--- a/src/HOL/Fun.thy Sun Jun 19 22:51:42 2016 +0200
+++ b/src/HOL/Fun.thy Mon Jun 20 17:03:50 2016 +0200
@@ -11,11 +11,10 @@
keywords "functor" :: thy_goal
begin
-lemma apply_inverse:
- "f x = u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u"
+lemma apply_inverse: "f x = u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u"
by auto
-text\<open>Uniqueness, so NOT the axiom of choice.\<close>
+text \<open>Uniqueness, so NOT the axiom of choice.\<close>
lemma uniq_choice: "\<forall>x. \<exists>!y. Q x y \<Longrightarrow> \<exists>f. \<forall>x. Q x (f x)"
by (force intro: theI')
@@ -24,8 +23,8 @@
subsection \<open>The Identity Function \<open>id\<close>\<close>
-definition id :: "'a \<Rightarrow> 'a" where
- "id = (\<lambda>x. x)"
+definition id :: "'a \<Rightarrow> 'a"
+ where "id = (\<lambda>x. x)"
lemma id_apply [simp]: "id x = x"
by (simp add: id_def)
@@ -51,55 +50,51 @@
notation (ASCII)
comp (infixl "o" 55)
-lemma comp_apply [simp]: "(f o g) x = f (g x)"
+lemma comp_apply [simp]: "(f \<circ> g) x = f (g x)"
by (simp add: comp_def)
-lemma comp_assoc: "(f o g) o h = f o (g o h)"
+lemma comp_assoc: "(f \<circ> g) \<circ> h = f \<circ> (g \<circ> h)"
by (simp add: fun_eq_iff)
-lemma id_comp [simp]: "id o g = g"
+lemma id_comp [simp]: "id \<circ> g = g"
by (simp add: fun_eq_iff)
-lemma comp_id [simp]: "f o id = f"
+lemma comp_id [simp]: "f \<circ> id = f"
by (simp add: fun_eq_iff)
lemma comp_eq_dest:
- "a o b = c o d \<Longrightarrow> a (b v) = c (d v)"
+ "a \<circ> b = c \<circ> d \<Longrightarrow> a (b v) = c (d v)"
by (simp add: fun_eq_iff)
lemma comp_eq_elim:
- "a o b = c o d \<Longrightarrow> ((\<And>v. a (b v) = c (d v)) \<Longrightarrow> R) \<Longrightarrow> R"
+ "a \<circ> b = c \<circ> d \<Longrightarrow> ((\<And>v. a (b v) = c (d v)) \<Longrightarrow> R) \<Longrightarrow> R"
by (simp add: fun_eq_iff)
-lemma comp_eq_dest_lhs: "a o b = c \<Longrightarrow> a (b v) = c v"
- by clarsimp
-
-lemma comp_eq_id_dest: "a o b = id o c \<Longrightarrow> a (b v) = c v"
+lemma comp_eq_dest_lhs: "a \<circ> b = c \<Longrightarrow> a (b v) = c v"
by clarsimp
-lemma image_comp:
- "f ` (g ` r) = (f o g) ` r"
+lemma comp_eq_id_dest: "a \<circ> b = id \<circ> c \<Longrightarrow> a (b v) = c v"
+ by clarsimp
+
+lemma image_comp: "f ` (g ` r) = (f \<circ> g) ` r"
by auto
-lemma vimage_comp:
- "f -` (g -` x) = (g \<circ> f) -` x"
+lemma vimage_comp: "f -` (g -` x) = (g \<circ> f) -` x"
by auto
-lemma image_eq_imp_comp: "f ` A = g ` B \<Longrightarrow> (h o f) ` A = (h o g) ` B"
+lemma image_eq_imp_comp: "f ` A = g ` B \<Longrightarrow> (h \<circ> f) ` A = (h \<circ> g) ` B"
by (auto simp: comp_def elim!: equalityE)
lemma image_bind: "f ` (Set.bind A g) = Set.bind A (op ` f \<circ> g)"
-by(auto simp add: Set.bind_def)
+ by (auto simp add: Set.bind_def)
lemma bind_image: "Set.bind (f ` A) g = Set.bind A (g \<circ> f)"
-by(auto simp add: Set.bind_def)
+ by (auto simp add: Set.bind_def)
-lemma (in group_add) minus_comp_minus [simp]:
- "uminus \<circ> uminus = id"
+lemma (in group_add) minus_comp_minus [simp]: "uminus \<circ> uminus = id"
by (simp add: fun_eq_iff)
-lemma (in boolean_algebra) minus_comp_minus [simp]:
- "uminus \<circ> uminus = id"
+lemma (in boolean_algebra) minus_comp_minus [simp]: "uminus \<circ> uminus = id"
by (simp add: fun_eq_iff)
code_printing
@@ -108,8 +103,8 @@
subsection \<open>The Forward Composition Operator \<open>fcomp\<close>\<close>
-definition fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>>" 60) where
- "f \<circ>> g = (\<lambda>x. g (f x))"
+definition fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>>" 60)
+ where "f \<circ>> g = (\<lambda>x. g (f x))"
lemma fcomp_apply [simp]: "(f \<circ>> g) x = g (f x)"
by (simp add: fcomp_def)
@@ -123,7 +118,7 @@
lemma fcomp_id [simp]: "f \<circ>> id = f"
by (simp add: fcomp_def)
-lemma fcomp_comp: "fcomp f g = comp g f"
+lemma fcomp_comp: "fcomp f g = comp g f"
by (simp add: ext)
code_printing
@@ -134,168 +129,143 @@
subsection \<open>Mapping functions\<close>
-definition map_fun :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'd" where
- "map_fun f g h = g \<circ> h \<circ> f"
+definition map_fun :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'd"
+ where "map_fun f g h = g \<circ> h \<circ> f"
-lemma map_fun_apply [simp]:
- "map_fun f g h x = g (h (f x))"
+lemma map_fun_apply [simp]: "map_fun f g h x = g (h (f x))"
by (simp add: map_fun_def)
subsection \<open>Injectivity and Bijectivity\<close>
-definition inj_on :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> bool" where \<comment> "injective"
- "inj_on f A \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
+definition inj_on :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> bool" \<comment> \<open>injective\<close>
+ where "inj_on f A \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
-definition bij_betw :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool" where \<comment> "bijective"
- "bij_betw f A B \<longleftrightarrow> inj_on f A \<and> f ` A = B"
+definition bij_betw :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool" \<comment> \<open>bijective\<close>
+ where "bij_betw f A B \<longleftrightarrow> inj_on f A \<and> f ` A = B"
-text\<open>A common special case: functions injective, surjective or bijective over
-the entire domain type.\<close>
+text \<open>A common special case: functions injective, surjective or bijective over
