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Mathlib.Algebra.Ring.NegOnePow

Integer powers of (-1) #

This file defines the map negOnePow : ℤ → ℤˣ which sends n to (-1 : ℤˣ) ^ n.

The definition of negOnePow and some lemmas first appeared in contributions by Johan Commelin to the Liquid Tensor Experiment.

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def Int.negOnePow (n : ℤ) :
ℤˣ

The map ℤ → ℤˣ which sends n to (-1 : ℤˣ) ^ n.

Equations
Instances For
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    theorem Int.negOnePow_def (n : ℤ) :
    n.negOnePow = (-1) ^ n
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    theorem Int.negOnePow_add (n₁ n₂ : ℤ) :
    (n₁ + n₂).negOnePow = n₁.negOnePow * n₂.negOnePow
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    @[simp]
    theorem Int.negOnePow_zero :
    negOnePow 0 = 1
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    @[simp]
    theorem Int.negOnePow_one :
    negOnePow 1 = -1
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    theorem Int.negOnePow_succ (n : ℤ) :
    (n + 1).negOnePow = -n.negOnePow
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    theorem Int.negOnePow_even (n : ℤ) (hn : Even n) :
    n.negOnePow = 1
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    @[simp]
    theorem Int.negOnePow_two_mul (n : ℤ) :
    (2 * n).negOnePow = 1
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    theorem Int.negOnePow_odd (n : ℤ) (hn : Odd n) :
    n.negOnePow = -1
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    @[simp]
    theorem Int.negOnePow_two_mul_add_one (n : ℤ) :
    (2 * n + 1).negOnePow = -1
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    theorem Int.negOnePow_eq_one_iff (n : ℤ) :
    n.negOnePow = 1 ↔ Even n
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    theorem Int.negOnePow_eq_neg_one_iff (n : ℤ) :
    n.negOnePow = -1 ↔ Odd n
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    @[simp]
    theorem Int.abs_negOnePow (n : ℤ) :
    |↑n.negOnePow| = 1
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    @[simp]
    theorem Int.negOnePow_neg (n : ℤ) :
    (-n).negOnePow = n.negOnePow
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    @[simp]
    theorem Int.negOnePow_abs (n : ℤ) :
    |n|.negOnePow = n.negOnePow
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    theorem Int.negOnePow_sub (n₁ n₂ : ℤ) :
    (n₁ - n₂).negOnePow = n₁.negOnePow * n₂.negOnePow
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    theorem Int.negOnePow_eq_iff (n₁ n₂ : ℤ) :
    n₁.negOnePow = n₂.negOnePow ↔ Even (n₁ - n₂)
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    @[simp]
    theorem Int.negOnePow_mul_self (n : ℤ) :
    (n * n).negOnePow = n.negOnePow
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    theorem Int.cast_negOnePow (K : Type u_1) (n : ℤ) [Field K] :
    ↑↑n.negOnePow = (-1) ^ n
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    @[deprecated Int.cast_negOnePow (since := "2024-10-20")]
    theorem Int.coe_negOnePow (K : Type u_1) (n : ℤ) [Field K] :
    ↑↑n.negOnePow = (-1) ^ n

    Alias of Int.cast_negOnePow.

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    theorem Int.cast_negOnePow_natCast (R : Type u_1) [Ring R] (n : ℕ) :
    ↑↑(↑n).negOnePow = (-1) ^ n
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    theorem Int.coe_negOnePow_natCast (n : ℕ) :
    ↑(↑n).negOnePow = (-1) ^ n