Theory Predicate

(*  Title:      HOL/Predicate.thy
    Author:     Lukas Bulwahn and Florian Haftmann, TU Muenchen
*)

section Predicates as enumerations

theory Predicate
imports String
begin

subsection The type of predicate enumerations (a monad)

datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: "'a  bool")

lemma pred_eqI:
  "(w. eval P w  eval Q w)  P = Q"
  by (cases P, cases Q) (auto simp add: fun_eq_iff)

lemma pred_eq_iff:
  "P = Q  (w. eval P w  eval Q w)"
  by (simp add: pred_eqI)

instantiation pred :: (type) complete_lattice
begin

definition
  "P  Q  eval P  eval Q"

definition
  "P < Q  eval P < eval Q"

definition
  " = Pred "

lemma eval_bot [simp]:
  "eval   = "
  by (simp add: bot_pred_def)

definition
  " = Pred "

lemma eval_top [simp]:
  "eval   = "
  by (simp add: top_pred_def)

definition
  "P  Q = Pred (eval P  eval Q)"

lemma eval_inf [simp]:
  "eval (P  Q) = eval P  eval Q"
  by (simp add: inf_pred_def)

definition
  "P  Q = Pred (eval P  eval Q)"

lemma eval_sup [simp]:
  "eval (P  Q) = eval P  eval Q"
  by (simp add: sup_pred_def)

definition
  "A = Pred ((eval ` A))"

lemma eval_Inf [simp]:
  "eval (A) = (eval ` A)"
  by (simp add: Inf_pred_def)

definition
  "A = Pred ((eval ` A))"

lemma eval_Sup [simp]:
  "eval (A) = (eval ` A)"
  by (simp add: Sup_pred_def)

instance proof
qed (auto intro!: pred_eqI simp add: less_eq_pred_def less_pred_def le_fun_def less_fun_def)

end

lemma eval_INF [simp]:
  "eval ((f ` A)) = ((eval  f) ` A)"
  by (simp add: image_comp)

lemma eval_SUP [simp]:
  "eval ((f ` A)) = ((eval  f) ` A)"
  by (simp add: image_comp)

instantiation pred :: (type) complete_boolean_algebra
begin

definition
  "- P = Pred (- eval P)"

lemma eval_compl [simp]:
  "eval (- P) = - eval P"
  by (simp add: uminus_pred_def)

definition
  "P - Q = Pred (eval P - eval Q)"

lemma eval_minus [simp]:
  "eval (P - Q) = eval P - eval Q"
  by (simp add: minus_pred_def)

instance proof
  fix A::"'a pred set set"
  show "(Sup ` A)  (Inf ` {f ` A |f. YA. f Y  Y})"
  proof (simp add: less_eq_pred_def Sup_fun_def Inf_fun_def, safe)
    fix w
    assume A: "xA. fx. eval f w"
    define F where "F = (λ x . SOME f . f  x  eval f w)"
    have [simp]: "(f (F ` A). eval f w)"
      by (metis (no_types, lifting) A F_def image_iff some_eq_ex)
    have "(f. F ` A = f ` A  (YA. f Y  Y))  (f(F ` A). eval f w)"
      using A by (simp, metis (no_types, lifting) F_def someI)+
    from this show "x. (f. x = f ` A  (YA. f Y  Y))  (fx. eval f w)"
      by (rule exI [of _ "F ` A"])
  qed
qed (auto intro!: pred_eqI)

end

definition single :: "'a  'a pred" where
  "single x = Pred ((=) x)"

lemma eval_single [simp]:
  "eval (single x) = (=) x"
  by (simp add: single_def)

definition bind :: "'a pred  ('a  'b pred)  'b pred" (infixl "" 70) where
  "P  f = ((f ` {x. eval P x}))"

lemma eval_bind [simp]:
  "eval (P  f) = eval ((f ` {x. eval P x}))"
  by (simp add: bind_def)

