# HG changeset patch
# User nipkow
# Date 860164408 -7200
# Node ID 3fac3e8d5d3e97cfe816582f70418c31e25cda8e
# Parent  8a680e310f041ec3024d86a4a599784e4f38d13e
moved inj and surj from Set to Fun and Inv -> inv.

diff -r 8a680e310f04 -r 3fac3e8d5d3e src/HOL/Fun.ML
--- a/src/HOL/Fun.ML	Fri Apr 04 16:27:39 1997 +0200
+++ b/src/HOL/Fun.ML	Fri Apr 04 16:33:28 1997 +0200
@@ -19,48 +19,6 @@
 qed "apply_inverse";
 
 
-(*** Image of a set under a function ***)
-
-(*Frequently b does not have the syntactic form of f(x).*)
-val prems = goalw Fun.thy [image_def] "[| b=f(x);  x:A |] ==> b : f``A";
-by (REPEAT (resolve_tac (prems @ [CollectI,bexI,prem]) 1));
-qed "image_eqI";
-
-bind_thm ("imageI", refl RS image_eqI);
-
-(*The eta-expansion gives variable-name preservation.*)
-val major::prems = goalw Fun.thy [image_def]
-    "[| b : (%x.f(x))``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P"; 
-by (rtac (major RS CollectD RS bexE) 1);
-by (REPEAT (ares_tac prems 1));
-qed "imageE";
-
-AddIs  [image_eqI];
-AddSEs [imageE]; 
-
-goalw Fun.thy [o_def] "(f o g)``r = f``(g``r)";
-by (Fast_tac 1);
-qed "image_compose";
-
-goal Fun.thy "f``(A Un B) = f``A Un f``B";
-by (Fast_tac 1);
-qed "image_Un";
-
-(*** Range of a function -- just a translation for image! ***)
-
-goal Fun.thy "!!b. b=f(x) ==> b : range(f)";
-by (EVERY1 [etac image_eqI, rtac UNIV_I]);
-bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
-
-bind_thm ("rangeI", UNIV_I RS imageI);
-
-val [major,minor] = goal Fun.thy 
-    "[| b : range(%x.f(x));  !!x. b=f(x) ==> P |] ==> P"; 
-by (rtac (major RS imageE) 1);
-by (etac minor 1);
-qed "rangeE";
-
-
 (*** inj(f): f is a one-to-one function ***)
 
 val prems = goalw Fun.thy [inj_def]
@@ -95,14 +53,14 @@
 qed "inj_select";
 
 (*A one-to-one function has an inverse (given using select).*)
-val [major] = goalw Fun.thy [Inv_def] "inj(f) ==> Inv f (f x) = x";
+val [major] = goalw Fun.thy [inv_def] "inj(f) ==> inv f (f x) = x";
 by (EVERY1 [rtac (major RS inj_select)]);
-qed "Inv_f_f";
+qed "inv_f_f";
 
 (* Useful??? *)
 val [oneone,minor] = goal Fun.thy
-    "[| inj(f); !!y. y: range(f) ==> P(Inv f y) |] ==> P(x)";
-by (res_inst_tac [("t", "x")] (oneone RS (Inv_f_f RS subst)) 1);
+    "[| inj(f); !!y. y: range(f) ==> P(inv f y) |] ==> P(x)";
+by (res_inst_tac [("t", "x")] (oneone RS (inv_f_f RS subst)) 1);
 by (rtac (rangeI RS minor) 1);
 qed "inj_transfer";
 
@@ -152,27 +110,22 @@
 by (fast_tac (!claset addIs [prem RS injD, inj_ontoI]) 1);
 qed "inj_imp";
 
