

    next

      case (Some e)

      note es = `es x = Some e`

      show ?thesis

      proof (cases xs)

        case Nil

        with asm Env Some have False by simp

        then show ?thesis ..

      next

        case (Cons x' xs')

        from asm Env es have "lookup e xs = None" by simp

        then have "update (xs @ y # ys) opt e = e" by (rule hyp)

        with Env es Cons show "update ((x # xs) @ y # ys) opt env = env"

          by (simp add: fun_upd_idem_iff)

      qed

    qed

  qed

qed

theorem update_append_some:

  assumes "lookup env xs = Some e"

  shows "lookup (update (xs @ y # ys) opt env) xs = Some (update (y # ys) opt e)"

  using assms

proof (induct xs arbitrary: env e)

  case Nil

  then have "env = e" by simp

  then show ?case by simp

next

  case (Cons x xs)

  note hyp = Cons.hyps

    and asm = `lookup env (x # xs) = Some e`

  show "lookup (update ((x # xs) @ y # ys) opt env) (x # xs) =

      Some (update (y # ys) opt e)"

  proof (cases env)

    case (Val a)

    with asm have False by simp

    then show ?thesis ..

  next

    case (Env b es)

    show ?thesis

    proof (cases "es x")

      case None

      with asm Env have False by simp

      then show ?thesis ..

    next

      case (Some e')

      note es = `es x = Some e'`

      show ?thesis

      proof (cases xs)

        case Nil

        with asm Env es have "e = e'" by simp

        with Env es Nil show ?thesis by simp

      next

        case (Cons x' xs')

        from asm Env es have "lookup e' xs = Some e" by simp

        then have "lookup (update (xs @ y # ys) opt e') xs =

          Some (update (y # ys) opt e)" by (rule hyp)

        with Env es Cons show ?thesis by simp

      qed

    qed

  qed

qed

text {*

  \medskip Apparently, @{term update} does not affect the result of

  subsequent @{term lookup} operations at independent positions, i.e.\

  in case that the paths for @{term update} and @{term lookup} fork at

  a certain point.

*}

theorem lookup_update_other:

  assumes neq: "y \<noteq> (z::'c)"

  shows "lookup (update (xs @ z # zs) opt env) (xs @ y # ys) =

    lookup env (xs @ y # ys)"

proof (induct xs arbitrary: env)

  case Nil

  show ?case

  proof (cases env)

    case Val

    then show ?thesis by simp

  next

    case Env

    show ?thesis

    proof (cases zs)

      case Nil

      with neq Env show ?thesis by simp

    next

      case Cons

      with neq Env show ?thesis by simp

    qed

  qed

next

  case (Cons x xs)

  note hyp = Cons.hyps

  show ?case

  proof (cases env)

    case Val

    then show ?thesis by simp

  next

    case (Env y es)

    show ?thesis

    proof (cases xs)

      case Nil

      show ?thesis

      proof (cases "es x")

        case None

        with Env Nil show ?thesis by simp

      next

        case Some

        with neq hyp and Env Nil show ?thesis by simp

      qed

    next

      case (Cons x' xs')

      show ?thesis

      proof (cases "es x")

        case None

        with Env Cons show ?thesis by simp

      next

        case Some

        with neq hyp and Env Cons show ?thesis by simp

      qed

    qed

  qed

qed

text {* Environments and code generation *}

lemma [code, code del]:

  "(HOL.equal :: (_, _, _) env \<Rightarrow> _) = HOL.equal" ..

lemma equal_env_code [code]:

  fixes x y :: "'a\<Colon>equal"

    and f g :: "'c\<Colon>{equal, finite} \<Rightarrow> ('b\<Colon>equal, 'a, 'c) env option"

  shows "HOL.equal (Env x f) (Env y g) \<longleftrightarrow>

  HOL.equal x y \<and> (\<forall>z\<in>UNIV. case f z

   of None \<Rightarrow> (case g z

        of None \<Rightarrow> True | Some _ \<Rightarrow> False)

    | Some a \<Rightarrow> (case g z

        of None \<Rightarrow> False | Some b \<Rightarrow> HOL.equal a b))" (is ?env)

    and "HOL.equal (Val a) (Val b) \<longleftrightarrow> HOL.equal a b"

    and "HOL.equal (Val a) (Env y g) \<longleftrightarrow> False"

    and "HOL.equal (Env x f) (Val b) \<longleftrightarrow> False"

proof (unfold equal)

  have "f = g \<longleftrightarrow> (\<forall>z. case f z

   of None \<Rightarrow> (case g z

        of None \<Rightarrow> True | Some _ \<Rightarrow> False)

    | Some a \<Rightarrow> (case g z

        of None \<Rightarrow> False | Some b \<Rightarrow> a = b))" (is "?lhs = ?rhs")

  proof

    assume ?lhs

    then show ?rhs by (auto split: option.splits)

  next

    assume assm: ?rhs (is "\<forall>z. ?prop z")

    show ?lhs

    proof

      fix z

      from assm have "?prop z" ..

      then show "f z = g z" by (auto split: option.splits)

    qed

  qed

  then show "Env x f = Env y g \<longleftrightarrow>

    x = y \<and> (\<forall>z\<in>UNIV. case f z

     of None \<Rightarrow> (case g z

          of None \<Rightarrow> True | Some _ \<Rightarrow> False)

      | Some a \<Rightarrow> (case g z

          of None \<Rightarrow> False | Some b \<Rightarrow> a = b))" by simp

qed simp_all

lemma [code nbe]:

  "HOL.equal (x :: (_, _, _) env) x \<longleftrightarrow> True"

  by (fact equal_refl)

lemma [code, code del]:

  "(Code_Evaluation.term_of :: ('a::{term_of, type}, 'b::{term_of, type}, 'c::{term_of, type}) env \<Rightarrow> term) = Code_Evaluation.term_of" ..

end