+ the entire domain type.\<close>
-abbreviation
- "inj f \<equiv> inj_on f UNIV"
+abbreviation "inj f \<equiv> inj_on f UNIV"
-abbreviation surj :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" where \<comment> "surjective"
- "surj f \<equiv> (range f = UNIV)"
+abbreviation surj :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" \<comment> "surjective"
+ where "surj f \<equiv> range f = UNIV"
-abbreviation
- "bij f \<equiv> bij_betw f UNIV UNIV"
+abbreviation "bij f \<equiv> bij_betw f UNIV UNIV"
-text\<open>The negated case:\<close>
+text \<open>The negated case:\<close>
translations
-"\<not> CONST surj f" <= "CONST range f \<noteq> CONST UNIV"
-
-lemma injI:
- assumes "\<And>x y. f x = f y \<Longrightarrow> x = y"
- shows "inj f"
- using assms unfolding inj_on_def by auto
-
-theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)"
- by (unfold inj_on_def, blast)
+ "\<not> CONST surj f" \<leftharpoondown> "CONST range f \<noteq> CONST UNIV"
-lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"
-by (simp add: inj_on_def)
-
-lemma inj_on_eq_iff: "\<lbrakk>inj_on f A; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> (f x = f y) = (x = y)"
-by (force simp add: inj_on_def)
+lemma injI: "(\<And>x y. f x = f y \<Longrightarrow> x = y) \<Longrightarrow> inj f"
+ unfolding inj_on_def by auto
-lemma inj_on_cong:
- "(\<And> a. a : A \<Longrightarrow> f a = g a) \<Longrightarrow> inj_on f A = inj_on g A"
-unfolding inj_on_def by auto
-
-lemma inj_on_strict_subset:
- "inj_on f B \<Longrightarrow> A \<subset> B \<Longrightarrow> f ` A \<subset> f ` B"
+theorem range_ex1_eq: "inj f \<Longrightarrow> b \<in> range f \<longleftrightarrow> (\<exists>!x. b = f x)"
unfolding inj_on_def by blast
-lemma inj_comp:
- "inj f \<Longrightarrow> inj g \<Longrightarrow> inj (f \<circ> g)"
+lemma injD: "inj f \<Longrightarrow> f x = f y \<Longrightarrow> x = y"
+ by (simp add: inj_on_def)
+
+lemma inj_on_eq_iff: "inj_on f A \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x = f y \<longleftrightarrow> x = y"
+ by (force simp add: inj_on_def)
+
+lemma inj_on_cong: "(\<And>a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> inj_on f A = inj_on g A"
+ unfolding inj_on_def by auto
+
+lemma inj_on_strict_subset: "inj_on f B \<Longrightarrow> A \<subset> B \<Longrightarrow> f ` A \<subset> f ` B"
+ unfolding inj_on_def by blast
+
+lemma inj_comp: "inj f \<Longrightarrow> inj g \<Longrightarrow> inj (f \<circ> g)"
by (simp add: inj_on_def)
lemma inj_fun: "inj f \<Longrightarrow> inj (\<lambda>x y. f x)"
by (simp add: inj_on_def fun_eq_iff)
-lemma inj_eq: "inj f ==> (f(x) = f(y)) = (x=y)"
-by (simp add: inj_on_eq_iff)
+lemma inj_eq: "inj f \<Longrightarrow> f x = f y \<longleftrightarrow> x = y"
+ by (simp add: inj_on_eq_iff)
lemma inj_on_id[simp]: "inj_on id A"
by (simp add: inj_on_def)
-lemma inj_on_id2[simp]: "inj_on (%x. x) A"
-by (simp add: inj_on_def)
+lemma inj_on_id2[simp]: "inj_on (\<lambda>x. x) A"
+ by (simp add: inj_on_def)
lemma inj_on_Int: "inj_on f A \<or> inj_on f B \<Longrightarrow> inj_on f (A \<inter> B)"
-unfolding inj_on_def by blast
+ unfolding inj_on_def by blast
lemma surj_id: "surj id"
-by simp
+ by simp
lemma bij_id[simp]: "bij id"
-by (simp add: bij_betw_def)
+ by (simp add: bij_betw_def)
-lemma bij_uminus:
- fixes x :: "'a :: ab_group_add"
- shows "bij (uminus :: 'a\<Rightarrow>'a)"
-unfolding bij_betw_def inj_on_def
-by (force intro: minus_minus [symmetric])
+lemma bij_uminus: "bij (uminus :: 'a \<Rightarrow> 'a::ab_group_add)"
+ unfolding bij_betw_def inj_on_def
+ by (force intro: minus_minus [symmetric])
-lemma inj_onI [intro?]:
- "(!! x y. [| x:A; y:A; f(x) = f(y) |] ==> x=y) ==> inj_on f A"
-by (simp add: inj_on_def)
+lemma inj_onI [intro?]: "(\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x = f y \<Longrightarrow> x = y) \<Longrightarrow> inj_on f A"
+ by (simp add: inj_on_def)
-lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"
-by (auto dest: arg_cong [of concl: g] simp add: inj_on_def)
-
-lemma inj_onD: "[| inj_on f A; f(x)=f(y); x:A; y:A |] ==> x=y"
-by (unfold inj_on_def, blast)
+lemma inj_on_inverseI: "(\<And>x. x \<in> A \<Longrightarrow> g (f x) = x) \<Longrightarrow> inj_on f A"
+ by (auto dest: arg_cong [of concl: g] simp add: inj_on_def)
-lemma comp_inj_on:
- "[| inj_on f A; inj_on g (f`A) |] ==> inj_on (g o f) A"
-by (simp add: comp_def inj_on_def)
+lemma inj_onD: "inj_on f A \<Longrightarrow> f x = f y \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x = y"
+ unfolding inj_on_def by blast
-lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
+lemma comp_inj_on: "inj_on f A \<Longrightarrow> inj_on g (f ` A) \<Longrightarrow> inj_on (g \<circ> f) A"
+ by (simp add: comp_def inj_on_def)
+
+lemma inj_on_imageI: "inj_on (g \<circ> f) A \<Longrightarrow> inj_on g (f ` A)"
by (auto simp add: inj_on_def)
-lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y);
- inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"
-unfolding inj_on_def by blast
+lemma inj_on_image_iff:
+ "\<forall>x\<in>A. \<forall>y\<in>A. g (f x) = g (f y) \<longleftrightarrow> g x = g y \<Longrightarrow> inj_on f A \<Longrightarrow> inj_on g (f ` A) = inj_on g A"
+ unfolding inj_on_def by blast
-lemma inj_on_contraD: "[| inj_on f A; ~x=y; x:A; y:A |] ==> ~ f(x)=f(y)"
-unfolding inj_on_def by blast
