(* Title: HOL/Predicate.thy
ID: $Id$
Author: Stefan Berghofer, TU Muenchen
*)
header {* Predicates *}
theory Predicate
imports Inductive Relation
begin
subsection {* Equality and Subsets *}
lemma pred_equals_eq [pred_set_conv]: "((\<lambda>x. x \<in> R) = (\<lambda>x. x \<in> S)) = (R = S)"
by (auto simp add: expand_fun_eq)
lemma pred_equals_eq2 [pred_set_conv]: "((\<lambda>x y. (x, y) \<in> R) = (\<lambda>x y. (x, y) \<in> S)) = (R = S)"
by (auto simp add: expand_fun_eq)
lemma pred_subset_eq [pred_set_conv]: "((\<lambda>x. x \<in> R) <= (\<lambda>x. x \<in> S)) = (R <= S)"
by fast
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
subsection {* 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)
subsection {* 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)
subsection {* 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
subsection {* 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
subsection {* 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)
subsection {* 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 {* Composition of two relations *}
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)
subsection {* 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)
subsection {* 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)
subsection {* 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)
subsection {* 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)
subsection {* The Powerset operator *}
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)
subsection {* Properties of relations - predicate versions *}
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}"
end