wenzelm@10358
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(* Title: HOL/Relation.thy
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paulson@1983
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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paulson@1983
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Copyright 1996 University of Cambridge
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nipkow@1128
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*)
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nipkow@1128
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berghofe@12905
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header {* Relations *}
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berghofe@12905
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nipkow@15131
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theory Relation
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haftmann@31011
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imports Finite_Set Datatype
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haftmann@31011
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(*FIXME order is important, otherwise merge problem for canonical interpretation of class monoid_mult wrt. power!*)
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nipkow@15131
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begin
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paulson@5978
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wenzelm@12913
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subsection {* Definitions *}
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wenzelm@12913
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wenzelm@19656
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definition
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wenzelm@21404
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converse :: "('a * 'b) set => ('b * 'a) set"
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wenzelm@21404
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("(_^-1)" [1000] 999) where
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wenzelm@10358
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"r^-1 == {(y, x). (x, y) : r}"
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paulson@5978
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wenzelm@21210
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notation (xsymbols)
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wenzelm@19656
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converse ("(_\<inverse>)" [1000] 999)
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wenzelm@19656
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wenzelm@19656
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definition
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wenzelm@21404
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rel_comp :: "[('b * 'c) set, ('a * 'b) set] => ('a * 'c) set"
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wenzelm@21404
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(infixr "O" 75) where
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wenzelm@12913
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"r O s == {(x,z). EX y. (x, y) : s & (y, z) : r}"
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wenzelm@12913
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wenzelm@21404
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definition
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wenzelm@21404
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Image :: "[('a * 'b) set, 'a set] => 'b set"
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wenzelm@21404
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(infixl "``" 90) where
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wenzelm@12913
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"r `` s == {y. EX x:s. (x,y):r}"
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paulson@7912
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wenzelm@21404
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definition
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wenzelm@21404
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Id :: "('a * 'a) set" where -- {* the identity relation *}
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wenzelm@12913
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"Id == {p. EX x. p = (x,x)}"
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paulson@7912
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wenzelm@21404
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definition
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nipkow@30198
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Id_on :: "'a set => ('a * 'a) set" where -- {* diagonal: identity over a set *}
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nipkow@30198
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"Id_on A == \<Union>x\<in>A. {(x,x)}"
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wenzelm@12913
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wenzelm@21404
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definition
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wenzelm@21404
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Domain :: "('a * 'b) set => 'a set" where
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wenzelm@12913
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"Domain r == {x. EX y. (x,y):r}"
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paulson@5978
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wenzelm@21404
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definition
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wenzelm@21404
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Range :: "('a * 'b) set => 'b set" where
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wenzelm@12913
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"Range r == Domain(r^-1)"
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paulson@5978
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wenzelm@21404
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definition
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wenzelm@21404
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Field :: "('a * 'a) set => 'a set" where
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paulson@13830
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"Field r == Domain r \<union> Range r"
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paulson@10786
