--- a/src/HOL/Predicate.thy Wed Jul 11 11:04:39 2007 +0200
+++ b/src/HOL/Predicate.thy Wed Jul 11 11:07:57 2007 +0200
@@ -9,202 +9,46 @@
imports Inductive Relation
begin
-subsection {* Converting between predicates and sets *}
-
-definition
- member :: "'a set => 'a => bool" where
- "member == %S x. x : S"
-
-lemma memberI[intro!, Pure.intro!]: "x : S ==> member S x"
- by (simp add: member_def)
-
-lemma memberD[dest!, Pure.dest!]: "member S x ==> x : S"
- by (simp add: member_def)
-
-lemma member_eq[simp]: "member S x = (x : S)"
- by (simp add: member_def)
-
-lemma member_Collect_eq[simp]: "member (Collect P) = P"
- by (simp add: member_def)
-
-lemma Collect_member_eq[simp]: "Collect (member S) = S"
- by (simp add: member_def)
-
-lemma split_set: "(!!S. PROP P S) == (!!S. PROP P (Collect S))"
-proof
- fix S
- assume "!!S. PROP P S"
- then show "PROP P (Collect S)" .
-next
- fix S
- assume "!!S. PROP P (Collect S)"
- then have "PROP P {x. x : S}" .
- thus "PROP P S" by simp
-qed
-
-lemma split_predicate: "(!!S. PROP P S) == (!!S. PROP P (member S))"
-proof
- fix S
- assume "!!S. PROP P S"
- then show "PROP P (member S)" .
-next
- fix S
- assume "!!S. PROP P (member S)"
- then have "PROP P (member {x. S x})" .
- thus "PROP P S" by simp
-qed
-
-lemma member_right_eq: "(x == member y) == (Collect x == y)"
- by (rule equal_intr_rule, simp, drule symmetric, simp)
-
-definition
- member2 :: "('a * 'b) set => 'a => 'b \<Rightarrow> bool" where
- "member2 == %S x y. (x, y) : S"
-
-definition
- Collect2 :: "('a => 'b => bool) => ('a * 'b) set" where
- "Collect2 == %P. {(x, y). P x y}"
-
-lemma member2I[intro!, Pure.intro!]: "(x, y) : S ==> member2 S x y"
- by (simp add: member2_def)
-
-lemma member2D[dest!, Pure.dest!]: "member2 S x y ==> (x, y) : S"
- by (simp add: member2_def)
-
-lemma member2_eq[simp]: "member2 S x y = ((x, y) : S)"
- by (simp add: member2_def)
-
-lemma Collect2I: "P x y ==> (x, y) : Collect2 P"
- by (simp add: Collect2_def)
-
-lemma Collect2D: "(x, y) : Collect2 P ==> P x y"
- by (simp add: Collect2_def)
-
-lemma member2_Collect2_eq[simp]: "member2 (Collect2 P) = P"
- by (simp add: member2_def Collect2_def)
-
-lemma Collect2_member2_eq[simp]: "Collect2 (member2 S) = S"
- by (auto simp add: member2_def Collect2_def)
-
-lemma mem_Collect2_eq[iff]: "((x, y) : Collect2 P) = P x y"
- by (iprover intro: Collect2I dest: Collect2D)
-
-lemma member2_Collect_split_eq [simp]: "member2 (Collect (split P)) = P"
- by (simp add: expand_fun_eq)
+subsection {* Equality and Subsets *}
-lemma split_set2: "(!!S. PROP P S) == (!!S. PROP P (Collect2 S))"
-proof
- fix S
- assume "!!S. PROP P S"
- then show "PROP P (Collect2 S)" .
-next
- fix S
- assume "!!S. PROP P (Collect2 S)"
- then have "PROP P (Collect2 (member2 S))" .
- thus "PROP P S" by simp
-qed
-
-lemma split_predicate2: "(!!S. PROP P S) == (!!S. PROP P (member2 S))"
-proof
- fix S
- assume "!!S. PROP P S"
- then show "PROP P (member2 S)" .
