(* 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
inf (infixl "\<sqinter>" 70) and
sup (infixl "\<squnion>" 65) and
Inf ("\<Sqinter>_" [900] 900) and
Sup ("\<Squnion>_" [900] 900) and
top ("\<top>") and
bot ("\<bottom>")
subsection {* Predicates as (complete) lattices *}
subsubsection {* @{const sup} on @{typ bool} *}
lemma sup_boolI1:
"P \<Longrightarrow> P \<squnion> Q"
by (simp add: sup_bool_eq)
lemma sup_boolI2:
"Q \<Longrightarrow> P \<squnion> Q"
by (simp add: sup_bool_eq)
lemma sup_boolE:
"P \<squnion> Q \<Longrightarrow> (P \<Longrightarrow> R) \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
by (auto simp add: sup_bool_eq)
subsubsection {* Equality and Subsets *}
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: expand_fun_eq mem_def)
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 top1I [intro!]: "top x"
by (simp add: top_fun_eq top_bool_eq)
lemma top2I [intro!]: "top x y"
by (simp add: top_fun_eq top_bool_eq)
lemma bot1E [elim!]: "bot x \<Longrightarrow> P"
by (simp add: bot_fun_eq bot_bool_eq)
lemma bot2E [elim!]: "bot x y \<Longrightarrow> P"
by (simp add: bot_fun_eq bot_bool_eq)
subsubsection {* The empty set *}
lemma bot_empty_eq: "bot = (\<lambda>x. x \<in> {})"
by (auto simp add: expand_fun_eq)
lemma bot_empty_eq2: "bot = (\<lambda>x y. (x, y) \<in> {})"
by (auto simp add: expand_fun_eq)
subsubsection {* Binary union *}
lemma sup1_iff [simp]: "sup A B x \<longleftrightarrow> A x | B x"
by (simp add: sup_fun_eq sup_bool_eq)
lemma sup2_iff [simp]: "sup A B x y \<longleftrightarrow> A x y | B x y"
by (simp add: sup_fun_eq sup_bool_eq)
lemma sup_Un_eq [pred_set_conv]: "sup (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<union> S)"
by (simp add: expand_fun_eq)
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: expand_fun_eq)
lemma sup1I1 [elim?]: "A x \<Longrightarrow> sup A B x"
by simp
lemma sup2I1 [elim?]: "A x y \<Longrightarrow> sup A B x y"
by simp
lemma sup1I2 [elim?]: "B x \<Longrightarrow> sup A B x"
by simp
lemma sup2I2 [elim?]: "B x y \<Longrightarrow> sup A B x y"
by simp
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
lemma sup2CI [intro!]: "(~ B x y ==> A x y) ==> sup A B x y"
by auto
lemma sup1E [elim!]: "sup A B x ==> (A x ==> P) ==> (B x ==> P) ==> P"
by simp iprover
lemma sup2E [elim!]: "sup A B x y ==> (A x y ==> P) ==> (B x y ==> P) ==> P"
by simp iprover
subsubsection {* Binary intersection *}
lemma inf1_iff [simp]: "inf A B x \<longleftrightarrow> A x \<and> B x"
by (simp add: inf_fun_eq inf_bool_eq)
lemma inf2_iff [simp]: "inf A B x y \<longleftrightarrow> A x y \<and> B x y"
by (simp add: inf_fun_eq inf_bool_eq)
lemma inf_Int_eq [pred_set_conv]: "inf (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
by (simp add: expand_fun_eq)
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: expand_fun_eq)
lemma inf1I [intro!]: "A x ==> B x ==> inf A B x"
by simp
lemma inf2I [intro!]: "A x y ==> B x y ==> inf A B x y"
by simp
lemma inf1D1: "inf A B x ==> A x"
by simp
lemma inf2D1: "inf A B x y ==> A x y"
by simp
lemma inf1D2: "inf A B x ==> B x"
by simp
lemma inf2D2: "inf A B x y ==> B x y"
by simp
lemma inf1E [elim!]: "inf A B x ==> (A x ==> B x ==> P) ==> P"
by simp
lemma inf2E [elim!]: "inf A B x y ==> (A x y ==> B x y ==> P) ==> P"
by simp
subsubsection {* Unions of families *}
lemma SUP1_iff [simp]: "(SUP x:A. B x) b = (EX x:A. B x b)"
by (simp add: SUPR_def Sup_fun_def Sup_bool_def) blast
lemma SUP2_iff [simp]: "(SUP x:A. B x) b c = (EX x:A. B x b c)"
by (simp add: SUPR_def Sup_fun_def Sup_bool_def) blast
lemma SUP1_I [intro]: "a : A ==> B a b ==> (SUP x:A. B x) b"
by auto
lemma SUP2_I [intro]: "a : A ==> B a b c ==> (SUP x:A. B x) b c"
by auto