lemma bind_bind:
  "(P  Q)  R = P  (λx. Q x  R)"
  by (rule pred_eqI) auto

lemma bind_single:
  "P  single = P"
  by (rule pred_eqI) auto

lemma single_bind:
  "single x  P = P x"
  by (rule pred_eqI) auto

lemma bottom_bind:
  "  P = "
  by (rule pred_eqI) auto

lemma sup_bind:
  "(P  Q)  R = P  R  Q  R"
  by (rule pred_eqI) auto

lemma Sup_bind:
  "(A  f) = ((λx. x  f) ` A)"
  by (rule pred_eqI) auto

lemma pred_iffI:
  assumes "x. eval A x  eval B x"
  and "x. eval B x  eval A x"
  shows "A = B"
  using assms by (auto intro: pred_eqI)
  
lemma singleI: "eval (single x) x"
  by simp

lemma singleI_unit: "eval (single ()) x"
  by simp

lemma singleE: "eval (single x) y  (y = x  P)  P"
  by simp

lemma singleE': "eval (single x) y  (x = y  P)  P"
  by simp

lemma bindI: "eval P x  eval (Q x) y  eval (P  Q) y"
  by auto

lemma bindE: "eval (R  Q) y  (x. eval R x  eval (Q x) y  P)  P"
  by auto

lemma botE: "eval  x  P"
  by auto

lemma supI1: "eval A x  eval (A  B) x"
  by auto

lemma supI2: "eval B x  eval (A  B) x" 
  by auto

lemma supE: "eval (A  B) x  (eval A x  P)  (eval B x  P)  P"
  by auto

lemma single_not_bot [simp]:
  "single x  "
  by (auto simp add: single_def bot_pred_def fun_eq_iff)

lemma not_bot:
  assumes "A  "
  obtains x where "eval A x"
  using assms by (cases A) (auto simp add: bot_pred_def)


subsection Emptiness check and definite choice

definition is_empty :: "'a pred  bool" where
  "is_empty A  A = "

lemma is_empty_bot:
  "is_empty "
  by (simp add: is_empty_def)

lemma not_is_empty_single:
  "¬ is_empty (single x)"
  by (auto simp add: is_empty_def single_def bot_pred_def fun_eq_iff)

lemma is_empty_sup:
  "is_empty (A  B)  is_empty A  is_empty B"
  by (auto simp add: is_empty_def)

definition singleton :: "(unit  'a)  'a pred  'a" where
  "singleton default A = (if ∃!x. eval A x then THE x. eval A x else default ())" for default

lemma singleton_eqI:
  "∃!x. eval A x  eval A x  singleton default A = x" for default
  by (auto simp add: singleton_def)

lemma eval_singletonI:
  "∃!x. eval A x  eval A (singleton default A)" for default
proof -
  assume assm: "∃!x. eval A x"
  then obtain x where x: "eval A x" ..
  with assm have "singleton default A = x" by (rule singleton_eqI)
  with x show ?thesis by simp
qed

lemma single_singleton:
  "∃!x. eval A x  single (singleton default A) = A" for default
proof -
  assume assm: "∃!x. eval A x"
  then have "eval A (singleton default A)"
    by (rule eval_singletonI)
  moreover from assm have "x. eval A x  singleton default A = x"
    by (rule singleton_eqI)
  ultimately have "eval (single (singleton default A)) = eval A"
    by (simp (no_asm_use) add: single_def fun_eq_iff) blast
  then have "x. eval (single (singleton default A)) x = eval A x"
    by simp
  then show ?thesis by (rule pred_eqI)
qed

lemma singleton_undefinedI:
  "¬ (∃!x. eval A x)  singleton default A = default ()" for default
  by (simp add: singleton_def)

lemma singleton_bot:
  "singleton default  = default ()" for default
  by (auto simp add: bot_pred_def intro: singleton_undefinedI)

lemma singleton_single:
  "singleton default (single x) = x" for default
  by (auto simp add: intro: singleton_eqI singleI elim: singleE)

lemma singleton_sup_single_single:
  "singleton default (single x  single y) = (if x = y then x else default ())" for default
proof (cases "x = y")
  case True then show ?thesis by (simp add: singleton_single)
next
  case False
  have "eval (single x  single y) x"
    and "eval (single x  single y) y"
  by (auto intro: supI1 supI2 singleI)
  with False have "¬ (∃!z. eval (single x  single y) z)"
    by blast
  then have "singleton default (single x  single y) = default ()"
    by (rule singleton_undefinedI)
  with False show ?thesis by simp
qed

