(*  Title:      HOL/Predicate.thy

     Author:     Stefan Berghofer and Lukas Bulwahn and Florian Haftmann, TU Muenchen

 *)

 header {* Predicates as relations and enumerations *}

 theory Predicate

 imports Inductive Relation

 begin

 notation

   bot ("\<bottom>") 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 \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)

   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)

   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)

   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)

 subsection {* Predicates as (complete) lattices *}

 text {*

   Handy introduction and elimination rules for @{text "\<le>"}

   on unary and binary predicates

 *}

 lemma predicate1I:

   assumes PQ: "\<And>x. P x \<Longrightarrow> Q x"

   shows "P \<le> Q"

   apply (rule le_funI)

   apply (rule le_boolI)

   apply (rule PQ)

   apply assumption

   done

 lemma predicate1D [Pure.dest?, dest?]:

   "P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x"

   apply (erule le_funE)

   apply (erule le_boolE)

   apply assumption+

   done

 lemma rev_predicate1D:

   "P x ==> P <= Q ==> Q x"

   by (rule predicate1D)

 lemma predicate2I [Pure.intro!, intro!]:

   assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y"

   shows "P \<le> Q"

   apply (rule le_funI)+

   apply (rule le_boolI)

   apply (rule PQ)

   apply assumption

   done

 lemma predicate2D [Pure.dest, dest]:

   "P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y"

   apply (erule le_funE)+

   apply (erule le_boolE)

   apply assumption+

   done

 lemma rev_predicate2D:

   "P x y ==> P <= Q ==> Q x y"

   by (rule predicate2D)

 subsubsection {* Equality *}

 lemma pred_equals_eq: "((\<lambda>x. x \<in> R) = (\<lambda>x. x \<in> S)) = (R = S)"

   by (simp add: mem_def)

 lemma pred_equals_eq2 [pred_set_conv]: "((\<lambda>x y. (x, y) \<in> R) = (\<lambda>x y. (x, y) \<in> S)) = (R = S)"

   by (simp add: fun_eq_iff mem_def)

 subsubsection {* Order relation *}

 lemma pred_subset_eq: "((\<lambda>x. x \<in> R) <= (\<lambda>x. x \<in> S)) = (R <= S)"

   by (simp add: mem_def)

 lemma pred_subset_eq2 [pred_set_conv]: "((\<lambda>x y. (x, y) \<in> R) <= (\<lambda>x y. (x, y) \<in> S)) = (R <= S)"

   by fast

 subsubsection {* Top and bottom elements *}

 lemma bot1E [no_atp, elim!]: "bot x \<Longrightarrow> P"

   by (simp add: bot_fun_def)

 lemma bot2E [elim!]: "bot x y \<Longrightarrow> P"

   by (simp add: bot_fun_def)

 lemma bot_empty_eq: "bot = (\<lambda>x. x \<in> {})"

   by (auto simp add: fun_eq_iff)

 lemma bot_empty_eq2: "bot = (\<lambda>x y. (x, y) \<in> {})"

   by (auto simp add: fun_eq_iff)

 lemma top1I [intro!]: "top x"

   by (simp add: top_fun_def)

 lemma top2I [intro!]: "top x y"

   by (simp add: top_fun_def)

 subsubsection {* Binary intersection *}

 lemma inf1I [intro!]: "A x ==> B x ==> inf A B x"

   by (simp add: inf_fun_def)

 lemma inf2I [intro!]: "A x y ==> B x y ==> inf A B x y"

   by (simp add: inf_fun_def)

 lemma inf1E [elim!]: "inf A B x ==> (A x ==> B x ==> P) ==> P"

   by (simp add: inf_fun_def)

 lemma inf2E [elim!]: "inf A B x y ==> (A x y ==> B x y ==> P) ==> P"

   by (simp add: inf_fun_def)

 lemma inf1D1: "inf A B x ==> A x"

   by (simp add: inf_fun_def)