lemma SUP1_E [elim!]: "(SUP x:A. B x) b ==> (!!x. x : A ==> B x b ==> R) ==> R"
by auto
lemma SUP2_E [elim!]: "(SUP x:A. B x) b c ==> (!!x. x : A ==> B x b c ==> R) ==> R"
by auto
lemma SUP_UN_eq: "(SUP i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (UN i. r i))"
by (simp add: expand_fun_eq)
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: expand_fun_eq)
subsubsection {* Intersections of families *}
lemma INF1_iff [simp]: "(INF x:A. B x) b = (ALL x:A. B x b)"
by (simp add: INFI_def Inf_fun_def Inf_bool_def) blast
lemma INF2_iff [simp]: "(INF x:A. B x) b c = (ALL x:A. B x b c)"
by (simp add: INFI_def Inf_fun_def Inf_bool_def) blast
lemma INF1_I [intro!]: "(!!x. x : A ==> B x b) ==> (INF x:A. B x) b"
by auto
lemma INF2_I [intro!]: "(!!x. x : A ==> B x b c) ==> (INF x:A. B x) b c"
by auto
lemma INF1_D [elim]: "(INF x:A. B x) b ==> a : A ==> B a b"
by auto
lemma INF2_D [elim]: "(INF x:A. B x) b c ==> a : A ==> B a b c"
by auto
lemma INF1_E [elim]: "(INF x:A. B x) b ==> (B a b ==> R) ==> (a ~: A ==> R) ==> R"
by auto
lemma INF2_E [elim]: "(INF x:A. B x) b c ==> (B a b c ==> R) ==> (a ~: A ==> R) ==> R"
by auto
lemma INF_INT_eq: "(INF i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (INT i. r i))"
by (simp add: expand_fun_eq)
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: expand_fun_eq)
subsection {* Predicates as relations *}
subsubsection {* Composition *}
inductive
pred_comp :: "['b => 'c => bool, 'a => 'b => bool] => 'a => 'c => bool"
(infixr "OO" 75)
for r :: "'b => 'c => bool" and s :: "'a => 'b => bool"
where
pred_compI [intro]: "s a b ==> r 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: expand_fun_eq 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: expand_fun_eq)
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_eq inf_bool_eq)
(iprover intro: conversepI ext dest: conversepD)
lemma converse_join: "(sup r s)^--1 = sup r^--1 s^--1"
by (simp add: sup_fun_eq sup_bool_eq)
(iprover intro: conversepI ext dest: conversepD)
lemma conversep_noteq [simp]: "(op ~=)^--1 = op ~="
by (auto simp add: expand_fun_eq)
lemma conversep_eq [simp]: "(op =)^--1 = op ="
by (auto simp add: expand_fun_eq)
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 expand_fun_eq)
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}"
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 eval_inject: "eval x = eval y \<longleftrightarrow> x = y"
by (cases x) auto
definition single :: "'a \<Rightarrow> 'a pred" where
"single x = Pred ((op =) x)"
definition bind :: "'a pred \<Rightarrow> ('a \<Rightarrow> 'b pred) \<Rightarrow> 'b pred" (infixl "\<guillemotright>=" 70) where
"P \<guillemotright>= f = Pred (\<lambda>x. (\<exists>y. eval P y \<and> eval (f y) x))"
instantiation pred :: (type) complete_lattice
begin
definition
"P \<le> Q \<longleftrightarrow> eval P \<le> eval Q"
definition
"P < Q \<longleftrightarrow> eval P < eval Q"
definition
"\<bottom> = Pred \<bottom>"
definition
"\<top> = Pred \<top>"
definition
"P \<sqinter> Q = Pred (eval P \<sqinter> eval Q)"
definition
"P \<squnion> Q = Pred (eval P \<squnion> eval Q)"
definition
"\<Sqinter>A = Pred (INFI A eval)"
definition
"\<Squnion>A = Pred (SUPR A eval)"
instance by default
(auto simp add: less_eq_pred_def less_pred_def
inf_pred_def sup_pred_def bot_pred_def top_pred_def
Inf_pred_def Sup_pred_def,
auto simp add: le_fun_def less_fun_def le_bool_def less_bool_def
eval_inject mem_def)
end
lemma bind_bind:
"(P \<guillemotright>= Q) \<guillemotright>= R = P \<guillemotright>= (\<lambda>x. Q x \<guillemotright>= R)"
by (auto simp add: bind_def expand_fun_eq)
lemma bind_single:
"P \<guillemotright>= single = P"
by (simp add: bind_def single_def)
lemma single_bind:
