src/HOL/Isar_examples/Peirce.thy

separate HOL-Main image

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

header {* Peirce's Law *}

theory Peirce imports Main begin

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 "(A --> B) --> A"
  show A
  proof (rule classical)
    assume "~ A"
    have "A --> B"
    proof
      assume A
      with `~ A` show B by contradiction
    qed
    with `(A --> B) --> A` 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 "(A --> B) --> A"
  show A
  proof (rule classical)
    presume "A --> B"
    with `(A --> B) --> A` show A ..
  next
    assume "~ A"
    show "A --> B"
    proof
      assume A
      with `~ A` show 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 text
 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 rules of the ``case-analysis'' type usually gives 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