--- a/src/HOL/Predicate.thy Tue Aug 23 03:34:17 2011 +0900
+++ b/src/HOL/Predicate.thy Mon Aug 22 22:00:36 2011 +0200
@@ -47,7 +47,7 @@
done
lemma rev_predicate1D:
- "P x ==> P <= Q ==> Q x"
+ "P x \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x"
by (rule predicate1D)
lemma predicate2I [Pure.intro!, intro!]:
@@ -67,7 +67,7 @@
done
lemma rev_predicate2D:
- "P x y ==> P <= Q ==> Q x y"
+ "P x y \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x y"
by (rule predicate2D)
@@ -82,85 +82,85 @@
subsubsection {* Order relation *}
-lemma pred_subset_eq: "((\<lambda>x. x \<in> R) <= (\<lambda>x. x \<in> S)) = (R <= S)"
+lemma pred_subset_eq: "((\<lambda>x. x \<in> R) \<le> (\<lambda>x. x \<in> S)) = (R \<subseteq> 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)"
+lemma pred_subset_eq2 [pred_set_conv]: "((\<lambda>x y. (x, y) \<in> R) \<le> (\<lambda>x y. (x, y) \<in> S)) = (R \<subseteq> S)"
by fast
subsubsection {* Top and bottom elements *}
-lemma bot1E [no_atp, elim!]: "bot x \<Longrightarrow> P"
+lemma bot1E [no_atp, elim!]: "\<bottom> x \<Longrightarrow> P"
by (simp add: bot_fun_def)
-lemma bot2E [elim!]: "bot x y \<Longrightarrow> P"
+lemma bot2E [elim!]: "\<bottom> x y \<Longrightarrow> P"
by (simp add: bot_fun_def)
-lemma bot_empty_eq: "bot = (\<lambda>x. x \<in> {})"
+lemma bot_empty_eq: "\<bottom> = (\<lambda>x. x \<in> {})"
by (auto simp add: fun_eq_iff)
-lemma bot_empty_eq2: "bot = (\<lambda>x y. (x, y) \<in> {})"
+lemma bot_empty_eq2: "\<bottom> = (\<lambda>x y. (x, y) \<in> {})"
by (auto simp add: fun_eq_iff)
-lemma top1I [intro!]: "top x"
+lemma top1I [intro!]: "\<top> x"
by (simp add: top_fun_def)
-lemma top2I [intro!]: "top x y"
+lemma top2I [intro!]: "\<top> x y"
by (simp add: top_fun_def)
subsubsection {* Binary intersection *}
-lemma inf1I [intro!]: "A x ==> B x ==> inf A B x"
+lemma inf1I [intro!]: "A x \<Longrightarrow> B x \<Longrightarrow> (A \<sqinter> B) x"
by (simp add: inf_fun_def)
-lemma inf2I [intro!]: "A x y ==> B x y ==> inf A B x y"
+lemma inf2I [intro!]: "A x y \<Longrightarrow> B x y \<Longrightarrow> (A \<sqinter> B) x y"
by (simp add: inf_fun_def)
-lemma inf1E [elim!]: "inf A B x ==> (A x ==> B x ==> P) ==> P"
+lemma inf1E [elim!]: "(A \<sqinter> B) x \<Longrightarrow> (A x \<Longrightarrow> B x \<Longrightarrow> P) \<Longrightarrow> P"
by (simp add: inf_fun_def)
-lemma inf2E [elim!]: "inf A B x y ==> (A x y ==> B x y ==> P) ==> P"
+lemma inf2E [elim!]: "(A \<sqinter> B) x y \<Longrightarrow> (A x y \<Longrightarrow> B x y \<Longrightarrow> P) \<Longrightarrow> P"
by (simp add: inf_fun_def)
-lemma inf1D1: "inf A B x ==> A x"
+lemma inf1D1: "(A \<sqinter> B) x \<Longrightarrow> A x"
by (simp add: inf_fun_def)
-lemma inf2D1: "inf A B x y ==> A x y"
+lemma inf2D1: "(A \<sqinter> B) x y \<Longrightarrow> A x y"
by (simp add: inf_fun_def)
-lemma inf1D2: "inf A B x ==> B x"
+lemma inf1D2: "(A \<sqinter> B) x \<Longrightarrow> B x"
by (simp add: inf_fun_def)
-lemma inf2D2: "inf A B x y ==> B x y"
+lemma inf2D2: "(A \<sqinter> B) x y \<Longrightarrow> B x y"
by (simp add: inf_fun_def)
-lemma inf_Int_eq: "inf (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
+lemma inf_Int_eq: "(\<lambda>x. x \<in> R) \<sqinter> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
by (simp add: inf_fun_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)"
+lemma inf_Int_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<sqinter> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<inter> S)"
by (simp add: inf_fun_def mem_def)
subsubsection {* Binary union *}
-lemma sup1I1 [elim?]: "A x \<Longrightarrow> sup A B x"
+lemma sup1I1 [elim?]: "A x \<Longrightarrow> (A \<squnion> B) x"
by (simp add: sup_fun_def)
-lemma sup2I1 [elim?]: "A x y \<Longrightarrow> sup A B x y"
+lemma sup2I1 [elim?]: "A x y \<Longrightarrow> (A \<squnion> B) x y"
by (simp add: sup_fun_def)
-lemma sup1I2 [elim?]: "B x \<Longrightarrow> sup A B x"
+lemma sup1I2 [elim?]: "B x \<Longrightarrow> (A \<squnion> B) x"
by (simp add: sup_fun_def)
-lemma sup2I2 [elim?]: "B x y \<Longrightarrow> sup A B x y"
+lemma sup2I2 [elim?]: "B x y \<Longrightarrow> (A \<squnion> B) x y"
by (simp add: sup_fun_def)
-lemma sup1E [elim!]: "sup A B x ==> (A x ==> P) ==> (B x ==> P) ==> P"
