src/HOL/Relation.thy
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`     1 (*  Title: 	Relation.thy`
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`     2     ID:         \$Id\$`
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`     3     Author: 	Riccardo Mattolini, Dip. Sistemi e Informatica`
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`     4         and	Lawrence C Paulson, Cambridge University Computer Laboratory`
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`     5     Copyright   1994 Universita' di Firenze`
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`     6     Copyright   1993  University of Cambridge`
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`     7 *)`
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`     8 `
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`     9 Relation = Prod +`
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`    10 consts`
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`    11     id	        :: "('a * 'a)set"               (*the identity relation*)`
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`    12     O	        :: "[('b * 'c)set, ('a * 'b)set] => ('a * 'c)set" (infixr 60)`
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`    13     trans       :: "('a * 'a)set => bool" 	(*transitivity predicate*)`
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`    14     converse    :: "('a*'a) set => ('a*'a) set"`
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`    15     "^^"        :: "[('a*'a) set,'a set] => 'a set" (infixl 90)`
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`    16     Domain      :: "('a*'a) set => 'a set"`
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`    17     Range       :: "('a*'a) set => 'a set"`
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`    18 defs`
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`    19     id_def	"id == {p. ? x. p = (x,x)}"`
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`    20     comp_def	(*composition of relations*)`
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`    21 		"r O s == {xz. ? x y z. xz = (x,z) & (x,y):s & (y,z):r}"`
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`    22     trans_def	  "trans(r) == (!x y z. (x,y):r --> (y,z):r --> (x,z):r)"`
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`    23     converse_def  "converse(r) == {z. (? w:r. ? x y. w=(x,y) & z=(y,x))}"`
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`    24     Domain_def    "Domain(r) == {z. ! x. (z=x --> (? y. (x,y):r))}"`
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`    25     Range_def     "Range(r) == Domain(converse(r))"`
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`    26     Image_def     "r ^^ s == {y. y:Range(r) &  (? x:s. (x,y):r)}"`
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`    27 end`