-next
- fix S
- assume "!!S. PROP P (member2 S)"
- then have "PROP P (member2 (Collect2 S))" .
- thus "PROP P S" by simp
-qed
-
-lemma member2_right_eq: "(x == member2 y) == (Collect2 x == y)"
- by (rule equal_intr_rule, simp, drule symmetric, simp)
-
-ML_setup {*
-local
-
-fun vars_of b (v as Var _) = if b then [] else [v]
- | vars_of b (t $ u) = vars_of true t union vars_of false u
- | vars_of b (Abs (_, _, t)) = vars_of false t
- | vars_of _ _ = [];
-
-fun rew ths1 ths2 th = Drule.forall_elim_vars 0
- (rewrite_rule ths2 (rewrite_rule ths1 (Drule.forall_intr_list
- (map (cterm_of (theory_of_thm th)) (vars_of false (prop_of th))) th)));
-
-val get_eq = Simpdata.mk_eq o thm;
+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)
-val split_predicate = get_eq "split_predicate";
-val split_predicate2 = get_eq "split_predicate2";
-val split_set = get_eq "split_set";
-val split_set2 = get_eq "split_set2";
-val member_eq = get_eq "member_eq";
-val member2_eq = get_eq "member2_eq";
-val member_Collect_eq = get_eq "member_Collect_eq";
-val member2_Collect2_eq = get_eq "member2_Collect2_eq";
-val mem_Collect2_eq = get_eq "mem_Collect2_eq";
-val member_right_eq = thm "member_right_eq";
-val member2_right_eq = thm "member2_right_eq";
-
-val rew' = Thm.symmetric o rew [split_set2] [split_set,
- member_right_eq, member2_right_eq, member_Collect_eq, member2_Collect2_eq];
-
-val rules1 = [split_predicate, split_predicate2, member_eq, member2_eq];
-val rules2 = [split_set, mk_meta_eq mem_Collect_eq, mem_Collect2_eq];
-
-structure PredSetConvData = GenericDataFun
-(
- type T = thm list;
- val empty = [];
- val extend = I;
- fun merge _ = Drule.merge_rules;
-);
-
-fun mk_attr ths1 ths2 f = Attrib.syntax (Attrib.thms >> (fn ths =>
- Thm.rule_attribute (fn ctxt => rew ths1 (map (f o Simpdata.mk_eq)
- (ths @ PredSetConvData.get ctxt) @ ths2))));
-
-val pred_set_conv_att = Attrib.no_args (Thm.declaration_attribute
- (Drule.add_rule #> PredSetConvData.map));
-
-in
-
-val _ = ML_Context.>> (
- Attrib.add_attributes
- [("pred_set_conv", pred_set_conv_att,
- "declare rules for converting between predicate and set notation"),
- ("to_set", mk_attr [] rules1 I, "convert rule to set notation"),
- ("to_pred", mk_attr [split_set2] rules2 rew',
- "convert rule to predicate notation")])
-
-end;
-*}
-
-lemma member_inject [pred_set_conv]: "(member R = member S) = (R = S)"
+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 member2_inject [pred_set_conv]: "(member2 R = member2 S) = (R = S)"
- by (auto simp add: expand_fun_eq)
-
-lemma member_mono [pred_set_conv]: "(member R <= member S) = (R <= S)"
+lemma pred_subset_eq [pred_set_conv]: "((\<lambda>x. x \<in> R) <= (\<lambda>x. x \<in> S)) = (R <= S)"
by fast
-lemma member2_mono [pred_set_conv]: "(member2 R <= member2 S) = (R <= S)"
+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
-lemma member_empty [pred_set_conv]: "(%x. False) = member {}"
- by (simp add: expand_fun_eq)
+
+subsection {* Top and bottom elements *}
+
+lemma top1I [intro!]: "top x"
+ by (simp add: top_fun_eq top_bool_eq)
-lemma member2_empty [pred_set_conv]: "(%x y. False) = member2 {}"
- by (simp add: expand_fun_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 {* Binary union *}
