# HG changeset patch
# User paulson
# Date 888487657 -3600
# Node ID 134d24ddaad3222de1eae5a93f2096891de94a71
# Parent  481628ea8eddca499ea8ede3de1c3792b2429771
Proved choice and bchoice; changed Fun.thy -> thy

diff -r 481628ea8edd -r 134d24ddaad3 src/HOL/Fun.ML
--- a/src/HOL/Fun.ML	Thu Feb 26 10:48:19 1998 +0100
+++ b/src/HOL/Fun.ML	Thu Feb 26 11:07:37 1998 +0100
@@ -6,59 +6,71 @@
 Lemmas about functions.
 *)
 
-goal Fun.thy "(f = g) = (!x. f(x)=g(x))";
+
+goal thy "(f = g) = (!x. f(x)=g(x))";
 by (rtac iffI 1);
 by (Asm_simp_tac 1);
 by (rtac ext 1 THEN Asm_simp_tac 1);
 qed "expand_fun_eq";
 
-val prems = goal Fun.thy
+val prems = goal thy
     "[| f(x)=u;  !!x. P(x) ==> g(f(x)) = x;  P(x) |] ==> x=g(u)";
 by (rtac (arg_cong RS box_equals) 1);
 by (REPEAT (resolve_tac (prems@[refl]) 1));
 qed "apply_inverse";
 
 
+(** "Axiom" of Choice, proved using the description operator **)
+
+goal HOL.thy "!!Q. ALL x. EX y. Q x y ==> EX f. ALL x. Q x (f x)";
+by (fast_tac (claset() addEs [selectI]) 1);
+qed "choice";
+
+goal Set.thy "!!S. ALL x:S. EX y. Q x y ==> EX f. ALL x:S. Q x (f x)";
+by (fast_tac (claset() addEs [selectI]) 1);
+qed "bchoice";
+
+
 (*** inj(f): f is a one-to-one function ***)
 
-val prems = goalw Fun.thy [inj_def]
+val prems = goalw thy [inj_def]
     "[| !! x y. f(x) = f(y) ==> x=y |] ==> inj(f)";
 by (blast_tac (claset() addIs prems) 1);
 qed "injI";
 
-val [major] = goal Fun.thy "(!!x. g(f(x)) = x) ==> inj(f)";
+val [major] = goal thy "(!!x. g(f(x)) = x) ==> inj(f)";
 by (rtac injI 1);
 by (etac (arg_cong RS box_equals) 1);
 by (rtac major 1);
 by (rtac major 1);
 qed "inj_inverseI";
 
-val [major,minor] = goalw Fun.thy [inj_def]
+val [major,minor] = goalw thy [inj_def]
     "[| inj(f); f(x) = f(y) |] ==> x=y";
 by (rtac (major RS spec RS spec RS mp) 1);
 by (rtac minor 1);
 qed "injD";
 
 (*Useful with the simplifier*)
-val [major] = goal Fun.thy "inj(f) ==> (f(x) = f(y)) = (x=y)";
+val [major] = goal thy "inj(f) ==> (f(x) = f(y)) = (x=y)";
 by (rtac iffI 1);
 by (etac (major RS injD) 1);
 by (etac arg_cong 1);
 qed "inj_eq";
 
-val [major] = goal Fun.thy "inj(f) ==> (@x. f(x)=f(y)) = y";
+val [major] = goal thy "inj(f) ==> (@x. f(x)=f(y)) = y";
 by (rtac (major RS injD) 1);
 by (rtac selectI 1);
 by (rtac refl 1);
 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 thy [inv_def] "inj(f) ==> inv f (f x) = x";
 by (EVERY1 [rtac (major RS inj_select)]);
 qed "inv_f_f";
 
 (* Useful??? *)
-val [oneone,minor] = goal Fun.thy
+val [oneone,minor] = goal 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);
 by (rtac (rangeI RS minor) 1);
@@ -67,36 +79,36 @@
 
 (*** inj_onto f A: f is one-to-one over A ***)
 
-val prems = goalw Fun.thy [inj_onto_def]
+val prems = goalw thy [inj_onto_def]
     "(!! x y. [| f(x) = f(y);  x:A;  y:A |] ==> x=y) ==> inj_onto f A";
 by (blast_tac (claset() addIs prems) 1);
 qed "inj_ontoI";
 
-val [major] = goal Fun.thy 
+val [major] = goal thy 
     "(!!x. x:A ==> g(f(x)) = x) ==> inj_onto f A";
 by (rtac inj_ontoI 1);
 by (etac (apply_inverse RS trans) 1);
 by (REPEAT (eresolve_tac [asm_rl,major] 1));
 qed "inj_onto_inverseI";
 
-val major::prems = goalw Fun.thy [inj_onto_def]
+val major::prems = goalw thy [inj_onto_def]
     "[| inj_onto f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y";
 by (rtac (major RS bspec RS bspec RS mp) 1);
 by (REPEAT (resolve_tac prems 1));
 qed "inj_ontoD";
 
-goal Fun.thy "!!x y.[| inj_onto f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)";
+goal thy "!!x y.[| inj_onto f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)";
 by (blast_tac (claset() addSDs [inj_ontoD]) 1);
 qed "inj_onto_iff";
 
-val major::prems = goal Fun.thy
+val major::prems = goal thy
     "[| inj_onto f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)";
 by (rtac contrapos 1);
 by (etac (major RS inj_ontoD) 2);
 by (REPEAT (resolve_tac prems 1));
 qed "inj_onto_contraD";
 
-goalw Fun.thy [inj_onto_def]
+goalw thy [inj_onto_def]
     "!!A B. [| A<=B; inj_onto f B |] ==> inj_onto f A";
 by (Blast_tac 1);
 qed "subset_inj_onto";
@@ -104,26 +116,26 @@
 
 (*** Lemmas about inj ***)
 
-goalw Fun.thy [o_def]
+goalw thy [o_def]
     "!!f g. [| inj(f);  inj_onto g (range f) |] ==> inj(g o f)";
 by (fast_tac (claset() addIs [injI] addEs [injD, inj_ontoD]) 1);
 qed "comp_inj";
 
-val [prem] = goal Fun.thy "inj(f) ==> inj_onto f A";
+val [prem] = goal thy "inj(f) ==> inj_onto f A";
 by (blast_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 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
+val prems = goal 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";
 
-goal Fun.thy "!!f. [| inj(f);  A<=range(f) |] ==> inj_onto (inv f) A";
+goal thy "!!f. [| inj(f);  A<=range(f) |] ==> inj_onto (inv f) A";
 by (fast_tac (claset() addIs [inj_ontoI] 
                       addEs [inv_injective,injD]) 1);
 qed "inj_onto_inv";