101 lines
2.4 KiB
Text
101 lines
2.4 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.bundled
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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} [inf_group A]
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local abbreviation G := trunc n A
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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 (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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induction g with p,
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exact tr p⁻¹
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end
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definition trunc_one [constructor] : G :=
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tr 1
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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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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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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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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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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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parameter (A)
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definition trunc_inf_group [constructor] [instance] : inf_group (trunc n A) :=
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⦃inf_group,
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mul := algebra.trunc_mul n,
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mul_assoc := algebra.trunc_mul_assoc n,
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one := algebra.trunc_one n,
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one_mul := algebra.trunc_one_mul n,
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mul_one := algebra.trunc_mul_one n,
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inv := algebra.trunc_inv n,
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mul_left_inv := algebra.trunc_mul_left_inv n⦄
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definition trunc_group [constructor] : group (trunc 0 A) :=
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group_of_inf_group _
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end
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section
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variables (n : trunc_index) {A : Type} [ab_inf_group A]
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theorem trunc_mul_comm (g h : trunc n A) : trunc_mul n g h = trunc_mul n 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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variable (A)
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definition trunc_ab_inf_group [constructor] [instance] : ab_inf_group (trunc n A) :=
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⦃ab_inf_group, trunc_inf_group n A, mul_comm := algebra.trunc_mul_comm n⦄
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definition trunc_ab_group [constructor] : ab_group (trunc 0 A) :=
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ab_group_of_ab_inf_group _
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end
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end algebra
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