+lemma sup1E [elim!]: "(A \<squnion> B) x \<Longrightarrow> (A x \<Longrightarrow> P) \<Longrightarrow> (B x \<Longrightarrow> P) \<Longrightarrow> P"
by (simp add: sup_fun_def) iprover
-lemma sup2E [elim!]: "sup A B x y ==> (A x y ==> P) ==> (B x y ==> P) ==> P"
+lemma sup2E [elim!]: "(A \<squnion> B) x y \<Longrightarrow> (A x y \<Longrightarrow> P) \<Longrightarrow> (B x y \<Longrightarrow> P) \<Longrightarrow> P"
by (simp add: sup_fun_def) iprover
text {*
@@ -168,76 +168,76 @@
@{text B}.
*}
-lemma sup1CI [intro!]: "(~ B x ==> A x) ==> sup A B x"
+lemma sup1CI [intro!]: "(\<not> B x \<Longrightarrow> A x) \<Longrightarrow> (A \<squnion> B) x"
by (auto simp add: sup_fun_def)
-lemma sup2CI [intro!]: "(~ B x y ==> A x y) ==> sup A B x y"
+lemma sup2CI [intro!]: "(\<not> B x y \<Longrightarrow> A x y) \<Longrightarrow> (A \<squnion> B) x y"
by (auto simp add: sup_fun_def)
-lemma sup_Un_eq: "sup (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<union> S)"
+lemma sup_Un_eq: "(\<lambda>x. x \<in> R) \<squnion> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<union> S)"
by (simp add: sup_fun_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)"
+lemma sup_Un_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<squnion> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<union> S)"
by (simp add: sup_fun_def mem_def)
subsubsection {* Intersections of families *}
-lemma INF1_iff: "(INF x:A. B x) b = (ALL x:A. B x b)"
+lemma INF1_iff: "(\<Sqinter>x\<in>A. B x) b = (\<forall>x\<in>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)"
+lemma INF2_iff: "(\<Sqinter>x\<in>A. B x) b c = (\<forall>x\<in>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"
+lemma INF1_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> B x b) \<Longrightarrow> (\<Sqinter>x\<in>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"
+lemma INF2_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> B x b c) \<Longrightarrow> (\<Sqinter>x\<in>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"
+lemma INF1_D [elim]: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> a \<in> A \<Longrightarrow> 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"
+lemma INF2_D [elim]: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> a \<in> A \<Longrightarrow> 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"
+lemma INF1_E [elim]: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> (B a b \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> 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"
+lemma INF2_E [elim]: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> (B a b c \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> 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))"
+lemma INF_INT_eq: "(\<Sqinter>i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Sqinter>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))"
+lemma INF_INT_eq2: "(\<Sqinter>i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Sqinter>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)"
+lemma SUP1_iff: "(\<Squnion>x\<in>A. B x) b = (\<exists>x\<in>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)"
+lemma SUP2_iff: "(\<Squnion>x\<in>A. B x) b c = (\<exists>x\<in>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"
+lemma SUP1_I [intro]: "a \<in> A \<Longrightarrow> B a b \<Longrightarrow> (\<Squnion>x\<in>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"
+lemma SUP2_I [intro]: "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>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"
+lemma SUP1_E [elim!]: "(\<Squnion>x\<in>A. B x) b \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b \<Longrightarrow> R) \<Longrightarrow> 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"
+lemma SUP2_E [elim!]: "(\<Squnion>x\<in>A. B x) b c \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b c \<Longrightarrow> R) \<Longrightarrow> 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))"
+lemma SUP_UN_eq: "(\<Squnion>i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Union>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))"
+lemma SUP_UN_eq2: "(\<Squnion>i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Union>i. r i))"
by (simp add: SUPR_apply fun_eq_iff)