+subsection {* The empty set *}
+
+lemma bot_empty_eq: "bot = (\<lambda>x. x \<in> {})"
+ by (auto simp add: expand_fun_eq)
-lemma member_Un [pred_set_conv]: "sup (member R) (member S) = member (R Un S)"
- by (simp add: expand_fun_eq sup_fun_eq sup_bool_eq)
+lemma bot_empty_eq2: "bot = (\<lambda>x y. (x, y) \<in> {})"
+ by (auto simp add: expand_fun_eq)
-lemma member2_Un [pred_set_conv]: "sup (member2 R) (member2 S) = member2 (R Un S)"
- by (simp add: expand_fun_eq sup_fun_eq sup_bool_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)
@@ -212,16 +56,22 @@
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 join1I2 [elim?]: "B x \<Longrightarrow> sup A B x"
+lemma sup1I2 [elim?]: "B x \<Longrightarrow> sup A B x"
by simp
-lemma sup1I2 [elim?]: "B x y \<Longrightarrow> sup A B x y"
+lemma sup2I2 [elim?]: "B x y \<Longrightarrow> sup A B x y"
by simp
text {*
@@ -242,13 +92,7 @@
by simp iprover
-subsubsection {* Binary intersection *}
-
-lemma member_Int [pred_set_conv]: "inf (member R) (member S) = member (R Int S)"
- by (simp add: expand_fun_eq inf_fun_eq inf_bool_eq)
-
-lemma member2_Int [pred_set_conv]: "inf (member2 R) (member2 S) = member2 (R Int S)"
- by (simp add: expand_fun_eq inf_fun_eq inf_bool_eq)
+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)
@@ -256,6 +100,12 @@
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
@@ -281,13 +131,7 @@
by simp
-subsubsection {* Unions of families *}
-
-lemma member_SUP: "(SUP i. member (r i)) = member (UN i. r i)"
- by (simp add: SUPR_def Sup_fun_eq Sup_bool_eq expand_fun_eq) blast
-
-lemma member2_SUP: "(SUP i. member2 (r i)) = member2 (UN i. r i)"
- by (simp add: SUPR_def Sup_fun_eq Sup_bool_eq expand_fun_eq) blast
+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_eq Sup_bool_eq) blast
@@ -307,14 +151,14 @@
lemma SUP2_E [elim!]: "(SUP x:A. B x) b c ==> (!!x. x : A ==> B x b c ==> R) ==> R"
by auto
-
-subsubsection {* Intersections of families *}
+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 member_INF: "(INF i. member (r i)) = member (INT i. r i)"
- by (simp add: INFI_def Inf_fun_def Inf_bool_def expand_fun_eq) blast
+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)
-lemma member2_INF: "(INF i. member2 (r i)) = member2 (INT i. r i)"
- by (simp add: INFI_def Inf_fun_def Inf_bool_def expand_fun_eq) blast
+
+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
@@ -340,26 +184,32 @@
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 *}
-inductive2
+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_cases2 pred_compE [elim!]: "(r OO s) a c"
+inductive_cases pred_compE [elim!]: "(r OO s) a c"
lemma pred_comp_rel_comp_eq [pred_set_conv]:
- "(member2 r OO member2 s) = member2 (r O s)"
+ "((\<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 *}
-inductive2
+inductive
conversep :: "('a => 'b => bool) => 'b => 'a => bool"
("(_^--1)" [1000] 1000)
for r :: "'a => 'b => bool"
@@ -378,7 +228,7 @@
by (iprover intro: conversepI dest: conversepD)
lemma conversep_converse_eq [pred_set_conv]:
- "(member2 r)^--1 = member2 (r^-1)"
+ "(\<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"
@@ -405,29 +255,29 @@
subsection {* Domain *}
-inductive2
+inductive
