src/HOL/Library/FuncSet.thy
 author wenzelm Tue, 16 May 2006 21:33:01 +0200 changeset 19656 09be06943252 parent 19536 1a3a3cf8b4fa child 19736 d8d0f8f51d69 permissions -rw-r--r--
tuned concrete syntax -- abbreviation/const_syntax;
```
(*  Title:      HOL/Library/FuncSet.thy
ID:         \$Id\$
Author:     Florian Kammueller and Lawrence C Paulson
*)

header {* Pi and Function Sets *}

theory FuncSet
imports Main
begin

constdefs
Pi :: "['a set, 'a => 'b set] => ('a => 'b) set"
"Pi A B == {f. \<forall>x. x \<in> A --> f x \<in> B x}"

extensional :: "'a set => ('a => 'b) set"
"extensional A == {f. \<forall>x. x~:A --> f x = arbitrary}"

"restrict" :: "['a => 'b, 'a set] => ('a => 'b)"
"restrict f A == (%x. if x \<in> A then f x else arbitrary)"

abbreviation
funcset :: "['a set, 'b set] => ('a => 'b) set"      (infixr "->" 60)
"A -> B == Pi A (%_. B)"

const_syntax (xsymbols)
funcset  (infixr "\<rightarrow>" 60)

syntax
"@Pi"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PI _:_./ _)" 10)
"@lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3%_:_./ _)" [0,0,3] 3)

syntax (xsymbols)
"@Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
"@lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)

syntax (HTML output)
"@Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
"@lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)

translations
"PI x:A. B" == "Pi A (%x. B)"
"%x:A. f" == "restrict (%x. f) A"

constdefs
"compose" :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)"
"compose A g f == \<lambda>x\<in>A. g (f x)"

subsection{*Basic Properties of @{term Pi}*}

lemma Pi_I: "(!!x. x \<in> A ==> f x \<in> B x) ==> f \<in> Pi A B"

lemma funcsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f \<in> A -> B"

lemma Pi_mem: "[|f: Pi A B; x \<in> A|] ==> f x \<in> B x"

lemma funcset_mem: "[|f \<in> A -> B; x \<in> A|] ==> f x \<in> B"

lemma funcset_image: "f \<in> A\<rightarrow>B ==> f ` A \<subseteq> B"

lemma Pi_eq_empty: "((PI x: A. B x) = {}) = (\<exists>x\<in>A. B(x) = {})"
txt{*Converse direction requires Axiom of Choice to exhibit a function
picking an element from each non-empty @{term "B x"}*}
apply (drule_tac x = "%u. SOME y. y \<in> B u" in spec, auto)
apply (cut_tac P= "%y. y \<in> B x" in some_eq_ex, auto)
done

lemma Pi_empty [simp]: "Pi {} B = UNIV"

lemma Pi_UNIV [simp]: "A -> UNIV = UNIV"

text{*Covariance of Pi-sets in their second argument*}
lemma Pi_mono: "(!!x. x \<in> A ==> B x <= C x) ==> Pi A B <= Pi A C"

text{*Contravariance of Pi-sets in their first argument*}
lemma Pi_anti_mono: "A' <= A ==> Pi A B <= Pi A' B"

subsection{*Composition With a Restricted Domain: @{term compose}*}

lemma funcset_compose:
"[| f \<in> A -> B; g \<in> B -> C |]==> compose A g f \<in> A -> C"
by (simp add: Pi_def compose_def restrict_def)

lemma compose_assoc:
"[| f \<in> A -> B; g \<in> B -> C; h \<in> C -> D |]
==> compose A h (compose A g f) = compose A (compose B h g) f"
by (simp add: expand_fun_eq Pi_def compose_def restrict_def)

lemma compose_eq: "x \<in> A ==> compose A g f x = g(f(x))"

lemma surj_compose: "[| f ` A = B; g ` B = C |] ==> compose A g f ` A = C"