lemma singleton_sup_aux:
  "singleton default (A  B) = (if A =  then singleton default B
    else if B =  then singleton default A
    else singleton default
      (single (singleton default A)  single (singleton default B)))" for default
proof (cases "(∃!x. eval A x)  (∃!y. eval B y)")
  case True then show ?thesis by (simp add: single_singleton)
next
  case False
  from False have A_or_B:
    "singleton default A = default ()  singleton default B = default ()"
    by (auto intro!: singleton_undefinedI)
  then have rhs: "singleton default
    (single (singleton default A)  single (singleton default B)) = default ()"
    by (auto simp add: singleton_sup_single_single singleton_single)
  from False have not_unique:
    "¬ (∃!x. eval A x)  ¬ (∃!y. eval B y)" by simp
  show ?thesis proof (cases "A    B  ")
    case True
    then obtain a b where a: "eval A a" and b: "eval B b"
      by (blast elim: not_bot)
    with True not_unique have "¬ (∃!x. eval (A  B) x)"
      by (auto simp add: sup_pred_def bot_pred_def)
    then have "singleton default (A  B) = default ()" by (rule singleton_undefinedI)
    with True rhs show ?thesis by simp
  next
    case False then show ?thesis by auto
  qed
qed

lemma singleton_sup:
  "singleton default (A  B) = (if A =  then singleton default B
    else if B =  then singleton default A
    else if singleton default A = singleton default B then singleton default A else default ())" for default
  using singleton_sup_aux [of default A B] by (simp only: singleton_sup_single_single)


subsection Derived operations

definition if_pred :: "bool  unit pred" where
  if_pred_eq: "if_pred b = (if b then single () else )"

definition holds :: "unit pred  bool" where
  holds_eq: "holds P = eval P ()"

definition not_pred :: "unit pred  unit pred" where
  not_pred_eq: "not_pred P = (if eval P () then  else single ())"

lemma if_predI: "P  eval (if_pred P) ()"
  unfolding if_pred_eq by (auto intro: singleI)

lemma if_predE: "eval (if_pred b) x  (b  x = ()  P)  P"
  unfolding if_pred_eq by (cases b) (auto elim: botE)

lemma not_predI: "¬ P  eval (not_pred (Pred (λu. P))) ()"
  unfolding not_pred_eq by (auto intro: singleI)

lemma not_predI': "¬ eval P ()  eval (not_pred P) ()"
  unfolding not_pred_eq by (auto intro: singleI)

lemma not_predE: "eval (not_pred (Pred (λu. P))) x  (¬ P  thesis)  thesis"
  unfolding not_pred_eq
  by (auto split: if_split_asm elim: botE)

lemma not_predE': "eval (not_pred P) x  (¬ eval P x  thesis)  thesis"
  unfolding not_pred_eq
  by (auto split: if_split_asm elim: botE)
lemma "f () = False  f () = True"
by simp

lemma closure_of_bool_cases [no_atp]:
  fixes f :: "unit  bool"
  assumes "f = (λu. False)  P f"
  assumes "f = (λu. True)  P f"
  shows "P f"
proof -
  have "f = (λu. False)  f = (λu. True)"
    apply (cases "f ()")
    apply (rule disjI2)
    apply (rule ext)
    apply (simp add: unit_eq)
    apply (rule disjI1)
    apply (rule ext)
    apply (simp add: unit_eq)
    done
  from this assms show ?thesis by blast
qed

lemma unit_pred_cases:
  assumes "P "
  assumes "P (single ())"
  shows "P Q"
using assms unfolding bot_pred_def bot_fun_def bot_bool_def empty_def single_def proof (cases Q)
  fix f
  assume "P (Pred (λu. False))" "P (Pred (λu. () = u))"
  then have "P (Pred f)" 
    by (cases _ f rule: closure_of_bool_cases) simp_all
  moreover assume "Q = Pred f"
  ultimately show "P Q" by simp
qed
  
lemma holds_if_pred:
  "holds (if_pred b) = b"
unfolding if_pred_eq holds_eq
by (cases b) (auto intro: singleI elim: botE)

lemma if_pred_holds:
  "if_pred (holds P) = P"
unfolding if_pred_eq holds_eq
by (rule unit_pred_cases) (auto intro: singleI elim: botE)

