src/HOL/AxClasses/Tutorial/Group.thy

 (*  Title:      HOL/AxClasses/Tutorial/Group.thy
     ID:         $Id$
     Author:     Markus Wenzel, TU Muenchen
 *)
 
-theory Group = Main:;
+theory Group = Main:
 
-subsection {* Monoids and Groups *};
+subsection {* Monoids and Groups *}
 
 consts
   times :: "'a => 'a => 'a"    (infixl "[*]" 70)
   inverse :: "'a => 'a"
-  one :: 'a;
+  one :: 'a
 
 
 axclass
   monoid < "term"
   assoc:      "(x [*] y) [*] z = x [*] (y [*] z)"
   left_unit:  "one [*] x = x"
-  right_unit: "x [*] one = x";
+  right_unit: "x [*] one = x"
 
 
 axclass
   semigroup < "term"
-  assoc: "(x [*] y) [*] z = x [*] (y [*] z)";
+  assoc: "(x [*] y) [*] z = x [*] (y [*] z)"
 
 axclass
   group < semigroup
   left_unit:    "one [*] x = x"
-  left_inverse: "inverse x [*] x = one";
+  left_inverse: "inverse x [*] x = one"
 
 axclass
   agroup < group
-  commute: "x [*] y = y [*] x";
+  commute: "x [*] y = y [*] x"
 
 
-subsection {* Abstract reasoning *};
+subsection {* Abstract reasoning *}
 
-theorem group_right_inverse: "x [*] inverse x = (one::'a::group)";
-proof -;
-  have "x [*] inverse x = one [*] (x [*] inverse x)";
-    by (simp only: group.left_unit);
-  also; have "... = one [*] x [*] inverse x";
-    by (simp only: semigroup.assoc);
-  also; have "... = inverse (inverse x) [*] inverse x [*] x [*] inverse x";
-    by (simp only: group.left_inverse);
-  also; have "... = inverse (inverse x) [*] (inverse x [*] x) [*] inverse x";
-    by (simp only: semigroup.assoc);
-  also; have "... = inverse (inverse x) [*] one [*] inverse x";
-    by (simp only: group.left_inverse);
-  also; have "... = inverse (inverse x) [*] (one [*] inverse x)";
-    by (simp only: semigroup.assoc);
-  also; have "... = inverse (inverse x) [*] inverse x";
-    by (simp only: group.left_unit);
-  also; have "... = one";
-    by (simp only: group.left_inverse);
-  finally; show ?thesis; .;
-qed;
+theorem group_right_inverse: "x [*] inverse x = (one::'a::group)"
+proof -
+  have "x [*] inverse x = one [*] (x [*] inverse x)"
+    by (simp only: group.left_unit)
+  also have "... = one [*] x [*] inverse x"
+    by (simp only: semigroup.assoc)
+  also have "... = inverse (inverse x) [*] inverse x [*] x [*] inverse x"
+    by (simp only: group.left_inverse)
+  also have "... = inverse (inverse x) [*] (inverse x [*] x) [*] inverse x"
+    by (simp only: semigroup.assoc)
+  also have "... = inverse (inverse x) [*] one [*] inverse x"
+    by (simp only: group.left_inverse)
+  also have "... = inverse (inverse x) [*] (one [*] inverse x)"
+    by (simp only: semigroup.assoc)
+  also have "... = inverse (inverse x) [*] inverse x"
+    by (simp only: group.left_unit)
+  also have "... = one"
+    by (simp only: group.left_inverse)
+  finally show ?thesis .
+qed
 
-theorem group_right_unit: "x [*] one = (x::'a::group)";
-proof -;
-  have "x [*] one = x [*] (inverse x [*] x)";
-    by (simp only: group.left_inverse);
-  also; have "... = x [*] inverse x [*] x";
-    by (simp only: semigroup.assoc);
-  also; have "... = one [*] x";
-    by (simp only: group_right_inverse);
-  also; have "... = x";
-    by (simp only: group.left_unit);
-  finally; show ?thesis; .;
-qed;
+theorem group_right_unit: "x [*] one = (x::'a::group)"
+proof -
+  have "x [*] one = x [*] (inverse x [*] x)"
+    by (simp only: group.left_inverse)
+  also have "... = x [*] inverse x [*] x"
+    by (simp only: semigroup.assoc)
+  also have "... = one [*] x"
+    by (simp only: group_right_inverse)
+  also have "... = x"
+    by (simp only: group.left_unit)
+  finally show ?thesis .
+qed
 
