--- a/src/HOL/Algebra/Congruence.thy Tue May 15 06:23:12 2018 +0200
+++ b/src/HOL/Algebra/Congruence.thy Tue May 15 11:33:43 2018 +0200
@@ -5,7 +5,8 @@
theory Congruence
imports
- Main HOL.FuncSet
+ Main
+ "HOL-Library.FuncSet"
begin
section \<open>Objects\<close>
--- a/src/HOL/Algebra/Group.thy Tue May 15 06:23:12 2018 +0200
+++ b/src/HOL/Algebra/Group.thy Tue May 15 11:33:43 2018 +0200
@@ -5,7 +5,7 @@
*)
theory Group
-imports Complete_Lattice HOL.FuncSet
+imports Complete_Lattice "HOL-Library.FuncSet"
begin
section \<open>Monoids and Groups\<close>
--- a/src/HOL/Analysis/Finite_Cartesian_Product.thy Tue May 15 06:23:12 2018 +0200
+++ b/src/HOL/Analysis/Finite_Cartesian_Product.thy Tue May 15 11:33:43 2018 +0200
@@ -10,6 +10,7 @@
L2_Norm
"HOL-Library.Numeral_Type"
"HOL-Library.Countable_Set"
+ "HOL-Library.FuncSet"
begin
subsection \<open>Finite Cartesian products, with indexing and lambdas\<close>
--- a/src/HOL/Analysis/Sigma_Algebra.thy Tue May 15 06:23:12 2018 +0200
+++ b/src/HOL/Analysis/Sigma_Algebra.thy Tue May 15 11:33:43 2018 +0200
@@ -11,6 +11,7 @@
imports
Complex_Main
"HOL-Library.Countable_Set"
+ "HOL-Library.FuncSet"
"HOL-Library.Indicator_Function"
"HOL-Library.Extended_Nonnegative_Real"
"HOL-Library.Disjoint_Sets"
--- a/src/HOL/Analysis/Topology_Euclidean_Space.thy Tue May 15 06:23:12 2018 +0200
+++ b/src/HOL/Analysis/Topology_Euclidean_Space.thy Tue May 15 11:33:43 2018 +0200
@@ -10,6 +10,7 @@
imports
"HOL-Library.Indicator_Function"
"HOL-Library.Countable_Set"
+ "HOL-Library.FuncSet"
Linear_Algebra
Norm_Arith
begin
--- a/src/HOL/FuncSet.thy Tue May 15 06:23:12 2018 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,568 +0,0 @@
-(* Title: HOL/FuncSet.thy
- Author: Florian Kammueller and Lawrence C Paulson, Lukas Bulwahn
-*)
-
-section \<open>Pi and Function Sets\<close>
-
-theory FuncSet
- imports Main
- abbrevs PiE = "Pi\<^sub>E"
- and PIE = "\<Pi>\<^sub>E"
-begin
-
-definition Pi :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<Rightarrow> 'b) set"
- where "Pi A B = {f. \<forall>x. x \<in> A \<longrightarrow> f x \<in> B x}"
-
-definition extensional :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) set"
- where "extensional A = {f. \<forall>x. x \<notin> A \<longrightarrow> f x = undefined}"
-
-definition "restrict" :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
- where "restrict f A = (\<lambda>x. if x \<in> A then f x else undefined)"
-
-abbreviation funcset :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set" (infixr "\<rightarrow>" 60)
- where "A \<rightarrow> B \<equiv> Pi A (\<lambda>_. B)"
-
-syntax
- "_Pi" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set" ("(3\<Pi> _\<in>_./ _)" 10)
- "_lam" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)" ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
-translations
- "\<Pi> x\<in>A. B" \<rightleftharpoons> "CONST Pi A (\<lambda>x. B)"
- "\<lambda>x\<in>A. f" \<rightleftharpoons> "CONST restrict (\<lambda>x. f) A"
-
-definition "compose" :: "'a set \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'c)"
- where "compose A g f = (\<lambda>x\<in>A. g (f x))"
-
-
-subsection \<open>Basic Properties of @{term Pi}\<close>
-
-lemma Pi_I[intro!]: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi A B"
- by (simp add: Pi_def)
-
-lemma Pi_I'[simp]: "(\<And>x. x \<in> A \<longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi A B"
- by (simp add:Pi_def)
-
-lemma funcsetI: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> f \<in> A \<rightarrow> B"
- by (simp add: Pi_def)
-
-lemma Pi_mem: "f \<in> Pi A B \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<in> B x"
- by (simp add: Pi_def)
-
-lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)"
- unfolding Pi_def by auto
-
-lemma PiE [elim]: "f \<in> Pi A B \<Longrightarrow> (f x \<in> B x \<Longrightarrow> Q) \<Longrightarrow> (x \<notin> A \<Longrightarrow> Q) \<Longrightarrow> Q"
- by (auto simp: Pi_def)
-
-lemma Pi_cong: "(\<And>w. w \<in> A \<Longrightarrow> f w = g w) \<Longrightarrow> f \<in> Pi A B \<longleftrightarrow> g \<in> Pi A B"
- by (auto simp: Pi_def)
-
-lemma funcset_id [simp]: "(\<lambda>x. x) \<in> A \<rightarrow> A"
- by auto
-
-lemma funcset_mem: "f \<in> A \<rightarrow> B \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<in> B"
- by (simp add: Pi_def)
-
-lemma funcset_image: "f \<in> A \<rightarrow> B \<Longrightarrow> f ` A \<subseteq> B"
- by auto
-
-lemma image_subset_iff_funcset: "F ` A \<subseteq> B \<longleftrightarrow> F \<in> A \<rightarrow> B"
- by auto
-
-lemma Pi_eq_empty[simp]: "(\<Pi> x \<in> A. B x) = {} \<longleftrightarrow> (\<exists>x\<in>A. B x = {})"
- apply (simp add: Pi_def)
- apply auto
- txt \<open>Converse direction requires Axiom of Choice to exhibit a function
- picking an element from each non-empty @{term "B x"}\<close>
- apply (drule_tac x = "\<lambda>u. SOME y. y \<in> B u" in spec)
- apply auto
- apply (cut_tac P = "\<lambda>y. y \<in> B x" in some_eq_ex)
- apply auto
- done
-
-lemma Pi_empty [simp]: "Pi {} B = UNIV"
- by (simp add: Pi_def)
-
-lemma Pi_Int: "Pi I E \<inter> Pi I F = (\<Pi> i\<in>I. E i \<inter> F i)"
- by auto
-
-lemma Pi_UN:
- fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set"
- assumes "finite I"
- and mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i"
- shows "(\<Union>n. Pi I (A n)) = (\<Pi> i\<in>I. \<Union>n. A n i)"
-proof (intro set_eqI iffI)
- fix f
- assume "f \<in> (\<Pi> i\<in>I. \<Union>n. A n i)"
- then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i"
- by auto
- from bchoice[OF this] obtain n where n: "f i \<in> A (n i) i" if "i \<in> I" for i
- by auto
- obtain k where k: "n i \<le> k" if "i \<in> I" for i
- using \<open>finite I\<close> finite_nat_set_iff_bounded_le[of "n`I"] by auto
- have "f \<in> Pi I (A k)"
- proof (intro Pi_I)
- fix i
- assume "i \<in> I"
- from mono[OF this, of "n i" k] k[OF this] n[OF this]
- show "f i \<in> A k i" by auto
- qed
- then show "f \<in> (\<Union>n. Pi I (A n))"
- by auto
-qed auto
-
-lemma Pi_UNIV [simp]: "A \<rightarrow> UNIV = UNIV"
- by (simp add: Pi_def)
-
-text \<open>Covariance of Pi-sets in their second argument\<close>
-lemma Pi_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> Pi A B \<subseteq> Pi A C"
- by auto
-
-text \<open>Contravariance of Pi-sets in their first argument\<close>
-lemma Pi_anti_mono: "A' \<subseteq> A \<Longrightarrow> Pi A B \<subseteq> Pi A' B"
- by auto
-
-lemma prod_final:
- assumes 1: "fst \<circ> f \<in> Pi A B"
- and 2: "snd \<circ> f \<in> Pi A C"
- shows "f \<in> (\<Pi> z \<in> A. B z \<times> C z)"
-proof (rule Pi_I)
- fix z
- assume z: "z \<in> A"
- have "f z = (fst (f z), snd (f z))"
- by simp
- also have "\<dots> \<in> B z \<times> C z"
- by (metis SigmaI PiE o_apply 1 2 z)
- finally show "f z \<in> B z \<times> C z" .
