--- a/NEWS Wed Oct 21 17:34:35 2009 +0200
+++ b/NEWS Thu Oct 22 09:27:48 2009 +0200
@@ -153,8 +153,8 @@
this. Fix using O_assoc[symmetric]. The same applies to the curried
version "R OO S".
-* Function "Inv" is renamed to "inv_onto" and function "inv" is now an
-abbreviation for "inv_onto UNIV". Lemmas are renamed accordingly.
+* Function "Inv" is renamed to "inv_into" and function "inv" is now an
+abbreviation for "inv_into UNIV". Lemmas are renamed accordingly.
INCOMPATIBILITY.
* ML antiquotation @{code_datatype} inserts definition of a datatype
--- a/doc-src/Main/Docs/Main_Doc.thy Wed Oct 21 17:34:35 2009 +0200
+++ b/doc-src/Main/Docs/Main_Doc.thy Thu Oct 22 09:27:48 2009 +0200
@@ -170,13 +170,13 @@
\smallskip
\begin{tabular}{@ {} l @ {~::~} l @ {}}
-@{const Hilbert_Choice.inv_onto} & @{term_type_only Hilbert_Choice.inv_onto "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)"}
+@{const Hilbert_Choice.inv_into} & @{term_type_only Hilbert_Choice.inv_into "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)"}
\end{tabular}
\subsubsection*{Syntax}
\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}}
-@{term inv} & @{term[source]"inv_onto UNIV"}
+@{term inv} & @{term[source]"inv_into UNIV"}
\end{tabular}
\section{Fixed Points}
--- a/doc-src/Main/Docs/document/Main_Doc.tex Wed Oct 21 17:34:35 2009 +0200
+++ b/doc-src/Main/Docs/document/Main_Doc.tex Thu Oct 22 09:27:48 2009 +0200
@@ -181,13 +181,13 @@
\smallskip
\begin{tabular}{@ {} l @ {~::~} l @ {}}
-\isa{inv{\isacharunderscore}onto} & \isa{{\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}a}
+\isa{inv{\isacharunderscore}into} & \isa{{\isacharprime}a\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}a}
\end{tabular}
\subsubsection*{Syntax}
\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}}
-\isa{inv} & \isa{{\isachardoublequote}inv{\isacharunderscore}onto\ UNIV{\isachardoublequote}}
+\isa{inv} & \isa{{\isachardoublequote}inv{\isacharunderscore}into\ UNIV{\isachardoublequote}}
\end{tabular}
\section{Fixed Points}
--- a/doc-src/TutorialI/Rules/rules.tex Wed Oct 21 17:34:35 2009 +0200
+++ b/doc-src/TutorialI/Rules/rules.tex Thu Oct 22 09:27:48 2009 +0200
@@ -1357,7 +1357,7 @@
some $x$ such that $P(x)$ is true, provided one exists.
Isabelle uses \sdx{SOME} for the Greek letter~$\varepsilon$.
-Here is the definition of~\cdx{inv},\footnote{In fact, \isa{inv} is defined via a second constant \isa{inv_onto}, which we ignore here.} which expresses inverses of
+Here is the definition of~\cdx{inv},\footnote{In fact, \isa{inv} is defined via a second constant \isa{inv_into}, which we ignore here.} which expresses inverses of
functions:
\begin{isabelle}
inv\ f\ \isasymequiv \ \isasymlambda y.\ SOME\ x.\ f\ x\ =\ y%
--- a/src/HOL/Algebra/Bij.thy Wed Oct 21 17:34:35 2009 +0200
+++ b/src/HOL/Algebra/Bij.thy Thu Oct 22 09:27:48 2009 +0200
@@ -31,8 +31,8 @@
subsection {*Bijections Form a Group *}
-lemma restrict_inv_onto_Bij: "f \<in> Bij S \<Longrightarrow> (\<lambda>x \<in> S. (inv_onto S f) x) \<in> Bij S"
