9a17a244c9
More results from the Spectral repository are moved to this library Also make various type-class arguments of truncatedness and equivalences which were hard to synthesize explicit
113 lines
2.9 KiB
Text
113 lines
2.9 KiB
Text
/-
|
|
Copyright (c) 2015 Floris van Doorn. All rights reserved.
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
|
Authors: Floris van Doorn
|
|
|
|
truncating an ∞-group to a group
|
|
-/
|
|
|
|
import hit.trunc algebra.bundled
|
|
|
|
open eq is_trunc trunc
|
|
|
|
namespace algebra
|
|
|
|
section
|
|
parameters (n : trunc_index) {A : Type} [inf_group A]
|
|
|
|
local abbreviation G := trunc n A
|
|
|
|
definition trunc_mul [unfold 9 10] (g h : G) : G :=
|
|
begin
|
|
induction g with p,
|
|
induction h with q,
|
|
exact tr (p * q)
|
|
end
|
|
|
|
definition trunc_inv [unfold 9] (g : G) : G :=
|
|
begin
|
|
induction g with p,
|
|
exact tr p⁻¹
|
|
end
|
|
|
|
definition trunc_one [constructor] : G :=
|
|
tr 1
|
|
|
|
local notation 1 := trunc_one
|
|
local postfix ⁻¹ := trunc_inv
|
|
local infix * := trunc_mul
|
|
|
|
theorem trunc_mul_assoc (g₁ g₂ g₃ : G) : g₁ * g₂ * g₃ = g₁ * (g₂ * g₃) :=
|
|
begin
|
|
induction g₁ with p₁,
|
|
induction g₂ with p₂,
|
|
induction g₃ with p₃,
|
|
exact ap tr !mul.assoc,
|
|
end
|
|
|
|
theorem trunc_one_mul (g : G) : 1 * g = g :=
|
|
begin
|
|
induction g with p,
|
|
exact ap tr !one_mul
|
|
end
|
|
|
|
theorem trunc_mul_one (g : G) : g * 1 = g :=
|
|
begin
|
|
induction g with p,
|
|
exact ap tr !mul_one
|
|
end
|
|
|
|
theorem trunc_mul_left_inv (g : G) : g⁻¹ * g = 1 :=
|
|
begin
|
|
induction g with p,
|
|
exact ap tr !mul.left_inv
|
|
end
|
|
|
|
parameter (A)
|
|
definition inf_group_trunc [constructor] [instance] : inf_group (trunc n A) :=
|
|
⦃inf_group,
|
|
mul := algebra.trunc_mul n,
|
|
mul_assoc := algebra.trunc_mul_assoc n,
|
|
one := algebra.trunc_one n,
|
|
one_mul := algebra.trunc_one_mul n,
|
|
mul_one := algebra.trunc_mul_one n,
|
|
inv := algebra.trunc_inv n,
|
|
mul_left_inv := algebra.trunc_mul_left_inv n⦄
|
|
|
|
definition group_trunc [constructor] : group (trunc 0 A) :=
|
|
group_of_inf_group _
|
|
|
|
end
|
|
|
|
definition igtrunc [constructor] (n : ℕ₋₂) (A : InfGroup) : InfGroup :=
|
|
InfGroup.mk (trunc n A) (inf_group_trunc n A)
|
|
|
|
definition gtrunc [constructor] (A : InfGroup) : Group :=
|
|
Group.mk (trunc 0 A) (group_trunc A)
|
|
|
|
section
|
|
variables (n : trunc_index) {A : Type} [ab_inf_group A]
|
|
|
|
theorem trunc_mul_comm (g h : trunc n A) : trunc_mul n g h = trunc_mul n h g :=
|
|
begin
|
|
induction g with p,
|
|
induction h with q,
|
|
exact ap tr !mul.comm
|
|
end
|
|
|
|
variable (A)
|
|
definition ab_inf_group_trunc [constructor] [instance] : ab_inf_group (trunc n A) :=
|
|
⦃ab_inf_group, inf_group_trunc n A, mul_comm := algebra.trunc_mul_comm n⦄
|
|
|
|
definition ab_group_trunc [constructor] : ab_group (trunc 0 A) :=
|
|
ab_group_of_ab_inf_group _
|
|
|
|
definition aigtrunc [constructor] (n : ℕ₋₂) (A : AbInfGroup) : AbInfGroup :=
|
|
AbInfGroup.mk (trunc n A) (ab_inf_group_trunc n A)
|
|
|
|
definition agtrunc [constructor] (A : AbInfGroup) : AbGroup :=
|
|
AbGroup.mk (trunc 0 A) (ab_group_trunc A)
|
|
|
|
end
|
|
|
|
end algebra
|