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wenzelm@21404
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definition
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nipkow@30198
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refl_on :: "['a set, ('a * 'a) set] => bool" where -- {* reflexivity over a set *}
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nipkow@30198
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"refl_on A r == r \<subseteq> A \<times> A & (ALL x: A. (x,x) : r)"
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paulson@6806
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nipkow@26297
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abbreviation
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nipkow@30198
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refl :: "('a * 'a) set => bool" where -- {* reflexivity over a type *}
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nipkow@30198
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"refl == refl_on UNIV"
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nipkow@26297
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wenzelm@21404
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definition
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wenzelm@21404
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sym :: "('a * 'a) set => bool" where -- {* symmetry predicate *}
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wenzelm@12913
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"sym r == ALL x y. (x,y): r --> (y,x): r"
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paulson@6806
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wenzelm@21404
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definition
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wenzelm@21404
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antisym :: "('a * 'a) set => bool" where -- {* antisymmetry predicate *}
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wenzelm@12913
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"antisym r == ALL x y. (x,y):r --> (y,x):r --> x=y"
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paulson@6806
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wenzelm@21404
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definition
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wenzelm@21404
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trans :: "('a * 'a) set => bool" where -- {* transitivity predicate *}
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wenzelm@12913
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"trans r == (ALL x y z. (x,y):r --> (y,z):r --> (x,z):r)"
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paulson@5978
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wenzelm@21404
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definition
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nipkow@29796
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irrefl :: "('a * 'a) set => bool" where
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nipkow@29796
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"irrefl r \<equiv> \<forall>x. (x,x) \<notin> r"
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nipkow@29796
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nipkow@29796
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definition
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nipkow@29796
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total_on :: "'a set => ('a * 'a) set => bool" where
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nipkow@29796
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"total_on A r \<equiv> \<forall>x\<in>A.\<forall>y\<in>A. x\<noteq>y \<longrightarrow> (x,y)\<in>r \<or> (y,x)\<in>r"
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nipkow@29796
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nipkow@29796
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abbreviation "total \<equiv> total_on UNIV"
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nipkow@29796
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nipkow@29796
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definition
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wenzelm@21404
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single_valued :: "('a * 'b) set => bool" where
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wenzelm@12913
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"single_valued r == ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z)"
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berghofe@7014
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wenzelm@21404
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definition
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wenzelm@21404
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inv_image :: "('b * 'b) set => ('a => 'b) => ('a * 'a) set" where
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wenzelm@12913
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"inv_image r f == {(x, y). (f x, f y) : r}"
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oheimb@11136
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berghofe@12905
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wenzelm@12913
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subsection {* The identity relation *}
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berghofe@12905
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berghofe@12905
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lemma IdI [intro]: "(a, a) : Id"
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nipkow@26271
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by (simp add: Id_def)
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berghofe@12905
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berghofe@12905
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lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P"
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nipkow@26271
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by (unfold Id_def) (iprover elim: CollectE)
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berghofe@12905
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berghofe@12905
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lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)"
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nipkow@26271
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by (unfold Id_def) blast
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berghofe@12905
