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open classical | |
variables (α : Type) (p q : α → Prop) | |
variable a : α | |
variable r : Prop | |
example : (∃ x, p x) ↔ ¬ (∀ x, ¬ p x) := | |
iff.intro | |
(λ ⟨a, hpa⟩, (λ ax, ax a hpa)) | |
(λ nanpx : (∀ x, p x → false) → false, | |
suffices hpa : p a, from ⟨a, hpa⟩, | |
by_contradiction | |
(λ hnpa, | |
nanpx (λ ha, hnpa)) | |
-- type mismatch at application | |
-- nanpx (λ (ha : α), hnpa) | |
-- term | |
-- λ (ha : α), hnpa | |
-- has type | |
-- α → ¬p a | |
-- but is expected to have type | |
-- ∀ (x : α), p x → false | |
) |
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