src/HOL/Isar_examples/Group.thy
 author wenzelm Wed, 14 Jul 1999 13:07:09 +0200 changeset 7005 cc778d613217 parent 6908 1bf0590f4790 child 7133 64c9f2364dae permissions -rw-r--r--
tuned comments;
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(*  Title:      HOL/Isar_examples/Group.thy
ID:         \$Id\$
Author:     Markus Wenzel, TU Muenchen
*)

theory Group = Main:;

title {* Basic group theory *};

section {* Groups *};

text {*
We define an axiomatic type class of general groups over signature
(op *, one, inv).
*};

consts
one :: "'a"
inv :: "'a => 'a";

axclass
group < times
group_assoc:         "(x * y) * z = x * (y * z)"
group_left_unit:     "one * x = x"
group_left_inverse:  "inv x * x = one";

text {*
The group axioms only state the properties of left unit and inverse,
the right versions are derivable as follows.  The calculational proof
style below closely follows typical presentations given in any basic
course on algebra.
*};

theorem group_right_inverse: "x * inv x = (one::'a::group)";
proof same;
have "x * inv x = one * (x * inv x)";
by (simp only: group_left_unit);
also; have "... = (one * x) * inv x";
by (simp only: group_assoc);
also; have "... = inv (inv x) * inv x * x * inv x";
by (simp only: group_left_inverse);
also; have "... = inv (inv x) * (inv x * x) * inv x";
by (simp only: group_assoc);
also; have "... = inv (inv x) * one * inv x";
by (simp only: group_left_inverse);
also; have "... = inv (inv x) * (one * inv x)";
by (simp only: group_assoc);
also; have "... = inv (inv x) * inv x";
by (simp only: group_left_unit);
also; have "... = one";
by (simp only: group_left_inverse);
finally; show ??thesis; .;
qed;

text {*
With group_right_inverse already at our disposal, group_right_unit is
now obtained much easier as follows.
*};

theorem group_right_unit: "x * one = (x::'a::group)";
proof same;
have "x * one = x * (inv x * x)";
by (simp only: group_left_inverse);
also; have "... = x * inv x * x";
by (simp only: group_assoc);
also; have "... = one * x";
by (simp only: group_right_inverse);
also; have "... = x";
by (simp only: group_left_unit);
finally; show ??thesis; .;
qed;

text {*
There are only two Isar language elements for calculational proofs:
'also' for initial or intermediate calculational steps, and 'finally'
for building the result of a calculation.  These constructs are not
hardwired into Isabelle/Isar, but defined on top of the basic Isar/VM
interpreter.  Expanding the 'also' or 'finally' derived language
elements, calculations may be simulated as demonstrated below.

Note that "..." is just a special term binding that happens to be
bound automatically to the argument of the last fact established by
assume or any local goal.  In contrast to ??thesis, "..." is bound
after the proof is finished.
*};

theorem "x * one = (x::'a::group)";
proof same;
have "x * one = x * (inv x * x)";
by (simp only: group_left_inverse);

note calculation = facts
-- {* first calculational step: init calculation register *};

have "... = x * inv x * x";
by (simp only: group_assoc);

note calculation = trans [OF calculation facts]
-- {* general calculational step: compose with transitivity rule *};

have "... = one * x";
by (simp only: group_right_inverse);

note calculation = trans [OF calculation facts]
-- {* general calculational step: compose with transitivity rule *};

have "... = x";
by (simp only: group_left_unit);

note calculation = trans [OF calculation facts]
-- {* final calculational step: compose with transitivity rule ... *};
from calculation
-- {* ... and pick up final result *};

show ??thesis; .;
qed;

section {* Groups and monoids *};

text {*
Monoids are usually defined like this.
*};

axclass monoid < times
monoid_assoc:       "(x * y) * z = x * (y * z)"
monoid_left_unit:   "one * x = x"
monoid_right_unit:  "x * one = x";

text {*
Groups are *not* yet monoids directly from the definition.  For
monoids, right_unit had to be included as an axiom, but for groups
both right_unit and right_inverse are derivable from the other
axioms.  With group_right_unit derived as a theorem of group theory
(see above), we may still instantiate group < monoid properly as
follows.
*};

instance group < monoid;
by (expand_classes,
rule group_assoc,
rule group_left_unit,
rule group_right_unit);

end;
```