+lemma inj_on_contraD: "inj_on f A \<Longrightarrow> x \<noteq> y \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x \<noteq> f y"
+ unfolding inj_on_def by blast
lemma inj_singleton [simp]: "inj_on (\<lambda>x. {x}) A"
by (simp add: inj_on_def)
lemma inj_on_empty[iff]: "inj_on f {}"
-by(simp add: inj_on_def)
-
-lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"
-unfolding inj_on_def by blast
+ by (simp add: inj_on_def)
-lemma inj_on_Un:
- "inj_on f (A Un B) =
- (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
-apply(unfold inj_on_def)
-apply (blast intro:sym)
-done
+lemma subset_inj_on: "inj_on f B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> inj_on f A"
+ unfolding inj_on_def by blast
+
+lemma inj_on_Un: "inj_on f (A \<union> B) \<longleftrightarrow> inj_on f A \<and> inj_on f B \<and> f ` (A - B) \<inter> f ` (B - A) = {}"
+ unfolding inj_on_def by (blast intro: sym)
-lemma inj_on_insert[iff]:
- "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
-apply(unfold inj_on_def)
-apply (blast intro:sym)
-done
+lemma inj_on_insert [iff]: "inj_on f (insert a A) \<longleftrightarrow> inj_on f A \<and> f a \<notin> f ` (A - {a})"
+ unfolding inj_on_def by (blast intro: sym)
+
+lemma inj_on_diff: "inj_on f A \<Longrightarrow> inj_on f (A - B)"
+ unfolding inj_on_def by blast
-lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
-apply(unfold inj_on_def)
-apply (blast)
-done
+lemma comp_inj_on_iff: "inj_on f A \<Longrightarrow> inj_on f' (f ` A) \<longleftrightarrow> inj_on (f' \<circ> f) A"
+ by (auto simp add: comp_inj_on inj_on_def)
-lemma comp_inj_on_iff:
- "inj_on f A \<Longrightarrow> inj_on f' (f ` A) \<longleftrightarrow> inj_on (f' o f) A"
-by(auto simp add: comp_inj_on inj_on_def)
-
-lemma inj_on_imageI2:
- "inj_on (f' o f) A \<Longrightarrow> inj_on f A"
-by(auto simp add: comp_inj_on inj_on_def)
+lemma inj_on_imageI2: "inj_on (f' \<circ> f) A \<Longrightarrow> inj_on f A"
+ by (auto simp add: comp_inj_on inj_on_def)
lemma inj_img_insertE:
assumes "inj_on f A"
- assumes "x \<notin> B" and "insert x B = f ` A"
- obtains x' A' where "x' \<notin> A'" and "A = insert x' A'"
- and "x = f x'" and "B = f ` A'"
+ assumes "x \<notin> B"
+ and "insert x B = f ` A"
+ obtains x' A' where "x' \<notin> A'" and "A = insert x' A'" and "x = f x'" and "B = f ` A'"
proof -
from assms have "x \<in> f ` A" by auto
then obtain x' where *: "x' \<in> A" "x = f x'" by auto
- then have "A = insert x' (A - {x'})" by auto
- with assms * have "B = f ` (A - {x'})"
- by (auto dest: inj_on_contraD)
+ then have A: "A = insert x' (A - {x'})" by auto
+ with assms * have B: "B = f ` (A - {x'})" by (auto dest: inj_on_contraD)
have "x' \<notin> A - {x'}" by simp
- from \<open>x' \<notin> A - {x'}\<close> \<open>A = insert x' (A - {x'})\<close> \<open>x = f x'\<close> \<open>B = image f (A - {x'})\<close>
- show ?thesis ..
+ from this A \<open>x = f x'\<close> B show ?thesis ..
qed
lemma linorder_injI:
- assumes hyp: "\<And>x y. x < (y::'a::linorder) \<Longrightarrow> f x \<noteq> f y"
+ assumes hyp: "\<And>x y::'a::linorder. x < y \<Longrightarrow> f x \<noteq> f y"
shows "inj f"
\<comment> \<open>Courtesy of Stephan Merz\<close>
proof (rule inj_onI)
@@ -307,7 +277,9 @@
lemma surj_def: "surj f \<longleftrightarrow> (\<forall>y. \<exists>x. y = f x)"
by auto
-lemma surjI: assumes *: "\<And> x. g (f x) = x" shows "surj g"
+lemma surjI:
+ assumes *: "\<And> x. g (f x) = x"
+ shows "surj g"
using *[symmetric] by auto
lemma surjD: "surj f \<Longrightarrow> \<exists>x. y = f x"
@@ -316,15 +288,17 @@
lemma surjE: "surj f \<Longrightarrow> (\<And>x. y = f x \<Longrightarrow> C) \<Longrightarrow> C"
by (simp add: surj_def, blast)
-lemma comp_surj: "[| surj f; surj g |] ==> surj (g o f)"
-apply (simp add: comp_def surj_def, clarify)
-apply (drule_tac x = y in spec, clarify)
-apply (drule_tac x = x in spec, blast)
-done
+lemma comp_surj: "surj f \<Longrightarrow> surj g \<Longrightarrow> surj (g \<circ> f)"
+ apply (simp add: comp_def surj_def)
+ apply clarify
+ apply (drule_tac x = y in spec)
+ apply clarify
+ apply (drule_tac x = x in spec)
+ apply blast
+ done
-lemma bij_betw_imageI:
- "\<lbrakk> inj_on f A; f ` A = B \<rbrakk> \<Longrightarrow> bij_betw f A B"
-unfolding bij_betw_def by clarify
+lemma bij_betw_imageI: "inj_on f A \<Longrightarrow> f ` A = B \<Longrightarrow> bij_betw f A B"
+ unfolding bij_betw_def by clarify
lemma bij_betw_imp_surj_on: "bij_betw f A B \<Longrightarrow> f ` A = B"
unfolding bij_betw_def by clarify
@@ -332,122 +306,119 @@
lemma bij_betw_imp_surj: "bij_betw f A UNIV \<Longrightarrow> surj f"
unfolding bij_betw_def by auto
-lemma bij_betw_empty1:
- assumes "bij_betw f {} A"
- shows "A = {}"
-using assms unfolding bij_betw_def by blast
+lemma bij_betw_empty1: "bij_betw f {} A \<Longrightarrow> A = {}"
+ unfolding bij_betw_def by blast
-lemma bij_betw_empty2:
- assumes "bij_betw f A {}"
- shows "A = {}"
-using assms unfolding bij_betw_def by blast
+lemma bij_betw_empty2: "bij_betw f A {} \<Longrightarrow> A = {}"
+ unfolding bij_betw_def by blast
-lemma inj_on_imp_bij_betw:
- "inj_on f A \<Longrightarrow> bij_betw f A (f ` A)"
-unfolding bij_betw_def by simp
+lemma inj_on_imp_bij_betw: "inj_on f A \<Longrightarrow> bij_betw f A (f ` A)"
+ unfolding bij_betw_def by simp
lemma bij_def: "bij f \<longleftrightarrow> inj f \<and> surj f"
unfolding bij_betw_def ..