-val [prem] = goalw Fun.thy [Inv_def] "y : range(f) ==> f(Inv f y) = y";
+val [prem] = goalw Fun.thy [inv_def] "y : range(f) ==> f(inv f y) = y";
 by (EVERY1 [rtac (prem RS rangeE), rtac selectI, etac sym]);
-qed "f_Inv_f";
-
-val prems = goal Fun.thy
-    "[| Inv f x=Inv f y; x: range(f);  y: range(f) |] ==> x=y";
-by (rtac (arg_cong RS box_equals) 1);
-by (REPEAT (resolve_tac (prems @ [f_Inv_f]) 1));
-qed "Inv_injective";
+qed "f_inv_f";
 
 val prems = goal Fun.thy
-    "[| inj(f);  A<=range(f) |] ==> inj_onto (Inv f) A";
+    "[| inv f x=inv f y; x: range(f);  y: range(f) |] ==> x=y";
+by (rtac (arg_cong RS box_equals) 1);
+by (REPEAT (resolve_tac (prems @ [f_inv_f]) 1));
+qed "inv_injective";
+
+val prems = goal Fun.thy
+    "[| inj(f);  A<=range(f) |] ==> inj_onto (inv f) A";
 by (cut_facts_tac prems 1);
 by (fast_tac (!claset addIs [inj_ontoI] 
-                      addEs [Inv_injective,injD]) 1);
-qed "inj_onto_Inv";
+                      addEs [inv_injective,injD]) 1);
+qed "inj_onto_inv";
 
 
-AddIs  [rangeI]; 
-AddSEs [rangeE]; 
-
 val set_cs = !claset delrules [equalityI];
-
-
diff -r 8a680e310f04 -r 3fac3e8d5d3e src/HOL/Fun.thy
--- a/src/HOL/Fun.thy	Fri Apr 04 16:27:39 1997 +0200
+++ b/src/HOL/Fun.thy	Fri Apr 04 16:33:28 1997 +0200
@@ -3,7 +3,22 @@
     Author:     Tobias Nipkow, Cambridge University Computer Laboratory
     Copyright   1994  University of Cambridge
 
-Lemmas about functions.
+Notions about functions.
 *)
 
-Fun = Set
+Fun = Set +
+
+consts
+
+  inj, surj     :: ('a => 'b) => bool                   (*inj/surjective*)
+  inj_onto      :: ['a => 'b, 'a set] => bool
+  inv           :: ('a => 'b) => ('b => 'a)
+
+defs
+
+  inj_def       "inj f          == ! x y. f(x)=f(y) --> x=y"
+  inj_onto_def  "inj_onto f A   == ! x:A. ! y:A. f(x)=f(y) --> x=y"
+  surj_def      "surj f         == ! y. ? x. y=f(x)"
+  inv_def       "inv(f::'a=>'b) == (% y. @x. f(x)=y)"
+
+end
diff -r 8a680e310f04 -r 3fac3e8d5d3e src/HOL/HOL.thy
--- a/src/HOL/HOL.thy	Fri Apr 04 16:27:39 1997 +0200
+++ b/src/HOL/HOL.thy	Fri Apr 04 16:33:28 1997 +0200
@@ -33,7 +33,6 @@
   Not           :: bool => bool                     ("~ _" [40] 40)
   True, False   :: bool
   If            :: [bool, 'a, 'a] => 'a   ("(if (_)/ then (_)/ else (_))" 10)
-  Inv           :: ('a => 'b) => ('b => 'a)
 
   (* Binders *)
 
@@ -170,7 +169,6 @@
   (* Misc Definitions *)
 
   Let_def       "Let s f == f(s)"
-  Inv_def       "Inv(f::'a=>'b)  == (% y. @x. f(x)=y)"
   o_def         "(f::'b=>'c) o g == (%(x::'a). f(g(x)))"
   if_def        "If P x y == @z::'a. (P=True --> z=x) & (P=False --> z=y)"
 
diff -r 8a680e310f04 -r 3fac3e8d5d3e src/HOL/Set.ML
--- a/src/HOL/Set.ML	Fri Apr 04 16:27:39 1997 +0200
+++ b/src/HOL/Set.ML	Fri Apr 04 16:33:28 1997 +0200
@@ -602,6 +602,51 @@
 AddEs  [InterD, InterE];
 