-qed
-
-lemma Pi_split_domain[simp]: "x \<in> Pi (I \<union> J) X \<longleftrightarrow> x \<in> Pi I X \<and> x \<in> Pi J X"
- by (auto simp: Pi_def)
-
-lemma Pi_split_insert_domain[simp]: "x \<in> Pi (insert i I) X \<longleftrightarrow> x \<in> Pi I X \<and> x i \<in> X i"
- by (auto simp: Pi_def)
-
-lemma Pi_cancel_fupd_range[simp]: "i \<notin> I \<Longrightarrow> x \<in> Pi I (B(i := b)) \<longleftrightarrow> x \<in> Pi I B"
- by (auto simp: Pi_def)
-
-lemma Pi_cancel_fupd[simp]: "i \<notin> I \<Longrightarrow> x(i := a) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
- by (auto simp: Pi_def)
-
-lemma Pi_fupd_iff: "i \<in> I \<Longrightarrow> f \<in> Pi I (B(i := A)) \<longleftrightarrow> f \<in> Pi (I - {i}) B \<and> f i \<in> A"
- apply auto
- apply (drule_tac x=x in Pi_mem)
- apply (simp_all split: if_split_asm)
- apply (drule_tac x=i in Pi_mem)
- apply (auto dest!: Pi_mem)
- done
-
-
-subsection \<open>Composition With a Restricted Domain: @{term compose}\<close>
-
-lemma funcset_compose: "f \<in> A \<rightarrow> B \<Longrightarrow> g \<in> B \<rightarrow> C \<Longrightarrow> compose A g f \<in> A \<rightarrow> C"
- by (simp add: Pi_def compose_def restrict_def)
-
-lemma compose_assoc:
- assumes "f \<in> A \<rightarrow> B"
- and "g \<in> B \<rightarrow> C"
- and "h \<in> C \<rightarrow> D"
- shows "compose A h (compose A g f) = compose A (compose B h g) f"
- using assms by (simp add: fun_eq_iff Pi_def compose_def restrict_def)
-
-lemma compose_eq: "x \<in> A \<Longrightarrow> compose A g f x = g (f x)"
- by (simp add: compose_def restrict_def)
-
-lemma surj_compose: "f ` A = B \<Longrightarrow> g ` B = C \<Longrightarrow> compose A g f ` A = C"
- by (auto simp add: image_def compose_eq)
-
-
-subsection \<open>Bounded Abstraction: @{term restrict}\<close>
-
-lemma restrict_cong: "I = J \<Longrightarrow> (\<And>i. i \<in> J =simp=> f i = g i) \<Longrightarrow> restrict f I = restrict g J"
- by (auto simp: restrict_def fun_eq_iff simp_implies_def)
-
-lemma restrict_in_funcset: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> (\<lambda>x\<in>A. f x) \<in> A \<rightarrow> B"
- by (simp add: Pi_def restrict_def)
-
-lemma restrictI[intro!]: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> (\<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_apply': "x \<in> A \<Longrightarrow> (\<lambda>y\<in>A. f y) x = f x"
- by simp
-
-lemma restrict_ext: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
- by (simp add: fun_eq_iff Pi_def restrict_def)
-
-lemma restrict_UNIV: "restrict f UNIV = f"
- by (simp add: 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 \<rightarrow> B \<Longrightarrow> f \<in> extensional A \<Longrightarrow> compose A (\<lambda>y\<in>B. y) f = f"
- by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)
-
-lemma compose_Id: "g \<in> A \<rightarrow> B \<Longrightarrow> g \<in> extensional A \<Longrightarrow> compose A g (\<lambda>x\<in>A. x) = g"
- by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)
-
-lemma image_restrict_eq [simp]: "(restrict f A) ` A = f ` A"
- by (auto simp add: restrict_def)
-
-lemma restrict_restrict[simp]: "restrict (restrict f A) B = restrict f (A \<inter> B)"
- unfolding restrict_def by (simp add: fun_eq_iff)
-
-lemma restrict_fupd[simp]: "i \<notin> I \<Longrightarrow> restrict (f (i := x)) I = restrict f I"
- by (auto simp: restrict_def)
-
-lemma restrict_upd[simp]: "i \<notin> I \<Longrightarrow> (restrict f I)(i := y) = restrict (f(i := y)) (insert i I)"
- by (auto simp: fun_eq_iff)
-
-lemma restrict_Pi_cancel: "restrict x I \<in> Pi I A \<longleftrightarrow> x \<in> Pi I A"
- by (auto simp: restrict_def Pi_def)
-
-
-subsection \<open>Bijections Between Sets\<close>
-
-text \<open>The definition of @{const bij_betw} is in \<open>Fun.thy\<close>, but most of
-the theorems belong here, or need at least @{term Hilbert_Choice}.\<close>
-
-lemma bij_betwI:
- assumes "f \<in> A \<rightarrow> B"
- and "g \<in> B \<rightarrow> A"
- and g_f: "\<And>x. x\<in>A \<Longrightarrow> g (f x) = x"
- and f_g: "\<And>y. y\<in>B \<Longrightarrow> f (g y) = y"
- shows "bij_betw f A B"
- unfolding bij_betw_def
-proof
- show "inj_on f A"
- by (metis g_f inj_on_def)
- have "f ` A \<subseteq> B"
- using \<open>f \<in> A \<rightarrow> B\<close> by auto
- moreover
- have "B \<subseteq> f ` A"
- by auto (metis Pi_mem \<open>g \<in> B \<rightarrow> A\<close> f_g image_iff)
- ultimately show "f ` A = B"
- by blast
-qed
-
-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 \<Longrightarrow> inj_on g B \<Longrightarrow> 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 \<Longrightarrow> bij_betw g B C \<Longrightarrow> 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 \<open>Extensionality\<close>
-
-lemma extensional_empty[simp]: "extensional {} = {\<lambda>x. undefined}"
- unfolding extensional_def by auto
-
-lemma extensional_arb: "f \<in> extensional A \<Longrightarrow> x \<notin> A \<Longrightarrow> 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:
- assumes "f \<in> extensional A"
- and "g \<in> extensional A"
- and "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
- shows "f = g"
- using assms by (force simp add: fun_eq_iff extensional_def)
-
-lemma extensional_restrict: "f \<in> extensional A \<Longrightarrow> restrict f A = f"
- by (rule extensionalityI[OF restrict_extensional]) auto
-
-lemma extensional_subset: "f \<in> extensional A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f \<in> extensional B"
- unfolding extensional_def by auto
-
-lemma inv_into_funcset: "f ` A = B \<Longrightarrow> (\<lambda>x\<in>B. inv_into A f x) \<in> B \<rightarrow> A"
- by (unfold inv_into_def) (fast intro: someI2)
-
-lemma compose_inv_into_id: "bij_betw f A B \<Longrightarrow> compose A (\<lambda>y\<in>B. inv_into 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_into: "f ` A = B \<Longrightarrow> compose B f (\<lambda>y\<in>B. inv_into A f y) = (\<lambda>x\<in>B. x)"
- apply (simp add: compose_def)
- apply (rule restrict_ext)
- apply (simp add: f_inv_into_f)
- done
-
-lemma extensional_insert[intro, simp]:
- assumes "a \<in> extensional (insert i I)"
- shows "a(i := b) \<in> extensional (insert i I)"
- using assms unfolding extensional_def by auto
-
