src/HOL/Library/FuncSet.thy
 author nipkow Thu Feb 21 17:34:09 2008 +0100 (2008-02-21) changeset 26106 be52145f482d parent 21404 eb85850d3eb7 child 27183 0fc4c0f97a1b permissions -rw-r--r--
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas
1 (*  Title:      HOL/Library/FuncSet.thy
2     ID:         \$Id\$
3     Author:     Florian Kammueller and Lawrence C Paulson
4 *)
6 header {* Pi and Function Sets *}
8 theory FuncSet
9 imports Main
10 begin
12 definition
13   Pi :: "['a set, 'a => 'b set] => ('a => 'b) set" where
14   "Pi A B = {f. \<forall>x. x \<in> A --> f x \<in> B x}"
16 definition
17   extensional :: "'a set => ('a => 'b) set" where
18   "extensional A = {f. \<forall>x. x~:A --> f x = arbitrary}"
20 definition
21   "restrict" :: "['a => 'b, 'a set] => ('a => 'b)" where
22   "restrict f A = (%x. if x \<in> A then f x else arbitrary)"
24 abbreviation
25   funcset :: "['a set, 'b set] => ('a => 'b) set"
26     (infixr "->" 60) where
27   "A -> B == Pi A (%_. B)"
29 notation (xsymbols)
30   funcset  (infixr "\<rightarrow>" 60)
32 syntax
33   "_Pi"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PI _:_./ _)" 10)
34   "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3%_:_./ _)" [0,0,3] 3)
36 syntax (xsymbols)
37   "_Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
38   "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
40 syntax (HTML output)
41   "_Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
42   "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
44 translations
45   "PI x:A. B" == "CONST Pi A (%x. B)"
46   "%x:A. f" == "CONST restrict (%x. f) A"
48 definition
49   "compose" :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)" where
50   "compose A g f = (\<lambda>x\<in>A. g (f x))"
53 subsection{*Basic Properties of @{term Pi}*}
55 lemma Pi_I: "(!!x. x \<in> A ==> f x \<in> B x) ==> f \<in> Pi A B"
58 lemma funcsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f \<in> A -> B"
61 lemma Pi_mem: "[|f: Pi A B; x \<in> A|] ==> f x \<in> B x"
64 lemma funcset_mem: "[|f \<in> A -> B; x \<in> A|] ==> f x \<in> B"
67 lemma funcset_image: "f \<in> A\<rightarrow>B ==> f ` A \<subseteq> B"
68   by (auto simp add: Pi_def)
70 lemma Pi_eq_empty: "((PI x: A. B x) = {}) = (\<exists>x\<in>A. B(x) = {})"
71 apply (simp add: Pi_def, auto)
72 txt{*Converse direction requires Axiom of Choice to exhibit a function
73 picking an element from each non-empty @{term "B x"}*}
74 apply (drule_tac x = "%u. SOME y. y \<in> B u" in spec, auto)
75 apply (cut_tac P= "%y. y \<in> B x" in some_eq_ex, auto)
76 done
78 lemma Pi_empty [simp]: "Pi {} B = UNIV"
81 lemma Pi_UNIV [simp]: "A -> UNIV = UNIV"
84 text{*Covariance of Pi-sets in their second argument*}
85 lemma Pi_mono: "(!!x. x \<in> A ==> B x <= C x) ==> Pi A B <= Pi A C"
86   by (simp add: Pi_def, blast)
88 text{*Contravariance of Pi-sets in their first argument*}
89 lemma Pi_anti_mono: "A' <= A ==> Pi A B <= Pi A' B"
90   by (simp add: Pi_def, blast)
93 subsection{*Composition With a Restricted Domain: @{term compose}*}
95 lemma funcset_compose:
96     "[| f \<in> A -> B; g \<in> B -> C |]==> compose A g f \<in> A -> C"
97   by (simp add: Pi_def compose_def restrict_def)
99 lemma compose_assoc:
100     "[| f \<in> A -> B; g \<in> B -> C; h \<in> C -> D |]
101       ==> compose A h (compose A g f) = compose A (compose B h g) f"
102   by (simp add: expand_fun_eq Pi_def compose_def restrict_def)
104 lemma compose_eq: "x \<in> A ==> compose A g f x = g(f(x))"
105   by (simp add: compose_def restrict_def)
107 lemma surj_compose: "[| f ` A = B; g ` B = C |] ==> compose A g f ` A = C"
108   by (auto simp add: image_def compose_eq)
111 subsection{*Bounded Abstraction: @{term restrict}*}
113 lemma restrict_in_funcset: "(!!x. x \<in> A ==> f x \<in> B) ==> (\<lambda>x\<in>A. f x) \<in> A -> B"