- by (simp add: Bij_def bij_betw_inv_onto)
+lemma restrict_inv_into_Bij: "f \<in> Bij S \<Longrightarrow> (\<lambda>x \<in> S. (inv_into S f) x) \<in> Bij S"
+ by (simp add: Bij_def bij_betw_inv_into)
lemma id_Bij: "(\<lambda>x\<in>S. x) \<in> Bij S "
by (auto simp add: Bij_def bij_betw_def inj_on_def)
@@ -41,8 +41,8 @@
by (auto simp add: Bij_def bij_betw_compose)
lemma Bij_compose_restrict_eq:
- "f \<in> Bij S \<Longrightarrow> compose S (restrict (inv_onto S f) S) f = (\<lambda>x\<in>S. x)"
- by (simp add: Bij_def compose_inv_onto_id)
+ "f \<in> Bij S \<Longrightarrow> compose S (restrict (inv_into S f) S) f = (\<lambda>x\<in>S. x)"
+ by (simp add: Bij_def compose_inv_into_id)
theorem group_BijGroup: "group (BijGroup S)"
apply (simp add: BijGroup_def)
@@ -52,19 +52,19 @@
apply (simp add: compose_Bij)
apply (blast intro: compose_assoc [symmetric] dest: Bij_imp_funcset)
apply (simp add: id_Bij Bij_imp_funcset Bij_imp_extensional, simp)
-apply (blast intro: Bij_compose_restrict_eq restrict_inv_onto_Bij)
+apply (blast intro: Bij_compose_restrict_eq restrict_inv_into_Bij)
done
subsection{*Automorphisms Form a Group*}
-lemma Bij_inv_onto_mem: "\<lbrakk> f \<in> Bij S; x \<in> S\<rbrakk> \<Longrightarrow> inv_onto S f x \<in> S"
-by (simp add: Bij_def bij_betw_def inv_onto_into)
+lemma Bij_inv_into_mem: "\<lbrakk> f \<in> Bij S; x \<in> S\<rbrakk> \<Longrightarrow> inv_into S f x \<in> S"
+by (simp add: Bij_def bij_betw_def inv_into_into)
-lemma Bij_inv_onto_lemma:
+lemma Bij_inv_into_lemma:
assumes eq: "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> h(g x y) = g (h x) (h y)"
shows "\<lbrakk>h \<in> Bij S; g \<in> S \<rightarrow> S \<rightarrow> S; x \<in> S; y \<in> S\<rbrakk>
- \<Longrightarrow> inv_onto S h (g x y) = g (inv_onto S h x) (inv_onto S h y)"
+ \<Longrightarrow> inv_into S h (g x y) = g (inv_into S h x) (inv_into S h y)"
apply (simp add: Bij_def bij_betw_def)
apply (subgoal_tac "\<exists>x'\<in>S. \<exists>y'\<in>S. x = h x' & y = h y'", clarify)
apply (simp add: eq [symmetric] inv_f_f funcset_mem [THEN funcset_mem], blast)
@@ -84,17 +84,17 @@
lemma (in group) mult_funcset: "mult G \<in> carrier G \<rightarrow> carrier G \<rightarrow> carrier G"
by (simp add: Pi_I group.axioms)
-lemma (in group) restrict_inv_onto_hom:
+lemma (in group) restrict_inv_into_hom:
"\<lbrakk>h \<in> hom G G; h \<in> Bij (carrier G)\<rbrakk>
- \<Longrightarrow> restrict (inv_onto (carrier G) h) (carrier G) \<in> hom G G"
- by (simp add: hom_def Bij_inv_onto_mem restrictI mult_funcset
- group.axioms Bij_inv_onto_lemma)
+ \<Longrightarrow> restrict (inv_into (carrier G) h) (carrier G) \<in> hom G G"
+ by (simp add: hom_def Bij_inv_into_mem restrictI mult_funcset
+ group.axioms Bij_inv_into_lemma)
lemma inv_BijGroup:
- "f \<in> Bij S \<Longrightarrow> m_inv (BijGroup S) f = (\<lambda>x \<in> S. (inv_onto S f) x)"
+ "f \<in> Bij S \<Longrightarrow> m_inv (BijGroup S) f = (\<lambda>x \<in> S. (inv_into S f) x)"
apply (rule group.inv_equality)
apply (rule group_BijGroup)
