`rify` tactic #

The rify tactic is used to shift propositions from ℕ, ℤ or ℚ to ℝ.

Although less useful than its cousins zify and qify, it can be useful when your goal or context already involves real numbers.

In the example below, assumption hn is about natural numbers, hk is about integers and involves casting a natural number to ℤ, and the conclusion is about real numbers. The proof uses rify to lift both assumptions to ℝ before calling linarith.

import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Rify

example {n : ℕ} {k : ℤ} (hn : 8 ≤ n) (hk : 2 * k ≤ n + 2) :
    (0 : ℝ) < n - k - 1 := by
  rify at hn hk
  linarith

TODO: Investigate whether we should generalize this to other fields.

source

def Mathlib.Tactic.Rify.rify :

Lean.ParserDescr

The rify tactic is used to shift propositions from ℕ, ℤ or ℚ to ℝ. Although less useful than its cousins zify and qify, it can be useful when your goal or context already involves real numbers.

example {n : ℕ} {k : ℤ} (hn : 8 ≤ n) (hk : 2 * k ≤ n + 2) :
    (0 : ℝ) < n - k - 1 := by
  rify at hn hk /- Now have hn : 8 ≤ (n : ℝ)   hk : 2 * (k : ℝ) ≤ (n : ℝ) + 2 -/
  linarith

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

rify 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 : ℕ) (h : a - b < c) (hab : b ≤ a) : a < b + c := by
  rify [hab] at h ⊢
  linarith

Note that zify or qify would work just as well in the above example (and zify is the natural choice since it is enough to get rid of the pathological ℕ subtraction).

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