lemma is_empty_holds:
  "is_empty P  ¬ holds P"
unfolding is_empty_def holds_eq
by (rule unit_pred_cases) (auto elim: botE intro: singleI)

definition map :: "('a  'b)  'a pred  'b pred" where
  "map f P = P  (single  f)"

lemma eval_map [simp]:
  "eval (map f P) = (x{x. eval P x}. (λy. f x = y))"
  by (simp add: map_def comp_def image_comp)

functor map: map
  by (rule ext, rule pred_eqI, auto)+


subsection Implementation

datatype (plugins only: code extraction) (dead 'a) seq =
  Empty
| Insert "'a" "'a pred"
| Join "'a pred" "'a seq"

primrec pred_of_seq :: "'a seq  'a pred" where
  "pred_of_seq Empty = "
| "pred_of_seq (Insert x P) = single x  P"
| "pred_of_seq (Join P xq) = P  pred_of_seq xq"

definition Seq :: "(unit  'a seq)  'a pred" where
  "Seq f = pred_of_seq (f ())"

code_datatype Seq

primrec member :: "'a seq  'a  bool"  where
  "member Empty x  False"
| "member (Insert y P) x  x = y  eval P x"
| "member (Join P xq) x  eval P x  member xq x"

lemma eval_member:
  "member xq = eval (pred_of_seq xq)"
proof (induct xq)
  case Empty show ?case
  by (auto simp add: fun_eq_iff elim: botE)
next
  case Insert show ?case
  by (auto simp add: fun_eq_iff elim: supE singleE intro: supI1 supI2 singleI)
next
  case Join then show ?case
  by (auto simp add: fun_eq_iff elim: supE intro: supI1 supI2)
qed

lemma eval_code [(* FIXME declare simp *)code]: "eval (Seq f) = member (f ())"
  unfolding Seq_def by (rule sym, rule eval_member)

lemma single_code [code]:
  "single x = Seq (λu. Insert x )"
  unfolding Seq_def by simp

primrec "apply" :: "('a  'b pred)  'a seq  'b seq" where
  "apply f Empty = Empty"
| "apply f (Insert x P) = Join (f x) (Join (P  f) Empty)"
| "apply f (Join P xq) = Join (P  f) (apply f xq)"

lemma apply_bind:
  "pred_of_seq (apply f xq) = pred_of_seq xq  f"
proof (induct xq)
  case Empty show ?case
    by (simp add: bottom_bind)
next
  case Insert show ?case
    by (simp add: single_bind sup_bind)
next
  case Join then show ?case
    by (simp add: sup_bind)
qed
  
lemma bind_code [code]:
  "Seq g  f = Seq (λu. apply f (g ()))"
  unfolding Seq_def by (rule sym, rule apply_bind)

lemma bot_set_code [code]:
  " = Seq (λu. Empty)"
  unfolding Seq_def by simp

primrec adjunct :: "'a pred  'a seq  'a seq" where
  "adjunct P Empty = Join P Empty"
| "adjunct P (Insert x Q) = Insert x (Q  P)"
| "adjunct P (Join Q xq) = Join Q (adjunct P xq)"

lemma adjunct_sup:
  "pred_of_seq (adjunct P xq) = P  pred_of_seq xq"
  by (induct xq) (simp_all add: sup_assoc sup_commute sup_left_commute)

lemma sup_code [code]:
  "Seq f  Seq g = Seq (λu. case f ()
    of Empty  g ()
     | Insert x P  Insert x (P  Seq g)
     | Join P xq  adjunct (Seq g) (Join P xq))"
proof (cases "f ()")
  case Empty
  thus ?thesis
    unfolding Seq_def by (simp add: sup_commute [of ""])
next
  case Insert
  thus ?thesis
    unfolding Seq_def by (simp add: sup_assoc)
next
  case Join
  thus ?thesis
    unfolding Seq_def
    by (simp add: adjunct_sup sup_assoc sup_commute sup_left_commute)
qed

primrec contained :: "'a seq  'a pred  bool" where
  "contained Empty Q  True"
| "contained (Insert x P) Q  eval Q x  P  Q"
| "contained (Join P xq) Q  P  Q  contained xq Q"