"single x \<guillemotright>= P = P x"
by (simp add: bind_def single_def)
lemma bottom_bind:
"\<bottom> \<guillemotright>= P = \<bottom>"
by (auto simp add: bot_pred_def bind_def expand_fun_eq)
lemma sup_bind:
"(P \<squnion> Q) \<guillemotright>= R = P \<guillemotright>= R \<squnion> Q \<guillemotright>= R"
by (auto simp add: bind_def sup_pred_def expand_fun_eq)
lemma Sup_bind: "(\<Squnion>A \<guillemotright>= f) = \<Squnion>((\<lambda>x. x \<guillemotright>= f) ` A)"
by (auto simp add: bind_def Sup_pred_def expand_fun_eq)
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"
proof -
from assms have "\<And>x. eval A x \<longleftrightarrow> eval B x" by blast
then show ?thesis by (cases A, cases B) (simp add: expand_fun_eq)
qed
lemma singleI: "eval (single x) x"
unfolding single_def by simp
lemma singleI_unit: "eval (single ()) x"
by simp (rule singleI)
lemma singleE: "eval (single x) y \<Longrightarrow> (y = x \<Longrightarrow> P) \<Longrightarrow> P"
unfolding single_def by simp
lemma singleE': "eval (single x) y \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
by (erule singleE) simp
lemma bindI: "eval P x \<Longrightarrow> eval (Q x) y \<Longrightarrow> eval (P \<guillemotright>= Q) y"
unfolding bind_def by auto
lemma bindE: "eval (R \<guillemotright>= Q) y \<Longrightarrow> (\<And>x. eval R x \<Longrightarrow> eval (Q x) y \<Longrightarrow> P) \<Longrightarrow> P"
unfolding bind_def by auto
lemma botE: "eval \<bottom> x \<Longrightarrow> P"
unfolding bot_pred_def by auto
lemma supI1: "eval A x \<Longrightarrow> eval (A \<squnion> B) x"
unfolding sup_pred_def by simp
lemma supI2: "eval B x \<Longrightarrow> eval (A \<squnion> B) x"
unfolding sup_pred_def by simp
lemma supE: "eval (A \<squnion> B) x \<Longrightarrow> (eval A x \<Longrightarrow> P) \<Longrightarrow> (eval B x \<Longrightarrow> P) \<Longrightarrow> P"
unfolding sup_pred_def by auto
subsubsection {* Derived operations *}
definition if_pred :: "bool \<Rightarrow> unit pred" where
if_pred_eq: "if_pred b = (if b then single () else \<bottom>)"
definition eq_pred :: "'a \<Rightarrow> 'a \<Rightarrow> unit pred" where
eq_pred_eq: "eq_pred a b = if_pred (a = b)"
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 eq_predI: "eval (eq_pred a a) ()"
unfolding eq_pred_eq if_pred_eq by (auto intro: singleI)
lemma eq_predE: "eval (eq_pred a b) x \<Longrightarrow> (a = b \<Longrightarrow> x = () \<Longrightarrow> P) \<Longrightarrow> P"
unfolding eq_pred_eq by (erule if_predE)
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)
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: expand_fun_eq elim: botE)
next
case Insert show ?case
by (auto simp add: expand_fun_eq elim: supE singleE intro: supI1 supI2 singleI)
next
case Join then show ?case
by (auto simp add: expand_fun_eq 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 Predicate.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
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> Join (Seq g) (Join P xq))" (*FIXME order!?*)
proof (cases "f ()")
case Empty
thus ?thesis
unfolding Seq_def by (simp add: sup_commute [of "\<bottom>"] sup_bot)
next
case Insert
thus ?thesis
unfolding Seq_def by (simp add: sup_assoc)
next
case Join
thus ?thesis
unfolding Seq_def by (simp add: sup_commute [of "pred_of_seq (g ())"] sup_assoc)
qed
declare eq_pred_def [code, code del]
no_notation
inf (infixl "\<sqinter>" 70) and
sup (infixl "\<squnion>" 65) and
Inf ("\<Sqinter>_" [900] 900) and
Sup ("\<Squnion>_" [900] 900) and
top ("\<top>") and
bot ("\<bottom>") and
bind (infixl "\<guillemotright>=" 70)
hide (open) type pred seq
hide (open) const Pred eval single bind if_pred eq_pred not_pred
Empty Insert Join Seq member "apply"
end