src/HOL/Library/FuncSet.thy
changeset 13593 e39f0751e4bf
parent 13586 0f339348df0e
child 13595 7e6cdcd113a2
     1.1 --- a/src/HOL/Library/FuncSet.thy	Fri Sep 27 10:35:10 2002 +0200
     1.2 +++ b/src/HOL/Library/FuncSet.thy	Fri Sep 27 10:36:21 2002 +0200
     1.3 @@ -50,26 +50,24 @@
     1.4  by (simp add: Pi_def)
     1.5  
     1.6  lemma Pi_mem: "[|f: Pi A B; x \<in> A|] ==> f x \<in> B x"
     1.7 -apply (simp add: Pi_def)
     1.8 -done
     1.9 +by (simp add: Pi_def)
    1.10  
    1.11  lemma funcset_mem: "[|f \<in> A -> B; x \<in> A|] ==> f x \<in> B"
    1.12  by (simp add: Pi_def)
    1.13  
    1.14  lemma Pi_eq_empty: "((PI x: A. B x) = {}) = (\<exists>x\<in>A. B(x) = {})"
    1.15 -apply (simp add: Pi_def)
    1.16 -apply auto
    1.17 +apply (simp add: Pi_def, auto)
    1.18  txt{*Converse direction requires Axiom of Choice to exhibit a function
    1.19  picking an element from each non-empty @{term "B x"}*}
    1.20 -apply (drule_tac x = "%u. SOME y. y \<in> B u" in spec) 
    1.21 -apply (auto );
    1.22 -apply (cut_tac P= "%y. y \<in> B x" in some_eq_ex)
    1.23 -apply (auto ); 
    1.24 +apply (drule_tac x = "%u. SOME y. y \<in> B u" in spec, auto)
    1.25 +apply (cut_tac P= "%y. y \<in> B x" in some_eq_ex, auto) 
    1.26  done
    1.27  
    1.28 -lemma Pi_empty: "Pi {} B = UNIV"
    1.29 -apply (simp add: Pi_def) 
    1.30 -done
    1.31 +lemma Pi_empty [simp]: "Pi {} B = UNIV"
    1.32 +by (simp add: Pi_def)
    1.33 +
    1.34 +lemma Pi_UNIV [simp]: "A -> UNIV = UNIV"
    1.35 +by (simp add: Pi_def)
    1.36  
    1.37  text{*Covariance of Pi-sets in their second argument*}
    1.38  lemma Pi_mono: "(!!x. x \<in> A ==> B x <= C x) ==> Pi A B <= Pi A C"
    1.39 @@ -92,12 +90,10 @@
    1.40  by (simp add: expand_fun_eq Pi_def compose_def restrict_def) 
    1.41  
    1.42  lemma compose_eq: "x \<in> A ==> compose A g f x = g(f(x))"
    1.43 -apply (simp add: compose_def restrict_def)
    1.44 -done
    1.45 +by (simp add: compose_def restrict_def)
    1.46  
    1.47  lemma surj_compose: "[| f ` A = B; g ` B = C |] ==> compose A g f ` A = C"
    1.48 -apply (auto simp add: image_def compose_eq)
    1.49 -done
    1.50 +by (auto simp add: image_def compose_eq)
    1.51  
    1.52  lemma inj_on_compose:
    1.53       "[| f ` A = B; inj_on f A; inj_on g B |] ==> inj_on (compose A g f) A"
    1.54 @@ -122,8 +118,7 @@
    1.55  by (simp add: expand_fun_eq Pi_def Pi_def restrict_def)
    1.56  
    1.57  lemma inj_on_restrict_eq: "inj_on (restrict f A) A = inj_on f A"
    1.58 -apply (simp add: inj_on_def restrict_def)
    1.59 -done
    1.60 +by (simp add: inj_on_def restrict_def)
    1.61  
    1.62  
    1.63  lemma Id_compose:
    1.64 @@ -138,8 +133,7 @@
    1.65  subsection{*Extensionality*}
    1.66  
    1.67  lemma extensional_arb: "[|f \<in> extensional A; x\<notin> A|] ==> f x = arbitrary"
    1.68 -apply (simp add: extensional_def)
    1.69 -done
    1.70 +by (simp add: extensional_def)
    1.71  
    1.72  lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
    1.73  by (simp add: restrict_def extensional_def)
    1.74 @@ -161,8 +155,7 @@
    1.75       "[| inj_on f A;  f ` A = B |]  
    1.76        ==> compose A (\<lambda>y\<in>B. Inv A f y) f = (\<lambda>x\<in>A. x)"
    1.77  apply (simp add: compose_def)
    1.78 -apply (rule restrict_ext)
    1.79 -apply auto
    1.80 +apply (rule restrict_ext, auto)
    1.81  apply (erule subst)
    1.82  apply (simp add: Inv_f_f)
    1.83  done