src/HOL/Library/FuncSet.thy
changeset 21404 eb85850d3eb7
parent 21210 c17fd2df4e9e
child 26106 be52145f482d
     1.1 --- a/src/HOL/Library/FuncSet.thy	Fri Nov 17 02:19:55 2006 +0100
     1.2 +++ b/src/HOL/Library/FuncSet.thy	Fri Nov 17 02:20:03 2006 +0100
     1.3 @@ -10,17 +10,20 @@
     1.4  begin
     1.5  
     1.6  definition
     1.7 -  Pi :: "['a set, 'a => 'b set] => ('a => 'b) set"
     1.8 +  Pi :: "['a set, 'a => 'b set] => ('a => 'b) set" where
     1.9    "Pi A B = {f. \<forall>x. x \<in> A --> f x \<in> B x}"
    1.10  
    1.11 -  extensional :: "'a set => ('a => 'b) set"
    1.12 +definition
    1.13 +  extensional :: "'a set => ('a => 'b) set" where
    1.14    "extensional A = {f. \<forall>x. x~:A --> f x = arbitrary}"
    1.15  
    1.16 -  "restrict" :: "['a => 'b, 'a set] => ('a => 'b)"
    1.17 +definition
    1.18 +  "restrict" :: "['a => 'b, 'a set] => ('a => 'b)" where
    1.19    "restrict f A = (%x. if x \<in> A then f x else arbitrary)"
    1.20  
    1.21  abbreviation
    1.22 -  funcset :: "['a set, 'b set] => ('a => 'b) set"      (infixr "->" 60)
    1.23 +  funcset :: "['a set, 'b set] => ('a => 'b) set"
    1.24 +    (infixr "->" 60) where
    1.25    "A -> B == Pi A (%_. B)"
    1.26  
    1.27  notation (xsymbols)
    1.28 @@ -43,7 +46,7 @@
    1.29    "%x:A. f" == "CONST restrict (%x. f) A"
    1.30  
    1.31  definition
    1.32 -  "compose" :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)"
    1.33 +  "compose" :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)" where
    1.34    "compose A g f = (\<lambda>x\<in>A. g (f x))"
    1.35  
    1.36  
    1.37 @@ -142,7 +145,7 @@
    1.38  the theorems belong here, or need at least @{term Hilbert_Choice}.*}
    1.39  
    1.40  definition
    1.41 -  bij_betw :: "['a => 'b, 'a set, 'b set] => bool"         -- {* bijective *}
    1.42 +  bij_betw :: "['a => 'b, 'a set, 'b set] => bool" where -- {* bijective *}
    1.43    "bij_betw f A B = (inj_on f A & f ` A = B)"
    1.44  
    1.45  lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"