|
1247
|
1 |
(* Title: HOL/AxClasses/Tutorial/Group.ML
|
|
|
2 |
ID: $Id$
|
|
|
3 |
Author: Markus Wenzel, TU Muenchen
|
|
|
4 |
|
|
|
5 |
Some basic theorems of group theory.
|
|
|
6 |
*)
|
|
|
7 |
|
|
|
8 |
open Group;
|
|
|
9 |
|
|
|
10 |
fun sub r = standard (r RS subst);
|
|
|
11 |
fun ssub r = standard (r RS ssubst);
|
|
|
12 |
|
|
|
13 |
|
|
|
14 |
goal Group.thy "x <*> inv x = (1::'a::group)";
|
|
|
15 |
br (sub left_unit) 1;
|
|
|
16 |
back();
|
|
|
17 |
br (sub assoc) 1;
|
|
|
18 |
br (sub left_inv) 1;
|
|
|
19 |
back();
|
|
|
20 |
back();
|
|
|
21 |
br (ssub assoc) 1;
|
|
|
22 |
back();
|
|
|
23 |
br (ssub left_inv) 1;
|
|
|
24 |
br (ssub assoc) 1;
|
|
|
25 |
br (ssub left_unit) 1;
|
|
|
26 |
br (ssub left_inv) 1;
|
|
|
27 |
br refl 1;
|
|
|
28 |
qed "right_inv";
|
|
|
29 |
|
|
|
30 |
|
|
|
31 |
goal Group.thy "x <*> 1 = (x::'a::group)";
|
|
|
32 |
br (sub left_inv) 1;
|
|
|
33 |
br (sub assoc) 1;
|
|
|
34 |
br (ssub right_inv) 1;
|
|
|
35 |
br (ssub left_unit) 1;
|
|
|
36 |
br refl 1;
|
|
|
37 |
qed "right_unit";
|
|
|
38 |
|
|
|
39 |
|
|
|
40 |
goal Group.thy "e <*> x = x --> e = (1::'a::group)";
|
|
|
41 |
br impI 1;
|
|
|
42 |
br (sub right_unit) 1;
|
|
|
43 |
back();
|
|
|
44 |
by (res_inst_tac [("x", "x")] (sub right_inv) 1);
|
|
|
45 |
br (sub assoc) 1;
|
|
|
46 |
br arg_cong 1;
|
|
|
47 |
back();
|
|
|
48 |
ba 1;
|
|
|
49 |
qed "strong_one_unit";
|
|
|
50 |
|
|
|
51 |
|
|
|
52 |
goal Group.thy "EX! e. ALL x. e <*> x = (x::'a::group)";
|
|
|
53 |
br ex1I 1;
|
|
|
54 |
br allI 1;
|
|
|
55 |
br left_unit 1;
|
|
|
56 |
br mp 1;
|
|
|
57 |
br strong_one_unit 1;
|
|
|
58 |
be allE 1;
|
|
|
59 |
ba 1;
|
|
|
60 |
qed "ex1_unit";
|
|
|
61 |
|
|
|
62 |
|
|
|
63 |
goal Group.thy "ALL x. EX! e. e <*> x = (x::'a::group)";
|
|
|
64 |
br allI 1;
|
|
|
65 |
br ex1I 1;
|
|
|
66 |
br left_unit 1;
|
|
|
67 |
br (strong_one_unit RS mp) 1;
|
|
|
68 |
ba 1;
|
|
|
69 |
qed "ex1_unit'";
|
|
|
70 |
|
|
|
71 |
|
|
|
72 |
goal Group.thy "y <*> x = 1 --> y = inv (x::'a::group)";
|
|
|
73 |
br impI 1;
|
|
|
74 |
br (sub right_unit) 1;
|
|
|
75 |
back();
|
|
|
76 |
back();
|
|
|
77 |
br (sub right_unit) 1;
|
|
|
78 |
by (res_inst_tac [("x", "x")] (sub right_inv) 1);
|
|
|
79 |
br (sub assoc) 1;
|
|
|
80 |
br (sub assoc) 1;
|
|
|
81 |
br arg_cong 1;
|
|
|
82 |
back();
|
|
|
83 |
br (ssub left_inv) 1;
|
|
|
84 |
ba 1;
|
|
|
85 |
qed "one_inv";
|
|
|
86 |
|
|
|
87 |
|
|
|
88 |
goal Group.thy "ALL x. EX! y. y <*> x = (1::'a::group)";
|
|
|
89 |
br allI 1;
|
|
|
90 |
br ex1I 1;
|
|
|
91 |
br left_inv 1;
|
|
|
92 |
br mp 1;
|
|
|
93 |
br one_inv 1;
|
|
|
94 |
ba 1;
|
|
|
95 |
qed "ex1_inv";
|
|
|
96 |
|
|
|
97 |
|
|
|
98 |
goal Group.thy "inv (x <*> y) = inv y <*> inv (x::'a::group)";
|
|
|
99 |
br sym 1;
|
|
|
100 |
br mp 1;
|
|
|
101 |
br one_inv 1;
|
|
|
102 |
br (ssub assoc) 1;
|
|
|
103 |
br (sub assoc) 1;
|
|
|
104 |
back();
|
|
|
105 |
br (ssub left_inv) 1;
|
|
|
106 |
br (ssub left_unit) 1;
|
|
|
107 |
br (ssub left_inv) 1;
|
|
|
108 |
br refl 1;
|
|
|
109 |
qed "inv_product";
|
|
|
110 |
|
|
|
111 |
|
|
|
112 |
goal Group.thy "inv (inv x) = (x::'a::group)";
|
|
|
113 |
br sym 1;
|
|
|
114 |
br (one_inv RS mp) 1;
|
|
|
115 |
br (ssub right_inv) 1;
|
|
|
116 |
br refl 1;
|
|
|
117 |
qed "inv_inv";
|
|
|
118 |
|