(*  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 bot_bool_def)

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

  by (simp add: bot_fun_def bot_bool_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 top_bool_def)

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

  by (simp add: top_fun_def top_bool_def)

subsubsection {* Binary intersection *}

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

  by (simp add: inf_fun_def inf_bool_def)

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

  by (simp add: inf_fun_def inf_bool_def)

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

  by (simp add: inf_fun_def inf_bool_def)

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

  by (simp add: inf_fun_def inf_bool_def)

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

  by (simp add: inf_fun_def inf_bool_def)

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

  by (simp add: inf_fun_def inf_bool_def)

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

  by (simp add: inf_fun_def inf_bool_def)

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

  by (simp add: inf_fun_def inf_bool_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 inf_bool_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 inf_bool_def mem_def)

subsubsection {* Binary union *}

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

  by (simp add: sup_fun_def sup_bool_def)

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

  by (simp add: sup_fun_def sup_bool_def)

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

  by (simp add: sup_fun_def sup_bool_def)

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

  by (simp add: sup_fun_def sup_bool_def)

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

  by (simp add: sup_fun_def sup_bool_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 sup_bool_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 sup_bool_def)

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

  by (auto simp add: sup_fun_def sup_bool_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 sup_bool_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 sup_bool_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 elim: pred_compE)

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 inf_bool_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 sup_bool_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]:

assumes "(f :: unit \<Rightarrow> bool) = (%u. False) \<Longrightarrow> P f"

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

shows "P f"

proof -

  have "f = (%u. False) \<or> f = (%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 prems 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

apply (cases Q)

apply simp

apply (rule_tac f="fun" in closure_of_bool_cases)

apply auto

apply (subgoal_tac "(%x. () = x) = (%x. True)") 

apply auto

done

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)