-lemma extensional_Int[simp]: "extensional I \<inter> extensional I' = extensional (I \<inter> I')"
- unfolding extensional_def by auto
-
-lemma extensional_UNIV[simp]: "extensional UNIV = UNIV"
- by (auto simp: extensional_def)
-
-lemma restrict_extensional_sub[intro]: "A \<subseteq> B \<Longrightarrow> restrict f A \<in> extensional B"
- unfolding restrict_def extensional_def by auto
-
-lemma extensional_insert_undefined[intro, simp]:
- "a \<in> extensional (insert i I) \<Longrightarrow> a(i := undefined) \<in> extensional I"
- unfolding extensional_def by auto
-
-lemma extensional_insert_cancel[intro, simp]:
- "a \<in> extensional I \<Longrightarrow> a \<in> extensional (insert i I)"
- unfolding extensional_def by auto
-
-
-subsection \<open>Cardinality\<close>
-
-lemma card_inj: "f \<in> A \<rightarrow> B \<Longrightarrow> inj_on f A \<Longrightarrow> finite B \<Longrightarrow> card A \<le> card B"
- by (rule card_inj_on_le) auto
-
-lemma card_bij:
- assumes "f \<in> A \<rightarrow> B" "inj_on f A"
- and "g \<in> B \<rightarrow> A" "inj_on g B"
- and "finite A" "finite B"
- shows "card A = card B"
- using assms by (blast intro: card_inj order_antisym)
-
-
-subsection \<open>Extensional Function Spaces\<close>
-
-definition PiE :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<Rightarrow> 'b) set"
- where "PiE S T = Pi S T \<inter> extensional S"
-
-abbreviation "Pi\<^sub>E A B \<equiv> PiE A B"
-
-syntax
- "_PiE" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set" ("(3\<Pi>\<^sub>E _\<in>_./ _)" 10)
-translations
- "\<Pi>\<^sub>E x\<in>A. B" \<rightleftharpoons> "CONST Pi\<^sub>E A (\<lambda>x. B)"
-
-abbreviation extensional_funcset :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set" (infixr "\<rightarrow>\<^sub>E" 60)
- where "A \<rightarrow>\<^sub>E B \<equiv> (\<Pi>\<^sub>E i\<in>A. B)"
-
-lemma extensional_funcset_def: "extensional_funcset S T = (S \<rightarrow> T) \<inter> extensional S"
- by (simp add: PiE_def)
-
-lemma PiE_empty_domain[simp]: "Pi\<^sub>E {} T = {\<lambda>x. undefined}"
- unfolding PiE_def by simp
-
-lemma PiE_UNIV_domain: "Pi\<^sub>E UNIV T = Pi UNIV T"
- unfolding PiE_def by simp
-
-lemma PiE_empty_range[simp]: "i \<in> I \<Longrightarrow> F i = {} \<Longrightarrow> (\<Pi>\<^sub>E i\<in>I. F i) = {}"
- unfolding PiE_def by auto
-
-lemma PiE_eq_empty_iff: "Pi\<^sub>E I F = {} \<longleftrightarrow> (\<exists>i\<in>I. F i = {})"
-proof
- assume "Pi\<^sub>E I F = {}"
- show "\<exists>i\<in>I. F i = {}"
- proof (rule ccontr)
- assume "\<not> ?thesis"
- then have "\<forall>i. \<exists>y. (i \<in> I \<longrightarrow> y \<in> F i) \<and> (i \<notin> I \<longrightarrow> y = undefined)"
- by auto
- from choice[OF this]
- obtain f where " \<forall>x. (x \<in> I \<longrightarrow> f x \<in> F x) \<and> (x \<notin> I \<longrightarrow> f x = undefined)" ..
- then have "f \<in> Pi\<^sub>E I F"
- by (auto simp: extensional_def PiE_def)
- with \<open>Pi\<^sub>E I F = {}\<close> show False
- by auto
- qed
-qed (auto simp: PiE_def)
-
-lemma PiE_arb: "f \<in> Pi\<^sub>E S T \<Longrightarrow> x \<notin> S \<Longrightarrow> f x = undefined"
- unfolding PiE_def by auto (auto dest!: extensional_arb)
-
-lemma PiE_mem: "f \<in> Pi\<^sub>E S T \<Longrightarrow> x \<in> S \<Longrightarrow> f x \<in> T x"
- unfolding PiE_def by auto
-
-lemma PiE_fun_upd: "y \<in> T x \<Longrightarrow> f \<in> Pi\<^sub>E S T \<Longrightarrow> f(x := y) \<in> Pi\<^sub>E (insert x S) T"
- unfolding PiE_def extensional_def by auto
-
-lemma fun_upd_in_PiE: "x \<notin> S \<Longrightarrow> f \<in> Pi\<^sub>E (insert x S) T \<Longrightarrow> f(x := undefined) \<in> Pi\<^sub>E S T"
- unfolding PiE_def extensional_def by auto
-
-lemma PiE_insert_eq: "Pi\<^sub>E (insert x S) T = (\<lambda>(y, g). g(x := y)) ` (T x \<times> Pi\<^sub>E S T)"
-proof -
- {
- fix f assume "f \<in> Pi\<^sub>E (insert x S) T" "x \<notin> S"
- then have "f \<in> (\<lambda>(y, g). g(x := y)) ` (T x \<times> Pi\<^sub>E S T)"
- by (auto intro!: image_eqI[where x="(f x, f(x := undefined))"] intro: fun_upd_in_PiE PiE_mem)
- }
- moreover
- {
- fix f assume "f \<in> Pi\<^sub>E (insert x S) T" "x \<in> S"
- then have "f \<in> (\<lambda>(y, g). g(x := y)) ` (T x \<times> Pi\<^sub>E S T)"
- by (auto intro!: image_eqI[where x="(f x, f)"] intro: fun_upd_in_PiE PiE_mem simp: insert_absorb)
- }
- ultimately show ?thesis
- by (auto intro: PiE_fun_upd)
-qed
-
-lemma PiE_Int: "Pi\<^sub>E I A \<inter> Pi\<^sub>E I B = Pi\<^sub>E I (\<lambda>x. A x \<inter> B x)"
- by (auto simp: PiE_def)
-
-lemma PiE_cong: "(\<And>i. i\<in>I \<Longrightarrow> A i = B i) \<Longrightarrow> Pi\<^sub>E I A = Pi\<^sub>E I B"
- unfolding PiE_def by (auto simp: Pi_cong)
-
-lemma PiE_E [elim]:
- assumes "f \<in> Pi\<^sub>E A B"
- obtains "x \<in> A" and "f x \<in> B x"
- | "x \<notin> A" and "f x = undefined"
- using assms by (auto simp: Pi_def PiE_def extensional_def)
-
-lemma PiE_I[intro!]:
- "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> (\<And>x. x \<notin> A \<Longrightarrow> f x = undefined) \<Longrightarrow> f \<in> Pi\<^sub>E A B"
- by (simp add: PiE_def extensional_def)
-
-lemma PiE_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> Pi\<^sub>E A B \<subseteq> Pi\<^sub>E A C"
- by auto
-
-lemma PiE_iff: "f \<in> Pi\<^sub>E I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i) \<and> f \<in> extensional I"
- by (simp add: PiE_def Pi_iff)
-
-lemma PiE_restrict[simp]: "f \<in> Pi\<^sub>E A B \<Longrightarrow> restrict f A = f"
- by (simp add: extensional_restrict PiE_def)
-
-lemma restrict_PiE[simp]: "restrict f I \<in> Pi\<^sub>E I S \<longleftrightarrow> f \<in> Pi I S"
- by (auto simp: PiE_iff)
-
-lemma PiE_eq_subset:
- assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
- and eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
- and "i \<in> I"
- shows "F i \<subseteq> F' i"
-proof
- fix x
- assume "x \<in> F i"
- with ne have "\<forall>j. \<exists>y. (j \<in> I \<longrightarrow> y \<in> F j \<and> (i = j \<longrightarrow> x = y)) \<and> (j \<notin> I \<longrightarrow> y = undefined)"
- by auto
- from choice[OF this] obtain f
- where f: " \<forall>j. (j \<in> I \<longrightarrow> f j \<in> F j \<and> (i = j \<longrightarrow> x = f j)) \<and> (j \<notin> I \<longrightarrow> f j = undefined)" ..
