(*  Title:      HOL/Examples/Peirce.thy
    Author:     Makarius
*)

section ‹Peirce's Law›

theory Peirce
  imports Main
begin

text ‹
 We consider Peirce's Law: ‹((A ⟶ B) ⟶ A) ⟶ 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.🚫‹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 🚫‹presume›). Before assuming the negated
 goal ‹¬ A›, its intended consequence ‹A ⟶ 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 ⟶ B› later on. The overall effect is that of a logical
 ∗‹cut›.

 Technically speaking, whenever some goal is solved by 🚫‹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
 🚫‹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 ∗‹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