Theory Predicate

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theory Predicate
imports List
`(*  Title:      HOL/Predicate.thy    Author:     Lukas Bulwahn and Florian Haftmann, TU Muenchen*)header {* Predicates as enumerations *}theory Predicateimports Listbeginnotation  bot ("⊥") and  top ("\<top>") and  inf (infixl "\<sqinter>" 70) and  sup (infixl "\<squnion>" 65) and  Inf ("\<Sqinter>_" [900] 900) and  Sup ("\<Squnion>_" [900] 900)syntax (xsymbols)  "_INF1"     :: "pttrns => 'b => 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)  "_INF"      :: "pttrn => 'a set => 'b => 'b"  ("(3\<Sqinter>_∈_./ _)" [0, 0, 10] 10)  "_SUP1"     :: "pttrns => 'b => 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)  "_SUP"      :: "pttrn => 'a set => 'b => 'b"  ("(3\<Squnion>_∈_./ _)" [0, 0, 10] 10)subsection {* The type of predicate enumerations (a monad) *}datatype 'a pred = Pred "'a => bool"primrec eval :: "'a pred => 'a => bool" where  eval_pred: "eval (Pred f) = f"lemma Pred_eval [simp]:  "Pred (eval x) = x"  by (cases x) simplemma 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_latticebegindefinition  "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  "\<top> = Pred \<top>"lemma eval_top [simp]:  "eval \<top>  = \<top>"  by (simp add: top_pred_def)definition  "P \<sqinter> Q = Pred (eval P \<sqinter> eval Q)"lemma eval_inf [simp]:  "eval (P \<sqinter> Q) = eval P \<sqinter> eval Q"  by (simp add: inf_pred_def)definition  "P \<squnion> Q = Pred (eval P \<squnion> eval Q)"lemma eval_sup [simp]:  "eval (P \<squnion> Q) = eval P \<squnion> eval Q"  by (simp add: sup_pred_def)definition  "\<Sqinter>A = Pred (INFI A eval)"lemma eval_Inf [simp]:  "eval (\<Sqinter>A) = INFI A eval"  by (simp add: Inf_pred_def)definition  "\<Squnion>A = Pred (SUPR A eval)"lemma eval_Sup [simp]:  "eval (\<Squnion>A) = SUPR A eval"  by (simp add: Sup_pred_def)instance proofqed (auto intro!: pred_eqI simp add: less_eq_pred_def less_pred_def le_fun_def less_fun_def)endlemma eval_INFI [simp]:  "eval (INFI A f) = INFI A (eval o f)"  by (simp only: INF_def eval_Inf image_compose)lemma eval_SUPR [simp]:  "eval (SUPR A f) = SUPR A (eval o f)"  by (simp only: SUP_def eval_Sup image_compose)instantiation pred :: (type) complete_boolean_algebrabegindefinition  "- 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 proofqed (auto intro!: pred_eqI)enddefinition single :: "'a => 'a pred" where  "single x = Pred ((op =) x)"lemma eval_single [simp]:  "eval (single x) = (op =) x"  by (simp add: single_def)definition bind :: "'a pred => ('a => 'b pred) => 'b pred" (infixl "»=" 70) where  "P »= f = (SUPR {x. eval P x} f)"lemma eval_bind [simp]:  "eval (P »= f) = eval (SUPR {x. eval P x} f)"  by (simp add: bind_def)lemma bind_bind:  "(P »= Q) »= R = P »= (λx. Q x »= R)"  by (rule pred_eqI) autolemma bind_single:  "P »= single = P"  by (rule pred_eqI) autolemma single_bind:  "single x »= P = P x"  by (rule pred_eqI) autolemma bottom_bind:  "⊥ »= P = ⊥"  by (rule pred_eqI) autolemma sup_bind:  "(P \<squnion> Q) »= R = P »= R \<squnion> Q »= R"  by (rule pred_eqI) autolemma Sup_bind:  "(\<Squnion>A »= f) = \<Squnion>((λx. x »= f) ` A)"  by (rule pred_eqI) autolemma 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 simplemma singleI_unit: "eval (single ()) x"  by simplemma singleE: "eval (single x) y ==> (y = x ==> P) ==> P"  by simplemma singleE': "eval (single x) y ==> (x = y ==> P) ==> P"  by simplemma bindI: "eval P x ==> eval (Q x) y ==> eval (P »= Q) y"  by autolemma bindE: "eval (R »= Q) y ==> (!!x. eval R x ==> eval (Q x) y ==> P) ==> P"  by autolemma botE: "eval ⊥ x ==> P"  by autolemma supI1: "eval A x ==> eval (A \<squnion> B) x"  by autolemma supI2: "eval B x ==> eval (A \<squnion> B) x"   by autolemma supE: "eval (A \<squnion> B) x ==> (eval A x ==> P) ==> (eval B x ==> P) ==> P"  by autolemma 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 \<squnion> B) <-> is_empty A ∧ is_empty B"  by (auto simp add: is_empty_def)definition singleton :: "(unit => 'a) => 'a pred => 'a" where  "singleton dfault A = (if ∃!x. eval A x then THE x. eval A x else dfault ())"lemma singleton_eqI:  "∃!x. eval A x ==> eval A x ==> singleton dfault A = x"  by (auto simp add: singleton_def)lemma eval_singletonI:  "∃!x. eval A x ==> eval A (singleton dfault A)"proof -  assume assm: "∃!x. eval A x"  then obtain x where "eval A x" ..  