-apply (simp_all add:BijGroup_def restrict_inv_onto_Bij Bij_compose_restrict_eq)
+apply (simp_all add:BijGroup_def restrict_inv_into_Bij Bij_compose_restrict_eq)
done
lemma (in group) subgroup_auto:
@@ -115,7 +115,7 @@
assume "x \<in> auto G"
thus "inv\<^bsub>BijGroup (carrier G)\<^esub> x \<in> auto G"
by (simp del: restrict_apply
- add: inv_BijGroup auto_def restrict_inv_onto_Bij restrict_inv_onto_hom)
+ add: inv_BijGroup auto_def restrict_inv_into_Bij restrict_inv_into_hom)
qed
theorem (in group) AutoGroup: "group (AutoGroup G)"
--- a/src/HOL/Algebra/Group.thy Wed Oct 21 17:34:35 2009 +0200
+++ b/src/HOL/Algebra/Group.thy Thu Oct 22 09:27:48 2009 +0200
@@ -553,11 +553,11 @@
by (simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def)
lemma (in group) iso_sym:
- "h \<in> G \<cong> H \<Longrightarrow> inv_onto (carrier G) h \<in> H \<cong> G"
-apply (simp add: iso_def bij_betw_inv_onto)
-apply (subgoal_tac "inv_onto (carrier G) h \<in> carrier H \<rightarrow> carrier G")
- prefer 2 apply (simp add: bij_betw_imp_funcset [OF bij_betw_inv_onto])
-apply (simp add: hom_def bij_betw_def inv_onto_f_eq f_inv_onto_f Pi_def)
+ "h \<in> G \<cong> H \<Longrightarrow> inv_into (carrier G) h \<in> H \<cong> G"
+apply (simp add: iso_def bij_betw_inv_into)
+apply (subgoal_tac "inv_into (carrier G) h \<in> carrier H \<rightarrow> carrier G")
+ prefer 2 apply (simp add: bij_betw_imp_funcset [OF bij_betw_inv_into])
+apply (simp add: hom_def bij_betw_def inv_into_f_eq f_inv_into_f Pi_def)
done
lemma (in group) iso_trans:
--- a/src/HOL/Finite_Set.thy Wed Oct 21 17:34:35 2009 +0200
+++ b/src/HOL/Finite_Set.thy Thu Oct 22 09:27:48 2009 +0200
@@ -162,9 +162,9 @@
from finite_imp_nat_seg_image_inj_on[OF `finite A`]
obtain f and n::nat where bij: "bij_betw f {i. i<n} A"
by (auto simp:bij_betw_def)
- let ?f = "the_inv_onto {i. i<n} f"
+ let ?f = "the_inv_into {i. i<n} f"
have "inj_on ?f A & ?f ` A = {i. i<n}"
- by (fold bij_betw_def) (rule bij_betw_the_inv_onto[OF bij])
+ by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
thus ?thesis by blast
qed
--- a/src/HOL/Fun.thy Wed Oct 21 17:34:35 2009 +0200
+++ b/src/HOL/Fun.thy Thu Oct 22 09:27:48 2009 +0200
@@ -508,65 +508,65 @@
subsection {* Inversion of injective functions *}
-definition the_inv_onto :: "'a set => ('a => 'b) => ('b => 'a)" where
-"the_inv_onto A f == %x. THE y. y : A & f y = x"
+definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
+"the_inv_into A f == %x. THE y. y : A & f y = x"
-lemma the_inv_onto_f_f:
- "[| inj_on f A; x : A |] ==> the_inv_onto A f (f x) = x"
-apply (simp add: the_inv_onto_def inj_on_def)
+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 intro: the_equality)
done
-lemma f_the_inv_onto_f:
- "inj_on f A ==> y : f`A ==> f (the_inv_onto A f y) = y"
-apply (simp add: the_inv_onto_def)
+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 the_inv_onto_into:
- "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_onto A f x : B"
-apply (simp add: the_inv_onto_def)
+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_onto_onto[simp]:
- "inj_on f A ==> the_inv_onto A f ` (f ` A) = A"
-by (fast intro:the_inv_onto_into the_inv_onto_f_f[symmetric])
+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_onto_f_eq:
- "[| inj_on f A; f x = y; x : A |] ==> the_inv_onto A f y = x"
+lemma the_inv_into_f_eq:
+ "[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x"
apply (erule subst)
- apply (erule the_inv_onto_f_f, assumption)
+ apply (erule the_inv_into_f_f, assumption)
done
-lemma the_inv_onto_comp:
+lemma the_inv_into_comp:
"[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
- the_inv_onto A (f o g) x = (the_inv_onto A g o the_inv_onto (g ` A) f) x"
-apply (rule the_inv_onto_f_eq)
+ 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_onto_f the_inv_onto_into)
-apply (simp add: the_inv_onto_into)
+ 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_onto:
- "inj_on f A \<Longrightarrow> inj_on (the_inv_onto A f) (f ` A)"
-by (auto intro: inj_onI simp: image_def the_inv_onto_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: image_def the_inv_into_f_f)
-lemma bij_betw_the_inv_onto:
- "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_onto A f) B A"
-by (auto simp add: bij_betw_def inj_on_the_inv_onto the_inv_onto_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_onto UNIV f"
+ "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_onto_f_f)
+ by (rule the_inv_into_f_f)
subsection {* Proof tool setup *}
--- a/src/HOL/Hilbert_Choice.thy Wed Oct 21 17:34:35 2009 +0200
+++ b/src/HOL/Hilbert_Choice.thy Thu Oct 22 09:27:48 2009 +0200
@@ -31,11 +31,11 @@
in Syntax.const "_Eps" $ x $ t end)]
*}
-definition inv_onto :: "'a set => ('a => 'b) => ('b => 'a)" where
-"inv_onto A f == %x. SOME y. y : A & f y = x"
+definition inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
+"inv_into A f == %x. SOME y. y : A & f y = x"
abbreviation inv :: "('a => 'b) => ('b => 'a)" where
-"inv == inv_onto UNIV"
+"inv == inv_into UNIV"
subsection {*Hilbert's Epsilon-operator*}
@@ -92,40 +92,40 @@
subsection {*Function Inverse*}
lemma inv_def: "inv f = (%y. SOME x. f x = y)"
-by(simp add: inv_onto_def)
+by(simp add: inv_into_def)
-lemma inv_onto_into: "x : f ` A ==> inv_onto A f x : A"
-apply (simp add: inv_onto_def)
+lemma inv_into_into: "x : f ` A ==> inv_into A f x : A"
+apply (simp add: inv_into_def)
apply (fast intro: someI2)
done
lemma inv_id [simp]: "inv id = id"
-by (simp add: inv_onto_def id_def)
+by (simp add: inv_into_def id_def)
-lemma inv_onto_f_f [simp]:
- "[| inj_on f A; x : A |] ==> inv_onto A f (f x) = x"
-apply (simp add: inv_onto_def inj_on_def)
+lemma inv_into_f_f [simp]:
+ "[| inj_on f A; x : A |] ==> inv_into A f (f x) = x"
+apply (simp add: inv_into_def inj_on_def)
apply (blast intro: someI2)
done
lemma inv_f_f: "inj f ==> inv f (f x) = x"
-by (simp add: inv_onto_f_f)
+by (simp add: inv_into_f_f)
-lemma f_inv_onto_f: "y : f`A ==> f (inv_onto A f y) = y"
-apply (simp add: inv_onto_def)
+lemma f_inv_into_f: "y : f`A ==> f (inv_into A f y) = y"
+apply (simp add: inv_into_def)
apply (fast intro: someI2)
