lean2/examples/lean/ex3.lean

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import macros.
theorem and_comm (a b : Bool) : (a ∧ b) ⇒ (b ∧ a)
:= assume H_ab, and::intro (and::elimr H_ab) (and::eliml H_ab).
theorem or_comm (a b : Bool) : (a b) ⇒ (b a)
:= assume H_ab,
or::elim H_ab
(λ H_a, or::intror b H_a)
(λ H_b, or::introl H_b a).
-- (em a) is the excluded middle a ¬a
-- (mt H H_na) is Modus Tollens with
-- H : (a ⇒ b) ⇒ a)
-- H_na : ¬a
-- produces
-- ¬(a ⇒ b)
--
-- not::imp::eliml applied to
-- (MT H H_na) : ¬(a ⇒ b)
-- produces
-- a
theorem pierce (a b : Bool) : ((a ⇒ b) ⇒ a) ⇒ a
:= assume H, or::elim (em a)
(λ H_a, H_a)
(λ H_na, not::imp::eliml (mt H H_na)).
print environment 3.