lemma single_less_eq_eval:
  "single x  P  eval P x"
  by (auto simp add: less_eq_pred_def le_fun_def)

lemma contained_less_eq:
  "contained xq Q  pred_of_seq xq  Q"
  by (induct xq) (simp_all add: single_less_eq_eval)

lemma less_eq_pred_code [code]:
  "Seq f  Q = (case f ()
   of Empty  True
    | Insert x P  eval Q x  P  Q
    | Join P xq  P  Q  contained xq Q)"
  by (cases "f ()")
    (simp_all add: Seq_def single_less_eq_eval contained_less_eq)

instantiation pred :: (type) equal
begin

definition equal_pred
  where [simp]: "HOL.equal P Q  P = (Q :: 'a pred)"

instance by standard simp

end
    
lemma [code]:
  "HOL.equal P Q  P  Q  Q  P" for P Q :: "'a pred"
  by auto

lemma [code nbe]:
  "HOL.equal P P  True" for P :: "'a pred"
  by (fact equal_refl)

lemma [code]:
  "case_pred f P = f (eval P)"
  by (fact pred.case_eq_if)

lemma [code]:
  "rec_pred f P = f (eval P)"
  by (cases P) simp

inductive eq :: "'a  'a  bool" where "eq x x"

lemma eq_is_eq: "eq x y  (x = y)"
  by (rule eq_reflection) (auto intro: eq.intros elim: eq.cases)

primrec null :: "'a seq  bool" where
  "null Empty  True"
| "null (Insert x P)  False"
| "null (Join P xq)  is_empty P  null xq"

lemma null_is_empty:
  "null xq  is_empty (pred_of_seq xq)"
  by (induct xq) (simp_all add: is_empty_bot not_is_empty_single is_empty_sup)

lemma is_empty_code [code]:
  "is_empty (Seq f)  null (f ())"
  by (simp add: null_is_empty Seq_def)

primrec the_only :: "(unit  'a)  'a seq  'a" where
  "the_only default Empty = default ()" for default
| "the_only default (Insert x P) =
    (if is_empty P then x else let y = singleton default P in if x = y then x else default ())" for default
| "the_only default (Join P xq) =
    (if is_empty P then the_only default xq else if null xq then singleton default P
       else let x = singleton default P; y = the_only default xq in
       if x = y then x else default ())" for default

lemma the_only_singleton:
  "the_only default xq = singleton default (pred_of_seq xq)" for default
  by (induct xq)
    (auto simp add: singleton_bot singleton_single is_empty_def
    null_is_empty Let_def singleton_sup)

lemma singleton_code [code]:
  "singleton default (Seq f) =
    (case f () of
      Empty  default ()
    | Insert x P  if is_empty P then x
        else let y = singleton default P in
          if x = y then x else default ()
    | Join P xq  if is_empty P then the_only default xq
        else if null xq then singleton default P
        else let x = singleton default P; y = the_only default xq in
          if x = y then x else default ())" for default
  by (cases "f ()")
   (auto simp add: Seq_def the_only_singleton is_empty_def
      null_is_empty singleton_bot singleton_single singleton_sup Let_def)

definition the :: "'a pred  'a" where
  "the A = (THE x. eval A x)"

lemma the_eqI:
  "(THE x. eval P x) = x  the P = x"
  by (simp add: the_def)

lemma the_eq [code]: "the A = singleton (λx. Code.abort (STR ''not_unique'') (λ_. the A)) A"
  by (rule the_eqI) (simp add: singleton_def the_def)

code_reflect Predicate
  datatypes pred = Seq and seq = Empty | Insert | Join

ML 
signature PREDICATE =
sig
  val anamorph: ('a -> ('b * 'a) option) -> int -> 'a -> 'b list * 'a
  datatype 'a pred = Seq of (unit -> 'a seq)
  and 'a seq = Empty | Insert of 'a * 'a pred | Join of 'a pred * 'a seq
  val map: ('a -> 'b) -> 'a pred -> 'b pred
  val yield: 'a pred -> ('a * 'a pred) option
  val yieldn: int -> 'a pred -> 'a list * 'a pred
end;

