src/HOL/Isar_examples/Peirce.thy

improved presentation;

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

header {* Peirce's Law *};

theory Peirce = Main:;

text {*
 We consider Peirce's Law: $((A \impl B) \impl A) \impl A$.  This is
 an inherently non-intuitionistic statement, so its proof will
 certainly involve some form of classical contradiction.

 The first proof is again a well-balanced combination of plain
 backward and forward reasoning.  The actual classical step is where
 the negated goal may be introduced as additional assumption.  This
 eventually leads to a contradiction.\footnote{The rule involved there
 is negation elimination; it holds in intuitionistic logic as well.}
*};

theorem "((A --> B) --> A) --> A";
proof;
  assume aba: "(A --> B) --> A";
  show A;
  proof (rule classical);
    assume "~ A";
    have "A --> B";
    proof;
      assume A;
      thus B; by contradiction;
    qed;
    with aba; show A; ..;
  qed;
qed;

text {*
 In the subsequent version the reasoning is rearranged by means of
 ``weak assumptions'' (as introduced by \isacommand{presume}).  Before
 assuming the negated goal $\neg A$, its intended consequence $A \impl
 B$ is put into place in order to solve the main problem.
 Nevertheless, we do not get anything for free, but have to establish
 $A \impl B$ later on.  The overall effect is that of a logical
 \emph{cut}.

 Technically speaking, whenever some goal is solved by
 \isacommand{show} in the context of weak assumptions then the latter
 give rise to new subgoals, which may be established separately.  In
 contrast, strong assumptions (as introduced by \isacommand{assume})
 are solved immediately.
*};

theorem "((A --> B) --> A) --> A";
proof;
  assume aba: "(A --> B) --> A";
  show A;
  proof (rule classical);
    presume "A --> B";
    with aba; show A; ..;
  next;
    assume "~ A";
    show "A --> B";
    proof;
      assume A;
      thus B; by contradiction;
    qed;
  qed;
qed;

text {*
 Note that the goals stemming from weak assumptions may be even left
 until qed time, where they get eventually solved ``by assumption'' as
 well.  In that case there is really no fundamental difference between
 the two kinds of assumptions, apart from the order of reducing the
 individual parts of the proof configuration.

 Nevertheless, the ``strong'' mode of plain assumptions is quite
 important in practice to achieve robustness of proof document
 interpretation.  By forcing both the conclusion \emph{and} the
 assumptions to unify with the pending goal to be solved, goal
 selection becomes quite deterministic.  For example, decomposition
 with ``case-analysis'' type rules usually give rise to several goals
 that only differ in there local contexts.  With strong assumptions
 these may be still solved in any order in a predictable way, while
 weak ones would quickly lead to great confusion, eventually demanding
 even some backtracking.
*};

end;