If the target of the main goal is a proposition p,
by_contra! reduces the goal to proving False using the additional hypothesis this : ¬ p.
by_contra! h can be used to name the hypothesis h : ¬ p.
The hypothesis ¬ p will be negation normalized using push_neg.
For instance, ¬ a < b will be changed to b ≤ a.
by_contra! h : q will normalize negations in ¬ p, normalize negations in q,
and then check that the two normalized forms are equal.
The resulting hypothesis is the pre-normalized form, q.
If the name h is not explicitly provided, then this will be used as name.
This tactic uses classical reasoning.
It is a variant on the tactic by_contra.
Examples:
example : 1 < 2 := by
by_contra! h
-- h : 2 ≤ 1 ⊢ False
example : 1 < 2 := by
by_contra! h : ¬ 1 < 2
-- h : ¬ 1 < 2 ⊢ False
Equations
- One or more equations did not get rendered due to their size.