# Theory Peirce

```(*  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

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
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
```