- then have "f \<in> Pi\<^sub>E I F"
- by (auto simp: extensional_def PiE_def)
- then have "f \<in> Pi\<^sub>E I F'"
- using assms by simp
- then show "x \<in> F' i"
- using f \<open>i \<in> I\<close> by (auto simp: PiE_def)
-qed
-
-lemma PiE_eq_iff_not_empty:
- assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
- shows "Pi\<^sub>E I F = Pi\<^sub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i)"
-proof (intro iffI ballI)
- fix i
- assume eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
- assume i: "i \<in> I"
- show "F i = F' i"
- using PiE_eq_subset[of I F F', OF ne eq i]
- using PiE_eq_subset[of I F' F, OF ne(2,1) eq[symmetric] i]
- by auto
-qed (auto simp: PiE_def)
-
-lemma PiE_eq_iff:
- "Pi\<^sub>E I F = Pi\<^sub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i) \<or> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
-proof (intro iffI disjCI)
- assume eq[simp]: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
- assume "\<not> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
- then have "(\<forall>i\<in>I. F i \<noteq> {}) \<and> (\<forall>i\<in>I. F' i \<noteq> {})"
- using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by auto
- with PiE_eq_iff_not_empty[of I F F'] show "\<forall>i\<in>I. F i = F' i"
- by auto
-next
- assume "(\<forall>i\<in>I. F i = F' i) \<or> (\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {})"
- then show "Pi\<^sub>E I F = Pi\<^sub>E I F'"
- using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by (auto simp: PiE_def)
-qed
-
-lemma extensional_funcset_fun_upd_restricts_rangeI:
- "\<forall>y \<in> S. f x \<noteq> f y \<Longrightarrow> f \<in> (insert x S) \<rightarrow>\<^sub>E T \<Longrightarrow> f(x := undefined) \<in> S \<rightarrow>\<^sub>E (T - {f x})"
- unfolding extensional_funcset_def extensional_def
- apply auto
- apply (case_tac "x = xa")
- apply auto
- done
-
-lemma extensional_funcset_fun_upd_extends_rangeI:
- assumes "a \<in> T" "f \<in> S \<rightarrow>\<^sub>E (T - {a})"
- shows "f(x := a) \<in> insert x S \<rightarrow>\<^sub>E T"
- using assms unfolding extensional_funcset_def extensional_def by auto
-
-
-subsubsection \<open>Injective Extensional Function Spaces\<close>
-
-lemma extensional_funcset_fun_upd_inj_onI:
- assumes "f \<in> S \<rightarrow>\<^sub>E (T - {a})"
- and "inj_on f S"
- shows "inj_on (f(x := a)) S"
- using assms
- unfolding extensional_funcset_def by (auto intro!: inj_on_fun_updI)
-
-lemma extensional_funcset_extend_domain_inj_on_eq:
- assumes "x \<notin> S"
- shows "{f. f \<in> (insert x S) \<rightarrow>\<^sub>E T \<and> inj_on f (insert x S)} =
- (\<lambda>(y, g). g(x:=y)) ` {(y, g). y \<in> T \<and> g \<in> S \<rightarrow>\<^sub>E (T - {y}) \<and> inj_on g S}"
- using assms
- apply (auto del: PiE_I PiE_E)
- apply (auto intro: extensional_funcset_fun_upd_inj_onI
- extensional_funcset_fun_upd_extends_rangeI del: PiE_I PiE_E)
- apply (auto simp add: image_iff inj_on_def)
- apply (rule_tac x="xa x" in exI)
- apply (auto intro: PiE_mem del: PiE_I PiE_E)
- apply (rule_tac x="xa(x := undefined)" in exI)
- apply (auto intro!: extensional_funcset_fun_upd_restricts_rangeI)
- apply (auto dest!: PiE_mem split: if_split_asm)
- done
-
-lemma extensional_funcset_extend_domain_inj_onI:
- assumes "x \<notin> S"
- shows "inj_on (\<lambda>(y, g). g(x := y)) {(y, g). y \<in> T \<and> g \<in> S \<rightarrow>\<^sub>E (T - {y}) \<and> inj_on g S}"
- using assms
- apply (auto intro!: inj_onI)
- apply (metis fun_upd_same)
- apply (metis assms PiE_arb fun_upd_triv fun_upd_upd)
- done
-
-
-subsubsection \<open>Cardinality\<close>
-
-lemma finite_PiE: "finite S \<Longrightarrow> (\<And>i. i \<in> S \<Longrightarrow> finite (T i)) \<Longrightarrow> finite (\<Pi>\<^sub>E i \<in> S. T i)"
- by (induct S arbitrary: T rule: finite_induct) (simp_all add: PiE_insert_eq)
-
-lemma inj_combinator: "x \<notin> S \<Longrightarrow> inj_on (\<lambda>(y, g). g(x := y)) (T x \<times> Pi\<^sub>E S T)"
-proof (safe intro!: inj_onI ext)
- fix f y g z
- assume "x \<notin> S"
- assume fg: "f \<in> Pi\<^sub>E S T" "g \<in> Pi\<^sub>E S T"
- assume "f(x := y) = g(x := z)"
- then have *: "\<And>i. (f(x := y)) i = (g(x := z)) i"
- unfolding fun_eq_iff by auto
- from this[of x] show "y = z" by simp
- fix i from *[of i] \<open>x \<notin> S\<close> fg show "f i = g i"
- by (auto split: if_split_asm simp: PiE_def extensional_def)
-qed
-
-lemma card_PiE: "finite S \<Longrightarrow> card (\<Pi>\<^sub>E i \<in> S. T i) = (\<Prod> i\<in>S. card (T i))"
-proof (induct rule: finite_induct)
- case empty
- then show ?case by auto
-next
- case (insert x S)
- then show ?case
- by (simp add: PiE_insert_eq inj_combinator card_image card_cartesian_product)
-qed
-
-end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/FuncSet.thy Tue May 15 11:33:43 2018 +0200
@@ -0,0 +1,568 @@
+(* Title: HOL/FuncSet.thy
+ Author: Florian Kammueller and Lawrence C Paulson, Lukas Bulwahn
+*)
+
+section \<open>Pi and Function Sets\<close>
+
+theory FuncSet
+ imports Main
+ abbrevs PiE = "Pi\<^sub>E"
+ and PIE = "\<Pi>\<^sub>E"
+begin
+
+definition Pi :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<Rightarrow> 'b) set"
+ where "Pi A B = {f. \<forall>x. x \<in> A \<longrightarrow> f x \<in> B x}"
+
+definition extensional :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) set"
+ where "extensional A = {f. \<forall>x. x \<notin> A \<longrightarrow> f x = undefined}"
+
+definition "restrict" :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
+ where "restrict f A = (\<lambda>x. if x \<in> A then f x else undefined)"
+
+abbreviation funcset :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set" (infixr "\<rightarrow>" 60)
+ where "A \<rightarrow> B \<equiv> Pi A (\<lambda>_. B)"
+
+syntax
+ "_Pi" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set" ("(3\<Pi> _\<in>_./ _)" 10)
+ "_lam" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)" ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
+translations
+ "\<Pi> x\<in>A. B" \<rightleftharpoons> "CONST Pi A (\<lambda>x. B)"
+ "\<lambda>x\<in>A. f" \<rightleftharpoons> "CONST restrict (\<lambda>x. f) A"
+
+definition "compose" :: "'a set \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'c)"
+ where "compose A g f = (\<lambda>x\<in>A. g (f x))"
+
+
+subsection \<open>Basic Properties of @{term Pi}\<close>
+