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nipkow@30198
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lemma refl_Id: "refl Id"
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nipkow@30198
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by (simp add: refl_on_def)
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berghofe@12905
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berghofe@12905
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lemma antisym_Id: "antisym Id"
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berghofe@12905
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-- {* A strange result, since @{text Id} is also symmetric. *}
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nipkow@26271
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by (simp add: antisym_def)
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berghofe@12905
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huffman@19228
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lemma sym_Id: "sym Id"
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nipkow@26271
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by (simp add: sym_def)
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huffman@19228
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berghofe@12905
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lemma trans_Id: "trans Id"
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nipkow@26271
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by (simp add: trans_def)
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berghofe@12905
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berghofe@12905
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wenzelm@12913
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subsection {* Diagonal: identity over a set *}
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berghofe@12905
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nipkow@30198
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lemma Id_on_empty [simp]: "Id_on {} = {}"
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nipkow@30198
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by (simp add: Id_on_def)
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paulson@13812
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nipkow@30198
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lemma Id_on_eqI: "a = b ==> a : A ==> (a, b) : Id_on A"
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nipkow@30198
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by (simp add: Id_on_def)
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berghofe@12905
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nipkow@30198
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lemma Id_onI [intro!,noatp]: "a : A ==> (a, a) : Id_on A"
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nipkow@30198
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by (rule Id_on_eqI) (rule refl)
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berghofe@12905
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nipkow@30198
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lemma Id_onE [elim!]:
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nipkow@30198
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"c : Id_on A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P"
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wenzelm@12913
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-- {* The general elimination rule. *}
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nipkow@30198
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by (unfold Id_on_def) (iprover elim!: UN_E singletonE)
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berghofe@12905
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nipkow@30198
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lemma Id_on_iff: "((x, y) : Id_on A) = (x = y & x : A)"
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nipkow@26271
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by blast
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berghofe@12905
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nipkow@30198
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lemma Id_on_subset_Times: "Id_on A \<subseteq> A \<times> A"
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nipkow@26271
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by blast
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berghofe@12905
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berghofe@12905
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berghofe@12905
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subsection {* Composition of two relations *}
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berghofe@12905
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wenzelm@12913
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lemma rel_compI [intro]:
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berghofe@12905
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"(a, b) : s ==> (b, c) : r ==> (a, c) : r O s"
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nipkow@26271
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by (unfold rel_comp_def) blast
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berghofe@12905
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wenzelm@12913
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lemma rel_compE [elim!]: "xz : r O s ==>
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berghofe@12905
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(!!x y z. xz = (x, z) ==> (x, y) : s ==> (y, z) : r ==> P) ==> P"
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nipkow@26271
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by (unfold rel_comp_def) (iprover elim!: CollectE splitE exE conjE)
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berghofe@12905
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berghofe@12905
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lemma rel_compEpair:
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berghofe@12905
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"(a, c) : r O s ==> (!!y. (a, y) : s ==> (y, c) : r ==> P) ==> P"
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nipkow@26271
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by (iprover elim: rel_compE Pair_inject ssubst)
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berghofe@12905
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berghofe@12905
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lemma R_O_Id [simp]: "R O Id = R"