 lemma inf2D1: "inf A B x y ==> A x y"

   by (simp add: inf_fun_def)

 lemma inf1D2: "inf A B x ==> B x"

   by (simp add: inf_fun_def)

 lemma inf2D2: "inf A B x y ==> B x y"

   by (simp add: inf_fun_def)

 lemma inf_Int_eq: "inf (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"

   by (simp add: inf_fun_def mem_def)

 lemma inf_Int_eq2 [pred_set_conv]: "inf (\<lambda>x y. (x, y) \<in> R) (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<inter> S)"

   by (simp add: inf_fun_def mem_def)

 subsubsection {* Binary union *}

 lemma sup1I1 [elim?]: "A x \<Longrightarrow> sup A B x"

   by (simp add: sup_fun_def)

 lemma sup2I1 [elim?]: "A x y \<Longrightarrow> sup A B x y"

   by (simp add: sup_fun_def)

 lemma sup1I2 [elim?]: "B x \<Longrightarrow> sup A B x"

   by (simp add: sup_fun_def)

 lemma sup2I2 [elim?]: "B x y \<Longrightarrow> sup A B x y"

   by (simp add: sup_fun_def)

 lemma sup1E [elim!]: "sup A B x ==> (A x ==> P) ==> (B x ==> P) ==> P"

   by (simp add: sup_fun_def) iprover

 lemma sup2E [elim!]: "sup A B x y ==> (A x y ==> P) ==> (B x y ==> P) ==> P"

   by (simp add: sup_fun_def) iprover

 text {*

   \medskip Classical introduction rule: no commitment to @{text A} vs

   @{text B}.

 *}

 lemma sup1CI [intro!]: "(~ B x ==> A x) ==> sup A B x"

   by (auto simp add: sup_fun_def)

 lemma sup2CI [intro!]: "(~ B x y ==> A x y) ==> sup A B x y"

   by (auto simp add: sup_fun_def)

 lemma sup_Un_eq: "sup (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<union> S)"

   by (simp add: sup_fun_def mem_def)

 lemma sup_Un_eq2 [pred_set_conv]: "sup (\<lambda>x y. (x, y) \<in> R) (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<union> S)"

   by (simp add: sup_fun_def mem_def)

 subsubsection {* Intersections of families *}

 lemma INF1_iff: "(INF x:A. B x) b = (ALL x:A. B x b)"

   by (simp add: INFI_apply)

 lemma INF2_iff: "(INF x:A. B x) b c = (ALL x:A. B x b c)"

   by (simp add: INFI_apply)

 lemma INF1_I [intro!]: "(!!x. x : A ==> B x b) ==> (INF x:A. B x) b"

   by (auto simp add: INFI_apply)

 lemma INF2_I [intro!]: "(!!x. x : A ==> B x b c) ==> (INF x:A. B x) b c"

   by (auto simp add: INFI_apply)

 lemma INF1_D [elim]: "(INF x:A. B x) b ==> a : A ==> B a b"

   by (auto simp add: INFI_apply)

 lemma INF2_D [elim]: "(INF x:A. B x) b c ==> a : A ==> B a b c"

   by (auto simp add: INFI_apply)

 lemma INF1_E [elim]: "(INF x:A. B x) b ==> (B a b ==> R) ==> (a ~: A ==> R) ==> R"

   by (auto simp add: INFI_apply)

 lemma INF2_E [elim]: "(INF x:A. B x) b c ==> (B a b c ==> R) ==> (a ~: A ==> R) ==> R"

   by (auto simp add: INFI_apply)

 lemma INF_INT_eq: "(INF i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (INT i. r i))"

   by (simp add: INFI_apply fun_eq_iff)

 lemma INF_INT_eq2: "(INF i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (INT i. r i))"

   by (simp add: INFI_apply fun_eq_iff)

 subsubsection {* Unions of families *}

 lemma SUP1_iff: "(SUP x:A. B x) b = (EX x:A. B x b)"