DomainP :: "('a => 'b => bool) => 'a => bool"
for r :: "'a => 'b => bool"
where
DomainPI [intro]: "r a b ==> DomainP r a"
-inductive_cases2 DomainPE [elim!]: "DomainP r a"
+inductive_cases DomainPE [elim!]: "DomainP r a"
-lemma member2_DomainP [pred_set_conv]: "DomainP (member2 r) = member (Domain r)"
+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 *}
-inductive2
+inductive
RangeP :: "('a => 'b => bool) => 'b => bool"
for r :: "'a => 'b => bool"
where
RangePI [intro]: "r a b ==> RangeP r b"
-inductive_cases2 RangePE [elim!]: "RangeP r b"
+inductive_cases RangePE [elim!]: "RangeP r b"
-lemma member2_RangeP [pred_set_conv]: "RangeP (member2 r) = member (Range r)"
+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)
@@ -437,77 +287,31 @@
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 (member2 r) f = member2 (inv_image r f)"
+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 (Collect2 r)"
+ "antisymP r == antisym {(x, y). r x y}"
abbreviation transP :: "('a => 'a => bool) => bool" where
- "transP r == trans (Collect2 r)"
+ "transP r == trans {(x, y). r x y}"
abbreviation single_valuedP :: "('a => 'b => bool) => bool" where
- "single_valuedP r == single_valued (Collect2 r)"
-
-
-subsection {* Bounded quantifiers for predicates *}
-
-text {*
-Bounded existential quantifier for predicates (executable).
-*}
-
-inductive2 bexp :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
- for P :: "'a \<Rightarrow> bool" and Q :: "'a \<Rightarrow> bool"
-where
- bexpI [intro]: "P x \<Longrightarrow> Q x \<Longrightarrow> bexp P Q"
-
-lemmas bexpE [elim!] = bexp.cases
-
-syntax
- Bexp :: "'a \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool \<Rightarrow> bool" ("(3\<exists>_\<triangleright>_./ _)" [0, 0, 10] 10)
-
-translations
- "\<exists>x\<triangleright>P. Q" \<rightleftharpoons> "CONST bexp P (\<lambda>x. Q)"
-
-constdefs
- ballp :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
- "ballp P Q \<equiv> \<forall>x. P x \<longrightarrow> Q x"
-
-syntax
- Ballp :: "'a \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool \<Rightarrow> bool" ("(3\<forall>_\<triangleright>_./ _)" [0, 0, 10] 10)
-
-translations
- "\<forall>x\<triangleright>P. Q" \<rightleftharpoons> "CONST ballp P (\<lambda>x. Q)"
-
-(* To avoid eta-contraction of body: *)
-print_translation {*
-let
- fun btr' syn [A,Abs abs] =
- let val (x,t) = atomic_abs_tr' abs
- in Syntax.const syn $ x $ A $ t end
-in
-[("ballp", btr' "Ballp"),("bexp", btr' "Bexp")]
-end
-*}
-
-lemma ballpI [intro!]: "(\<And>x. A x \<Longrightarrow> P x) \<Longrightarrow> \<forall>x\<triangleright>A. P x"
- by (simp add: ballp_def)
-
-lemma bspecp [dest?]: "\<forall>x\<triangleright>A. P x \<Longrightarrow> A x \<Longrightarrow> P x"
- by (simp add: ballp_def)
-
-lemma ballpE [elim]: "\<forall>x\<triangleright>A. P x \<Longrightarrow> (P x \<Longrightarrow> Q) \<Longrightarrow> (\<not> A x \<Longrightarrow> Q) \<Longrightarrow> Q"
- by (unfold ballp_def) blast
-
-lemma ballp_not_bexp_eq: "(\<forall>x\<triangleright>P. Q x) = (\<not> (\<exists>x\<triangleright>P. \<not> Q x))"
- by (blast dest: bspecp)
-
-declare ballp_not_bexp_eq [THEN eq_reflection, code unfold]
+ "single_valuedP r == single_valued {(x, y). r x y}"
end