done
-lemma inv_onto_f_eq: "[| inj_on f A; x : A; f x = y |] ==> inv_onto A f y = x"
+lemma inv_into_f_eq: "[| inj_on f A; x : A; f x = y |] ==> inv_into A f y = x"
apply (erule subst)
-apply (fast intro: inv_onto_f_f)
+apply (fast intro: inv_into_f_f)
done
lemma inv_f_eq: "[| inj f; f x = y |] ==> inv f y = x"
-by (simp add:inv_onto_f_eq)
+by (simp add:inv_into_f_eq)
lemma inj_imp_inv_eq: "[| inj f; ALL x. f(g x) = x |] ==> inv f = g"
-by (blast intro: ext inv_onto_f_eq)
+by (blast intro: ext inv_into_f_eq)
text{*But is it useful?*}
lemma inj_transfer:
@@ -134,12 +134,12 @@
proof -
have "f x \<in> range f" by auto
hence "P(inv f (f x))" by (rule minor)
- thus "P x" by (simp add: inv_onto_f_f [OF injf])
+ thus "P x" by (simp add: inv_into_f_f [OF injf])
qed
lemma inj_iff: "(inj f) = (inv f o f = id)"
apply (simp add: o_def expand_fun_eq)
-apply (blast intro: inj_on_inverseI inv_onto_f_f)
+apply (blast intro: inj_on_inverseI inv_into_f_f)
done
lemma inv_o_cancel[simp]: "inj f ==> inv f o f = id"
@@ -148,34 +148,34 @@
lemma o_inv_o_cancel[simp]: "inj f ==> g o inv f o f = g"
by (simp add: o_assoc[symmetric])
-lemma inv_onto_image_cancel[simp]:
- "inj_on f A ==> S <= A ==> inv_onto A f ` f ` S = S"
+lemma inv_into_image_cancel[simp]:
+ "inj_on f A ==> S <= A ==> inv_into A f ` f ` S = S"
by(fastsimp simp: image_def)
lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
-by (blast intro: surjI inv_onto_f_f)
+by (blast intro: surjI inv_into_f_f)
lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
-by (simp add: f_inv_onto_f surj_range)
+by (simp add: f_inv_into_f surj_range)
-lemma inv_onto_injective:
- assumes eq: "inv_onto A f x = inv_onto A f y"
+lemma inv_into_injective:
+ assumes eq: "inv_into A f x = inv_into A f y"
and x: "x: f`A"
and y: "y: f`A"
shows "x=y"
proof -
- have "f (inv_onto A f x) = f (inv_onto A f y)" using eq by simp
- thus ?thesis by (simp add: f_inv_onto_f x y)
+ have "f (inv_into A f x) = f (inv_into A f y)" using eq by simp
+ thus ?thesis by (simp add: f_inv_into_f x y)
qed
-lemma inj_on_inv_onto: "B <= f`A ==> inj_on (inv_onto A f) B"
-by (blast intro: inj_onI dest: inv_onto_injective injD)
+lemma inj_on_inv_into: "B <= f`A ==> inj_on (inv_into A f) B"
+by (blast intro: inj_onI dest: inv_into_injective injD)
-lemma bij_betw_inv_onto: "bij_betw f A B ==> bij_betw (inv_onto A f) B A"
-by (auto simp add: bij_betw_def inj_on_inv_onto)
+lemma bij_betw_inv_into: "bij_betw f A B ==> bij_betw (inv_into A f) B A"
+by (auto simp add: bij_betw_def inj_on_inv_into)
lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
-by (simp add: inj_on_inv_onto surj_range)
+by (simp add: inj_on_inv_into surj_range)
lemma surj_iff: "(surj f) = (f o inv f = id)"
apply (simp add: o_def expand_fun_eq)
@@ -193,7 +193,7 @@
lemma inv_equality: "[| !!x. g (f x) = x; !!y. f (g y) = y |] ==> inv f = g"
apply (rule ext)
-apply (auto simp add: inv_onto_def)
+apply (auto simp add: inv_into_def)
done
lemma inv_inv_eq: "bij f ==> inv (inv f) = f"
@@ -208,13 +208,13 @@
inv(inv f)=f all fail.