 
+(*** Image of a set under a function ***)
+
+(*Frequently b does not have the syntactic form of f(x).*)
+val prems = goalw thy [image_def] "[| b=f(x);  x:A |] ==> b : f``A";
+by (REPEAT (resolve_tac (prems @ [CollectI,bexI,prem]) 1));
+qed "image_eqI";
+
+bind_thm ("imageI", refl RS image_eqI);
+
+(*The eta-expansion gives variable-name preservation.*)
+val major::prems = goalw thy [image_def]
+    "[| b : (%x.f(x))``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P"; 
+by (rtac (major RS CollectD RS bexE) 1);
+by (REPEAT (ares_tac prems 1));
+qed "imageE";
+
+AddIs  [image_eqI];
+AddSEs [imageE]; 
+
+goalw thy [o_def] "(f o g)``r = f``(g``r)";
+by (Fast_tac 1);
+qed "image_compose";
+
+goal thy "f``(A Un B) = f``A Un f``B";
+by (Fast_tac 1);
+qed "image_Un";
+
+
+(*** Range of a function -- just a translation for image! ***)
+
+goal thy "!!b. b=f(x) ==> b : range(f)";
+by (EVERY1 [etac image_eqI, rtac UNIV_I]);
+bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
+
+bind_thm ("rangeI", UNIV_I RS imageI);
+
+val [major,minor] = goal thy 
+    "[| b : range(%x.f(x));  !!x. b=f(x) ==> P |] ==> P"; 
+by (rtac (major RS imageE) 1);
+by (etac minor 1);
+qed "rangeE";
+
+AddIs  [rangeI]; 
+AddSEs [rangeE]; 
+
 
 (*** Set reasoning tools ***)
 
diff -r 8a680e310f04 -r 3fac3e8d5d3e src/HOL/Set.thy
--- a/src/HOL/Set.thy	Fri Apr 04 16:27:39 1997 +0200
+++ b/src/HOL/Set.thy	Fri Apr 04 16:33:28 1997 +0200
@@ -32,8 +32,6 @@
   Pow           :: 'a set => 'a set set                 (*powerset*)
   range         :: ('a => 'b) => 'b set                 (*of function*)
   Ball, Bex     :: ['a set, 'a => bool] => bool         (*bounded quantifiers*)
-  inj, surj     :: ('a => 'b) => bool                   (*inj/surjective*)
-  inj_onto      :: ['a => 'b, 'a set] => bool
   "``"          :: ['a => 'b, 'a set] => ('b set)   (infixr 90)
   (*membership*)
   "op :"        :: ['a, 'a set] => bool             ("(_/ : _)" [50, 51] 50)
@@ -149,9 +147,6 @@
   empty_def     "{}             == {x. False}"
   insert_def    "insert a B     == {x.x=a} Un B"
   image_def     "f``A           == {y. ? x:A. y=f(x)}"
-  inj_def       "inj f          == ! x y. f(x)=f(y) --> x=y"
-  inj_onto_def  "inj_onto f A   == ! x:A. ! y:A. f(x)=f(y) --> x=y"
-  surj_def      "surj f         == ! y. ? x. y=f(x)"
 
 end
 
diff -r 8a680e310f04 -r 3fac3e8d5d3e src/HOL/equalities.ML
--- a/src/HOL/equalities.ML	Fri Apr 04 16:27:39 1997 +0200
+++ b/src/HOL/equalities.ML	Fri Apr 04 16:33:28 1997 +0200
@@ -412,7 +412,7 @@
 by (Blast_tac 1);
 qed "Int_Union";
 
-(* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: 
+(* Devlin, Setdamentals of Contemporary Set Theory, page 12, exercise 5: 
    Union of a family of unions **)
 goal Set.thy "(UN x:C. A(x) Un B(x)) = Union(A``C)  Un  Union(B``C)";
 by (Blast_tac 1);