Documentation

Mathlib.Algebra.CharP.Two

Lemmas about rings of characteristic two #

This file contains results about CharP R 2, in the CharTwo namespace.

The lemmas in this file with a _sq suffix are just special cases of the _pow_char lemmas elsewhere, with a shorter name for ease of discovery, and no need for a [Fact (Prime 2)] argument.

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theorem CharTwo.two_eq_zero {R : Type u_1} [Semiring R] [CharP R 2] :
2 = 0
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@[simp]
theorem CharTwo.add_self_eq_zero {R : Type u_1} [Semiring R] [CharP R 2] (x : R) :
x + x = 0
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@[simp]
theorem CharTwo.bit0_eq_zero {R : Type u_1} [Semiring R] [CharP R 2] :
bit0 = 0
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theorem CharTwo.bit0_apply_eq_zero {R : Type u_1} [Semiring R] [CharP R 2] (x : R) :
bit0 x = 0
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@[simp]
theorem CharTwo.bit1_eq_one {R : Type u_1} [Semiring R] [CharP R 2] :
bit1 = 1
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theorem CharTwo.bit1_apply_eq_one {R : Type u_1} [Semiring R] [CharP R 2] (x : R) :
bit1 x = 1
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@[simp]
theorem CharTwo.neg_eq {R : Type u_1} [Ring R] [CharP R 2] (x : R) :
-x = x
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theorem CharTwo.neg_eq' {R : Type u_1} [Ring R] [CharP R 2] :
Neg.neg = id
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@[simp]
theorem CharTwo.sub_eq_add {R : Type u_1} [Ring R] [CharP R 2] (x : R) (y : R) :
x - y = x + y
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theorem CharTwo.sub_eq_add' {R : Type u_1} [Ring R] [CharP R 2] :
HSub.hSub = fun (x x_1 : R) => x + x_1
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theorem CharTwo.add_sq {R : Type u_1} [CommSemiring R] [CharP R 2] (x : R) (y : R) :
(x + y) ^ 2 = x ^ 2 + y ^ 2
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theorem CharTwo.add_mul_self {R : Type u_1} [CommSemiring R] [CharP R 2] (x : R) (y : R) :
(x + y) * (x + y) = x * x + y * y
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theorem CharTwo.list_sum_sq {R : Type u_1} [CommSemiring R] [CharP R 2] (l : List R) :
l.sum ^ 2 = (List.map (fun (x : R) => x ^ 2) l).sum
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theorem CharTwo.list_sum_mul_self {R : Type u_1} [CommSemiring R] [CharP R 2] (l : List R) :
l.sum * l.sum = (List.map (fun (x : R) => x * x) l).sum
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theorem CharTwo.multiset_sum_sq {R : Type u_1} [CommSemiring R] [CharP R 2] (l : Multiset R) :
l.sum ^ 2 = (Multiset.map (fun (x : R) => x ^ 2) l).sum
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theorem CharTwo.multiset_sum_mul_self {R : Type u_1} [CommSemiring R] [CharP R 2] (l : Multiset R) :
l.sum * l.sum = (Multiset.map (fun (x : R) => x * x) l).sum
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theorem CharTwo.sum_sq {R : Type u_1} {ι : Type u_2} [CommSemiring R] [CharP R 2] (s : Finset ι) (f : ι → R) :
(s.sum fun (i : ι) => f i) ^ 2 = s.sum fun (i : ι) => f i ^ 2
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theorem CharTwo.sum_mul_self {R : Type u_1} {ι : Type u_2} [CommSemiring R] [CharP R 2] (s : Finset ι) (f : ι → R) :
((s.sum fun (i : ι) => f i) * s.sum fun (i : ι) => f i) = s.sum fun (i : ι) => f i * f i
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theorem neg_one_eq_one_iff {R : Type u_1} [Ring R] [Nontrivial R] :
-1 = 1 ↔ ringChar R = 2
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@[simp]
theorem orderOf_neg_one {R : Type u_1} [Ring R] [Nontrivial R] :
orderOf (-1) = if ringChar R = 2 then 1 else 2