Inv -> inv_onto, inv abbr. inv_onto UNIV.
(* Title: HOL/Library/FuncSet.thy
Author: Florian Kammueller and Lawrence C Paulson
*)
header {* Pi and Function Sets *}
theory FuncSet
imports Hilbert_Choice Main
begin
definition
Pi :: "['a set, 'a => 'b set] => ('a => 'b) set" where
"Pi A B = {f. \<forall>x. x \<in> A --> f x \<in> B x}"
definition
extensional :: "'a set => ('a => 'b) set" where
"extensional A = {f. \<forall>x. x~:A --> f x = undefined}"
definition
"restrict" :: "['a => 'b, 'a set] => ('a => 'b)" where
"restrict f A = (%x. if x \<in> A then f x else undefined)"
abbreviation
funcset :: "['a set, 'b set] => ('a => 'b) set"
(infixr "->" 60) where
"A -> B == Pi A (%_. B)"
notation (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" == "CONST Pi A (%x. B)"
"%x:A. f" == "CONST restrict (%x. f) A"
definition
"compose" :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)" where
"compose A g f = (\<lambda>x\<in>A. g (f x))"
subsection{*Basic Properties of @{term Pi}*}
lemma Pi_I[intro!]: "(!!x. x \<in> A ==> f x \<in> B x) ==> f \<in> Pi A B"
by (simp add: Pi_def)
lemma Pi_I'[simp]: "(!!x. x : A --> f x : B x) ==> f : Pi A B"
by(simp add:Pi_def)
lemma funcsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f \<in> A -> B"
by (simp add: Pi_def)
lemma Pi_mem: "[|f: Pi A B; x \<in> A|] ==> f x \<in> B x"
by (simp add: Pi_def)
lemma PiE [elim]:
"f : Pi A B ==> (f x : B x ==> Q) ==> (x ~: A ==> Q) ==> Q"
by(auto simp: Pi_def)
lemma funcset_id [simp]: "(\<lambda>x. x) \<in> A \<rightarrow> A"
by (auto intro: Pi_I)
lemma funcset_mem: "[|f \<in> A -> B; x \<in> A|] ==> f x \<in> B"
by (simp add: Pi_def)
lemma funcset_image: "f \<in> A\<rightarrow>B ==> f ` A \<subseteq> B"
by auto
lemma Pi_eq_empty[simp]: "((PI x: A. B x) = {}) = (\<exists>x\<in>A. B(x) = {})"
apply (simp add: Pi_def, auto)
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"
by (simp add: Pi_def)
lemma Pi_UNIV [simp]: "A -> UNIV = UNIV"
by (simp add: Pi_def)
(*
lemma funcset_id [simp]: "(%x. x): A -> A"
by (simp add: Pi_def)
*)
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"
by auto
text{*Contravariance of Pi-sets in their first argument*}
lemma Pi_anti_mono: "A' <= A ==> Pi A B <= Pi A' B"
by auto
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))"
by (simp add: compose_def restrict_def)
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"
by (simp add: Pi_def restrict_def)
lemma restrictI[intro!]: "(!!x. x \<in> A ==> f x \<in> B x) ==> (\<lambda>x\<in>A. f x) \<in> Pi A B"
by (simp add: Pi_def restrict_def)
lemma restrict_apply [simp]:
"(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else undefined)"
by (simp add: restrict_def)
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 restrict_def)
lemma inj_on_restrict_eq [simp]: "inj_on (restrict f A) A = inj_on f A"
by (simp add: inj_on_def restrict_def)
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"
by (auto simp add: restrict_def)
subsection{*Bijections Between Sets*}
text{*The definition of @{const bij_betw} is in @{text "Fun.thy"}, but most of
the theorems belong here, or need at least @{term Hilbert_Choice}.*}
lemma bij_betw_imp_funcset: "bij_betw f A B \<Longrightarrow> f \<in> A \<rightarrow> B"
by (auto simp add: bij_betw_def)
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"
by (simp add: bij_betw_def)
subsection{*Extensionality*}
lemma extensional_arb: "[|f \<in> extensional A; x\<notin> A|] ==> f x = undefined"
by (simp add: extensional_def)
lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
by (simp add: restrict_def extensional_def)
lemma compose_extensional [simp]: "compose A f g \<in> extensional A"
by (simp add: compose_def)
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_onto_funcset: "f ` A = B ==> (\<lambda>x\<in>B. inv_onto A f x) : B -> A"
by (unfold inv_onto_def) (fast intro: someI2)
lemma compose_inv_onto_id:
"bij_betw f A B ==> compose A (\<lambda>y\<in>B. inv_onto A f y) f = (\<lambda>x\<in>A. x)"
apply (simp add: bij_betw_def compose_def)
apply (rule restrict_ext, auto)
done
lemma compose_id_inv_onto:
"f ` A = B ==> compose B f (\<lambda>y\<in>B. inv_onto A f y) = (\<lambda>x\<in>B. x)"
apply (simp add: compose_def)
apply (rule restrict_ext)
apply (simp add: f_inv_onto_f)
done
subsection{*Cardinality*}
lemma card_inj: "[|f \<in> A\<rightarrow>B; inj_on f A; finite B|] ==> card(A) \<le> card(B)"
by (rule card_inj_on_le) auto
lemma card_bij:
"[|f \<in> A\<rightarrow>B; inj_on f A;
g \<in> B\<rightarrow>A; inj_on g B; finite A; finite B|] ==> card(A) = card(B)"
by (blast intro: card_inj order_antisym)
end