   by (simp add: SUPR_apply)

 lemma SUP2_iff: "(SUP x:A. B x) b c = (EX x:A. B x b c)"

   by (simp add: SUPR_apply)

 lemma SUP1_I [intro]: "a : A ==> B a b ==> (SUP x:A. B x) b"

   by (auto simp add: SUPR_apply)

 lemma SUP2_I [intro]: "a : A ==> B a b c ==> (SUP x:A. B x) b c"

   by (auto simp add: SUPR_apply)

 lemma SUP1_E [elim!]: "(SUP x:A. B x) b ==> (!!x. x : A ==> B x b ==> R) ==> R"

   by (auto simp add: SUPR_apply)

 lemma SUP2_E [elim!]: "(SUP x:A. B x) b c ==> (!!x. x : A ==> B x b c ==> R) ==> R"

   by (auto simp add: SUPR_apply)

 lemma SUP_UN_eq: "(SUP i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (UN i. r i))"

   by (simp add: SUPR_apply fun_eq_iff)

 lemma SUP_UN_eq2: "(SUP i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (UN i. r i))"

   by (simp add: SUPR_apply fun_eq_iff)

 subsection {* Predicates as relations *}

 subsubsection {* Composition  *}

 inductive

   pred_comp  :: "['a => 'b => bool, 'b => 'c => bool] => 'a => 'c => bool"

     (infixr "OO" 75)

   for r :: "'a => 'b => bool" and s :: "'b => 'c => bool"

 where

   pred_compI [intro]: "r a b ==> s b c ==> (r OO s) a c"

 inductive_cases pred_compE [elim!]: "(r OO s) a c"

 lemma pred_comp_rel_comp_eq [pred_set_conv]:

   "((\<lambda>x y. (x, y) \<in> r) OO (\<lambda>x y. (x, y) \<in> s)) = (\<lambda>x y. (x, y) \<in> r O s)"

   by (auto simp add: fun_eq_iff)

 subsubsection {* Converse *}

 inductive

   conversep :: "('a => 'b => bool) => 'b => 'a => bool"

     ("(_^--1)" [1000] 1000)

   for r :: "'a => 'b => bool"

 where

   conversepI: "r a b ==> r^--1 b a"

 notation (xsymbols)

   conversep  ("(_\<inverse>\<inverse>)" [1000] 1000)

 lemma conversepD:

   assumes ab: "r^--1 a b"

   shows "r b a" using ab

   by cases simp

 lemma conversep_iff [iff]: "r^--1 a b = r b a"

   by (iprover intro: conversepI dest: conversepD)

 lemma conversep_converse_eq [pred_set_conv]:

   "(\<lambda>x y. (x, y) \<in> r)^--1 = (\<lambda>x y. (x, y) \<in> r^-1)"

   by (auto simp add: fun_eq_iff)

 lemma conversep_conversep [simp]: "(r^--1)^--1 = r"

   by (iprover intro: order_antisym conversepI dest: conversepD)

 lemma converse_pred_comp: "(r OO s)^--1 = s^--1 OO r^--1"

   by (iprover intro: order_antisym conversepI pred_compI

     elim: pred_compE dest: conversepD)

 lemma converse_meet: "(inf r s)^--1 = inf r^--1 s^--1"

   by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD)

 lemma converse_join: "(sup r s)^--1 = sup r^--1 s^--1"

   by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD)

 lemma conversep_noteq [simp]: "(op ~=)^--1 = op ~="

   by (auto simp add: fun_eq_iff)

 lemma conversep_eq [simp]: "(op =)^--1 = op ="

   by (auto simp add: fun_eq_iff)

 subsubsection {* Domain *}

 inductive

   DomainP :: "('a => 'b => bool) => 'a => bool"

   for r :: "'a => 'b => bool"

 where

   DomainPI [intro]: "r a b ==> DomainP r a"