**)
-lemma inv_onto_comp:
+lemma inv_into_comp:
"[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
- inv_onto A (f o g) x = (inv_onto A g o inv_onto (g ` A) f) x"
-apply (rule inv_onto_f_eq)
+ inv_into A (f o g) x = (inv_into A g o inv_into (g ` A) f) x"
+apply (rule inv_into_f_eq)
apply (fast intro: comp_inj_on)
- apply (simp add: inv_onto_into)
-apply (simp add: f_inv_onto_f inv_onto_into)
+ apply (simp add: inv_into_into)
+apply (simp add: f_inv_into_f inv_into_into)
done
lemma o_inv_distrib: "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"
@@ -239,7 +239,7 @@
lemma bij_vimage_eq_inv_image: "bij f ==> f -` A = inv f ` A"
apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
-apply (blast intro: bij_is_inj [THEN inv_onto_f_f, symmetric])
+apply (blast intro: bij_is_inj [THEN inv_into_f_f, symmetric])
done
lemma finite_fun_UNIVD1:
@@ -256,7 +256,7 @@
moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1)"
proof (rule UNIV_eq_I)
fix x :: 'a
- from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1" by (simp add: inv_onto_def)
+ from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1" by (simp add: inv_into_def)
thus "x \<in> range (\<lambda>f\<Colon>'a \<Rightarrow> 'b. inv f b1)" by blast
qed
ultimately show "finite (UNIV :: 'a set)" by simp
--- a/src/HOL/Library/FuncSet.thy Wed Oct 21 17:34:35 2009 +0200
+++ b/src/HOL/Library/FuncSet.thy Thu Oct 22 09:27:48 2009 +0200
@@ -190,20 +190,20 @@
!!x. x\<in>A ==> f x = g x |] ==> f = g"
by (force simp add: expand_fun_eq extensional_def)
-lemma inv_onto_funcset: "f ` A = B ==> (\<lambda>x\<in>B. inv_onto A f x) : B -> A"
-by (unfold inv_onto_def) (fast intro: someI2)
+lemma inv_into_funcset: "f ` A = B ==> (\<lambda>x\<in>B. inv_into A f x) : B -> A"
+by (unfold inv_into_def) (fast intro: someI2)
-lemma compose_inv_onto_id:
- "bij_betw f A B ==> compose A (\<lambda>y\<in>B. inv_onto A f y) f = (\<lambda>x\<in>A. x)"
+lemma compose_inv_into_id:
+ "bij_betw f A B ==> 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_onto:
- "f ` A = B ==> compose B f (\<lambda>y\<in>B. inv_onto A f y) = (\<lambda>x\<in>B. x)"
+lemma compose_id_inv_into:
+ "f ` A = B ==> 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_onto_f)
+apply (simp add: f_inv_into_f)
done
--- a/src/HOL/Library/Permutations.thy Wed Oct 21 17:34:35 2009 +0200
+++ b/src/HOL/Library/Permutations.thy Thu Oct 22 09:27:48 2009 +0200
@@ -83,7 +83,7 @@
unfolding permutes_def by simp
lemma permutes_inv_eq: "p permutes S ==> inv p y = x \<longleftrightarrow> p x = y"
- unfolding permutes_def inv_onto_def apply auto
+ unfolding permutes_def inv_def apply auto
apply (erule allE[where x=y])
apply (erule allE[where x=y])
apply (rule someI_ex) apply blast
--- a/src/HOL/SizeChange/Correctness.thy Wed Oct 21 17:34:35 2009 +0200
+++ b/src/HOL/SizeChange/Correctness.thy Thu Oct 22 09:27:48 2009 +0200
@@ -1146,7 +1146,7 @@
assumes "finite S"
shows "set (set2list S) = S"
unfolding set2list_def
-proof (rule f_inv_onto_f)
+proof (rule f_inv_into_f)
from `finite S` have "\<exists>l. set l = S"
by (rule finite_list)
thus "S \<in> range set"
--- a/src/HOL/Tools/Datatype/datatype_rep_proofs.ML Wed Oct 21 17:34:35 2009 +0200
+++ b/src/HOL/Tools/Datatype/datatype_rep_proofs.ML Thu Oct 22 09:27:48 2009 +0200
@@ -481,7 +481,7 @@