-lemma bijI: "[| inj f; surj f |] ==> bij f"
-by (simp add: bij_def)
+lemma bijI: "inj f \<Longrightarrow> surj f \<Longrightarrow> bij f"
+ by (simp add: bij_def)
-lemma bij_is_inj: "bij f ==> inj f"
-by (simp add: bij_def)
+lemma bij_is_inj: "bij f \<Longrightarrow> inj f"
+ by (simp add: bij_def)
-lemma bij_is_surj: "bij f ==> surj f"
-by (simp add: bij_def)
+lemma bij_is_surj: "bij f \<Longrightarrow> surj f"
+ by (simp add: bij_def)
lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
-by (simp add: bij_betw_def)
+ by (simp add: bij_betw_def)
-lemma bij_betw_trans:
- "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"
-by(auto simp add:bij_betw_def comp_inj_on)
+lemma bij_betw_trans: "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g \<circ> f) A C"
+ by (auto simp add:bij_betw_def comp_inj_on)
-lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
+lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g \<circ> f)"
by (rule bij_betw_trans)
-lemma bij_betw_comp_iff:
- "bij_betw f A A' \<Longrightarrow> bij_betw f' A' A'' \<longleftrightarrow> bij_betw (f' o f) A A''"
-by(auto simp add: bij_betw_def inj_on_def)
+lemma bij_betw_comp_iff: "bij_betw f A A' \<Longrightarrow> bij_betw f' A' A'' \<longleftrightarrow> bij_betw (f' \<circ> f) A A''"
+ by (auto simp add: bij_betw_def inj_on_def)
lemma bij_betw_comp_iff2:
- assumes BIJ: "bij_betw f' A' A''" and IM: "f ` A \<le> A'"
- shows "bij_betw f A A' \<longleftrightarrow> bij_betw (f' o f) A A''"
-using assms
-proof(auto simp add: bij_betw_comp_iff)
+ assumes bij: "bij_betw f' A' A''"
+ and img: "f ` A \<le> A'"
+ shows "bij_betw f A A' \<longleftrightarrow> bij_betw (f' \<circ> f) A A''"
+ using assms
+proof (auto simp add: bij_betw_comp_iff)
assume *: "bij_betw (f' \<circ> f) A A''"
- thus "bij_betw f A A'"
- using IM
- proof(auto simp add: bij_betw_def)
+ then show "bij_betw f A A'"
+ using img
+ proof (auto simp add: bij_betw_def)
assume "inj_on (f' \<circ> f) A"
- thus "inj_on f A" using inj_on_imageI2 by blast
+ then show "inj_on f A" using inj_on_imageI2 by blast
next
- fix a' assume **: "a' \<in> A'"
- hence "f' a' \<in> A''" using BIJ unfolding bij_betw_def by auto
- then obtain a where 1: "a \<in> A \<and> f'(f a) = f' a'" using *
- unfolding bij_betw_def by force
- hence "f a \<in> A'" using IM by auto
- hence "f a = a'" using BIJ ** 1 unfolding bij_betw_def inj_on_def by auto
- thus "a' \<in> f ` A" using 1 by auto
+ fix a'
+ assume **: "a' \<in> A'"
+ then have "f' a' \<in> A''" using bij unfolding bij_betw_def by auto
+ then obtain a where 1: "a \<in> A \<and> f'(f a) = f' a'"
+ using * unfolding bij_betw_def by force
+ then have "f a \<in> A'" using img by auto
+ then have "f a = a'"
+ using bij ** 1 unfolding bij_betw_def inj_on_def by auto
+ then show "a' \<in> f ` A"
+ using 1 by auto
qed
qed
-lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
+lemma bij_betw_inv:
+ assumes "bij_betw f A B"
+ shows "\<exists>g. bij_betw g B A"
proof -
have i: "inj_on f A" and s: "f ` A = B"
- using assms by(auto simp:bij_betw_def)
- let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"
- { fix a b assume P: "?P b a"
- hence ex1: "\<exists>a. ?P b a" using s by blast
- hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])
- hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp
- } note g = this
+ using assms by (auto simp: bij_betw_def)
+ let ?P = "\<lambda>b a. a \<in> A \<and> f a = b"
+ let ?g = "\<lambda>b. The (?P b)"
+ have g: "?g b = a" if P: "?P b a" for a b
+ proof -
+ from that have ex1: "\<exists>a. ?P b a" using s by blast
+ then have uex1: "\<exists>!a. ?P b a" by (blast dest:inj_onD[OF i])
+ then show ?thesis using the1_equality[OF uex1, OF P] P by simp
+ qed
have "inj_on ?g B"
- proof(rule inj_onI)
- fix x y assume "x:B" "y:B" "?g x = ?g y"
- from s \<open>x:B\<close> obtain a1 where a1: "?P x a1" by blast
- from s \<open>y:B\<close> obtain a2 where a2: "?P y a2" by blast
- from g[OF a1] a1 g[OF a2] a2 \<open>?g x = ?g y\<close> show "x=y" by simp
+ proof (rule inj_onI)
+ fix x y
+ assume "x \<in> B" "y \<in> B" "?g x = ?g y"
+ from s \<open>x \<in> B\<close> obtain a1 where a1: "?P x a1" by blast
+ from s \<open>y \<in> B\<close> obtain a2 where a2: "?P y a2" by blast
+ from g [OF a1] a1 g [OF a2] a2 \<open>?g x = ?g y\<close> show "x = y" by simp
qed
moreover have "?g ` B = A"
- proof(auto simp: image_def)
- fix b assume "b:B"
+ proof (auto simp: image_def)
+ fix b
+ assume "b \<in> B"
with s obtain a where P: "?P b a" by blast
- thus "?g b \<in> A" using g[OF P] by auto
+ then show "?g b \<in> A" using g[OF P] by auto
next
- fix a assume "a:A"
+ fix a
+ assume "a \<in> A"
then obtain b where P: "?P b a" using s by blast
- then have "b:B" using s by blast
+ then have "b \<in> B" using s by blast
with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast
qed
- ultimately show ?thesis by(auto simp:bij_betw_def)
+ ultimately show ?thesis by (auto simp: bij_betw_def)
qed
-lemma bij_betw_cong:
- "(\<And> a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> bij_betw f A A' = bij_betw g A A'"
-unfolding bij_betw_def inj_on_def by force
+lemma bij_betw_cong: "(\<And> a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> bij_betw f A A' = bij_betw g A A'"
+ unfolding bij_betw_def inj_on_def by force
-lemma bij_betw_id[intro, simp]:
- "bij_betw id A A"
-unfolding bij_betw_def id_def by auto
+lemma bij_betw_id[intro, simp]: "bij_betw id A A"
+ unfolding bij_betw_def id_def by auto
-lemma bij_betw_id_iff:
- "bij_betw id A B \<longleftrightarrow> A = B"
-by(auto simp add: bij_betw_def)
+lemma bij_betw_id_iff: "bij_betw id A B \<longleftrightarrow> A = B"
+ by (auto simp add: bij_betw_def)
lemma bij_betw_combine:
assumes "bij_betw f A B" "bij_betw f C D" "B \<inter> D = {}"
shows "bij_betw f (A \<union> C) (B \<union> D)"
using assms unfolding bij_betw_def inj_on_Un image_Un by auto
-lemma bij_betw_subset:
- assumes BIJ: "bij_betw f A A'" and
- SUB: "B \<le> A" and IM: "f ` B = B'"
- shows "bij_betw f B B'"
-using assms
-by(unfold bij_betw_def inj_on_def, auto simp add: inj_on_def)
+lemma bij_betw_subset: "bij_betw f A A' \<Longrightarrow> B \<le> A \<Longrightarrow> f ` B = B' \<Longrightarrow> bij_betw f B B'"
+ by (auto simp add: bij_betw_def inj_on_def)
lemma bij_pointE:
assumes "bij f"
@@ -460,85 +431,77 @@
with that show thesis by blast
qed
-lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
-by simp
+lemma surj_image_vimage_eq: "surj f \<Longrightarrow> f ` (f -` A) = A"
+ by simp
lemma surj_vimage_empty:
- assumes "surj f" shows "f -` A = {} \<longleftrightarrow> A = {}"
- using surj_image_vimage_eq[OF \<open>surj f\<close>, of A]
+ assumes "surj f"
+ shows "f -` A = {} \<longleftrightarrow> A = {}"
+ using surj_image_vimage_eq [OF \<open>surj f\<close>, of A]
by (intro iffI) fastforce+
-lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
-by (simp add: inj_on_def, blast)
+lemma inj_vimage_image_eq: "inj f \<Longrightarrow> f -` (f ` A) = A"
+ unfolding inj_on_def by blast
-lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
-by (blast intro: sym)
+lemma vimage_subsetD: "surj f \<Longrightarrow> f -` B \<subseteq> A \<Longrightarrow> B \<subseteq> f ` A"
+ by (blast intro: sym)
-lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
-by (unfold inj_on_def, blast)
+lemma vimage_subsetI: "inj f \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> f -` B \<subseteq> A"
+ unfolding inj_on_def by blast
-lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
-apply (unfold bij_def)
-apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
-done
+lemma vimage_subset_eq: "bij f \<Longrightarrow> f -` B \<subseteq> A \<longleftrightarrow> B \<subseteq> f ` A"
+ unfolding bij_def by (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
-lemma inj_on_image_eq_iff: "\<lbrakk> inj_on f C; A \<subseteq> C; B \<subseteq> C \<rbrakk> \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
-by(fastforce simp add: inj_on_def)
+lemma inj_on_image_eq_iff: "inj_on f C \<Longrightarrow> A \<subseteq> C \<Longrightarrow> B \<subseteq> C \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
+ by (fastforce simp add: inj_on_def)
lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
-by(erule inj_on_image_eq_iff) simp_all
+ by (erule inj_on_image_eq_iff) simp_all
-lemma inj_on_image_Int:
- "[| inj_on f C; A<=C; B<=C |] ==> f`(A Int B) = f`A Int f`B"
- by (simp add: inj_on_def, blast)
+lemma inj_on_image_Int: "inj_on f C \<Longrightarrow> A \<subseteq> C \<Longrightarrow> B \<subseteq> C \<Longrightarrow> f ` (A \<inter> B) = f ` A \<inter> f ` B"
+ unfolding inj_on_def by blast
+
+lemma inj_on_image_set_diff: "inj_on f C \<Longrightarrow> A - B \<subseteq> C \<Longrightarrow> B \<subseteq> C \<Longrightarrow> f ` (A - B) = f ` A - f ` B"
+ unfolding inj_on_def by blast
-lemma inj_on_image_set_diff:
- "[| inj_on f C; A-B \<subseteq> C; B \<subseteq> C |] ==> f`(A-B) = f`A - f`B"
- by (simp add: inj_on_def, blast)
+lemma image_Int: "inj f \<Longrightarrow> f ` (A \<inter> B) = f ` A \<inter> f ` B"
+ unfolding inj_on_def by blast
-lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
- by (simp add: inj_on_def, blast)
+lemma image_set_diff: "inj f \<Longrightarrow> f ` (A - B) = f ` A - f ` B"
+ unfolding inj_on_def by blast
-lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
-by (simp add: inj_on_def, blast)
-
-lemma inj_on_image_mem_iff: "\<lbrakk>inj_on f B; a \<in> B; A \<subseteq> B\<rbrakk> \<Longrightarrow> f a \<in> f`A \<longleftrightarrow> a \<in> A"
+lemma inj_on_image_mem_iff: "inj_on f B \<Longrightarrow> a \<in> B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f a \<in> f ` A \<longleftrightarrow> a \<in> A"
by (auto simp: inj_on_def)
(*FIXME DELETE*)
-lemma inj_on_image_mem_iff_alt: "inj_on f B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f a \<in> f`A \<Longrightarrow> a \<in> B \<Longrightarrow> a \<in> A"
+lemma inj_on_image_mem_iff_alt: "inj_on f B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f a \<in> f ` A \<Longrightarrow> a \<in> B \<Longrightarrow> a \<in> A"
by (blast dest: inj_onD)
-lemma inj_image_mem_iff: "inj f \<Longrightarrow> f a \<in> f`A \<longleftrightarrow> a \<in> A"
+lemma inj_image_mem_iff: "inj f \<Longrightarrow> f a \<in> f ` A \<longleftrightarrow> a \<in> A"
by (blast dest: injD)
-lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
+lemma inj_image_subset_iff: "inj f \<Longrightarrow> f ` A \<subseteq> f ` B \<longleftrightarrow> A \<subseteq> B"
by (blast dest: injD)
-lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
+lemma inj_image_eq_iff: "inj f \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
by (blast dest: injD)
-lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
-by auto
-
-lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
-by (auto simp add: inj_on_def)
+lemma surj_Compl_image_subset: "surj f \<Longrightarrow> - (f ` A) \<subseteq> f ` (- A)"
+ by auto
-lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
-apply (simp add: bij_def)
-apply (rule equalityI)
-apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
-done
+lemma inj_image_Compl_subset: "inj f \<Longrightarrow> f ` (- A) \<subseteq> - (f ` A)"
+ by (auto simp add: inj_on_def)
+
+lemma bij_image_Compl_eq: "bij f \<Longrightarrow> f ` (- A) = - (f ` A)"
+ by (simp add: bij_def inj_image_Compl_subset surj_Compl_image_subset equalityI)