moreover with assm have "singleton dfault A = x" by (rule singleton_eqI)  ultimately show ?thesis by simp qedlemma single_singleton:  "∃!x. eval A x ==> single (singleton dfault A) = A"proof -  assume assm: "∃!x. eval A x"  then have "eval A (singleton dfault A)"    by (rule eval_singletonI)  moreover from assm have "!!x. eval A x ==> singleton dfault A = x"    by (rule singleton_eqI)  ultimately have "eval (single (singleton dfault A)) = eval A"    by (simp (no_asm_use) add: single_def fun_eq_iff) blast  then have "!!x. eval (single (singleton dfault A)) x = eval A x"    by simp  then show ?thesis by (rule pred_eqI)qedlemma singleton_undefinedI:  "¬ (∃!x. eval A x) ==> singleton dfault A = dfault ()"  by (simp add: singleton_def)lemma singleton_bot:  "singleton dfault ⊥ = dfault ()"  by (auto simp add: bot_pred_def intro: singleton_undefinedI)lemma singleton_single:  "singleton dfault (single x) = x"  by (auto simp add: intro: singleton_eqI singleI elim: singleE)lemma singleton_sup_single_single:  "singleton dfault (single x \<squnion> single y) = (if x = y then x else dfault ())"proof (cases "x = y")  case True then show ?thesis by (simp add: singleton_single)next  case False  have "eval (single x \<squnion> single y) x"    and "eval (single x \<squnion> single y) y"  by (auto intro: supI1 supI2 singleI)  with False have "¬ (∃!z. eval (single x \<squnion> single y) z)"    by blast  then have "singleton dfault (single x \<squnion> single y) = dfault ()"    by (rule singleton_undefinedI)  with False show ?thesis by simpqedlemma singleton_sup_aux:  "singleton dfault (A \<squnion> B) = (if A = ⊥ then singleton dfault B    else if B = ⊥ then singleton dfault A    else singleton dfault      (single (singleton dfault A) \<squnion> single (singleton dfault B)))"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 dfault A = dfault () ∨ singleton dfault B = dfault ()"    by (auto intro!: singleton_undefinedI)  then have rhs: "singleton dfault    (single (singleton dfault A) \<squnion> single (singleton dfault B)) = dfault ()"    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 \<squnion> B) x)"      by (auto simp add: sup_pred_def bot_pred_def)    then have "singleton dfault (A \<squnion> B) = dfault ()" by (rule singleton_undefinedI)    with True rhs show ?thesis by simp  next    case False then show ?thesis by auto  qedqedlemma singleton_sup:  "singleton dfault (A \<squnion> B) = (if A = ⊥ then singleton dfault B    else if B = ⊥ then singleton dfault A    else if singleton dfault A = singleton dfault B then singleton dfault A else dfault ())"using singleton_sup_aux [of dfault 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 eval_pred 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: split_if_asm elim: botE)lemma not_predE': "eval (not_pred P) x ==> (¬ eval P x ==> thesis) ==> thesis"  unfolding not_pred_eq  by (auto split: split_if_asm elim: botE)lemma "f () = False ∨ f () = True"by simplemma 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 blastqedlemma 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 simpqed  lemma holds_if_pred:  "holds (if_pred b) = b"unfolding if_pred_eq holds_eqby (cases b) (auto intro: singleI elim: botE)lemma if_pred_holds:  "if_pred (holds P) = P"unfolding if_pred_eq holds_eqby (rule unit_pred_cases) (auto intro: singleI elim: botE)lemma is_empty_holds:  "is_empty P <-> ¬ holds P"unfolding is_empty_def holds_eqby (rule unit_pred_cases) (auto elim: botE intro: singleI)definition map :: "('a => 'b) => 'a pred => 'b pred" where  "map f P = P »= (single o f)"lemma eval_map [simp]:  "eval (map f P) = (\<Squnion>x∈{x. eval P x}. (λy. f x = y))"  by (auto simp add: map_def comp_def)enriched_type map: map  by (rule ext, rule pred_eqI, auto)+subsection {* Implementation *}datatype '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 \<squnion> P"| "pred_of_seq (Join P xq) = P \<squnion> pred_of_seq xq"definition Seq :: "(unit => 'a seq) => 'a pred" where  "Seq f = pred_of_seq (f ())"code_datatype Seqprimrec 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)qedlemma 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 simpprimrec "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 simpprimrec adjunct :: "'a pred => 'a seq => 'a seq" where  "adjunct P Empty = Join P Empty"| "adjunct P (Insert x Q) = Insert x (Q \<squnion> P)"| "adjunct