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nipkow@26271
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by fast
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berghofe@12905
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berghofe@12905
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lemma Id_O_R [simp]: "Id O R = R"
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nipkow@26271
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by fast
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berghofe@12905
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krauss@23185
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lemma rel_comp_empty1[simp]: "{} O R = {}"
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nipkow@26271
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by blast
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krauss@23185
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krauss@23185
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lemma rel_comp_empty2[simp]: "R O {} = {}"
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nipkow@26271
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by blast
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krauss@23185
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berghofe@12905
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lemma O_assoc: "(R O S) O T = R O (S O T)"
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nipkow@26271
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by blast
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berghofe@12905
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wenzelm@12913
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lemma trans_O_subset: "trans r ==> r O r \<subseteq> r"
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nipkow@26271
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by (unfold trans_def) blast
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berghofe@12905
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wenzelm@12913
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lemma rel_comp_mono: "r' \<subseteq> r ==> s' \<subseteq> s ==> (r' O s') \<subseteq> (r O s)"
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nipkow@26271
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by blast
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berghofe@12905
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berghofe@12905
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lemma rel_comp_subset_Sigma:
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wenzelm@12913
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"s \<subseteq> A \<times> B ==> r \<subseteq> B \<times> C ==> (r O s) \<subseteq> A \<times> C"
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nipkow@26271
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by blast
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berghofe@12905
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krauss@28008
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lemma rel_comp_distrib[simp]: "R O (S \<union> T) = (R O S) \<union> (R O T)"
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krauss@28008
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by auto
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krauss@28008
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krauss@28008
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lemma rel_comp_distrib2[simp]: "(S \<union> T) O R = (S O R) \<union> (T O R)"
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krauss@28008
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by auto
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krauss@28008
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berghofe@12905
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wenzelm@12913
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subsection {* Reflexivity *}
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wenzelm@12913
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nipkow@30198
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lemma refl_onI: "r \<subseteq> A \<times> A ==> (!!x. x : A ==> (x, x) : r) ==> refl_on A r"
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nipkow@30198
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by (unfold refl_on_def) (iprover intro!: ballI)
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berghofe@12905
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nipkow@30198
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lemma refl_onD: "refl_on A r ==> a : A ==> (a, a) : r"
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nipkow@30198
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by (unfold refl_on_def) blast
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berghofe@12905
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nipkow@30198
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lemma refl_onD1: "refl_on A r ==> (x, y) : r ==> x : A"
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nipkow@30198
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by (unfold refl_on_def) blast
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huffman@19228
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nipkow@30198
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lemma refl_onD2: "refl_on A r ==> (x, y) : r ==> y : A"
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nipkow@30198
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by (unfold refl_on_def) blast
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huffman@19228
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nipkow@30198
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lemma refl_on_Int: "refl_on A r ==> refl_on B s ==> refl_on (A \<inter> B) (r \<inter> s)"
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nipkow@30198
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by (unfold refl_on_def) blast
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huffman@19228
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nipkow@30198
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lemma refl_on_Un: "refl_on A r ==> refl_on B s ==> refl_on (A \<union> B) (r \<union> s)"
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nipkow@30198
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by (unfold refl_on_def) blast
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huffman@19228
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nipkow@30198
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lemma refl_on_INTER:
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nipkow@30198