structure Predicate : PREDICATE =
struct

fun anamorph f k x =
 (if k = 0 then ([], x)
  else case f x
   of NONE => ([], x)
    | SOME (v, y) => let
        val k' = k - 1;
        val (vs, z) = anamorph f k' y
      in (v :: vs, z) end);

datatype pred = datatype Predicate.pred
datatype seq = datatype Predicate.seq

fun map f = @{code Predicate.map} f;

fun yield (Seq f) = next (f ())
and next Empty = NONE
  | next (Insert (x, P)) = SOME (x, P)
  | next (Join (P, xq)) = (case yield P
     of NONE => next xq
      | SOME (x, Q) => SOME (x, Seq (fn _ => Join (Q, xq))));

fun yieldn k = anamorph yield k;

end;


text Conversion from and to sets

definition pred_of_set :: "'a set  'a pred" where
  "pred_of_set = Pred  (λA x. x  A)"

lemma eval_pred_of_set [simp]:
  "eval (pred_of_set A) x  x A"
  by (simp add: pred_of_set_def)

definition set_of_pred :: "'a pred  'a set" where
  "set_of_pred = Collect  eval"

lemma member_set_of_pred [simp]:
  "x  set_of_pred P  Predicate.eval P x"
  by (simp add: set_of_pred_def)

definition set_of_seq :: "'a seq  'a set" where
  "set_of_seq = set_of_pred  pred_of_seq"

lemma member_set_of_seq [simp]:
  "x  set_of_seq xq = Predicate.member xq x"
  by (simp add: set_of_seq_def eval_member)

lemma of_pred_code [code]:
  "set_of_pred (Predicate.Seq f) = (case f () of
     Predicate.Empty  {}
   | Predicate.Insert x P  insert x (set_of_pred P)
   | Predicate.Join P xq  set_of_pred P  set_of_seq xq)"
  by (auto split: seq.split simp add: eval_code)

lemma of_seq_code [code]:
  "set_of_seq Predicate.Empty = {}"
  "set_of_seq (Predicate.Insert x P) = insert x (set_of_pred P)"
  "set_of_seq (Predicate.Join P xq) = set_of_pred P  set_of_seq xq"
  by auto

text Lazy Evaluation of an indexed function

function iterate_upto :: "(natural  'a)  natural  natural  'a Predicate.pred"
where
  "iterate_upto f n m =
    Predicate.Seq (%u. if n > m then Predicate.Empty
     else Predicate.Insert (f n) (iterate_upto f (n + 1) m))"
by pat_completeness auto

termination by (relation "measure (%(f, n, m). nat_of_natural (m + 1 - n))")
  (auto simp add: less_natural_def)

text Misc

declare Inf_set_fold [where 'a = "'a Predicate.pred", code]
declare Sup_set_fold [where 'a = "'a Predicate.pred", code]

(* FIXME: better implement conversion by bisection *)

lemma pred_of_set_fold_sup:
  assumes "finite A"
  shows "pred_of_set A = Finite_Set.fold sup bot (Predicate.single ` A)" (is "?lhs = ?rhs")
proof (rule sym)
  interpret comp_fun_idem "sup :: 'a Predicate.pred  'a Predicate.pred  'a Predicate.pred"
    by (fact comp_fun_idem_sup)
  from finite A show "?rhs = ?lhs" by (induct A) (auto intro!: pred_eqI)
qed

lemma pred_of_set_set_fold_sup:
  "pred_of_set (set xs) = fold sup (List.map Predicate.single xs) bot"
proof -
  interpret comp_fun_idem "sup :: 'a Predicate.pred  'a Predicate.pred  'a Predicate.pred"
    by (fact comp_fun_idem_sup)
  show ?thesis by (simp add: pred_of_set_fold_sup fold_set_fold [symmetric])
qed

lemma pred_of_set_set_foldr_sup [code]:
  "pred_of_set (set xs) = foldr sup (List.map Predicate.single xs) bot"
  by (simp add: pred_of_set_set_fold_sup ac_simps foldr_fold fun_eq_iff)

no_notation
  bind (infixl "" 70)

hide_type (open) pred seq
hide_const (open) Pred eval single bind is_empty singleton if_pred not_pred holds
  Empty Insert Join Seq member pred_of_seq "apply" adjunct null the_only eq map the
  iterate_upto
hide_fact (open) null_def member_def

end