qify tactic

The qify tactic is used to shift propositions from ℕ or ℤ to ℚ. This is often useful since ℚ has well-behaved division.

example (a b c x y z : ℕ) (h : ¬ x*y*z < 0) : c < a + 3*b := by
  qify
  qify at h
  /-
  h : ¬↑x * ↑y * ↑z < 0
  ⊢ ↑c < ↑a + 3 * ↑b
  -/
  sorry

def Mathlib.Tactic.Qify.qify : Lean.ParserDescr

The qify tactic is used to shift propositions from ℕ or ℤ to ℚ. This is often useful since ℚ has well-behaved division.

example (a b c x y z : ℕ) (h : ¬ x*y*z < 0) : c < a + 3*b := by
  qify
  qify at h
  /-
  h : ¬↑x * ↑y * ↑z < 0
  ⊢ ↑c < ↑a + 3 * ↑b
  -/
  sorry

qify can be given extra lemmas to use in simplification. This is especially useful in the presence of nat subtraction: passing ≤ arguments will allow push_cast to do more work.

example (a b c : ℤ) (h : a / b = c) (hab : b ∣ a) (hb : b ≠ 0) : a = c * b := by
  qify [hab] at h hb ⊢
  exact (div_eq_iff hb).1 h

qify makes use of the @[zify_simps] and @[qify_simps] attributes to move propositions, and the push_cast tactic to simplify the ℚ-valued expressions.

Equations

• One or more equations did not get rendered due to their size.

theorem Mathlib.Tactic.Qify.int_cast_eq (a : ℤ) (b : ℤ) : a = b ↔ ↑a = ↑b

theorem Mathlib.Tactic.Qify.int_cast_le (a : ℤ) (b : ℤ) : a ≤ b ↔ ↑a ≤ ↑b

theorem Mathlib.Tactic.Qify.int_cast_lt (a : ℤ) (b : ℤ) : a < b ↔ ↑a < ↑b

theorem Mathlib.Tactic.Qify.int_cast_ne (a : ℤ) (b : ℤ) : a ≠ b ↔ ↑a ≠ ↑b