+lemma Pi_I[intro!]: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi A B"
+ by (simp add: Pi_def)
+
+lemma Pi_I'[simp]: "(\<And>x. x \<in> A \<longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi A B"
+ by (simp add:Pi_def)
+
+lemma funcsetI: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> f \<in> A \<rightarrow> B"
+ by (simp add: Pi_def)
+
+lemma Pi_mem: "f \<in> Pi A B \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<in> B x"
+ by (simp add: Pi_def)
+
+lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)"
+ unfolding Pi_def by auto
+
+lemma PiE [elim]: "f \<in> Pi A B \<Longrightarrow> (f x \<in> B x \<Longrightarrow> Q) \<Longrightarrow> (x \<notin> A \<Longrightarrow> Q) \<Longrightarrow> Q"
+ by (auto simp: Pi_def)
+
+lemma Pi_cong: "(\<And>w. w \<in> A \<Longrightarrow> f w = g w) \<Longrightarrow> f \<in> Pi A B \<longleftrightarrow> g \<in> Pi A B"
+ by (auto simp: Pi_def)
+
+lemma funcset_id [simp]: "(\<lambda>x. x) \<in> A \<rightarrow> A"
+ by auto
+
+lemma funcset_mem: "f \<in> A \<rightarrow> B \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<in> B"
+ by (simp add: Pi_def)
+
+lemma funcset_image: "f \<in> A \<rightarrow> B \<Longrightarrow> f ` A \<subseteq> B"
+ by auto
+
+lemma image_subset_iff_funcset: "F ` A \<subseteq> B \<longleftrightarrow> F \<in> A \<rightarrow> B"
+ by auto
+
+lemma Pi_eq_empty[simp]: "(\<Pi> x \<in> A. B x) = {} \<longleftrightarrow> (\<exists>x\<in>A. B x = {})"
+ apply (simp add: Pi_def)
+ apply auto
+ txt \<open>Converse direction requires Axiom of Choice to exhibit a function
+ picking an element from each non-empty @{term "B x"}\<close>
+ apply (drule_tac x = "\<lambda>u. SOME y. y \<in> B u" in spec)
+ apply auto
+ apply (cut_tac P = "\<lambda>y. y \<in> B x" in some_eq_ex)
+ apply auto
+ done
+
+lemma Pi_empty [simp]: "Pi {} B = UNIV"
+ by (simp add: Pi_def)
+
+lemma Pi_Int: "Pi I E \<inter> Pi I F = (\<Pi> i\<in>I. E i \<inter> F i)"
+ by auto
+
+lemma Pi_UN:
+ fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set"
+ assumes "finite I"
+ and mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i"
+ shows "(\<Union>n. Pi I (A n)) = (\<Pi> i\<in>I. \<Union>n. A n i)"
+proof (intro set_eqI iffI)
+ fix f
+ assume "f \<in> (\<Pi> i\<in>I. \<Union>n. A n i)"
+ then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i"
+ by auto
+ from bchoice[OF this] obtain n where n: "f i \<in> A (n i) i" if "i \<in> I" for i
+ by auto
+ obtain k where k: "n i \<le> k" if "i \<in> I" for i
+ using \<open>finite I\<close> finite_nat_set_iff_bounded_le[of "n`I"] by auto
+ have "f \<in> Pi I (A k)"
+ proof (intro Pi_I)
+ fix i
+ assume "i \<in> I"
+ from mono[OF this, of "n i" k] k[OF this] n[OF this]
+ show "f i \<in> A k i" by auto
+ qed
+ then show "f \<in> (\<Union>n. Pi I (A n))"
+ by auto
+qed auto
+
+lemma Pi_UNIV [simp]: "A \<rightarrow> UNIV = UNIV"
+ by (simp add: Pi_def)
+
+text \<open>Covariance of Pi-sets in their second argument\<close>
+lemma Pi_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> Pi A B \<subseteq> Pi A C"
+ by auto
+
+text \<open>Contravariance of Pi-sets in their first argument\<close>
+lemma Pi_anti_mono: "A' \<subseteq> A \<Longrightarrow> Pi A B \<subseteq> Pi A' B"
+ by auto
+
+lemma prod_final:
+ assumes 1: "fst \<circ> f \<in> Pi A B"
+ and 2: "snd \<circ> f \<in> Pi A C"
+ shows "f \<in> (\<Pi> z \<in> A. B z \<times> C z)"
+proof (rule Pi_I)
+ fix z
+ assume z: "z \<in> A"
+ have "f z = (fst (f z), snd (f z))"
+ by simp
+ also have "\<dots> \<in> B z \<times> C z"
+ by (metis SigmaI PiE o_apply 1 2 z)
+ finally show "f z \<in> B z \<times> C z" .
+qed
+
+lemma Pi_split_domain[simp]: "x \<in> Pi (I \<union> J) X \<longleftrightarrow> x \<in> Pi I X \<and> x \<in> Pi J X"
+ by (auto simp: Pi_def)
+
+lemma Pi_split_insert_domain[simp]: "x \<in> Pi (insert i I) X \<longleftrightarrow> x \<in> Pi I X \<and> x i \<in> X i"
+ by (auto simp: Pi_def)
+
+lemma Pi_cancel_fupd_range[simp]: "i \<notin> I \<Longrightarrow> x \<in> Pi I (B(i := b)) \<longleftrightarrow> x \<in> Pi I B"
+ by (auto simp: Pi_def)
+
+lemma Pi_cancel_fupd[simp]: "i \<notin> I \<Longrightarrow> x(i := a) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
+ by (auto simp: Pi_def)
+
+lemma Pi_fupd_iff: "i \<in> I \<Longrightarrow> f \<in> Pi I (B(i := A)) \<longleftrightarrow> f \<in> Pi (I - {i}) B \<and> f i \<in> A"
+ apply auto
+ apply (drule_tac x=x in Pi_mem)
+ apply (simp_all split: if_split_asm)
+ apply (drule_tac x=i in Pi_mem)
+ apply (auto dest!: Pi_mem)
+ done
+
+
+subsection \<open>Composition With a Restricted Domain: @{term compose}\<close>
+
+lemma funcset_compose: "f \<in> A \<rightarrow> B \<Longrightarrow> g \<in> B \<rightarrow> C \<Longrightarrow> compose A g f \<in> A \<rightarrow> C"
+ by (simp add: Pi_def compose_def restrict_def)
+
+lemma compose_assoc:
+ assumes "f \<in> A \<rightarrow> B"
+ and "g \<in> B \<rightarrow> C"
+ and "h \<in> C \<rightarrow> D"
+ shows "compose A h (compose A g f) = compose A (compose B h g) f"
+ using assms by (simp add: fun_eq_iff Pi_def compose_def restrict_def)
+
+lemma compose_eq: "x \<in> A \<Longrightarrow> compose A g f x = g (f x)"
+ by (simp add: compose_def restrict_def)
+
+lemma surj_compose: "f ` A = B \<Longrightarrow> g ` B = C \<Longrightarrow> compose A g f ` A = C"
+ by (auto simp add: image_def compose_eq)
+
+
+subsection \<open>Bounded Abstraction: @{term restrict}\<close>
+
+lemma restrict_cong: "I = J \<Longrightarrow> (\<And>i. i \<in> J =simp=> f i = g i) \<Longrightarrow> restrict f I = restrict g J"
+ by (auto simp: restrict_def fun_eq_iff simp_implies_def)
+
+lemma restrict_in_funcset: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> (\<lambda>x\<in>A. f x) \<in> A \<rightarrow> B"
+ by (simp add: Pi_def restrict_def)
+
+lemma restrictI[intro!]: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> (\<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_apply': "x \<in> A \<Longrightarrow> (\<lambda>y\<in>A. f y) x = f x"
+ by simp
+
+lemma restrict_ext: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
+ by (simp add: fun_eq_iff Pi_def restrict_def)
+
+lemma restrict_UNIV: "restrict f UNIV = f"
+ by (simp add: 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 \<rightarrow> B \<Longrightarrow> f \<in> extensional A \<Longrightarrow> compose A (\<lambda>y\<in>B. y) f = f"