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"ALL x:S. refl_on (A x) (r x) ==> refl_on (INTER S A) (INTER S r)"
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nipkow@30198
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by (unfold refl_on_def) fast
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huffman@19228
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nipkow@30198
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lemma refl_on_UNION:
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nipkow@30198
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"ALL x:S. refl_on (A x) (r x) \<Longrightarrow> refl_on (UNION S A) (UNION S r)"
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nipkow@30198
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by (unfold refl_on_def) blast
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huffman@19228
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nipkow@30198
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lemma refl_on_empty[simp]: "refl_on {} {}"
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nipkow@30198
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by(simp add:refl_on_def)
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nipkow@26297
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nipkow@30198
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lemma refl_on_Id_on: "refl_on A (Id_on A)"
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nipkow@30198
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by (rule refl_onI [OF Id_on_subset_Times Id_onI])
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huffman@19228
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wenzelm@12913
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wenzelm@12913
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subsection {* Antisymmetry *}
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berghofe@12905
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berghofe@12905
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lemma antisymI:
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berghofe@12905
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"(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r"
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nipkow@26271
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by (unfold antisym_def) iprover
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berghofe@12905
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berghofe@12905
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lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b"
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nipkow@26271
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by (unfold antisym_def) iprover
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berghofe@12905
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huffman@19228
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lemma antisym_subset: "r \<subseteq> s ==> antisym s ==> antisym r"
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nipkow@26271
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by (unfold antisym_def) blast
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wenzelm@12913
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huffman@19228
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lemma antisym_empty [simp]: "antisym {}"
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nipkow@26271
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by (unfold antisym_def) blast
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huffman@19228
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nipkow@30198
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lemma antisym_Id_on [simp]: "antisym (Id_on A)"
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nipkow@26271
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by (unfold antisym_def) blast
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huffman@19228
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huffman@19228
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239 |
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huffman@19228
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240 |
subsection {* Symmetry *}
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huffman@19228
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241 |
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huffman@19228
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lemma symI: "(!!a b. (a, b) : r ==> (b, a) : r) ==> sym r"
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nipkow@26271
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by (unfold sym_def) iprover
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paulson@15177
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paulson@15177
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245 |
lemma symD: "sym r ==> (a, b) : r ==> (b, a) : r"
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nipkow@26271
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246 |
by (unfold sym_def, blast)
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berghofe@12905
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247 |
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huffman@19228
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lemma sym_Int: "sym r ==> sym s ==> sym (r \<inter> s)"
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nipkow@26271
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by (fast intro: symI dest: symD)
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huffman@19228
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huffman@19228
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lemma sym_Un: "sym r ==> sym s ==> sym (r \<union> s)"
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nipkow@26271
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by (fast intro: symI dest: symD)
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huffman@19228
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huffman@19228
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254 |
lemma sym_INTER: "ALL x:S. sym (r x) ==> sym (INTER S r)"
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nipkow@26271
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by (fast intro: symI dest: symD)
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huffman@19228
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huffman@19228
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257 |
lemma sym_UNION: "ALL x:S. sym (r x) ==> sym (UNION S r)"
