src/HOL/Relation.thy
author berghofe
Fri, 16 Jul 1999 12:09:48 +0200
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permissions -rw-r--r--
Added some definitions and theorems needed for the construction of datatypes involving function types.
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(*  Title:      Relation.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1996  University of Cambridge
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*)
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Relation = Prod +
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consts
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  O            :: "[('b * 'c)set, ('a * 'b)set] => ('a * 'c)set" (infixr 60)
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  converse     :: "('a*'b) set => ('b*'a) set"     ("(_^-1)" [1000] 999)
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  "^^"         :: "[('a*'b) set,'a set] => 'b set" (infixl 90)
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defs
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  comp_def         "r O s == {(x,z). ? y. (x,y):s & (y,z):r}"
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  converse_def     "r^-1 == {(y,x). (x,y):r}"
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  Image_def        "r ^^ s == {y. ? x:s. (x,y):r}"
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constdefs
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  Id     :: "('a * 'a)set"                 (*the identity relation*)
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      "Id == {p. ? x. p = (x,x)}"
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  diag   :: "'a set => ('a * 'a)set"
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    "diag(A) == UN x:A. {(x,x)}"
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  Domain :: "('a*'b) set => 'a set"
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    "Domain(r) == {x. ? y. (x,y):r}"
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  Range  :: "('a*'b) set => 'b set"
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    "Range(r) == Domain(r^-1)"
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  refl   :: "['a set, ('a*'a) set] => bool" (*reflexivity over a set*)
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    "refl A r == r <= A Times A & (ALL x: A. (x,x) : r)"
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  sym    :: "('a*'a) set=>bool"             (*symmetry predicate*)
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    "sym(r) == ALL x y. (x,y): r --> (y,x): r"
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  antisym:: "('a * 'a)set => bool"          (*antisymmetry predicate*)
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    "antisym(r) == ALL x y. (x,y):r --> (y,x):r --> x=y"
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  trans  :: "('a * 'a)set => bool"          (*transitivity predicate*)
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    "trans(r) == (!x y z. (x,y):r --> (y,z):r --> (x,z):r)"
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  Univalent :: "('a * 'b)set => bool"
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    "Univalent r == !x y. (x,y):r --> (!z. (x,z):r --> y=z)"
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  fun_rel_comp :: "['a => 'b, ('b * 'c) set] => ('a => 'c) set"
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    "fun_rel_comp f R == {g. !x. (f x, g x) : R}"
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syntax
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  reflexive :: "('a * 'a)set => bool"       (*reflexivity over a type*)
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translations
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  "reflexive" == "refl UNIV"
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end