author paulson Fri Sep 27 10:36:21 2002 +0200 (2002-09-27) changeset 13593 e39f0751e4bf parent 13592 dfe0c7191125 child 13594 c2ee8f5a5652
Tidied. New Pi-theorem.
```     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
```