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nipkow@26271
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258 |
by (fast intro: symI dest: symD)
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huffman@19228
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259 |
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nipkow@30198
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lemma sym_Id_on [simp]: "sym (Id_on A)"
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nipkow@26271
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by (rule symI) clarify
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huffman@19228
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huffman@19228
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263 |
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huffman@19228
|
264 |
subsection {* Transitivity *}
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huffman@19228
|
265 |
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berghofe@12905
|
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lemma transI:
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berghofe@12905
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267 |
"(!!x y z. (x, y) : r ==> (y, z) : r ==> (x, z) : r) ==> trans r"
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nipkow@26271
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by (unfold trans_def) iprover
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berghofe@12905
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berghofe@12905
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lemma transD: "trans r ==> (a, b) : r ==> (b, c) : r ==> (a, c) : r"
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nipkow@26271
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by (unfold trans_def) iprover
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berghofe@12905
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272 |
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huffman@19228
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273 |
lemma trans_Int: "trans r ==> trans s ==> trans (r \<inter> s)"
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nipkow@26271
|
274 |
by (fast intro: transI elim: transD)
|
huffman@19228
|
275 |
|
huffman@19228
|
276 |
lemma trans_INTER: "ALL x:S. trans (r x) ==> trans (INTER S r)"
|
nipkow@26271
|
277 |
by (fast intro: transI elim: transD)
|
huffman@19228
|
278 |
|
nipkow@30198
|
279 |
lemma trans_Id_on [simp]: "trans (Id_on A)"
|
nipkow@26271
|
280 |
by (fast intro: transI elim: transD)
|
huffman@19228
|
281 |
|
nipkow@29796
|
282 |
lemma trans_diff_Id: " trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r-Id)"
|
nipkow@29796
|
283 |
unfolding antisym_def trans_def by blast
|
nipkow@29796
|
284 |
|
nipkow@29796
|
285 |
subsection {* Irreflexivity *}
|
nipkow@29796
|
286 |
|
nipkow@29796
|
287 |
lemma irrefl_diff_Id[simp]: "irrefl(r-Id)"
|
nipkow@29796
|
288 |
by(simp add:irrefl_def)
|
nipkow@29796
|
289 |
|
nipkow@29796
|
290 |
subsection {* Totality *}
|
nipkow@29796
|
291 |
|
nipkow@29796
|
292 |
lemma total_on_empty[simp]: "total_on {} r"
|
nipkow@29796
|
293 |
by(simp add:total_on_def)
|
nipkow@29796
|
294 |
|
nipkow@29796
|
295 |
lemma total_on_diff_Id[simp]: "total_on A (r-Id) = total_on A r"
|
nipkow@29796
|
296 |
by(simp add: total_on_def)
|
berghofe@12905
|
297 |
|
wenzelm@12913
|
298 |
subsection {* Converse *}
|
wenzelm@12913
|
299 |
|
wenzelm@12913
|
300 |
lemma converse_iff [iff]: "((a,b): r^-1) = ((b,a) : r)"
|
nipkow@26271
|
301 |
by (simp add: converse_def)
|
berghofe@12905
|
302 |
|
nipkow@13343
|
303 |
lemma converseI[sym]: "(a, b) : r ==> (b, a) : r^-1"
|
nipkow@26271
|
304 |
by (simp add: converse_def)
|
berghofe@12905
|
305 |
|
nipkow@13343
|
306 |
lemma converseD[sym]: "(a,b) : r^-1 ==> (b, a) : r"
|
nipkow@26271
|
307 |
by (simp add: converse_def)
|
berghofe@12905
|
308 |
|
berghofe@12905
|
309 |
lemma converseE [elim!]:
|
berghofe@12905
|
310 |
"yx : r^-1 ==> (!!x y. yx = (y, x) ==> (x, y) : r ==> P) ==> P"
|
wenzelm@12913
|
311 |
-- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *}
|
nipkow@26271
|
312 |
by (unfold converse_def) (iprover elim!: CollectE splitE bexE)
|
berghofe@12905
|
313 |
|
berghofe@12905
|
314 |
lemma converse_converse [simp]: "(r^-1)^-1 = r"
|
nipkow@26271
|
315 |
by (unfold converse_def) blast
|
berghofe@12905
|
316 |
|
berghofe@12905
|
317 |
lemma converse_rel_comp: "(r O s)^-1 = s^-1 O r^-1"
|
nipkow@26271
|
318 |
by blast
|
berghofe@12905
|
319 |
|
huffman@19228
|
320 |
lemma converse_Int: "(r \<inter> s)^-1 = r^-1 \<inter> s^-1"
|
nipkow@26271
|
321 |
by blast
|
huffman@19228
|
322 |
|
huffman@19228
|
323 |
lemma converse_Un: "(r \<union> s)^-1 = r^-1 \<union> s^-1"
|
nipkow@26271
|
324 |
by blast
|
huffman@19228
|
325 |
|
huffman@19228
|
326 |
lemma converse_INTER: "(INTER S r)^-1 = (INT x:S. (r x)^-1)"
|
nipkow@26271
|
327 |
by fast
|
huffman@19228
|
328 |
|
huffman@19228
|
329 |
lemma converse_UNION: "(UNION S r)^-1 = (UN x:S. (r x)^-1)"
|
nipkow@26271
|
330 |
by blast
|
huffman@19228
|
331 |
|
berghofe@12905
|
332 |
lemma converse_Id [simp]: "Id^-1 = Id"
|
nipkow@26271
|
333 |
by blast
|
berghofe@12905
|
334 |
|
nipkow@30198
|
335 |
lemma converse_Id_on [simp]: "(Id_on A)^-1 = Id_on A"
|
nipkow@26271
|
336 |
by blast
|
berghofe@12905
|
337 |
|
nipkow@30198
|
338 |
lemma refl_on_converse [simp]: "refl_on A (converse r) = refl_on A r"
|
nipkow@30198
|
339 |
by (unfold refl_on_def) auto
|
berghofe@12905
|
340 |
|
huffman@19228
|
341 |
lemma sym_converse [simp]: "sym (converse r) = sym r"
|
nipkow@26271
|
342 |
by (unfold sym_def) blast
|
huffman@19228
|
343 |
|
huffman@19228
|
344 |
lemma antisym_converse [simp]: "antisym (converse r) = antisym r"
|
nipkow@26271
|
345 |
by (unfold antisym_def) blast
|
berghofe@12905
|
346 |
|
huffman@19228
|
347 |
lemma trans_converse [simp]: "trans (converse r) = trans r"
|
nipkow@26271
|
348 |
by (unfold trans_def) blast
|
berghofe@12905
|
349 |
|
huffman@19228
|
350 |
lemma sym_conv_converse_eq: "sym r = (r^-1 = r)"
|
nipkow@26271
|
351 |
by (unfold sym_def) fast
|
huffman@19228
|
352 |
|
huffman@19228
|
353 |
lemma sym_Un_converse: "sym (r \<union> r^-1)"
|
nipkow@26271
|
354 |
by (unfold sym_def) blast
|
huffman@19228
|
355 |
|
huffman@19228
|
356 |
lemma sym_Int_converse: "sym (r \<inter> r^-1)"
|
nipkow@26271
|
357 |
by (unfold sym_def) blast
|
huffman@19228
|
358 |
|
nipkow@29796
|
359 |
lemma total_on_converse[simp]: "total_on A (r^-1) = total_on A r"
|
nipkow@29796
|
360 |
by (auto simp: total_on_def)
|
nipkow@29796
|
361 |
|
wenzelm@12913
|
362 |
|
berghofe@12905
|
363 |
subsection {* Domain *}
|
berghofe@12905
