104 lines
2.7 KiB
Text
104 lines
2.7 KiB
Text
/-
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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 (n : trunc_index) {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 n A
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include mul
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definition trunc_mul [unfold 9 10] (g h : G) : G :=
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begin
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induction g with p,
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induction h with q,
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exact tr (mul p q)
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end
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omit mul include inv
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definition trunc_inv [unfold 9] (g : G) : G :=
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begin
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induction g with p,
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exact tr (inv p)
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end
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omit inv include one
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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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parameters {mul} {inv} {one}
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omit one include mul_assoc
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theorem trunc_mul_assoc (g₁ g₂ g₃ : G) : g₁ * g₂ * g₃ = g₁ * (g₂ * g₃) :=
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begin
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induction g₁ with p₁,
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induction g₂ with p₂,
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induction g₃ with p₃,
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exact ap tr !mul_assoc,
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end
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omit mul_assoc include one_mul
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theorem trunc_one_mul (g : G) : 1 * g = g :=
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begin
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induction g with p,
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exact ap tr !one_mul
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end
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omit one_mul include mul_one
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theorem trunc_mul_one (g : G) : g * 1 = g :=
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begin
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induction g with p,
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exact ap tr !mul_one
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end
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omit mul_one include mul_left_inv
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theorem trunc_mul_left_inv (g : G) : g⁻¹ * g = 1 :=
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begin
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induction g with p,
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exact ap tr !mul_left_inv
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end
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omit mul_left_inv
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theorem trunc_mul_comm (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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induction g with p,
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induction h with q,
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exact ap tr !mul_comm
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end
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parameters (mul) (inv) (one)
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definition trunc_group [constructor] : group (trunc 0 A) :=
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⦃group,
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mul := algebra.trunc_mul 0 mul,
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mul_assoc := algebra.trunc_mul_assoc 0 mul_assoc,
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one := algebra.trunc_one 0 one,
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one_mul := algebra.trunc_one_mul 0 one_mul,
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mul_one := algebra.trunc_mul_one 0 mul_one,
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inv := algebra.trunc_inv 0 inv,
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mul_left_inv := algebra.trunc_mul_left_inv 0 mul_left_inv,
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is_set_carrier := _⦄
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definition trunc_ab_group [constructor] (mul_comm : ∀a b, mul a b = mul b a)
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: ab_group (trunc 0 A) :=
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⦃ab_group, trunc_group, mul_comm := algebra.trunc_mul_comm 0 mul_comm⦄
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end
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end algebra
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