 inductive_cases DomainPE [elim!]: "DomainP r a"

 lemma DomainP_Domain_eq [pred_set_conv]: "DomainP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Domain r)"

   by (blast intro!: Orderings.order_antisym predicate1I)

 subsubsection {* Range *}

 inductive

   RangeP :: "('a => 'b => bool) => 'b => bool"

   for r :: "'a => 'b => bool"

 where

   RangePI [intro]: "r a b ==> RangeP r b"

 inductive_cases RangePE [elim!]: "RangeP r b"

 lemma RangeP_Range_eq [pred_set_conv]: "RangeP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Range r)"

   by (blast intro!: Orderings.order_antisym predicate1I)

 subsubsection {* Inverse image *}

 definition

   inv_imagep :: "('b => 'b => bool) => ('a => 'b) => 'a => 'a => bool" where

   "inv_imagep r f == %x y. r (f x) (f y)"

 lemma [pred_set_conv]: "inv_imagep (\<lambda>x y. (x, y) \<in> r) f = (\<lambda>x y. (x, y) \<in> inv_image r f)"

   by (simp add: inv_image_def inv_imagep_def)

 lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)"

   by (simp add: inv_imagep_def)

 subsubsection {* Powerset *}

 definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool" where

   "Powp A == \<lambda>B. \<forall>x \<in> B. A x"

 lemma Powp_Pow_eq [pred_set_conv]: "Powp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> Pow A)"

   by (auto simp add: Powp_def fun_eq_iff)

 lemmas Powp_mono [mono] = Pow_mono [to_pred pred_subset_eq]

 subsubsection {* Properties of relations *}

 abbreviation antisymP :: "('a => 'a => bool) => bool" where

   "antisymP r == antisym {(x, y). r x y}"

 abbreviation transP :: "('a => 'a => bool) => bool" where

   "transP r == trans {(x, y). r x y}"

 abbreviation single_valuedP :: "('a => 'b => bool) => bool" where

   "single_valuedP r == single_valued {(x, y). r x y}"

 (*FIXME inconsistencies: abbreviations vs. definitions, suffix `P` vs. suffix `p`*)

 definition reflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where

   "reflp r \<longleftrightarrow> refl {(x, y). r x y}"

 definition symp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where

   "symp r \<longleftrightarrow> sym {(x, y). r x y}"

 definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where

   "transp r \<longleftrightarrow> trans {(x, y). r x y}"

 lemma reflpI:

   "(\<And>x. r x x) \<Longrightarrow> reflp r"

   by (auto intro: refl_onI simp add: reflp_def)

 lemma reflpE:

   assumes "reflp r"

   obtains "r x x"

   using assms by (auto dest: refl_onD simp add: reflp_def)

 lemma sympI:

   "(\<And>x y. r x y \<Longrightarrow> r y x) \<Longrightarrow> symp r"

   by (auto intro: symI simp add: symp_def)

 lemma sympE:

   assumes "symp r" and "r x y"

   obtains "r y x"

   using assms by (auto dest: symD simp add: symp_def)

 lemma transpI:

   "(\<And>x y z. r x y \<Longrightarrow> r y z \<Longrightarrow> r x z) \<Longrightarrow> transp r"

   by (auto intro: transI simp add: transp_def)

 lemma transpE:

   assumes "transp r" and "r x y" and "r y z"

   obtains "r x z"

   using assms by (auto dest: transD simp add: transp_def)

 subsection {* Predicates as enumerations *}

 subsubsection {* The type of predicate enumerations (a monad) *}

 datatype 'a pred = Pred "'a \<Rightarrow> bool"

 primrec eval :: "'a pred \<Rightarrow> 'a \<Rightarrow> bool" where

   eval_pred: "eval (Pred f) = f"

 lemma Pred_eval [simp]:

   "Pred (eval x) = x"

   by (cases x) simp

 lemma pred_eqI:

   "(\<And>w. eval P w \<longleftrightarrow> eval Q w) \<Longrightarrow> P = Q"

   by (cases P, cases Q) (auto simp add: fun_eq_iff)

 lemma eval_mem [simp]:

   "x \<in> eval P \<longleftrightarrow> eval P x"

   by (simp add: mem_def)

 lemma eq_mem [simp]:

   "x \<in> (op =) y \<longleftrightarrow> x = y"

   by (auto simp add: mem_def)

 instantiation pred :: (type) "{complete_lattice, boolean_algebra}"

 begin

 definition

   "P \<le> Q \<longleftrightarrow> eval P \<le> eval Q"

 definition

   "P < Q \<longleftrightarrow> eval P < eval Q"

 definition

   "\<bottom> = Pred \<bottom>"

 lemma eval_bot [simp]:

   "eval \<bottom>  = \<bottom>"

   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)

 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

 qed (auto intro!: pred_eqI simp add: less_eq_pred_def less_pred_def uminus_apply minus_apply)

 end

 lemma eval_INFI [simp]:

   "eval (INFI A f) = INFI A (eval \<circ> f)"

   by (unfold INFI_def) simp

 lemma eval_SUPR [simp]:

   "eval (SUPR A f) = SUPR A (eval \<circ> f)"

   by (unfold SUPR_def) simp

 definition single :: "'a \<Rightarrow> '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 \<Rightarrow> ('a \<Rightarrow> 'b pred) \<Rightarrow> 'b pred" (infixl "\<guillemotright>=" 70) where

   "P \<guillemotright>= f = (SUPR {x. eval P x} f)"

 lemma eval_bind [simp]:

   "eval (P \<guillemotright>= f) = eval (SUPR {x. eval P x} f)"

   by (simp add: bind_def)

 lemma bind_bind:

   "(P \<guillemotright>= Q) \<guillemotright>= R = P \<guillemotright>= (\<lambda>x. Q x \<guillemotright>= R)"

   by (rule pred_eqI) auto

 lemma bind_single:

   "P \<guillemotright>= single = P"

   by (rule pred_eqI) auto

 lemma single_bind:

   "single x \<guillemotright>= P = P x"

   by (rule pred_eqI) auto

 lemma bottom_bind:

   "\<bottom> \<guillemotright>= P = \<bottom>"

   by (rule pred_eqI) auto

 lemma sup_bind:

   "(P \<squnion> Q) \<guillemotright>= R = P \<guillemotright>= R \<squnion> Q \<guillemotright>= R"

   by (rule pred_eqI) auto

 lemma Sup_bind:

   "(\<Squnion>A \<guillemotright>= f) = \<Squnion>((\<lambda>x. x \<guillemotright>= f) ` A)"

   by (rule pred_eqI) auto

 lemma pred_iffI:

   assumes "\<And>x. eval A x \<Longrightarrow> eval B x"

   and "\<And>x. eval B x \<Longrightarrow> 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 \<Longrightarrow> (y = x \<Longrightarrow> P) \<Longrightarrow> P"

   by simp

 lemma singleE': "eval (single x) y \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"

   by simp

 lemma bindI: "eval P x \<Longrightarrow> eval (Q x) y \<Longrightarrow> eval (P \<guillemotright>= Q) y"

   by auto

 lemma bindE: "eval (R \<guillemotright>= Q) y \<Longrightarrow> (\<And>x. eval R x \<Longrightarrow> eval (Q x) y \<Longrightarrow> P) \<Longrightarrow> P"

   by auto

 lemma botE: "eval \<bottom> x \<Longrightarrow> P"

   by auto

 lemma supI1: "eval A x \<Longrightarrow> eval (A \<squnion> B) x"

   by auto

 lemma supI2: "eval B x \<Longrightarrow> eval (A \<squnion> B) x" 

   by auto

 lemma supE: "eval (A \<squnion> B) x \<Longrightarrow> (eval A x \<Longrightarrow> P) \<Longrightarrow> (eval B x \<Longrightarrow> P) \<Longrightarrow> P"

   by auto

 lemma single_not_bot [simp]:

   "single x \<noteq> \<bottom>"

   by (auto simp add: single_def bot_pred_def fun_eq_iff)

 lemma not_bot:

   assumes "A \<noteq> \<bottom>"

   obtains x where "eval A x"

   using assms by (cases A)

     (auto simp add: bot_pred_def, auto simp add: mem_def)

 subsubsection {* Emptiness check and definite choice *}

 definition is_empty :: "'a pred \<Rightarrow> bool" where

   "is_empty A \<longleftrightarrow> A = \<bottom>"

 lemma is_empty_bot:

   "is_empty \<bottom>"

   by (simp add: is_empty_def)

 lemma not_is_empty_single:

   "\<not> 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) \<longleftrightarrow> is_empty A \<and> is_empty B"

   by (auto simp add: is_empty_def)

 definition singleton :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a pred \<Rightarrow> 'a" where

   "singleton dfault A = (if \<exists>!x. eval A x then THE x. eval A x else dfault ())"

 lemma singleton_eqI:

   "\<exists>!x. eval A x \<Longrightarrow> eval A x \<Longrightarrow> singleton dfault A = x"

   by (auto simp add: singleton_def)

 lemma eval_singletonI:

   "\<exists>!x. eval A x \<Longrightarrow> eval A (singleton dfault A)"

 proof -

   assume assm: "\<exists>!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 

 qed

 lemma single_singleton:

   "\<exists>!x. eval A x \<Longrightarrow> single (singleton dfault A) = A"

 proof -

   assume assm: "\<exists>!x. eval A x"

   then have "eval A (singleton dfault A)"

     by (rule eval_singletonI)

   moreover from assm have "\<And>x. eval A x \<Longrightarrow> 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 "\<And>x. eval (single (singleton dfault A)) x = eval A x"

     by simp

   then show ?thesis by (rule pred_eqI)

 qed

 lemma singleton_undefinedI:

   "\<not> (\<exists>!x. eval A x) \<Longrightarrow> singleton dfault A = dfault ()"

   by (simp add: singleton_def)

 lemma singleton_bot:

   "singleton dfault \<bottom> = 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 "\<not> (\<exists>!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 simp

 qed

 lemma singleton_sup_aux:

   "singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B

     else if B = \<bottom> then singleton dfault A

     else singleton dfault

       (single (singleton dfault A) \<squnion> single (singleton dfault B)))"

 proof (cases "(\<exists>!x. eval A x) \<and> (\<exists>!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 () \<or> 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:

     "\<not> (\<exists>!x. eval A x) \<or> \<not> (\<exists>!y. eval B y)" by simp

   show ?thesis proof (cases "A \<noteq> \<bottom> \<and> B \<noteq> \<bottom>")

     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 "\<not> (\<exists>!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

   qed

 qed

 lemma singleton_sup:

   "singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B

     else if B = \<bottom> 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)

 subsubsection {* Derived operations *}

 definition if_pred :: "bool \<Rightarrow> unit pred" where

   if_pred_eq: "if_pred b = (if b then single () else \<bottom>)"

 definition holds :: "unit pred \<Rightarrow> bool" where

   holds_eq: "holds P = eval P ()"

 definition not_pred :: "unit pred \<Rightarrow> unit pred" where

   not_pred_eq: "not_pred P = (if eval P () then \<bottom> else single ())"

 lemma if_predI: "P \<Longrightarrow> eval (if_pred P) ()"

   unfolding if_pred_eq by (auto intro: singleI)

 lemma if_predE: "eval (if_pred b) x \<Longrightarrow> (b \<Longrightarrow> x = () \<Longrightarrow> P) \<Longrightarrow> P"

   unfolding if_pred_eq by (cases b) (auto elim: botE)

