lean2/examples/lean/ex3.lean

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import macros.
theorem my_and_comm (a b : Bool) : (a ∧ b) → (b ∧ a)
:= assume H_ab, and_intro (and_elimr H_ab) (and_eliml H_ab).
theorem my_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.