(* 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)
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