lemma inj_vimage_singleton: "inj f \<Longrightarrow> f -` {a} \<subseteq> {THE x. f x = a}"
- \<comment> \<open>The inverse image of a singleton under an injective function
- is included in a singleton.\<close>
+ \<comment> \<open>The inverse image of a singleton under an injective function is included in a singleton.\<close>
apply (auto simp add: inj_on_def)
apply (blast intro: the_equality [symmetric])
done
-lemma inj_on_vimage_singleton:
- "inj_on f A \<Longrightarrow> f -` {a} \<inter> A \<subseteq> {THE x. x \<in> A \<and> f x = a}"
+lemma inj_on_vimage_singleton: "inj_on f A \<Longrightarrow> f -` {a} \<inter> A \<subseteq> {THE x. x \<in> A \<and> f x = a}"
by (auto simp add: inj_on_def intro: the_equality [symmetric])
lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A"
@@ -548,84 +511,92 @@
by (auto intro!: inj_onI dest: strict_mono_eq)
lemma bij_betw_byWitness:
-assumes LEFT: "\<forall>a \<in> A. f'(f a) = a" and
- RIGHT: "\<forall>a' \<in> A'. f(f' a') = a'" and
- IM1: "f ` A \<le> A'" and IM2: "f' ` A' \<le> A"
-shows "bij_betw f A A'"
-using assms
-proof(unfold bij_betw_def inj_on_def, safe)
- fix a b assume *: "a \<in> A" "b \<in> A" and **: "f a = f b"
- have "a = f'(f a) \<and> b = f'(f b)" using * LEFT by simp
+ assumes left: "\<forall>a \<in> A. f' (f a) = a"
+ and right: "\<forall>a' \<in> A'. f (f' a') = a'"
+ and "f ` A \<le> A'"
+ and img2: "f' ` A' \<le> A"
+ shows "bij_betw f A A'"
+ using assms
+proof (unfold bij_betw_def inj_on_def, safe)
+ fix a b
+ assume *: "a \<in> A" "b \<in> A" and **: "f a = f b"
+ have "a = f' (f a) \<and> b = f'(f b)" using * left by simp
with ** show "a = b" by simp
next
fix a' assume *: "a' \<in> A'"
- hence "f' a' \<in> A" using IM2 by blast
+ hence "f' a' \<in> A" using img2 by blast
moreover
- have "a' = f(f' a')" using * RIGHT by simp
+ have "a' = f (f' a')" using * right by simp
ultimately show "a' \<in> f ` A" by blast
qed
corollary notIn_Un_bij_betw:
-assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'" and
- BIJ: "bij_betw f A A'"
-shows "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
-proof-
+ assumes "b \<notin> A"
+ and "f b \<notin> A'"
+ and "bij_betw f A A'"
+ shows "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
+proof -
have "bij_betw f {b} {f b}"
- unfolding bij_betw_def inj_on_def by simp
+ unfolding bij_betw_def inj_on_def by simp
with assms show ?thesis
- using bij_betw_combine[of f A A' "{b}" "{f b}"] by blast
+ using bij_betw_combine[of f A A' "{b}" "{f b}"] by blast
qed
lemma notIn_Un_bij_betw3:
-assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'"
-shows "bij_betw f A A' = bij_betw f (A \<union> {b}) (A' \<union> {f b})"
+ assumes "b \<notin> A"
+ and "f b \<notin> A'"
+ shows "bij_betw f A A' = bij_betw f (A \<union> {b}) (A' \<union> {f b})"
proof
assume "bij_betw f A A'"
- thus "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
- using assms notIn_Un_bij_betw[of b A f A'] by blast
+ then show "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
+ using assms notIn_Un_bij_betw [of b A f A'] by blast
next
assume *: "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
have "f ` A = A'"
- proof(auto)
- fix a assume **: "a \<in> A"
- hence "f a \<in> A' \<union> {f b}" using * unfolding bij_betw_def by blast
+ proof auto
+ fix a
+ assume **: "a \<in> A"
+ then have "f a \<in> A' \<union> {f b}"
+ using * unfolding bij_betw_def by blast
moreover
- {assume "f a = f b"
- hence "a = b" using * ** unfolding bij_betw_def inj_on_def by blast
- with NIN ** have False by blast
- }
+ have False if "f a = f b"
+ proof -
+ have "a = b" using * ** that unfolding bij_betw_def inj_on_def by blast
+ with \<open>b \<notin> A\<close> ** show ?thesis by blast
+ qed
ultimately show "f a \<in> A'" by blast
next
- fix a' assume **: "a' \<in> A'"
- hence "a' \<in> f`(A \<union> {b})"
- using * by (auto simp add: bij_betw_def)
+ fix a'
+ assume **: "a' \<in> A'"
+ then have "a' \<in> f ` (A \<union> {b})"
+ using * by (auto simp add: bij_betw_def)
then obtain a where 1: "a \<in> A \<union> {b} \<and> f a = a'" by blast
moreover
- {assume "a = b" with 1 ** NIN' have False by blast
- }
+ have False if "a = b" using 1 ** \<open>f b \<notin> A'\<close> that by blast
ultimately have "a \<in> A" by blast
with 1 show "a' \<in> f ` A" by blast
qed
- thus "bij_betw f A A'" using * bij_betw_subset[of f "A \<union> {b}" _ A] by blast
+ then show "bij_betw f A A'"
+ using * bij_betw_subset[of f "A \<union> {b}" _ A] by blast
qed
-subsection\<open>Function Updating\<close>
+subsection \<open>Function Updating\<close>
-definition fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where
- "fun_upd f a b == % x. if x=a then b else f x"
+definition fun_upd :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow> 'b)"
+ where "fun_upd f a b \<equiv> \<lambda>x. if x = a then b else f x"
nonterminal updbinds and updbind
syntax
- "_updbind" :: "['a, 'a] => updbind" ("(2_ :=/ _)")
- "" :: "updbind => updbinds" ("_")
- "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
- "_Update" :: "['a, updbinds] => 'a" ("_/'((_)')" [1000, 0] 900)
+ "_updbind" :: "'a \<Rightarrow> 'a \<Rightarrow> updbind" ("(2_ :=/ _)")
+ "" :: "updbind \<Rightarrow> updbinds" ("_")
+ "_updbinds":: "updbind \<Rightarrow> updbinds \<Rightarrow> updbinds" ("_,/ _")
+ "_Update" :: "'a \<Rightarrow> updbinds \<Rightarrow> 'a" ("_/'((_)')" [1000, 0] 900)
translations
- "_Update f (_updbinds b bs)" == "_Update (_Update f b) bs"
- "f(x:=y)" == "CONST fun_upd f x y"
+ "_Update f (_updbinds b bs)" \<rightleftharpoons> "_Update (_Update f b) bs"
+ "f(x:=y)" \<rightleftharpoons> "CONST fun_upd f x y"
(* Hint: to define the sum of two functions (or maps), use case_sum.