|
364 |
|
paulson@24286
|
365 |
declare Domain_def [noatp]
|
paulson@24286
|
366 |
|
berghofe@12905
|
367 |
lemma Domain_iff: "(a : Domain r) = (EX y. (a, y) : r)"
|
nipkow@26271
|
368 |
by (unfold Domain_def) blast
|
berghofe@12905
|
369 |
|
berghofe@12905
|
370 |
lemma DomainI [intro]: "(a, b) : r ==> a : Domain r"
|
nipkow@26271
|
371 |
by (iprover intro!: iffD2 [OF Domain_iff])
|
berghofe@12905
|
372 |
|
berghofe@12905
|
373 |
lemma DomainE [elim!]:
|
berghofe@12905
|
374 |
"a : Domain r ==> (!!y. (a, y) : r ==> P) ==> P"
|
nipkow@26271
|
375 |
by (iprover dest!: iffD1 [OF Domain_iff])
|
berghofe@12905
|
376 |
|
berghofe@12905
|
377 |
lemma Domain_empty [simp]: "Domain {} = {}"
|
nipkow@26271
|
378 |
by blast
|
berghofe@12905
|
379 |
|
berghofe@12905
|
380 |
lemma Domain_insert: "Domain (insert (a, b) r) = insert a (Domain r)"
|
nipkow@26271
|
381 |
by blast
|
berghofe@12905
|
382 |
|
berghofe@12905
|
383 |
lemma Domain_Id [simp]: "Domain Id = UNIV"
|
nipkow@26271
|
384 |
by blast
|
berghofe@12905
|
385 |
|
nipkow@30198
|
386 |
lemma Domain_Id_on [simp]: "Domain (Id_on A) = A"
|
nipkow@26271
|
387 |
by blast
|
berghofe@12905
|
388 |
|
paulson@13830
|
389 |
lemma Domain_Un_eq: "Domain(A \<union> B) = Domain(A) \<union> Domain(B)"
|
nipkow@26271
|
390 |
by blast
|
berghofe@12905
|
391 |
|
paulson@13830
|
392 |
lemma Domain_Int_subset: "Domain(A \<inter> B) \<subseteq> Domain(A) \<inter> Domain(B)"
|
nipkow@26271
|
393 |
by blast
|
berghofe@12905
|
394 |
|
wenzelm@12913
|
395 |
lemma Domain_Diff_subset: "Domain(A) - Domain(B) \<subseteq> Domain(A - B)"
|
nipkow@26271
|
396 |
by blast
|
berghofe@12905
|
397 |
|
paulson@13830
|
398 |
lemma Domain_Union: "Domain (Union S) = (\<Union>A\<in>S. Domain A)"
|
nipkow@26271
|
399 |
by blast
|
nipkow@26271
|
400 |
|
nipkow@26271
|
401 |
lemma Domain_converse[simp]: "Domain(r^-1) = Range r"
|
nipkow@26271
|
402 |
by(auto simp:Range_def)
|
berghofe@12905
|
403 |
|
wenzelm@12913
|
404 |
lemma Domain_mono: "r \<subseteq> s ==> Domain r \<subseteq> Domain s"
|
nipkow@26271
|
405 |
by blast
|
berghofe@12905
|
406 |
|
paulson@22172
|
407 |
lemma fst_eq_Domain: "fst ` R = Domain R";
|
nipkow@26271
|
408 |
by (auto intro!:image_eqI)
|
paulson@22172
|
409 |
|
haftmann@29609
|
410 |
lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)"
|
haftmann@29609
|
411 |
by auto
|
haftmann@29609
|
412 |
|
haftmann@29609
|
413 |
lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)"
|
haftmann@29609
|
414 |
by auto
|
haftmann@29609
|
415 |
|
berghofe@12905
|
416 |
|
berghofe@12905
|
417 |
subsection {* Range *}
|
berghofe@12905
|
418 |
|
berghofe@12905
|
419 |
lemma Range_iff: "(a : Range r) = (EX y. (y, a) : r)"
|
nipkow@26271
|
420 |
by (simp add: Domain_def Range_def)
|
berghofe@12905
|
421 |
|
berghofe@12905
|
422 |
lemma RangeI [intro]: "(a, b) : r ==> b : Range r"
|
nipkow@26271
|
423 |
by (unfold Range_def) (iprover intro!: converseI DomainI)
|
berghofe@12905
|
424 |
|
berghofe@12905
|
425 |
lemma RangeE [elim!]: "b : Range r ==> (!!x. (x, b) : r ==> P) ==> P"
|
nipkow@26271
|
426 |
by (unfold Range_def) (iprover elim!: DomainE dest!: converseD)
|
berghofe@12905
|
427 |
|
berghofe@12905
|
428 |
lemma Range_empty [simp]: "Range {} = {}"
|
nipkow@26271
|
429 |
by blast
|
berghofe@12905
|
430 |
|
berghofe@12905
|
431 |
lemma Range_insert: "Range (insert (a, b) r) = insert b (Range r)"
|
nipkow@26271
|
432 |
by blast
|
berghofe@12905
|
433 |
|
berghofe@12905
|
434 |
lemma Range_Id [simp]: "Range Id = UNIV"
|
nipkow@26271
|
435 |
by blast
|
berghofe@12905
|
436 |
|
nipkow@30198
|
437 |
lemma Range_Id_on [simp]: "Range (Id_on A) = A"
|
nipkow@26271
|
438 |
by auto
|
berghofe@12905
|
439 |
|
paulson@13830
|
440 |
lemma Range_Un_eq: "Range(A \<union> B) = Range(A) \<union> Range(B)"
|
nipkow@26271
|
441 |
by blast
|
berghofe@12905
|
442 |
|
paulson@13830
|
443 |
lemma Range_Int_subset: "Range(A \<inter> B) \<subseteq> Range(A) \<inter> Range(B)"
|
nipkow@26271
|
444 |
by blast
|
berghofe@12905
|
445 |
|
wenzelm@12913
|
446 |
lemma Range_Diff_subset: "Range(A) - Range(B) \<subseteq> Range(A - B)"
|
nipkow@26271
|
447 |
by blast
|
berghofe@12905
|
448 |
|
paulson@13830
|
449 |
lemma Range_Union: "Range (Union S) = (\<Union>A\<in>S. Range A)"
|
nipkow@26271
|
450 |
by blast
|
nipkow@26271
|
451 |
|
nipkow@26271
|
452 |
lemma Range_converse[simp]: "Range(r^-1) = Domain r"
|
nipkow@26271
|
453 |
by blast
|
berghofe@12905
|
454 |
|
paulson@22172
|
455 |
lemma snd_eq_Range: "snd ` R = Range R";
|
nipkow@26271
|
456 |
by (auto intro!:image_eqI)
|
nipkow@26271
|
457 |
|
nipkow@26271
|
458 |
|
nipkow@26271
|
459 |
subsection {* Field *}
|
nipkow@26271
|
460 |
|
nipkow@26271
|
461 |
lemma mono_Field: "r \<subseteq> s \<Longrightarrow> Field r \<subseteq> Field s"
|
nipkow@26271
|
462 |
by(auto simp:Field_def Domain_def Range_def)
|
nipkow@26271
|
463 |
|
nipkow@26271
|
464 |
lemma Field_empty[simp]: "Field {} = {}"
|
nipkow@26271
|
465 |
by(auto simp:Field_def)
|
nipkow@26271
|
466 |
|
nipkow@26271
|
467 |
lemma Field_insert[simp]: "Field (insert (a,b) r) = {a,b} \<union> Field r"
|
nipkow@26271
|
468 |
by(auto simp:Field_def)
|
nipkow@26271
|
469 |
|
nipkow@26271
|
470 |
lemma Field_Un[simp]: "Field (r \<union> s) = Field r \<union> Field s"
|
nipkow@26271
|
471 |
by(auto simp:Field_def)
|
nipkow@26271
|
472 |
|
nipkow@26271
|
473 |
lemma Field_Union[simp]: "Field (\<Union>R) = \<Union>(Field ` R)"
|
nipkow@26271
|
474 |
by(auto simp:Field_def)
|
nipkow@26271
|
475 |
|
nipkow@26271
|
476 |
lemma Field_converse[simp]: "Field(r^-1) = Field r"
|
nipkow@26271
|
477 |
by(auto simp:Field_def)
|
paulson@22172
|
478 |
|
berghofe@12905
|
479 |
|
berghofe@12905
|
480 |
subsection {* Image of a set under a relation *}
|
berghofe@12905
|
481 |
|
paulson@24286
|
482 |
declare Image_def [noatp]
|
paulson@24286
|
483 |
|
wenzelm@12913
|
484 |
lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)"
|
nipkow@26271
|
485 |
by (simp add: Image_def)
|
berghofe@12905
|
486 |
|
wenzelm@12913
|
487 |
lemma Image_singleton: "r``{a} = {b. (a, b) : r}"
|
nipkow@26271
|
488 |
by (simp add: Image_def)
|
berghofe@12905
|
489 |
|
wenzelm@12913
|
490 |
lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)"
|
nipkow@26271
|
491 |
by (rule Image_iff [THEN trans]) simp
|
berghofe@12905
|
492 |
|
paulson@24286
|
493 |
lemma ImageI [intro,noatp]: "(a, b) : r ==> a : A ==> b : r``A"
|
nipkow@26271
|
494 |
by (unfold Image_def) blast
|
berghofe@12905
|
495 |
|
berghofe@12905
|
496 |
lemma ImageE [elim!]:
|
wenzelm@12913
|
497 |
"b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P"
|
nipkow@26271
|
498 |
by (unfold Image_def) (iprover elim!: CollectE bexE)
|
berghofe@12905
|
499 |
|
berghofe@12905