val Abs_inverse_thms' =
map #1 newT_iso_axms @
- map2 (fn r_inj => fn r => @{thm f_the_inv_onto_f} OF [r_inj, r RS mp])
+ map2 (fn r_inj => fn r => @{thm f_the_inv_into_f} OF [r_inj, r RS mp])
iso_inj_thms_unfolded iso_thms;
val Abs_inverse_thms = maps (mk_funs_inv thy5) Abs_inverse_thms';
@@ -571,7 +571,7 @@
val Abs_t = if i < length newTs then
Const (Sign.intern_const thy6
("Abs_" ^ (nth new_type_names i)), Univ_elT --> T)
- else Const (@{const_name the_inv_onto},
+ else Const (@{const_name the_inv_into},
[HOLogic.mk_setT T, T --> Univ_elT, Univ_elT] ---> T) $
HOLogic.mk_UNIV T $ Const (nth all_rep_names i, T --> Univ_elT)
--- a/src/HOL/UNITY/Extend.thy Wed Oct 21 17:34:35 2009 +0200
+++ b/src/HOL/UNITY/Extend.thy Thu Oct 22 09:27:48 2009 +0200
@@ -121,7 +121,7 @@
assumes surj_h: "surj h"
and prem: "!! x y. g (h(x,y)) = x"
shows "fst (inv h z) = g z"
-by (metis UNIV_I f_inv_onto_f pair_collapse prem surj_h surj_range)
+by (metis UNIV_I f_inv_into_f pair_collapse prem surj_h surj_range)
subsection{*Trivial properties of f, g, h*}
--- a/src/HOL/ZF/HOLZF.thy Wed Oct 21 17:34:35 2009 +0200
+++ b/src/HOL/ZF/HOLZF.thy Thu Oct 22 09:27:48 2009 +0200
@@ -626,7 +626,7 @@
lemma nat2Nat_Nat2nat[simp]: "Elem n Nat \<Longrightarrow> nat2Nat (Nat2nat n) = n"
apply (simp add: Nat2nat_def)
- apply (rule_tac f_inv_onto_f)
+ apply (rule_tac f_inv_into_f)
apply (auto simp add: image_def Nat_def Sep)
done
--- a/src/HOL/ZF/Zet.thy Wed Oct 21 17:34:35 2009 +0200
+++ b/src/HOL/ZF/Zet.thy Thu Oct 22 09:27:48 2009 +0200
@@ -35,7 +35,7 @@
lemma image_zet_rep: "A \<in> zet \<Longrightarrow> ? z . g ` A = explode z"
apply (auto simp add: zet_def')
- apply (rule_tac x="Repl z (g o (inv_onto A f))" in exI)
+ apply (rule_tac x="Repl z (g o (inv_into A f))" in exI)
apply (simp add: explode_Repl_eq)
apply (subgoal_tac "explode z = f ` A")
apply (simp_all add: comp_image_eq)
@@ -49,10 +49,10 @@
by (auto simp add: zet_def')
then obtain f where injf: "inj_on (f :: _ \<Rightarrow> ZF) A"
by auto
- let ?w = "f o (inv_onto A g)"
- have subset: "(inv_onto A g) ` (g ` A) \<subseteq> A"
- by (auto simp add: inv_onto_into)
- have "inj_on (inv_onto A g) (g ` A)" by (simp add: inj_on_inv_onto)
+ let ?w = "f o (inv_into A g)"
+ have subset: "(inv_into A g) ` (g ` A) \<subseteq> A"
+ by (auto simp add: inv_into_into)
+ have "inj_on (inv_into A g) (g ` A)" by (simp add: inj_on_inv_into)
then have injw: "inj_on ?w (g ` A)"
apply (rule comp_inj_on)
apply (rule subset_inj_on[where B=A])
@@ -86,7 +86,7 @@
lemma zexplode_zimplode: "zexplode (zimplode A) = A"
apply (simp add: zimplode_def zexplode_def)
apply (simp add: implode_def)
- apply (subst f_inv_onto_f[where y="Rep_zet A"])
+ apply (subst f_inv_into_f[where y="Rep_zet A"])
apply (auto simp add: Rep_zet_inverse Rep_zet_eq_explode image_def)
done
--- a/src/HOL/ex/set.thy Wed Oct 21 17:34:35 2009 +0200
+++ b/src/HOL/ex/set.thy Thu Oct 22 09:27:48 2009 +0200
@@ -104,7 +104,7 @@
--{*The term above can be synthesized by a sufficiently detailed proof.*}
apply (rule bij_if_then_else)
apply (rule_tac [4] refl)
- apply (rule_tac [2] inj_on_inv_onto)
+ apply (rule_tac [2] inj_on_inv_into)
apply (erule subset_inj_on [OF _ subset_UNIV])
apply blast
apply (erule ssubst, subst double_complement, erule inv_image_comp [symmetric])