A nice infix syntax could be defined by
@@ -633,69 +604,69 @@
case_sum (infixr "'(+')"80)
*)
-lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
-apply (simp add: fun_upd_def, safe)
-apply (erule subst)
-apply (rule_tac [2] ext, auto)
-done
+lemma fun_upd_idem_iff: "f(x:=y) = f \<longleftrightarrow> f x = y"
+ unfolding fun_upd_def
+ apply safe
+ apply (erule subst)
+ apply (rule_tac [2] ext)
+ apply auto
+ done
-lemma fun_upd_idem: "f x = y ==> f(x:=y) = f"
+lemma fun_upd_idem: "f x = y \<Longrightarrow> f(x := y) = f"
by (simp only: fun_upd_idem_iff)
lemma fun_upd_triv [iff]: "f(x := f x) = f"
by (simp only: fun_upd_idem)
-lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
-by (simp add: fun_upd_def)
+lemma fun_upd_apply [simp]: "(f(x := y)) z = (if z = x then y else f z)"
+ by (simp add: fun_upd_def)
-(* fun_upd_apply supersedes these two, but they are useful
+(* fun_upd_apply supersedes these two, but they are useful
if fun_upd_apply is intentionally removed from the simpset *)
-lemma fun_upd_same: "(f(x:=y)) x = y"
-by simp
+lemma fun_upd_same: "(f(x := y)) x = y"
+ by simp
-lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
-by simp
-
-lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
-by (simp add: fun_eq_iff)
+lemma fun_upd_other: "z \<noteq> x \<Longrightarrow> (f(x := y)) z = f z"
+ by simp
-lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
-by (rule ext, auto)
+lemma fun_upd_upd [simp]: "f(x := y, x := z) = f(x := z)"
+ by (simp add: fun_eq_iff)
-lemma inj_on_fun_updI:
- "inj_on f A \<Longrightarrow> y \<notin> f ` A \<Longrightarrow> inj_on (f(x := y)) A"
+lemma fun_upd_twist: "a \<noteq> c \<Longrightarrow> (m(a := b))(c := d) = (m(c := d))(a := b)"
+ by (rule ext) auto
+
+lemma inj_on_fun_updI: "inj_on f A \<Longrightarrow> y \<notin> f ` A \<Longrightarrow> inj_on (f(x := y)) A"
by (fastforce simp: inj_on_def)
-lemma fun_upd_image:
- "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
-by auto
+lemma fun_upd_image: "f(x := y) ` A = (if x \<in> A then insert y (f ` (A - {x})) else f ` A)"
+ by auto
lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
by auto
lemma fun_upd_eqD: "f(x := y) = g(x := z) \<Longrightarrow> y = z"
-by(simp add: fun_eq_iff split: if_split_asm)
+ by (simp add: fun_eq_iff split: if_split_asm)
+
subsection \<open>\<open>override_on\<close>\<close>
-definition override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b" where
- "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
+definition override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
+ where "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
lemma override_on_emptyset[simp]: "override_on f g {} = f"
-by(simp add:override_on_def)
+ by (simp add:override_on_def)
-lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
-by(simp add:override_on_def)
+lemma override_on_apply_notin[simp]: "a \<notin> A \<Longrightarrow> (override_on f g A) a = f a"
+ by (simp add:override_on_def)
-lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
-by(simp add:override_on_def)
+lemma override_on_apply_in[simp]: "a \<in> A \<Longrightarrow> (override_on f g A) a = g a"
+ by (simp add:override_on_def)
subsection \<open>\<open>swap\<close>\<close>
definition swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
-where
- "swap a b f = f (a := f b, b:= f a)"
+ where "swap a b f = f (a := f b, b:= f a)"
lemma swap_apply [simp]:
"swap a b f a = f b"
@@ -703,20 +674,16 @@
"c \<noteq> a \<Longrightarrow> c \<noteq> b \<Longrightarrow> swap a b f c = f c"
by (simp_all add: swap_def)
-lemma swap_self [simp]:
- "swap a a f = f"
+lemma swap_self [simp]: "swap a a f = f"
by (simp add: swap_def)
-lemma swap_commute:
- "swap a b f = swap b a f"
+lemma swap_commute: "swap a b f = swap b a f"
by (simp add: fun_upd_def swap_def fun_eq_iff)
-lemma swap_nilpotent [simp]:
- "swap a b (swap a b f) = f"
+lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
by (rule ext, simp add: fun_upd_def swap_def)
-lemma swap_comp_involutory [simp]:
- "swap a b \<circ> swap a b = id"
+lemma swap_comp_involutory [simp]: "swap a b \<circ> swap a b = id"
by (rule ext) simp
lemma swap_triple:
@@ -725,10 +692,11 @@
using assms by (simp add: fun_eq_iff swap_def)
lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)"
- by (rule ext, simp add: fun_upd_def swap_def)
+ by (rule ext) (simp add: fun_upd_def swap_def)
lemma swap_image_eq [simp]:
- assumes "a \<in> A" "b \<in> A" shows "swap a b f ` A = f ` A"
+ assumes "a \<in> A" "b \<in> A"
+ shows "swap a b f ` A = f ` A"
proof -
have subset: "\<And>f. swap a b f ` A \<subseteq> f ` A"
using assms by (auto simp: image_iff swap_def)
@@ -736,20 +704,21 @@
with subset[of f] show ?thesis by auto
qed
-lemma inj_on_imp_inj_on_swap:
- "\<lbrakk>inj_on f A; a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> inj_on (swap a b f) A"
- by (simp add: inj_on_def swap_def, blast)
+lemma inj_on_imp_inj_on_swap: "inj_on f A \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> inj_on (swap a b f) A"
+ by (auto simp add: inj_on_def swap_def)
lemma inj_on_swap_iff [simp]:
- assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A \<longleftrightarrow> inj_on f A"
+ assumes A: "a \<in> A" "b \<in> A"
+ shows "inj_on (swap a b f) A \<longleftrightarrow> inj_on f A"
proof
assume "inj_on (swap a b f) A"
with A have "inj_on (swap a b (swap a b f)) A"
by (iprover intro: inj_on_imp_inj_on_swap)
- thus "inj_on f A" by simp
+ then show "inj_on f A" by simp
next
assume "inj_on f A"
- with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)
+ with A show "inj_on (swap a b f) A"
+ by (iprover intro: inj_on_imp_inj_on_swap)
qed
lemma surj_imp_surj_swap: "surj f \<Longrightarrow> surj (swap a b f)"
@@ -758,8 +727,7 @@
lemma surj_swap_iff [simp]: "surj (swap a b f) \<longleftrightarrow> surj f"
by simp
-lemma bij_betw_swap_iff [simp]:
- "\<lbrakk> x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> bij_betw (swap x y f) A B \<longleftrightarrow> bij_betw f A B"
+lemma bij_betw_swap_iff [simp]: "x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> bij_betw (swap x y f) A B \<longleftrightarrow> bij_betw f A B"
by (auto simp: bij_betw_def)
lemma bij_swap_iff [simp]: "bij (swap a b f) \<longleftrightarrow> bij f"
@@ -770,114 +738,107 @@
subsection \<open>Inversion of injective functions\<close>
-definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
- "the_inv_into A f == %x. THE y. y : A & f y = x"
+definition the_inv_into :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)"
+ where "the_inv_into A f \<equiv> \<lambda>x. THE y. y \<in> A \<and> f y = x"
+
+lemma the_inv_into_f_f: "inj_on f A \<Longrightarrow> x \<in> A \<Longrightarrow> the_inv_into A f (f x) = x"
+ unfolding the_inv_into_def inj_on_def by blast
-lemma the_inv_into_f_f:
- "[| inj_on f A; x : A |] ==> the_inv_into A f (f x) = x"
-apply (simp add: the_inv_into_def inj_on_def)
-apply blast
-done
-
-lemma f_the_inv_into_f:
- "inj_on f A ==> y : f`A ==> f (the_inv_into A f y) = y"
-apply (simp add: the_inv_into_def)
-apply (rule the1I2)
- apply(blast dest: inj_onD)
-apply blast
-done
+lemma f_the_inv_into_f: "inj_on f A \<Longrightarrow> y \<in> f ` A \<Longrightarrow> f (the_inv_into A f y) = y"
+ apply (simp add: the_inv_into_def)
+ apply (rule the1I2)
+ apply(blast dest: inj_onD)
+ apply blast
+ done
-lemma the_inv_into_into:
- "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B"
-apply (simp add: the_inv_into_def)
-apply (rule the1I2)
- apply(blast dest: inj_onD)
-apply blast
-done
+lemma the_inv_into_into: "inj_on f A \<Longrightarrow> x \<in> f ` A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> the_inv_into A f x \<in> B"
+ apply (simp add: the_inv_into_def)
+ apply (rule the1I2)
+ apply(blast dest: inj_onD)
+ apply blast
+ done
-lemma the_inv_into_onto[simp]:
- "inj_on f A ==> the_inv_into A f ` (f ` A) = A"
-by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])
+lemma the_inv_into_onto [simp]: "inj_on f A \<Longrightarrow> the_inv_into A f ` (f ` A) = A"
+ by (fast intro: the_inv_into_into the_inv_into_f_f [symmetric])
-lemma the_inv_into_f_eq:
- "[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x"
+lemma the_inv_into_f_eq: "inj_on f A \<Longrightarrow> f x = y \<Longrightarrow> x \<in> A \<Longrightarrow> the_inv_into A f y = x"
apply (erule subst)
- apply (erule the_inv_into_f_f, assumption)
+ apply (erule the_inv_into_f_f)
+ apply assumption
done
lemma the_inv_into_comp:
- "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
- the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x"
-apply (rule the_inv_into_f_eq)
- apply (fast intro: comp_inj_on)
- apply (simp add: f_the_inv_into_f the_inv_into_into)
-apply (simp add: the_inv_into_into)
-done
+ "inj_on f (g ` A) \<Longrightarrow> inj_on g A \<Longrightarrow> x \<in> f ` g ` A \<Longrightarrow>
+ the_inv_into A (f \<circ> g) x = (the_inv_into A g \<circ> the_inv_into (g ` A) f) x"
+ apply (rule the_inv_into_f_eq)
+ apply (fast intro: comp_inj_on)
+ apply (simp add: f_the_inv_into_f the_inv_into_into)
+ apply (simp add: the_inv_into_into)
+ done
-lemma inj_on_the_inv_into:
- "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
-by (auto intro: inj_onI simp: the_inv_into_f_f)
+lemma inj_on_the_inv_into: "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
+ by (auto intro: inj_onI simp: the_inv_into_f_f)
-lemma bij_betw_the_inv_into:
- "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"
-by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)
+lemma bij_betw_the_inv_into: "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"
+ by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)
-abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
- "the_inv f \<equiv> the_inv_into UNIV f"
+abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)"
+ where "the_inv f \<equiv> the_inv_into UNIV f"
lemma the_inv_f_f:
assumes "inj f"
- shows "the_inv f (f x) = x" using assms UNIV_I
- by (rule the_inv_into_f_f)
+ shows "the_inv f (f x) = x"
+ using assms UNIV_I by (rule the_inv_into_f_f)
subsection \<open>Cantor's Paradox\<close>
-lemma Cantors_paradox:
- "\<not>(\<exists>f. f ` A = Pow A)"
+lemma Cantors_paradox: "\<not> (\<exists>f. f ` A = Pow A)"
proof clarify
- fix f assume "f ` A = Pow A" hence *: "Pow A \<le> f ` A" by blast
+ fix f
+ assume "f ` A = Pow A"
+ then have *: "Pow A \<subseteq> f ` A" by blast
let ?X = "{a \<in> A. a \<notin> f a}"
have "?X \<in> Pow A" unfolding Pow_def by auto
with * obtain x where "x \<in> A \<and> f x = ?X" by blast
- thus False by best
+ then show False by best
qed
+
subsection \<open>Setup\<close>
subsubsection \<open>Proof tools\<close>
-text \<open>simplifies terms of the form
- f(...,x:=y,...,x:=z,...) to f(...,x:=z,...)\<close>
+text \<open>Simplify terms of the form f(...,x:=y,...,x:=z,...) to f(...,x:=z,...)\<close>
simproc_setup fun_upd2 ("f(v := w, x := y)") = \<open>fn _ =>
-let
- fun gen_fun_upd NONE T _ _ = NONE
- | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y)
- fun dest_fun_T1 (Type (_, T :: Ts)) = T
- fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) =
- let
- fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) =
- if v aconv x then SOME g else gen_fun_upd (find g) T v w
- | find t = NONE
- in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
+ let
+ fun gen_fun_upd NONE T _ _ = NONE
+ | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y)
+ fun dest_fun_T1 (Type (_, T :: Ts)) = T
+ fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) =
+ let
+ fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) =
+ if v aconv x then SOME g else gen_fun_upd (find g) T v w
+ | find t = NONE
+ in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
- val ss = simpset_of @{context}
+ val ss = simpset_of @{context}
- fun proc ctxt ct =
- let
- val t = Thm.term_of ct
- in
- case find_double t of
- (T, NONE) => NONE
- | (T, SOME rhs) =>
- SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
- (fn _ =>
- resolve_tac ctxt [eq_reflection] 1 THEN
- resolve_tac ctxt @{thms ext} 1 THEN
- simp_tac (put_simpset ss ctxt) 1))
- end
-in proc end
+ fun proc ctxt ct =
+ let
+ val t = Thm.term_of ct
+ in
+ case find_double t of
+ (T, NONE) => NONE
+ | (T, SOME rhs) =>
+ SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
+ (fn _ =>
+ resolve_tac ctxt [eq_reflection] 1 THEN
+ resolve_tac ctxt @{thms ext} 1 THEN
+ simp_tac (put_simpset ss ctxt) 1))
+ end
+ in proc end
\<close>
@@ -891,6 +852,7 @@
functor vimage
by (simp_all add: fun_eq_iff vimage_comp)
+
text \<open>Legacy theorem names\<close>
lemmas o_def = comp_def
@@ -904,4 +866,3 @@
lemmas o_eq_id_dest = comp_eq_id_dest
end
-