P (Join Q xq) = Join Q (adjunct P xq)"lemma adjunct_sup:  "pred_of_seq (adjunct P xq) = P \<squnion> pred_of_seq xq"  by (induct xq) (simp_all add: sup_assoc sup_commute sup_left_commute)lemma sup_code [code]:  "Seq f \<squnion> Seq g = Seq (λu. case f ()    of Empty => g ()     | Insert x P => Insert x (P \<squnion> 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)qedlemma [code]:  "size (P :: 'a Predicate.pred) = 0" by (cases P) simplemma [code]:  "pred_size f P = 0" by (cases P) simpprimrec 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)lemma eq_pred_code [code]:  fixes P Q :: "'a pred"  shows "HOL.equal P Q <-> P ≤ Q ∧ Q ≤ P"  by (auto simp add: equal)lemma [code nbe]:  "HOL.equal (x :: 'a pred) x <-> True"  by (fact equal_refl)lemma [code]:  "pred_case f P = f (eval P)"  by (cases P) simplemma [code]:  "pred_rec f P = f (eval P)"  by (cases P) simpinductive 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  [code del]: "the_only dfault Empty = dfault ()"| "the_only dfault (Insert x P) = (if is_empty P then x else let y = singleton dfault P in if x = y then x else dfault ())"| "the_only dfault (Join P xq) = (if is_empty P then the_only dfault xq else if null xq then singleton dfault P       else let x = singleton dfault P; y = the_only dfault xq in       if x = y then x else dfault ())"lemma the_only_singleton:  "the_only dfault xq = singleton dfault (pred_of_seq xq)"  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 dfault (Seq f) = (case f ()   of Empty => dfault ()    | Insert x P => if is_empty P then x        else let y = singleton dfault P in          if x = y then x else dfault ()    | Join P xq => if is_empty P then the_only dfault xq        else if null xq then singleton dfault P        else let x = singleton dfault P; y = the_only dfault xq in          if x = y then x else dfault ())"  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)definition not_unique :: "'a pred => 'a" where  [code del]: "not_unique A = (THE x. eval A x)"code_abort not_uniquelemma the_eq [code]: "the A = singleton (λx. not_unique A) A"  by (rule the_eqI) (simp add: singleton_def not_unique_def)code_reflect Predicate  datatypes pred = Seq and seq = Empty | Insert | Join  functions mapML {*signature PREDICATE =sig  datatype 'a pred = Seq of (unit -> 'a seq)  and 'a seq = Empty | Insert of 'a * 'a pred | Join of 'a pred * 'a seq  val yield: 'a pred -> ('a * 'a pred) option  val yieldn: int -> 'a pred -> 'a list * 'a pred  val map: ('a -> 'b) -> 'a pred -> 'b predend;structure Predicate : PREDICATE =structdatatype pred = datatype Predicate.preddatatype seq = datatype Predicate.seqfun map f = 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 anamorph f k x = (if k = 0 then ([], x)  else case f x   of NONE => ([], x)    | SOME (v, y) => let        val (vs, z) = anamorph f (k - 1) y      in (v :: vs, z) end);fun yieldn P = anamorph yield P;end;*}text {* Conversion from and to sets *}definition pred_of_set :: "'a set => 'a pred" where  "pred_of_set = Pred o (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 o 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 o 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 autotext {* Lazy Evaluation of an indexed function *}function iterate_upto :: "(code_numeral => 'a) => code_numeral => code_numeral => '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 autotermination by (relation "measure (%(f, n, m). Code_Numeral.nat_of (m + 1 - n))") autotext {* 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)qedlemma 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])qedlemma 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  bot ("⊥") and  top ("\<top>") and  inf (infixl "\<sqinter>" 70) and  sup (infixl "\<squnion>" 65) and  Inf ("\<Sqinter>_" [900] 900) and  Sup ("\<Squnion>_" [900] 900) and  bind (infixl "»=" 70)no_syntax (xsymbols)  "_INF1"     :: "pttrns => 'b => 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)  "_INF"      :: "pttrn => 'a set => 'b => 'b"  ("(3\<Sqinter>_∈_./ _)" [0, 0, 10] 10)  "_SUP1"     :: "pttrns => 'b => 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)  "_SUP"      :: "pttrn => 'a set => 'b => 'b"  ("(3\<Squnion>_∈_./ _)" [0, 0, 10] 10)hide_type (open) pred seqhide_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 not_unique the  iterate_uptohide_fact (open) null_def member_defend`