|
500 |
lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A"
|
berghofe@12905
|
501 |
-- {* This version's more effective when we already have the required @{text a} *}
|
nipkow@26271
|
502 |
by blast
|
berghofe@12905
|
503 |
|
berghofe@12905
|
504 |
lemma Image_empty [simp]: "R``{} = {}"
|
nipkow@26271
|
505 |
by blast
|
berghofe@12905
|
506 |
|
berghofe@12905
|
507 |
lemma Image_Id [simp]: "Id `` A = A"
|
nipkow@26271
|
508 |
by blast
|
berghofe@12905
|
509 |
|
nipkow@30198
|
510 |
lemma Image_Id_on [simp]: "Id_on A `` B = A \<inter> B"
|
nipkow@26271
|
511 |
by blast
|
berghofe@12905
|
512 |
|
paulson@13830
|
513 |
lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
|
nipkow@26271
|
514 |
by blast
|
berghofe@12905
|
515 |
|
paulson@13830
|
516 |
lemma Image_Int_eq:
|
paulson@13830
|
517 |
"single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B"
|
nipkow@26271
|
518 |
by (simp add: single_valued_def, blast)
|
paulson@13830
|
519 |
|
paulson@13830
|
520 |
lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"
|
nipkow@26271
|
521 |
by blast
|
berghofe@12905
|
522 |
|
paulson@13812
|
523 |
lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A"
|
nipkow@26271
|
524 |
by blast
|
paulson@13812
|
525 |
|
wenzelm@12913
|
526 |
lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B"
|
nipkow@26271
|
527 |
by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
|
berghofe@12905
|
528 |
|
paulson@13830
|
529 |
lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})"
|
berghofe@12905
|
530 |
-- {* NOT suitable for rewriting *}
|
nipkow@26271
|
531 |
by blast
|
berghofe@12905
|
532 |
|
wenzelm@12913
|
533 |
lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)"
|
nipkow@26271
|
534 |
by blast
|
berghofe@12905
|
535 |
|
paulson@13830
|
536 |
lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))"
|
nipkow@26271
|
537 |
by blast
|
berghofe@12905
|
538 |
|
paulson@13830
|
539 |
lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))"
|
nipkow@26271
|
540 |
by blast
|
berghofe@12905
|
541 |
|
paulson@13830
|
542 |
text{*Converse inclusion requires some assumptions*}
|
paulson@13830
|
543 |
lemma Image_INT_eq:
|
paulson@13830
|
544 |
"[|single_valued (r\<inverse>); A\<noteq>{}|] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)"
|
paulson@13830
|
545 |
apply (rule equalityI)
|
paulson@13830
|
546 |
apply (rule Image_INT_subset)
|
paulson@13830
|
547 |
apply (simp add: single_valued_def, blast)
|
paulson@13830
|
548 |
done
|
paulson@13830
|
549 |
|
wenzelm@12913
|
550 |
lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))"
|
nipkow@26271
|
551 |
by blast
|
berghofe@12905
|
552 |
|
berghofe@12905
|
553 |
|
wenzelm@12913
|
554 |
subsection {* Single valued relations *}
|
wenzelm@12913
|
555 |
|
wenzelm@12913
|
556 |
lemma single_valuedI:
|
berghofe@12905
|
557 |
"ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r"
|
nipkow@26271
|
558 |
by (unfold single_valued_def)
|
berghofe@12905
|
559 |
|
berghofe@12905
|
560 |
lemma single_valuedD:
|
berghofe@12905
|
561 |
"single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z"
|
nipkow@26271
|
562 |
by (simp add: single_valued_def)
|
berghofe@12905
|
563 |
|
huffman@19228
|
564 |
lemma single_valued_rel_comp:
|
huffman@19228
|
565 |
"single_valued r ==> single_valued s ==> single_valued (r O s)"
|
nipkow@26271
|
566 |
by (unfold single_valued_def) blast
|
huffman@19228
|
567 |
|
huffman@19228
|
568 |
lemma single_valued_subset:
|
huffman@19228
|
569 |
"r \<subseteq> s ==> single_valued s ==> single_valued r"
|
nipkow@26271
|
570 |
by (unfold single_valued_def) blast
|
huffman@19228
|
571 |
|
huffman@19228
|
572 |
lemma single_valued_Id [simp]: "single_valued Id"
|
nipkow@26271
|
573 |
by (unfold single_valued_def) blast
|
huffman@19228
|
574 |
|
nipkow@30198
|
575 |
lemma single_valued_Id_on [simp]: "single_valued (Id_on A)"
|
nipkow@26271
|
576 |
by (unfold single_valued_def) blast
|
huffman@19228
|
577 |
|
berghofe@12905
|
578 |
|
berghofe@12905
|
579 |
subsection {* Graphs given by @{text Collect} *}
|
berghofe@12905
|
580 |
|
berghofe@12905
|
581 |
lemma Domain_Collect_split [simp]: "Domain{(x,y). P x y} = {x. EX y. P x y}"
|
nipkow@26271
|
582 |
by auto
|
berghofe@12905
|
583 |
|
berghofe@12905
|
584 |
lemma Range_Collect_split [simp]: "Range{(x,y). P x y} = {y. EX x. P x y}"
|
nipkow@26271
|
585 |
by auto
|
berghofe@12905
|
586 |
|
berghofe@12905
|
587 |
lemma Image_Collect_split [simp]: "{(x,y). P x y} `` A = {y. EX x:A. P x y}"
|
nipkow@26271
|
588 |
by auto
|
berghofe@12905
|
589 |
|
berghofe@12905
|
590 |
|
wenzelm@12913
|
591 |
subsection {* Inverse image *}
|
berghofe@12905
|
592 |
|
huffman@19228
|
593 |
lemma sym_inv_image: "sym r ==> sym (inv_image r f)"
|
nipkow@26271
|
594 |
by (unfold sym_def inv_image_def) blast
|
huffman@19228
|
595 |
|
wenzelm@12913
|
596 |
lemma trans_inv_image: "trans r ==> trans (inv_image r f)"
|
berghofe@12905
|
597 |
apply (unfold trans_def inv_image_def)
|
berghofe@12905
|
598 |
apply (simp (no_asm))
|
berghofe@12905
|
599 |
apply blast
|
berghofe@12905
|
600 |
done
|
berghofe@12905
|
601 |
|
haftmann@23709
|
602 |
|
haftmann@29609
|
603 |
subsection {* Finiteness *}
|
haftmann@29609
|
604 |
|
haftmann@29609
|
605 |
lemma finite_converse [iff]: "finite (r^-1) = finite r"
|
haftmann@29609
|
606 |
apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
|
haftmann@29609
|
607 |
apply simp
|
haftmann@29609
|
608 |
apply (rule iffI)
|
haftmann@29609
|
609 |
apply (erule finite_imageD [unfolded inj_on_def])
|
haftmann@29609
|
610 |
apply (simp split add: split_split)
|
haftmann@29609
|
611 |
apply (erule finite_imageI)
|
haftmann@29609
|
612 |
apply (simp add: converse_def image_def, auto)
|
haftmann@29609
|
613 |
apply (rule bexI)
|
haftmann@29609
|
614 |
prefer 2 apply assumption
|
haftmann@29609
|
615 |
apply simp
|
haftmann@29609
|
616 |
done
|
haftmann@29609
|
617 |
|
haftmann@29609
|
618 |
text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi
|
haftmann@29609
|
619 |
Ehmety) *}
|
haftmann@29609
|
620 |
|
haftmann@29609
|
621 |
lemma finite_Field: "finite r ==> finite (Field r)"
|
haftmann@29609
|
622 |
-- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
|
haftmann@29609
|
623 |
apply (induct set: finite)
|
haftmann@29609
|
624 |
apply (auto simp add: Field_def Domain_insert Range_insert)
|
haftmann@29609
|
625 |
done
|
haftmann@29609
|
626 |
|
haftmann@29609
|
627 |
|
haftmann@23709
|
628 |
subsection {* Version of @{text lfp_induct} for binary relations *}
|
haftmann@23709
|
629 |
|
haftmann@23709
|
630 |
lemmas lfp_induct2 =
|
haftmann@23709
|
631 |
lfp_induct_set [of "(a, b)", split_format (complete)]
|
haftmann@23709
|
632 |
|
nipkow@1128
|
633 |
end
|