src/HOL/Isar_examples/Group.thy

improved presentation;

(*  Title:      HOL/Isar_examples/Group.thy
    ID:         $Id$
    Author:     Markus Wenzel, TU Muenchen
*)

header {* Basic group theory *};

theory Group = Main:;

subsection {* Groups and calculational reasoning *}; 

text {*
 Groups over signature $({\times} :: \alpha \To \alpha \To \alpha,
 \idt{one} :: \alpha, \idt{inv} :: \alpha \To \alpha)$ are defined as
 an axiomatic type class as follows.  Note that the parent class
 $\idt{times}$ is provided by the basic HOL theory.
*};

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 may be derived as follows.
*};

theorem group_right_inverse: "x * inv x = (one::'a::group)";
proof -;
  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 \name{group-right-inverse} already at our disposal,
 \name{group-right-unit} is now obtained much easier.
*};

theorem group_right_unit: "x * one = (x::'a::group)";
proof -;
  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 {*
 \medskip The calculational proof style above follows typical
 presentations given in any introductory course on algebra.  The basic
 technique is to form a transitive chain of equations, which in turn
 are established by simplifying with appropriate rules.  The low-level
 logical parts of equational reasoning are left implicit.

 Note that ``$\dots$'' is just a special term variable that happens to
 be bound automatically to the argument\footnote{The argument of a
 curried infix expression happens to be its right-hand side.} of the
 last fact achieved by any local assumption or proven statement.  In
 contrast to $\var{thesis}$, the ``$\dots$'' variable is bound
 \emph{after} the proof is finished.

 There are only two separate Isar language elements for calculational
 proofs: ``\isakeyword{also}'' for initial or intermediate
 calculational steps, and ``\isakeyword{finally}'' for exhibiting 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 \isakeyword{also} and \isakeyword{finally} derived
 language elements, calculations may be simulated as demonstrated
 below.
*};

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

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

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

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

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

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

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

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

  show ?thesis; .;
qed;

text {*
 Note that this scheme of calculations is not restricted to plain
 transitivity.  Rules like anti-symmetry, or even forward and backward
 substitution work as well.  For the actual \isacommand{also} and
 \isacommand{finally} commands, Isabelle/Isar maintains separate
 context information of ``transitivity'' rules.  Rule selection takes
 place automatically by higher-order unification.
*};


subsection {* Groups as monoids *};

text {*
 Monoids over signature $({\times} :: \alpha \To \alpha \To \alpha,
 \idt{one} :: \alpha)$ are 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 \emph{not} yet monoids directly from the definition.  For
 monoids, \name{right-unit} had to be included as an axiom, but for
 groups both \name{right-unit} and \name{right-inverse} are
 derivable from the other axioms.  With \name{group-right-unit}
 derived as a theorem of group theory (see above), we may still
 instantiate $\idt{group} \subset \idt{monoid}$ properly as follows.
*};

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

text {*
 The \isacommand{instance} command actually is a version of
 \isacommand{theorem}, setting up a goal that reflects the intended
 class relation (or type constructor arity).  Thus any Isar proof
 language element may be involved to establish this statement.  When
 concluding the proof, the result is transformed into the original
 type signature extension behind the scenes.
*};

end;