by (auto simp add: image_def compose_eq)

subsection{*Bounded Abstraction: @{term restrict}*}

lemma restrict_in_funcset: "(!!x. x \<in> A ==> f x \<in> B) ==> (\<lambda>x\<in>A. f x) \<in> A -> B"

lemma restrictI: "(!!x. x \<in> A ==> f x \<in> B x) ==> (\<lambda>x\<in>A. f x) \<in> Pi A B"

lemma restrict_apply [simp]:
"(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else arbitrary)"

lemma restrict_ext:
"(!!x. x \<in> A ==> f x = g x) ==> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
by (simp add: expand_fun_eq Pi_def Pi_def restrict_def)

lemma inj_on_restrict_eq [simp]: "inj_on (restrict f A) A = inj_on f A"

lemma Id_compose:
"[|f \<in> A -> B;  f \<in> extensional A|] ==> compose A (\<lambda>y\<in>B. y) f = f"
by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)

lemma compose_Id:
"[|g \<in> A -> B;  g \<in> extensional A|] ==> compose A g (\<lambda>x\<in>A. x) = g"
by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)

lemma image_restrict_eq [simp]: "(restrict f A) ` A = f ` A"

subsection{*Bijections Between Sets*}

text{*The basic definition could be moved to @{text "Fun.thy"}, but most of
the theorems belong here, or need at least @{term Hilbert_Choice}.*}

constdefs
bij_betw :: "['a => 'b, 'a set, 'b set] => bool"         (*bijective*)
"bij_betw f A B == inj_on f A & f ` A = B"

lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"

lemma bij_betw_imp_funcset: "bij_betw f A B \<Longrightarrow> f \<in> A \<rightarrow> B"
by (auto simp add: bij_betw_def inj_on_Inv Pi_def)

lemma bij_betw_Inv: "bij_betw f A B \<Longrightarrow> bij_betw (Inv A f) B A"
apply (auto simp add: bij_betw_def inj_on_Inv Inv_mem)
apply (simp add: image_compose [symmetric] o_def)
done

lemma inj_on_compose:
"[| bij_betw f A B; inj_on g B |] ==> inj_on (compose A g f) A"
by (auto simp add: bij_betw_def inj_on_def compose_eq)

lemma bij_betw_compose:
"[| bij_betw f A B; bij_betw g B C |] ==> bij_betw (compose A g f) A C"
apply (simp add: bij_betw_def compose_eq inj_on_compose)
apply (auto simp add: compose_def image_def)
done

lemma bij_betw_restrict_eq [simp]:
"bij_betw (restrict f A) A B = bij_betw f A B"

subsection{*Extensionality*}

lemma extensional_arb: "[|f \<in> extensional A; x\<notin> A|] ==> f x = arbitrary"

lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"

lemma compose_extensional [simp]: "compose A f g \<in> extensional A"

lemma extensionalityI:
"[| f \<in> extensional A; g \<in> extensional A;
!!x. x\<in>A ==> f x = g x |] ==> f = g"
by (force simp add: expand_fun_eq extensional_def)

lemma Inv_funcset: "f ` A = B ==> (\<lambda>x\<in>B. Inv A f x) : B -> A"
by (unfold Inv_def) (fast intro: restrict_in_funcset someI2)

lemma compose_Inv_id:
"bij_betw f A B ==> compose A (\<lambda>y\<in>B. Inv A f y) f = (\<lambda>x\<in>A. x)"
apply (rule restrict_ext, auto)
apply (erule subst)
done

lemma compose_id_Inv:
"f ` A = B ==> compose B f (\<lambda>y\<in>B. Inv A f y) = (\<lambda>x\<in>B. x)"
apply (rule restrict_ext)
done

subsection{*Cardinality*}

lemma card_inj: "[|f \<in> A\<rightarrow>B; inj_on f A; finite B|] ==> card(A) \<le> card(B)"
apply (rule card_inj_on_le)