+ by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)
+
+lemma compose_Id: "g \<in> A \<rightarrow> B \<Longrightarrow> g \<in> extensional A \<Longrightarrow> compose A g (\<lambda>x\<in>A. x) = g"
+ by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)
+
+lemma image_restrict_eq [simp]: "(restrict f A) ` A = f ` A"
+ by (auto simp add: restrict_def)
+
+lemma restrict_restrict[simp]: "restrict (restrict f A) B = restrict f (A \<inter> B)"
+ unfolding restrict_def by (simp add: fun_eq_iff)
+
+lemma restrict_fupd[simp]: "i \<notin> I \<Longrightarrow> restrict (f (i := x)) I = restrict f I"
+ by (auto simp: restrict_def)
+
+lemma restrict_upd[simp]: "i \<notin> I \<Longrightarrow> (restrict f I)(i := y) = restrict (f(i := y)) (insert i I)"
+ by (auto simp: fun_eq_iff)
+
+lemma restrict_Pi_cancel: "restrict x I \<in> Pi I A \<longleftrightarrow> x \<in> Pi I A"
+ by (auto simp: restrict_def Pi_def)
+
+
+subsection \<open>Bijections Between Sets\<close>
+
+text \<open>The definition of @{const bij_betw} is in \<open>Fun.thy\<close>, but most of
+the theorems belong here, or need at least @{term Hilbert_Choice}.\<close>
+
+lemma bij_betwI:
+ assumes "f \<in> A \<rightarrow> B"
+ and "g \<in> B \<rightarrow> A"
+ and g_f: "\<And>x. x\<in>A \<Longrightarrow> g (f x) = x"
+ and f_g: "\<And>y. y\<in>B \<Longrightarrow> f (g y) = y"
+ shows "bij_betw f A B"
+ unfolding bij_betw_def
+proof
+ show "inj_on f A"
+ by (metis g_f inj_on_def)
+ have "f ` A \<subseteq> B"
+ using \<open>f \<in> A \<rightarrow> B\<close> by auto
+ moreover
+ have "B \<subseteq> f ` A"
+ by auto (metis Pi_mem \<open>g \<in> B \<rightarrow> A\<close> f_g image_iff)
+ ultimately show "f ` A = B"
+ by blast
+qed
+
+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 \<Longrightarrow> inj_on g B \<Longrightarrow> 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 \<Longrightarrow> bij_betw g B C \<Longrightarrow> 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 \<open>Extensionality\<close>
+
+lemma extensional_empty[simp]: "extensional {} = {\<lambda>x. undefined}"
+ unfolding extensional_def by auto
+
+lemma extensional_arb: "f \<in> extensional A \<Longrightarrow> x \<notin> A \<Longrightarrow> 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:
+ assumes "f \<in> extensional A"
+ and "g \<in> extensional A"
+ and "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
+ shows "f = g"
+ using assms by (force simp add: fun_eq_iff extensional_def)
+
+lemma extensional_restrict: "f \<in> extensional A \<Longrightarrow> restrict f A = f"
+ by (rule extensionalityI[OF restrict_extensional]) auto
+
+lemma extensional_subset: "f \<in> extensional A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f \<in> extensional B"
+ unfolding extensional_def by auto
+
+lemma inv_into_funcset: "f ` A = B \<Longrightarrow> (\<lambda>x\<in>B. inv_into A f x) \<in> B \<rightarrow> A"
+ by (unfold inv_into_def) (fast intro: someI2)
+
+lemma compose_inv_into_id: "bij_betw f A B \<Longrightarrow> compose A (\<lambda>y\<in>B. inv_into 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_into: "f ` A = B \<Longrightarrow> compose B f (\<lambda>y\<in>B. inv_into A f y) = (\<lambda>x\<in>B. x)"
+ apply (simp add: compose_def)
+ apply (rule restrict_ext)
+ apply (simp add: f_inv_into_f)
+ done
+
+lemma extensional_insert[intro, simp]:
+ assumes "a \<in> extensional (insert i I)"
+ shows "a(i := b) \<in> extensional (insert i I)"
+ using assms unfolding extensional_def by auto
+
+lemma extensional_Int[simp]: "extensional I \<inter> extensional I' = extensional (I \<inter> I')"
+ unfolding extensional_def by auto
+
+lemma extensional_UNIV[simp]: "extensional UNIV = UNIV"
+ by (auto simp: extensional_def)
+
+lemma restrict_extensional_sub[intro]: "A \<subseteq> B \<Longrightarrow> restrict f A \<in> extensional B"
+ unfolding restrict_def extensional_def by auto
+
+lemma extensional_insert_undefined[intro, simp]:
+ "a \<in> extensional (insert i I) \<Longrightarrow> a(i := undefined) \<in> extensional I"
+ unfolding extensional_def by auto
+
+lemma extensional_insert_cancel[intro, simp]:
+ "a \<in> extensional I \<Longrightarrow> a \<in> extensional (insert i I)"
+ unfolding extensional_def by auto
+
+
+subsection \<open>Cardinality\<close>
+
+lemma card_inj: "f \<in> A \<rightarrow> B \<Longrightarrow> inj_on f A \<Longrightarrow> finite B \<Longrightarrow> card A \<le> card B"
+ by (rule card_inj_on_le) auto
+
+lemma card_bij:
+ assumes "f \<in> A \<rightarrow> B" "inj_on f A"
+ and "g \<in> B \<rightarrow> A" "inj_on g B"
+ and "finite A" "finite B"
+ shows "card A = card B"
+ using assms by (blast intro: card_inj order_antisym)
+
+
+subsection \<open>Extensional Function Spaces\<close>
+
+definition PiE :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<Rightarrow> 'b) set"
+ where "PiE S T = Pi S T \<inter> extensional S"
+
+abbreviation "Pi\<^sub>E A B \<equiv> PiE A B"
+
+syntax
+ "_PiE" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set" ("(3\<Pi>\<^sub>E _\<in>_./ _)" 10)
+translations
+ "\<Pi>\<^sub>E x\<in>A. B" \<rightleftharpoons> "CONST Pi\<^sub>E A (\<lambda>x. B)"
+
+abbreviation extensional_funcset :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set" (infixr "\<rightarrow>\<^sub>E" 60)
+ where "A \<rightarrow>\<^sub>E B \<equiv> (\<Pi>\<^sub>E i\<in>A. B)"
+
+lemma extensional_funcset_def: "extensional_funcset S T = (S \<rightarrow> T) \<inter> extensional S"
+ by (simp add: PiE_def)
+
+lemma PiE_empty_domain[simp]: "Pi\<^sub>E {} T = {\<lambda>x. undefined}"
+ unfolding PiE_def by simp
+
+lemma PiE_UNIV_domain: "Pi\<^sub>E UNIV T = Pi UNIV T"
+ unfolding PiE_def by simp
+
+lemma PiE_empty_range[simp]: "i \<in> I \<Longrightarrow> F i = {} \<Longrightarrow> (\<Pi>\<^sub>E i\<in>I. F i) = {}"
+ unfolding PiE_def by auto
+
+lemma PiE_eq_empty_iff: "Pi\<^sub>E I F = {} \<longleftrightarrow> (\<exists>i\<in>I. F i = {})"
+proof
+ assume "Pi\<^sub>E I F = {}"
+ show "\<exists>i\<in>I. F i = {}"
+ proof (rule ccontr)
+ assume "\<not> ?thesis"
+ then have "\<forall>i. \<exists>y. (i \<in> I \<longrightarrow> y \<in> F i) \<and> (i \<notin> I \<longrightarrow> y = undefined)"
+ by auto
+ from choice[OF this]
+ obtain f where " \<forall>x. (x \<in> I \<longrightarrow> f x \<in> F x) \<and> (x \<notin> I \<longrightarrow> f x = undefined)" ..
