# mathlibdocumentation

tactic.push_neg

theorem push_neg.not_not_eq (p : Prop) :
(¬¬p) = p

theorem push_neg.not_and_eq (p q : Prop) :
(¬(p q)) = (p → ¬q)

theorem push_neg.not_or_eq (p q : Prop) :
(¬(p q)) = (¬p ¬q)

theorem push_neg.not_forall_eq {α : Sort u} (s : α → Prop) :
(¬∀ (x : α), s x) = ∃ (x : α), ¬s x

theorem push_neg.not_exists_eq {α : Sort u} (s : α → Prop) :
(¬∃ (x : α), s x) = ∀ (x : α), ¬s x

theorem push_neg.not_implies_eq (p q : Prop) :
(¬(p → q)) = (p ¬q)

theorem push_neg.classical.implies_iff_not_or (p q : Prop) :
p → q ¬p q

theorem push_neg.not_eq {α : Sort u} (a b : α) :
¬a = b a b

theorem push_neg.not_le_eq {β : Type u} [linear_order β] (a b : β) :
(¬a b) = (b < a)

theorem push_neg.not_lt_eq {β : Type u} [linear_order β] (a b : β) :
(¬a < b) = (b a)

Push negations in the goal of some assumption.

For instance, a hypothesis h : ¬ ∀ x, ∃ y, x ≤ y will be transformed by push_neg at h into h : ∃ x, ∀ y, y < x. Variables names are conserved.

This tactic pushes negations inside expressions. For instance, given an assumption

h : ¬ ∀ ε > 0, ∃ δ > 0, ∀ x, |x - x₀| ≤ δ → |f x - y₀| ≤ ε)


writing push_neg at h will turn h into

h : ∃ ε, ε > 0 ∧ ∀ δ, δ > 0 → (∃ x, |x - x₀| ≤ δ ∧ ε < |f x - y₀|),


(the pretty printer does not use the abreviations ∀ δ > 0 and ∃ ε > 0 but this issue has nothing to do with push_neg). Note that names are conserved by this tactic, contrary to what would happen with simp using the relevant lemmas. One can also use this tactic at the goal using push_neg, at every assumption and the goal using push_neg at * or at selected assumptions and the goal using say push_neg at h h' ⊢ as usual.

theorem imp_of_not_imp_not (P Q : Prop) :
(¬Q → ¬P)P → Q

meta def name_with_opt  :

Matches either an identifier "h" or a pair of identifiers "h with k"

Transforms the goal into its contrapositive.

• contrapose turns a goal P → Q into ¬ Q → ¬ P
• contrapose! turns a goal P → Q into ¬ Q → ¬ P and pushes negations inside P and Q using push_neg
• contrapose h first reverts the local assumption h, and then uses contrapose and intro h
• contrapose! h first reverts the local assumption h, and then uses contrapose! and intro h
• contrapose h with new_h uses the name new_h for the introduced hypothesis