lean2/hott/algebra/trunc_group.hlean
2016-03-06 13:03:31 -05:00

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/-
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.group
open eq is_trunc trunc
namespace algebra
section
parameters (n : trunc_index) {A : Type} (mul : A → A → A) (inv : A → A) (one : A)
(mul_assoc : ∀a b c, mul (mul a b) c = mul a (mul b c))
(one_mul : ∀a, mul one a = a) (mul_one : ∀a, mul a one = a)
(mul_left_inv : ∀a, mul (inv a) a = one)
local abbreviation G := trunc n A
include mul
definition trunc_mul [unfold 9 10] (g h : G) : G :=
begin
induction g with p,
induction h with q,
exact tr (mul p q)
end
omit mul include inv
definition trunc_inv [unfold 9] (g : G) : G :=
begin
induction g with p,
exact tr (inv p)
end
omit inv include one
definition trunc_one [constructor] : G :=
tr one
local notation 1 := trunc_one
local postfix ⁻¹ := trunc_inv
local infix * := trunc_mul
parameters {mul} {inv} {one}
omit one include mul_assoc
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
omit mul_assoc include one_mul
theorem trunc_one_mul (g : G) : 1 * g = g :=
begin
induction g with p,
exact ap tr !one_mul
end
omit one_mul include mul_one
theorem trunc_mul_one (g : G) : g * 1 = g :=
begin
induction g with p,
exact ap tr !mul_one
end
omit mul_one include mul_left_inv
theorem trunc_mul_left_inv (g : G) : g⁻¹ * g = 1 :=
begin
induction g with p,
exact ap tr !mul_left_inv
end
omit mul_left_inv
theorem trunc_mul_comm (mul_comm : ∀a b, mul a b = mul b a) (g h : G)
: g * h = h * g :=
begin
induction g with p,
induction h with q,
exact ap tr !mul_comm
end
parameters (mul) (inv) (one)
definition trunc_group [constructor] : group (trunc 0 A) :=
⦃group,
mul := algebra.trunc_mul 0 mul,
mul_assoc := algebra.trunc_mul_assoc 0 mul_assoc,
one := algebra.trunc_one 0 one,
one_mul := algebra.trunc_one_mul 0 one_mul,
mul_one := algebra.trunc_mul_one 0 mul_one,
inv := algebra.trunc_inv 0 inv,
mul_left_inv := algebra.trunc_mul_left_inv 0 mul_left_inv,
is_set_carrier := _⦄
definition trunc_comm_group [constructor] (mul_comm : ∀a b, mul a b = mul b a)
: comm_group (trunc 0 A) :=
⦃comm_group, trunc_group, mul_comm := algebra.trunc_mul_comm 0 mul_comm⦄
end
end algebra