 lemma not_predI: "\<not> P \<Longrightarrow> eval (not_pred (Pred (\<lambda>u. P))) ()"

   unfolding not_pred_eq eval_pred by (auto intro: singleI)

 lemma not_predI': "\<not> eval P () \<Longrightarrow> eval (not_pred P) ()"

   unfolding not_pred_eq by (auto intro: singleI)

 lemma not_predE: "eval (not_pred (Pred (\<lambda>u. P))) x \<Longrightarrow> (\<not> P \<Longrightarrow> thesis) \<Longrightarrow> thesis"

   unfolding not_pred_eq

   by (auto split: split_if_asm elim: botE)

 lemma not_predE': "eval (not_pred P) x \<Longrightarrow> (\<not> eval P x \<Longrightarrow> thesis) \<Longrightarrow> thesis"

   unfolding not_pred_eq

   by (auto split: split_if_asm elim: botE)

 lemma "f () = False \<or> f () = True"

 by simp

 lemma closure_of_bool_cases [no_atp]:

   fixes f :: "unit \<Rightarrow> bool"

   assumes "f = (\<lambda>u. False) \<Longrightarrow> P f"

   assumes "f = (\<lambda>u. True) \<Longrightarrow> P f"

   shows "P f"

 proof -

   have "f = (\<lambda>u. False) \<or> f = (\<lambda>u. 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 \<bottom>"

   assumes "P (single ())"

   shows "P Q"

 using assms unfolding bot_pred_def Collect_def empty_def single_def proof (cases Q)

   fix f

   assume "P (Pred (\<lambda>u. False))" "P (Pred (\<lambda>u. () = 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 \<longleftrightarrow> \<not> holds P"

 unfolding is_empty_def holds_eq

 by (rule unit_pred_cases) (auto elim: botE intro: singleI)

 definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pred \<Rightarrow> 'b pred" where

   "map f P = P \<guillemotright>= (single o f)"

 lemma eval_map [simp]:

   "eval (map f P) = image f (eval P)"

   by (auto simp add: map_def)

 enriched_type map: map

   by (auto intro!: pred_eqI simp add: fun_eq_iff image_compose)

 subsubsection {* Implementation *}

 datatype 'a seq = Empty | Insert "'a" "'a pred" | Join "'a pred" "'a seq"

 primrec pred_of_seq :: "'a seq \<Rightarrow> 'a pred" where

     "pred_of_seq Empty = \<bottom>"

   | "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 \<Rightarrow> 'a seq) \<Rightarrow> 'a pred" where

   "Seq f = pred_of_seq (f ())"

 code_datatype Seq

 primrec member :: "'a seq \<Rightarrow> 'a \<Rightarrow> bool"  where

   "member Empty x \<longleftrightarrow> False"

   | "member (Insert y P) x \<longleftrightarrow> x = y \<or> eval P x"

   | "member (Join P xq) x \<longleftrightarrow> eval P x \<or> 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 [code]: "eval (Seq f) = member (f ())"

   unfolding Seq_def by (rule sym, rule eval_member)

 lemma single_code [code]:

   "single x = Seq (\<lambda>u. Insert x \<bottom>)"

   unfolding Seq_def by simp

 primrec "apply" :: "('a \<Rightarrow> 'b pred) \<Rightarrow> 'a seq \<Rightarrow> 'b seq" where

     "apply f Empty = Empty"

   | "apply f (Insert x P) = Join (f x) (Join (P \<guillemotright>= f) Empty)"

   | "apply f (Join P xq) = Join (P \<guillemotright>= f) (apply f xq)"

 lemma apply_bind:

   "pred_of_seq (apply f xq) = pred_of_seq xq \<guillemotright>= 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 \<guillemotright>= f = Seq (\<lambda>u. apply f (g ()))"

   unfolding Seq_def by (rule sym, rule apply_bind)

 lemma bot_set_code [code]:

   "\<bottom> = Seq (\<lambda>u. Empty)"

   unfolding Seq_def by simp

 primrec adjunct :: "'a pred \<Rightarrow> 'a seq \<Rightarrow> '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 (\<lambda>u. case f ()

     of Empty \<Rightarrow> g ()

      | Insert x P \<Rightarrow> Insert x (P \<squnion> Seq g)

      | Join P xq \<Rightarrow> adjunct (Seq g) (Join P xq))"

 proof (cases "f ()")

   case Empty

   thus ?thesis

     unfolding Seq_def by (simp add: sup_commute [of "\<bottom>"])

 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 \<Rightarrow> 'a pred \<Rightarrow> bool" where

     "contained Empty Q \<longleftrightarrow> True"

   | "contained (Insert x P) Q \<longleftrightarrow> eval Q x \<and> P \<le> Q"

   | "contained (Join P xq) Q \<longleftrightarrow> P \<le> Q \<and> contained xq Q"

 lemma single_less_eq_eval:

   "single x \<le> P \<longleftrightarrow> eval P x"

   by (auto simp add: single_def less_eq_pred_def mem_def)

 lemma contained_less_eq:

   "contained xq Q \<longleftrightarrow> pred_of_seq xq \<le> Q"

   by (induct xq) (simp_all add: single_less_eq_eval)

 lemma less_eq_pred_code [code]:

   "Seq f \<le> Q = (case f ()

    of Empty \<Rightarrow> True

     | Insert x P \<Rightarrow> eval Q x \<and> P \<le> Q

     | Join P xq \<Rightarrow> P \<le> Q \<and> 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 \<longleftrightarrow> P \<le> Q \<and> Q \<le> P"

   by (auto simp add: equal)

 lemma [code nbe]:

   "HOL.equal (x :: 'a pred) x \<longleftrightarrow> True"

   by (fact equal_refl)

 lemma [code]:

   "pred_case f P = f (eval P)"

   by (cases P) simp

 lemma [code]:

   "pred_rec f P = f (eval P)"

   by (cases P) simp

 inductive eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "eq x x"

 lemma eq_is_eq: "eq x y \<equiv> (x = y)"

   by (rule eq_reflection) (auto intro: eq.intros elim: eq.cases)

 primrec null :: "'a seq \<Rightarrow> bool" where

     "null Empty \<longleftrightarrow> True"

   | "null (Insert x P) \<longleftrightarrow> False"

   | "null (Join P xq) \<longleftrightarrow> is_empty P \<and> null xq"

 lemma null_is_empty:

   "null xq \<longleftrightarrow> 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) \<longleftrightarrow> null (f ())"

   by (simp add: null_is_empty Seq_def)

 primrec the_only :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a seq \<Rightarrow> '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 \<Rightarrow> dfault ()

     | Insert x P \<Rightarrow> if is_empty P then x

         else let y = singleton dfault P in

           if x = y then x else dfault ()

     | Join P xq \<Rightarrow> 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 not_unique :: "'a pred => 'a"

 where

   [code del]: "not_unique A = (THE x. eval A x)"

 definition the :: "'a pred => 'a"

 where

   "the A = (THE x. eval A x)"

 lemma the_eqI:

   "(THE x. eval P x) = x \<Longrightarrow> the P = x"

   by (simp add: the_def)

 lemma the_eq [code]: "the A = singleton (\<lambda>x. not_unique A) A"

   by (rule the_eqI) (simp add: singleton_def not_unique_def)

 code_abort not_unique

 code_reflect Predicate

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

   functions map

 ML {*

 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 pred

 end;

 structure Predicate : PREDICATE =

 struct

 datatype pred = datatype Predicate.pred

 datatype seq = datatype Predicate.seq

 fun 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;

 *}

 no_notation

   bot ("\<bottom>") 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 "\<guillemotright>=" 70)

 no_syntax (xsymbols)

   "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)

   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)

   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)

   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)

 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 not_unique the

 end