@@ -245,12 +245,9 @@
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 pred_comp :: "['a \<Rightarrow> 'b \<Rightarrow> bool, 'b \<Rightarrow> 'c \<Rightarrow> bool] \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool" (infixr "OO" 75)
+ for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and s :: "'b \<Rightarrow> 'c \<Rightarrow> bool" where
+ pred_compI [intro]: "r a b \<Longrightarrow> s b c \<Longrightarrow> (r OO s) a c"
inductive_cases pred_compE [elim!]: "(r OO s) a c"
@@ -261,12 +258,9 @@
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"
+inductive conversep :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(_^--1)" [1000] 1000)
+ for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where
+ conversepI: "r a b \<Longrightarrow> r^--1 b a"
notation (xsymbols)
conversep ("(_\<inverse>\<inverse>)" [1000] 1000)
@@ -290,13 +284,13 @@
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"
+lemma converse_meet: "(r \<sqinter> s)^--1 = r^--1 \<sqinter> s^--1"
by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD)
-lemma converse_join: "(sup r s)^--1 = sup r^--1 s^--1"
+lemma converse_join: "(r \<squnion> s)^--1 = r^--1 \<squnion> s^--1"
by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD)
-lemma conversep_noteq [simp]: "(op ~=)^--1 = op ~="
+lemma conversep_noteq [simp]: "(op \<noteq>)^--1 = op \<noteq>"
by (auto simp add: fun_eq_iff)
lemma conversep_eq [simp]: "(op =)^--1 = op ="
@@ -305,11 +299,9 @@
subsubsection {* Domain *}
-inductive
- DomainP :: "('a => 'b => bool) => 'a => bool"
- for r :: "'a => 'b => bool"
-where
- DomainPI [intro]: "r a b ==> DomainP r a"
+inductive DomainP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
+ for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where
+ DomainPI [intro]: "r a b \<Longrightarrow> DomainP r a"
inductive_cases DomainPE [elim!]: "DomainP r a"
@@ -319,11 +311,9 @@
subsubsection {* Range *}
-inductive
- RangeP :: "('a => 'b => bool) => 'b => bool"
- for r :: "'a => 'b => bool"
-where
- RangePI [intro]: "r a b ==> RangeP r b"
+inductive RangeP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"
+ for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where
+ RangePI [intro]: "r a b \<Longrightarrow> RangeP r b"
inductive_cases RangePE [elim!]: "RangeP r b"
@@ -333,9 +323,8 @@
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)"
+definition inv_imagep :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" where
+ "inv_imagep r f = (\<lambda>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)
@@ -347,7 +336,7 @@
subsubsection {* Powerset *}
definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool" where
- "Powp A == \<lambda>B. \<forall>x \<in> B. A x"
+ "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)
@@ -357,14 +346,14 @@
subsubsection {* Properties of relations *}
-abbreviation antisymP :: "('a => 'a => bool) => bool" where
- "antisymP r == antisym {(x, y). r x y}"
+abbreviation antisymP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
+ "antisymP r \<equiv> antisym {(x, y). r x y}"
-abbreviation transP :: "('a => 'a => bool) => bool" where
- "transP r == trans {(x, y). r x y}"
+abbreviation transP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
+ "transP r \<equiv> trans {(x, y). r x y}"
-abbreviation single_valuedP :: "('a => 'b => bool) => bool" where
- "single_valuedP r == single_valued {(x, y). r x y}"
+abbreviation single_valuedP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
+ "single_valuedP r \<equiv> single_valued {(x, y). r x y}"
(*FIXME inconsistencies: abbreviations vs. definitions, suffix `P` vs. suffix `p`*)
@@ -801,9 +790,9 @@
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"
+ "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 ())"
@@ -812,8 +801,8 @@
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"
+| "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)"
@@ -973,23 +962,21 @@
(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
+definition the :: "'a pred \<Rightarrow> '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)
+definition not_unique :: "'a pred \<Rightarrow> 'a" where
+ [code del]: "not_unique A = (THE x. eval A x)"
+
+code_abort not_unique
+
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