Isabelle中矛盾的惯用语证明？

Question

So far I wrote proofs by contradiction in the following style in Isabelle (using a pattern by Jeremy Siek ):

lemma "<expression>"
proof -
  {
    assume "¬ <expression>"
    then have False sorry
  }
  then show ?thesis by blast
qed

Is there a way that works without the nested raw proof block { ... } ?

Answer 1

There is the rule ccontr for classical proofs by contradiction:

have "<expression>"
proof (rule ccontr)
  assume "¬ <expression>"
  then show False sorry
qed

It may sometimes help to use by contradiction to prove the last step.

There is also the rule classical (which looks less intuitive):

have "<expression>"
proof (rule classical)
  assume "¬ <expression>"
  then show "<expression>" sorry
qed

For further examples using classical , see $ISABELLE_HOME/src/HOL/Isar_Examples/Drinker.thy

Answer 2

For better understanding of rule classical it can be printed in structured Isar style like this:

print_statement classical

Output:

theorem classical:
  obtains "¬ thesis"

Thus the pure evil to intuitionists appears a bit more intuitive: in order to prove some arbitrary thesis, we may assume that its negation holds.

The corresponding canonical proof pattern is this:

notepad
begin
  have A
  proof (rule classical)
    assume "¬ ?thesis"
    then show ?thesis sorry
  qed
end

Here ?thesis is the concrete thesis of the above claim of A , which may be an arbitrarily complex statement. This quasi abstraction via the abbreviation ?thesis is typical for idiomatic Isar, to emphasize the structure of reasoning.