78 lines
2.6 KiB
Text
78 lines
2.6 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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Author: Floris van Doorn
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Theorems about algebra specific to HoTT
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-/
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import .group arity types.pi hprop_trunc types.unit
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open equiv eq equiv.ops is_trunc
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namespace algebra
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open Group has_mul has_inv
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-- we prove under which conditions two groups are equal
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-- group and has_mul are classes. So, lean does not automatically generate
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-- coercions between them anymore.
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-- So, an application such as (@mul A G g h) in the following definition
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-- is type incorrect if the coercion is not added (explicitly or implicitly).
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local attribute group.to.has_mul [coercion]
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local attribute group.to_has_inv [coercion]
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universe variable l
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variables {A B : Type.{l}}
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definition group_eq {G H : group A} (same_mul' : Π(g h : A), @mul A G g h = @mul A H g h)
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: G = H :=
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begin
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have foo : Π(g : A), @inv A G g = (@inv A G g * g) * @inv A H g,
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from λg, !mul_inv_cancel_right⁻¹,
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cases G with Gm Gs Gh1 G1 Gh2 Gh3 Gi Gh4,
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cases H with Hm Hs Hh1 H1 Hh2 Hh3 Hi Hh4,
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rewrite [↑[semigroup.to_has_mul,group.to_has_inv] at (same_mul,foo)],
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have same_mul : Gm = Hm, from eq_of_homotopy2 same_mul',
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cases same_mul,
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have same_one : G1 = H1, from calc
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G1 = Hm G1 H1 : Hh3
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... = H1 : Gh2,
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have same_inv : Gi = Hi, from eq_of_homotopy (take g, calc
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Gi g = Hm (Hm (Gi g) g) (Hi g) : foo
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... = Hm G1 (Hi g) : by rewrite Gh4
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... = Hi g : Gh2),
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cases same_one, cases same_inv,
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have ps : Gs = Hs, from !is_hprop.elim,
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have ph1 : Gh1 = Hh1, from !is_hprop.elim,
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have ph2 : Gh2 = Hh2, from !is_hprop.elim,
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have ph3 : Gh3 = Hh3, from !is_hprop.elim,
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have ph4 : Gh4 = Hh4, from !is_hprop.elim,
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cases ps, cases ph1, cases ph2, cases ph3, cases ph4, reflexivity
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end
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definition group_pathover {G : group A} {H : group B} {f : A ≃ B}
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: (Π(g h : A), f (g * h) = f g * f h) → G =[ua f] H :=
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begin
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revert H,
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eapply (rec_on_ua_idp' f),
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intros H resp_mul,
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esimp [equiv.refl] at resp_mul, esimp,
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apply pathover_idp_of_eq, apply group_eq,
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exact resp_mul
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end
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definition Group_eq {G H : Group} (f : carrier G ≃ carrier H)
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(resp_mul : Π(g h : G), f (g * h) = f g * f h) : G = H :=
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begin
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cases G with Gc G, cases H with Hc H,
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apply (apo011 mk (ua f)),
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apply group_pathover, exact resp_mul
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end
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definition trivial_group_of_is_contr (G : Group) [H : is_contr G] : G = G0 :=
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begin
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fapply Group_eq,
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{ apply equiv_unit_of_is_contr},
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{ intros, reflexivity}
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end
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end algebra
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