(* Title: HOL/Isar_Examples/Peirce.thy
Author: Markus Wenzel, TU Muenchen
*)
section \<open>Peirce's Law\<close>
theory Peirce
imports Main
begin
text \<open>We consider Peirce's Law: \<open>((A \<longrightarrow> B) \<longrightarrow> A) \<longrightarrow> A\<close>. 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.}\<close>
theorem "((A \<longrightarrow> B) \<longrightarrow> A) \<longrightarrow> A"
proof
assume "(A \<longrightarrow> B) \<longrightarrow> A"
show A
proof (rule classical)
assume "\<not> A"
have "A \<longrightarrow> B"
proof
assume A
with \<open>\<not> A\<close> show B by contradiction
qed
with \<open>(A \<longrightarrow> B) \<longrightarrow> A\<close> show A ..
qed
qed
text \<open>In the subsequent version the reasoning is rearranged by means of
``weak assumptions'' (as introduced by \isacommand{presume}). Before
assuming the negated goal \<open>\<not> A\<close>, its intended consequence \<open>A \<longrightarrow> B\<close> is put
into place in order to solve the main problem. Nevertheless, we do not get
anything for free, but have to establish \<open>A \<longrightarrow> B\<close> later on. The overall
effect is that of a logical \<^emph>\<open>cut\<close>.
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.\<close>
theorem "((A \<longrightarrow> B) \<longrightarrow> A) \<longrightarrow> A"
proof
assume "(A \<longrightarrow> B) \<longrightarrow> A"
show A
proof (rule classical)
presume "A \<longrightarrow> B"
with \<open>(A \<longrightarrow> B) \<longrightarrow> A\<close> show A ..
next
assume "\<not> A"
show "A \<longrightarrow> B"
proof
assume A
with \<open>\<not> A\<close> show B by contradiction
qed
qed
qed
text \<open>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>\<open>and\<close> 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.\<close>
end