 
-subsection {* Abstract instantiation *};
+subsection {* Abstract instantiation *}
 
-instance monoid < semigroup;
-proof intro_classes;
-  fix x y z :: "'a::monoid";
-  show "x [*] y [*] z = x [*] (y [*] z)";
-    by (rule monoid.assoc);
-qed;
+instance monoid < semigroup
+proof intro_classes
+  fix x y z :: "'a::monoid"
+  show "x [*] y [*] z = x [*] (y [*] z)"
+    by (rule monoid.assoc)
+qed
 
-instance group < monoid;
-proof intro_classes;
-  fix x y z :: "'a::group";
-  show "x [*] y [*] z = x [*] (y [*] z)";
-    by (rule semigroup.assoc);
-  show "one [*] x = x";
-    by (rule group.left_unit);
-  show "x [*] one = x";
-    by (rule group_right_unit);
-qed;
+instance group < monoid
+proof intro_classes
+  fix x y z :: "'a::group"
+  show "x [*] y [*] z = x [*] (y [*] z)"
+    by (rule semigroup.assoc)
+  show "one [*] x = x"
+    by (rule group.left_unit)
+  show "x [*] one = x"
+    by (rule group_right_unit)
+qed
 
 
-subsection {* Concrete instantiation *};
+subsection {* Concrete instantiation *}
 
 defs (overloaded)
   times_bool_def:   "x [*] y == x ~= (y::bool)"
   inverse_bool_def: "inverse x == x::bool"
-  unit_bool_def:    "one == False";
+  unit_bool_def:    "one == False"
 
-instance bool :: agroup;
+instance bool :: agroup
 proof (intro_classes,
-    unfold times_bool_def inverse_bool_def unit_bool_def);
-  fix x y z;
-  show "((x ~= y) ~= z) = (x ~= (y ~= z))"; by blast;
-  show "(False ~= x) = x"; by blast;
-  show "(x ~= x) = False"; by blast;
-  show "(x ~= y) = (y ~= x)"; by blast;
-qed;
+    unfold times_bool_def inverse_bool_def unit_bool_def)
+  fix x y z
+  show "((x ~= y) ~= z) = (x ~= (y ~= z))" by blast
+  show "(False ~= x) = x" by blast
+  show "(x ~= x) = False" by blast
+  show "(x ~= y) = (y ~= x)" by blast
+qed
 
 
-subsection {* Lifting and Functors *};
+subsection {* Lifting and Functors *}
 
 defs (overloaded)
-  times_prod_def: "p [*] q == (fst p [*] fst q, snd p [*] snd q)";
+  times_prod_def: "p [*] q == (fst p [*] fst q, snd p [*] snd q)"
 
-instance * :: (semigroup, semigroup) semigroup;
-proof (intro_classes, unfold times_prod_def);
-  fix p q r :: "'a::semigroup * 'b::semigroup";
+instance * :: (semigroup, semigroup) semigroup
+proof (intro_classes, unfold times_prod_def)
+  fix p q r :: "'a::semigroup * 'b::semigroup"
   show
     "(fst (fst p [*] fst q, snd p [*] snd q) [*] fst r,
       snd (fst p [*] fst q, snd p [*] snd q) [*] snd r) =
        (fst p [*] fst (fst q [*] fst r, snd q [*] snd r),
-        snd p [*] snd (fst q [*] fst r, snd q [*] snd r))";
-    by (simp add: semigroup.assoc);
-qed;
+        snd p [*] snd (fst q [*] fst r, snd q [*] snd r))"
+    by (simp add: semigroup.assoc)
+qed
 
-end;
+end