diff -r 706cfddca75c -r 18b1441fb603 src/HOL/AxClasses/Tutorial/Group.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/AxClasses/Tutorial/Group.ML	Fri Sep 01 14:27:36 1995 +0200
@@ -0,0 +1,118 @@
+(*  Title:      HOL/AxClasses/Tutorial/Group.ML
+    ID:         $Id$
+    Author:     Markus Wenzel, TU Muenchen
+
+Some basic theorems of group theory.
+*)
+
+open Group;
+
+fun sub r = standard (r RS subst);
+fun ssub r = standard (r RS ssubst);
+
+
+goal Group.thy "x <*> inv x = (1::'a::group)";
+br (sub left_unit) 1;
+back();
+br (sub assoc) 1;
+br (sub left_inv) 1;
+back();
+back();
+br (ssub assoc) 1;
+back();
+br (ssub left_inv) 1;
+br (ssub assoc) 1;
+br (ssub left_unit) 1;
+br (ssub left_inv) 1;
+br refl 1;
+qed "right_inv";
+
+
+goal Group.thy "x <*> 1 = (x::'a::group)";
+br (sub left_inv) 1;
+br (sub assoc) 1;
+br (ssub right_inv) 1;
+br (ssub left_unit) 1;
+br refl 1;
+qed "right_unit";
+
+
+goal Group.thy "e <*> x = x --> e = (1::'a::group)";
+br impI 1;
+br (sub right_unit) 1;
+back();
+by (res_inst_tac [("x", "x")] (sub right_inv) 1);
+br (sub assoc) 1;
+br arg_cong 1;
+back();
+ba 1;
+qed "strong_one_unit";
+
+
+goal Group.thy "EX! e. ALL x. e <*> x = (x::'a::group)";
+br ex1I 1;
+br allI 1;
+br left_unit 1;
+br mp 1;
+br strong_one_unit 1;
+be allE 1;
+ba 1;
+qed "ex1_unit";
+
+
+goal Group.thy "ALL x. EX! e. e <*> x = (x::'a::group)";
+br allI 1;
+br ex1I 1;
+br left_unit 1;
+br (strong_one_unit RS mp) 1;
+ba 1;
+qed "ex1_unit'";
+
+
+goal Group.thy "y <*> x = 1 --> y = inv (x::'a::group)";
+br impI 1;
+br (sub right_unit) 1;
+back();
+back();
+br (sub right_unit) 1;
+by (res_inst_tac [("x", "x")] (sub right_inv) 1);
+br (sub assoc) 1;
+br (sub assoc) 1;
+br arg_cong 1;
+back();
+br (ssub left_inv) 1;
+ba 1;
+qed "one_inv";
+
+
+goal Group.thy "ALL x. EX! y. y <*> x = (1::'a::group)";
+br allI 1;
+br ex1I 1;
+br left_inv 1;
+br mp 1;
+br one_inv 1;
+ba 1;
+qed "ex1_inv";
+
+
+goal Group.thy "inv (x <*> y) = inv y <*> inv (x::'a::group)";
+br sym 1;
+br mp 1;
+br one_inv 1;
+br (ssub assoc) 1;
+br (sub assoc) 1;
+back();
+br (ssub left_inv) 1;
+br (ssub left_unit) 1;
+br (ssub left_inv) 1;
+br refl 1;
+qed "inv_product";
+
+
+goal Group.thy "inv (inv x) = (x::'a::group)";
+br sym 1;
+br (one_inv RS mp) 1;
+br (ssub right_inv) 1;
+br refl 1;
+qed "inv_inv";
+