+ then have "f \<in> Pi\<^sub>E I F"
+ by (auto simp: extensional_def PiE_def)
+ with \<open>Pi\<^sub>E I F = {}\<close> show False
+ by auto
+ qed
+qed (auto simp: PiE_def)
+
+lemma PiE_arb: "f \<in> Pi\<^sub>E S T \<Longrightarrow> x \<notin> S \<Longrightarrow> f x = undefined"
+ unfolding PiE_def by auto (auto dest!: extensional_arb)
+
+lemma PiE_mem: "f \<in> Pi\<^sub>E S T \<Longrightarrow> x \<in> S \<Longrightarrow> f x \<in> T x"
+ unfolding PiE_def by auto
+
+lemma PiE_fun_upd: "y \<in> T x \<Longrightarrow> f \<in> Pi\<^sub>E S T \<Longrightarrow> f(x := y) \<in> Pi\<^sub>E (insert x S) T"
+ unfolding PiE_def extensional_def by auto
+
+lemma fun_upd_in_PiE: "x \<notin> S \<Longrightarrow> f \<in> Pi\<^sub>E (insert x S) T \<Longrightarrow> f(x := undefined) \<in> Pi\<^sub>E S T"
+ unfolding PiE_def extensional_def by auto
+
+lemma PiE_insert_eq: "Pi\<^sub>E (insert x S) T = (\<lambda>(y, g). g(x := y)) ` (T x \<times> Pi\<^sub>E S T)"
+proof -
+ {
+ fix f assume "f \<in> Pi\<^sub>E (insert x S) T" "x \<notin> S"
+ then have "f \<in> (\<lambda>(y, g). g(x := y)) ` (T x \<times> Pi\<^sub>E S T)"
+ by (auto intro!: image_eqI[where x="(f x, f(x := undefined))"] intro: fun_upd_in_PiE PiE_mem)
+ }
+ moreover
+ {
+ fix f assume "f \<in> Pi\<^sub>E (insert x S) T" "x \<in> S"
+ then have "f \<in> (\<lambda>(y, g). g(x := y)) ` (T x \<times> Pi\<^sub>E S T)"
+ by (auto intro!: image_eqI[where x="(f x, f)"] intro: fun_upd_in_PiE PiE_mem simp: insert_absorb)
+ }
+ ultimately show ?thesis
+ by (auto intro: PiE_fun_upd)
+qed
+
+lemma PiE_Int: "Pi\<^sub>E I A \<inter> Pi\<^sub>E I B = Pi\<^sub>E I (\<lambda>x. A x \<inter> B x)"
+ by (auto simp: PiE_def)
+
+lemma PiE_cong: "(\<And>i. i\<in>I \<Longrightarrow> A i = B i) \<Longrightarrow> Pi\<^sub>E I A = Pi\<^sub>E I B"
+ unfolding PiE_def by (auto simp: Pi_cong)
+
+lemma PiE_E [elim]:
+ assumes "f \<in> Pi\<^sub>E A B"
+ obtains "x \<in> A" and "f x \<in> B x"
+ | "x \<notin> A" and "f x = undefined"
+ using assms by (auto simp: Pi_def PiE_def extensional_def)
+
+lemma PiE_I[intro!]:
+ "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> (\<And>x. x \<notin> A \<Longrightarrow> f x = undefined) \<Longrightarrow> f \<in> Pi\<^sub>E A B"
+ by (simp add: PiE_def extensional_def)
+
+lemma PiE_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> Pi\<^sub>E A B \<subseteq> Pi\<^sub>E A C"
+ by auto
+
+lemma PiE_iff: "f \<in> Pi\<^sub>E I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i) \<and> f \<in> extensional I"
+ by (simp add: PiE_def Pi_iff)
+
+lemma PiE_restrict[simp]: "f \<in> Pi\<^sub>E A B \<Longrightarrow> restrict f A = f"
+ by (simp add: extensional_restrict PiE_def)
+
+lemma restrict_PiE[simp]: "restrict f I \<in> Pi\<^sub>E I S \<longleftrightarrow> f \<in> Pi I S"
+ by (auto simp: PiE_iff)
+
+lemma PiE_eq_subset:
+ assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
+ and eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
+ and "i \<in> I"
+ shows "F i \<subseteq> F' i"
+proof
+ fix x
+ assume "x \<in> F i"
+ with ne have "\<forall>j. \<exists>y. (j \<in> I \<longrightarrow> y \<in> F j \<and> (i = j \<longrightarrow> x = y)) \<and> (j \<notin> I \<longrightarrow> y = undefined)"
+ by auto
+ from choice[OF this] obtain f
+ where f: " \<forall>j. (j \<in> I \<longrightarrow> f j \<in> F j \<and> (i = j \<longrightarrow> x = f j)) \<and> (j \<notin> I \<longrightarrow> f j = undefined)" ..
+ then have "f \<in> Pi\<^sub>E I F"
+ by (auto simp: extensional_def PiE_def)
+ then have "f \<in> Pi\<^sub>E I F'"
+ using assms by simp
+ then show "x \<in> F' i"
+ using f \<open>i \<in> I\<close> by (auto simp: PiE_def)
+qed
+
+lemma PiE_eq_iff_not_empty:
+ assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
+ shows "Pi\<^sub>E I F = Pi\<^sub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i)"
+proof (intro iffI ballI)
+ fix i
+ assume eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
+ assume i: "i \<in> I"
+ show "F i = F' i"
+ using PiE_eq_subset[of I F F', OF ne eq i]
+ using PiE_eq_subset[of I F' F, OF ne(2,1) eq[symmetric] i]
+ by auto
+qed (auto simp: PiE_def)
+
+lemma PiE_eq_iff:
+ "Pi\<^sub>E I F = Pi\<^sub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i) \<or> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
+proof (intro iffI disjCI)
+ assume eq[simp]: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
+ assume "\<not> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
+ then have "(\<forall>i\<in>I. F i \<noteq> {}) \<and> (\<forall>i\<in>I. F' i \<noteq> {})"
+ using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by auto
+ with PiE_eq_iff_not_empty[of I F F'] show "\<forall>i\<in>I. F i = F' i"
+ by auto
+next
+ assume "(\<forall>i\<in>I. F i = F' i) \<or> (\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {})"
+ then show "Pi\<^sub>E I F = Pi\<^sub>E I F'"
+ using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by (auto simp: PiE_def)
+qed
+
+lemma extensional_funcset_fun_upd_restricts_rangeI:
+ "\<forall>y \<in> S. f x \<noteq> f y \<Longrightarrow> f \<in> (insert x S) \<rightarrow>\<^sub>E T \<Longrightarrow> f(x := undefined) \<in> S \<rightarrow>\<^sub>E (T - {f x})"
+ unfolding extensional_funcset_def extensional_def
+ apply auto
+ apply (case_tac "x = xa")
+ apply auto
+ done
+
+lemma extensional_funcset_fun_upd_extends_rangeI:
+ assumes "a \<in> T" "f \<in> S \<rightarrow>\<^sub>E (T - {a})"
+ shows "f(x := a) \<in> insert x S \<rightarrow>\<^sub>E T"
+ using assms unfolding extensional_funcset_def extensional_def by auto
+
+
+subsubsection \<open>Injective Extensional Function Spaces\<close>
+
+lemma extensional_funcset_fun_upd_inj_onI:
+ assumes "f \<in> S \<rightarrow>\<^sub>E (T - {a})"
+ and "inj_on f S"
+ shows "inj_on (f(x := a)) S"
+ using assms
+ unfolding extensional_funcset_def by (auto intro!: inj_on_fun_updI)
+
+lemma extensional_funcset_extend_domain_inj_on_eq:
+ assumes "x \<notin> S"
+ shows "{f. f \<in> (insert x S) \<rightarrow>\<^sub>E T \<and> inj_on f (insert x S)} =
+ (\<lambda>(y, g). g(x:=y)) ` {(y, g). y \<in> T \<and> g \<in> S \<rightarrow>\<^sub>E (T - {y}) \<and> inj_on g S}"
+ using assms
+ apply (auto del: PiE_I PiE_E)
+ apply (auto intro: extensional_funcset_fun_upd_inj_onI
+ extensional_funcset_fun_upd_extends_rangeI del: PiE_I PiE_E)
+ apply (auto simp add: image_iff inj_on_def)
+ apply (rule_tac x="xa x" in exI)
+ apply (auto intro: PiE_mem del: PiE_I PiE_E)
+ apply (rule_tac x="xa(x := undefined)" in exI)
+ apply (auto intro!: extensional_funcset_fun_upd_restricts_rangeI)
+ apply (auto dest!: PiE_mem split: if_split_asm)
+ done
+
+lemma extensional_funcset_extend_domain_inj_onI:
+ assumes "x \<notin> S"
+ shows "inj_on (\<lambda>(y, g). g(x := y)) {(y, g). y \<in> T \<and> g \<in> S \<rightarrow>\<^sub>E (T - {y}) \<and> inj_on g S}"
+ using assms
