97 lines
2.5 KiB
Text
97 lines
2.5 KiB
Text
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/-
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Copyright (c) 2015 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn
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truncating an ∞-group to a group
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-/
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import hit.trunc algebra.group
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open eq is_trunc trunc
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namespace algebra
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section
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parameters (A : Type) (mul : A → A → A) (inv : A → A) (one : A)
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{mul_assoc : ∀a b c, mul (mul a b) c = mul a (mul b c)}
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{one_mul : ∀a, mul one a = a} {mul_one : ∀a, mul a one = a}
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{mul_left_inv : ∀a, mul (inv a) a = one}
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local abbreviation G := trunc 0 A
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include mul_assoc one_mul mul_one mul_left_inv
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definition trunc_mul [unfold 9 10] (g h : G) : G :=
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begin
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apply trunc.rec_on g, intro p,
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apply trunc.rec_on h, intro q,
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exact tr (mul p q)
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end
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definition trunc_inv [unfold 9] (g : G) : G :=
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begin
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apply trunc.rec_on g, intro p,
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exact tr (inv p)
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end
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definition trunc_one [constructor] : G :=
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tr one
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local notation 1 := trunc_one
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local postfix ⁻¹ := trunc_inv
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local infix * := trunc_mul
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definition trunc_mul_assoc [unfold 9 10 11] (g₁ g₂ g₃ : G) : g₁ * g₂ * g₃ = g₁ * (g₂ * g₃) :=
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begin
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apply trunc.rec_on g₁, intro p₁,
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apply trunc.rec_on g₂, intro p₂,
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apply trunc.rec_on g₃, intro p₃,
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exact ap tr !mul_assoc,
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end
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definition trunc_one_mul [unfold 9] (g : G) : 1 * g = g :=
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begin
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apply trunc.rec_on g, intro p,
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exact ap tr !one_mul
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end
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definition trunc_mul_one [unfold 9] (g : G) : g * 1 = g :=
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begin
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apply trunc.rec_on g, intro p,
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exact ap tr !mul_one
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end
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definition trunc_mul_left_inv [unfold 9] (g : G) : g⁻¹ * g = 1 :=
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begin
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apply trunc.rec_on g, intro p,
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exact ap tr !mul_left_inv
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end
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definition trunc_mul_comm [unfold 10 11] (mul_comm : ∀a b, mul a b = mul b a) (g h : G)
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: g * h = h * g :=
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begin
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apply trunc.rec_on g, intro p,
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apply trunc.rec_on h, intro q,
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exact ap tr !mul_comm
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end
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parameters (mul_assoc) (one_mul) (mul_one) (mul_left_inv) {A}
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definition trunc_group [constructor] : group G :=
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⦃group,
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mul := trunc_mul,
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mul_assoc := trunc_mul_assoc,
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one := trunc_one,
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one_mul := trunc_one_mul,
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mul_one := trunc_mul_one,
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inv := trunc_inv,
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mul_left_inv := trunc_mul_left_inv,
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is_hset_carrier := _⦄
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definition trunc_comm_group [constructor] (mul_comm : ∀a b, mul a b = mul b a) : comm_group G :=
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⦃comm_group, trunc_group, mul_comm := trunc_mul_comm mul_comm⦄
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end
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end algebra
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