114   by (simp add: Pi_def restrict_def)
116 lemma restrictI: "(!!x. x \<in> A ==> f x \<in> B x) ==> (\<lambda>x\<in>A. f x) \<in> Pi A B"
117   by (simp add: Pi_def restrict_def)
119 lemma restrict_apply [simp]:
120     "(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else arbitrary)"
123 lemma restrict_ext:
124     "(!!x. x \<in> A ==> f x = g x) ==> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
125   by (simp add: expand_fun_eq Pi_def Pi_def restrict_def)
127 lemma inj_on_restrict_eq [simp]: "inj_on (restrict f A) A = inj_on f A"
128   by (simp add: inj_on_def restrict_def)
130 lemma Id_compose:
131     "[|f \<in> A -> B;  f \<in> extensional A|] ==> compose A (\<lambda>y\<in>B. y) f = f"
132   by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)
134 lemma compose_Id:
135     "[|g \<in> A -> B;  g \<in> extensional A|] ==> compose A g (\<lambda>x\<in>A. x) = g"
136   by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)
138 lemma image_restrict_eq [simp]: "(restrict f A) ` A = f ` A"
139   by (auto simp add: restrict_def)
142 subsection{*Bijections Between Sets*}
144 text{*The definition of @{const bij_betw} is in @{text "Fun.thy"}, but most of
145 the theorems belong here, or need at least @{term Hilbert_Choice}.*}
147 lemma bij_betw_imp_funcset: "bij_betw f A B \<Longrightarrow> f \<in> A \<rightarrow> B"
148   by (auto simp add: bij_betw_def inj_on_Inv Pi_def)
150 lemma inj_on_compose:
151     "[| bij_betw f A B; inj_on g B |] ==> inj_on (compose A g f) A"
152   by (auto simp add: bij_betw_def inj_on_def compose_eq)
154 lemma bij_betw_compose:
155     "[| bij_betw f A B; bij_betw g B C |] ==> bij_betw (compose A g f) A C"
156   apply (simp add: bij_betw_def compose_eq inj_on_compose)
157   apply (auto simp add: compose_def image_def)
158   done
160 lemma bij_betw_restrict_eq [simp]:
161      "bij_betw (restrict f A) A B = bij_betw f A B"
165 subsection{*Extensionality*}
167 lemma extensional_arb: "[|f \<in> extensional A; x\<notin> A|] ==> f x = arbitrary"
170 lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
171   by (simp add: restrict_def extensional_def)
173 lemma compose_extensional [simp]: "compose A f g \<in> extensional A"
176 lemma extensionalityI:
177     "[| f \<in> extensional A; g \<in> extensional A;
178       !!x. x\<in>A ==> f x = g x |] ==> f = g"
179   by (force simp add: expand_fun_eq extensional_def)
181 lemma Inv_funcset: "f ` A = B ==> (\<lambda>x\<in>B. Inv A f x) : B -> A"
182   by (unfold Inv_def) (fast intro: restrict_in_funcset someI2)
184 lemma compose_Inv_id:
185     "bij_betw f A B ==> compose A (\<lambda>y\<in>B. Inv A f y) f = (\<lambda>x\<in>A. x)"
186   apply (simp add: bij_betw_def compose_def)
187   apply (rule restrict_ext, auto)
188   apply (erule subst)
190   done
192 lemma compose_id_Inv:
193     "f ` A = B ==> compose B f (\<lambda>y\<in>B. Inv A f y) = (\<lambda>x\<in>B. x)"
195   apply (rule restrict_ext)
197   done
200 subsection{*Cardinality*}
202 lemma card_inj: "[|f \<in> A\<rightarrow>B; inj_on f A; finite B|] ==> card(A) \<le> card(B)"
203   apply (rule card_inj_on_le)
204     apply (auto simp add: Pi_def)
205   done
207 lemma card_bij:
208      "[|f \<in> A\<rightarrow>B; inj_on f A;
209         g \<in> B\<rightarrow>A; inj_on g B; finite A; finite B|] ==> card(A) = card(B)"
210   by (blast intro: card_inj order_antisym)
213 (*The following declarations generate polymorphic Skolem functions for
214   these theorems. Eventually they should become redundant, once this
215   is done automatically.*)
217 declare FuncSet.Pi_I [skolem]
218 declare FuncSet.Pi_mono [skolem]
219 declare FuncSet.extensionalityI [skolem]
220 declare FuncSet.funcsetI [skolem]
221 declare FuncSet.restrictI [skolem]
222 declare FuncSet.restrict_in_funcset [skolem]
224 end