+ apply (auto intro!: inj_onI)
+ apply (metis fun_upd_same)
+ apply (metis assms PiE_arb fun_upd_triv fun_upd_upd)
+ done
+
+
+subsubsection \<open>Cardinality\<close>
+
+lemma finite_PiE: "finite S \<Longrightarrow> (\<And>i. i \<in> S \<Longrightarrow> finite (T i)) \<Longrightarrow> finite (\<Pi>\<^sub>E i \<in> S. T i)"
+ by (induct S arbitrary: T rule: finite_induct) (simp_all add: PiE_insert_eq)
+
+lemma inj_combinator: "x \<notin> S \<Longrightarrow> inj_on (\<lambda>(y, g). g(x := y)) (T x \<times> Pi\<^sub>E S T)"
+proof (safe intro!: inj_onI ext)
+ fix f y g z
+ assume "x \<notin> S"
+ assume fg: "f \<in> Pi\<^sub>E S T" "g \<in> Pi\<^sub>E S T"
+ assume "f(x := y) = g(x := z)"
+ then have *: "\<And>i. (f(x := y)) i = (g(x := z)) i"
+ unfolding fun_eq_iff by auto
+ from this[of x] show "y = z" by simp
+ fix i from *[of i] \<open>x \<notin> S\<close> fg show "f i = g i"
+ by (auto split: if_split_asm simp: PiE_def extensional_def)
+qed
+
+lemma card_PiE: "finite S \<Longrightarrow> card (\<Pi>\<^sub>E i \<in> S. T i) = (\<Prod> i\<in>S. card (T i))"
+proof (induct rule: finite_induct)
+ case empty
+ then show ?case by auto
+next
+ case (insert x S)
+ then show ?case
+ by (simp add: PiE_insert_eq inj_combinator card_image card_cartesian_product)
+qed
+
+end
--- a/src/HOL/Library/Library.thy Tue May 15 06:23:12 2018 +0200
+++ b/src/HOL/Library/Library.thy Tue May 15 11:33:43 2018 +0200
@@ -30,6 +30,7 @@
Finite_Map
Float
FSet
+ FuncSet
Function_Division
Fun_Lexorder
Going_To_Filter
--- a/src/HOL/Metis_Examples/Abstraction.thy Tue May 15 06:23:12 2018 +0200
+++ b/src/HOL/Metis_Examples/Abstraction.thy Tue May 15 11:33:43 2018 +0200
@@ -8,7 +8,7 @@
section \<open>Example Featuring Metis's Support for Lambda-Abstractions\<close>
theory Abstraction
-imports HOL.FuncSet
+imports "HOL-Library.FuncSet"
begin
(* For Christoph Benzmüller *)
--- a/src/HOL/Metis_Examples/Tarski.thy Tue May 15 06:23:12 2018 +0200
+++ b/src/HOL/Metis_Examples/Tarski.thy Tue May 15 11:33:43 2018 +0200
@@ -8,7 +8,7 @@
section \<open>Metis Example Featuring the Full Theorem of Tarski\<close>
theory Tarski
-imports Main HOL.FuncSet
+imports Main "HOL-Library.FuncSet"
begin
declare [[metis_new_skolem]]
--- a/src/HOL/Number_Theory/Prime_Powers.thy Tue May 15 06:23:12 2018 +0200
+++ b/src/HOL/Number_Theory/Prime_Powers.thy Tue May 15 11:33:43 2018 +0200
@@ -6,7 +6,7 @@
*)
section \<open>Prime powers\<close>
theory Prime_Powers
- imports Complex_Main "HOL-Computational_Algebra.Primes"
+ imports Complex_Main "HOL-Computational_Algebra.Primes" "HOL-Library.FuncSet"
begin
definition aprimedivisor :: "'a :: normalization_semidom \<Rightarrow> 'a" where
--- a/src/HOL/Vector_Spaces.thy Tue May 15 06:23:12 2018 +0200
+++ b/src/HOL/Vector_Spaces.thy Tue May 15 11:33:43 2018 +0200
@@ -9,7 +9,7 @@
section \<open>Vector Spaces\<close>
theory Vector_Spaces
- imports Modules FuncSet
+ imports Modules
begin
lemma isomorphism_expand:
@@ -847,7 +847,7 @@
lemma linear_exists_left_inverse_on:
assumes lf: "linear s1 s2 f"
assumes V: "vs1.subspace V" and f: "inj_on f V"
- shows "\<exists>g\<in>UNIV \<rightarrow> V. linear s2 s1 g \<and> (\<forall>v\<in>V. g (f v) = v)"
+ shows "\<exists>g. g ` UNIV \<subseteq> V \<and> linear s2 s1 g \<and> (\<forall>v\<in>V. g (f v) = v)"
proof -
interpret linear s1 s2 f by fact
obtain B where V_eq: "V = vs1.span B" and B: "vs1.independent B"
@@ -856,7 +856,7 @@
have f: "inj_on f (vs1.span B)"
using f unfolding V_eq .
show ?thesis
- proof (intro bexI ballI conjI)
+ proof (intro exI ballI conjI)
interpret p: vector_space_pair s2 s1 by unfold_locales
have fB: "vs2.independent (f ` B)"
using independent_injective_image[OF B f] .
@@ -868,7 +868,7 @@
moreover have "the_inv_into B f ` f ` B = B"
by (auto simp: image_comp comp_def the_inv_into_f_f inj_on_subset[OF f vs1.span_superset]
cong: image_cong)
- ultimately show "?g \<in> UNIV \<rightarrow> V"
+ ultimately show "?g ` UNIV \<subseteq> V"
by (auto simp: V_eq)
have "(?g \<circ> f) v = id v" if "v \<in> vs1.span B" for v
proof (rule vector_space_pair.linear_eq_on[where x=v])
@@ -890,7 +890,7 @@
lemma linear_exists_right_inverse_on:
assumes lf: "linear s1 s2 f"
assumes "vs1.subspace V"
- shows "\<exists>g\<in>UNIV \<rightarrow> V. linear s2 s1 g \<and> (\<forall>v\<in>f ` V. f (g v) = v)"
+ shows "\<exists>g. g ` UNIV \<subseteq> V \<and> linear s2 s1 g \<and> (\<forall>v\<in>f ` V. f (g v) = v)"
proof -
obtain B where V_eq: "V = vs1.span B" and B: "vs1.independent B"
using vs1.maximal_independent_subset[of V] vs1.span_minimal[OF _ \<open>vs1.subspace V\<close>]
@@ -900,7 +900,7 @@
then have "\<forall>v\<in>C. \<exists>b\<in>B. v = f b" by auto
then obtain g where g: "\<And>v. v \<in> C \<Longrightarrow> g v \<in> B" "\<And>v. v \<in> C \<Longrightarrow> f (g v) = v" by metis
show ?thesis
- proof (intro bexI ballI conjI)
+ proof (intro exI ballI conjI)
interpret p: vector_space_pair s2 s1 by unfold_locales
let ?g = "p.construct C g"
show "linear ( *b) ( *a) ?g"
@@ -908,7 +908,7 @@
have "?g v \<in> vs1.span (g ` C)" for v
by (rule p.construct_in_span[OF C])
also have "\<dots> \<subseteq> V" unfolding V_eq using g by (intro vs1.span_mono) auto
- finally show "?g \<in> UNIV \<rightarrow> V" by auto
+ finally show "?g ` UNIV \<subseteq> V" by auto
have "(f \<circ> ?g) v = id v" if v: "v \<in> f ` V" for v
proof (rule vector_space_pair.linear_eq_on[where x=v])
show "vector_space_pair ( *b) ( *b)" by unfold_locales
@@ -946,7 +946,7 @@
assumes sf: "vs2.span T \<subseteq> f`vs1.span S"
shows "\<exists>g. range g \<subseteq> vs1.span S \<and> linear s2 s1 g \<and> (\<forall>x\<in>vs2.span T. f (g x) = x)"
using linear_exists_right_inverse_on[OF lf vs1.subspace_span, of S] sf
- by (auto simp: linear_iff_module_hom)
+ by (force simp: linear_iff_module_hom)
lemma linear_surjective_right_inverse: "linear s1 s2 f \<Longrightarrow> surj f \<Longrightarrow> \<exists>g. linear s2 s1 g \<and> f \<circ> g = id"
using linear_surj_right_inverse[of f UNIV UNIV]
--- a/src/HOL/ex/Ballot.thy Tue May 15 06:23:12 2018 +0200
+++ b/src/HOL/ex/Ballot.thy Tue May 15 11:33:43 2018 +0200
@@ -8,6 +8,7 @@
theory Ballot
imports
Complex_Main
+ "HOL-Library.FuncSet"
begin
subsection \<open>Preliminaries\<close>
--- a/src/HOL/ex/Birthday_Paradox.thy Tue May 15 06:23:12 2018 +0200
+++ b/src/HOL/ex/Birthday_Paradox.thy Tue May 15 11:33:43 2018 +0200
@@ -5,7 +5,7 @@
section \<open>A Formulation of the Birthday Paradox\<close>
theory Birthday_Paradox
-imports Main HOL.FuncSet
+imports Main "HOL-Library.FuncSet"
begin
section \<open>Cardinality\<close>
--- a/src/HOL/ex/Tarski.thy Tue May 15 06:23:12 2018 +0200
+++ b/src/HOL/ex/Tarski.thy Tue May 15 11:33:43 2018 +0200
@@ -5,7 +5,7 @@
section \<open>The Full Theorem of Tarski\<close>
theory Tarski
-imports Main HOL.FuncSet
+imports Main "HOL-Library.FuncSet"
begin
text \<open>