wip
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Steve Awodey
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3 changed files with 57 additions and 8 deletions
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Steve@steveawodeysAir.wv.cc.cmu.edu.268
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@ -35,12 +35,20 @@ definition homology {B : AbGroup} (d : B →g B) (H : is_differential d) : AbGro
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definition homology_ugly {B : AbGroup} (d : B →g B) (H : is_differential d) : AbGroup :=
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@quotient_ab_group (ab_kernel d) (image_subgroup (ab_subgroup_of_subgroup_incl (diff_im_in_ker d H)))
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definition homology_iso_ugly {B : AbGroup} (d : B →g B) (H : is_differential d) : (homology d H) ≃g (homology_ugly d H) :=
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begin
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fapply @iso_of_ab_qg_group (ab_kernel d) (image_subgroup (ab_subgroup_of_subgroup_incl (diff_im_in_ker d H))) (image_subgroup_of_diff d H),
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fapply @iso_of_ab_qg_group (ab_kernel d),
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intro a,
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intro p, induction p with f, induction f with b p,
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fapply tr, fapply fiber.mk, fapply sigma.mk, exact d b, fapply tr, fapply fiber.mk, exact b, reflexivity,
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induction a with c q, fapply subtype_eq, refine p ⬝ _, reflexivity,
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intro b p, induction p with f, induction f with c p, induction p,
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induction c with a q, induction q with f, induction f with a' p, induction p,
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fapply tr, fapply fiber.mk, exact a', reflexivity
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end
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definition SES_iso_C {A B C C' : AbGroup} (ses : SES A B C) (k : C ≃g C') : SES A B C' :=
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begin
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fapply SES.mk,
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@ -68,6 +76,7 @@ definition SES_iso_C {A B C C' : AbGroup} (ses : SES A B C) (k : C ≃g C') : SE
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exact to_respect_one k⁻¹ᵍ
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end
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definition SES_of_differential_ugly {B : AbGroup} (d : B →g B) (H : is_differential d) : SES (ab_image d) (ab_kernel d) (homology_ugly d H) :=
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begin
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exact SES_of_inclusion (ab_subgroup_of_subgroup_incl (diff_im_in_ker d H)) (is_embedding_ab_subgroup_of_subgroup_incl (diff_im_in_ker d H)),
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@ -75,8 +84,9 @@ definition SES_of_differential_ugly {B : AbGroup} (d : B →g B) (H : is_differe
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definition SES_of_differential {B : AbGroup} (d : B →g B) (H : is_differential d) : SES (ab_image d) (ab_kernel d) (homology d H) :=
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begin
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exact SES_of_inclusion (ab_subgroup_of_subgroup_incl (diff_im_in_ker d H)) (is_embedding_ab_subgroup_of_subgroup_incl (diff_im_in_ker d H)),
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exact sorry,
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fapply SES_iso_C,
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fapply SES_of_inclusion (ab_subgroup_of_subgroup_incl (diff_im_in_ker d H)) (is_embedding_ab_subgroup_of_subgroup_incl (diff_im_in_ker d H)),
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exact (homology_iso_ugly d H)⁻¹ᵍ
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end
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structure exact_couple (A B : AbGroup) : Type :=
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@ -98,17 +108,38 @@ definition differential_is_differential {A B : AbGroup} (EC : exact_couple A B)
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section derived_couple
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variables {A B : AbGroup} (EC : exact_couple A B)
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parameters {A B : AbGroup} (EC : exact_couple A B)
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local abbreviation i := exact_couple.i EC
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local abbreviation j := exact_couple.j EC
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local abbreviation k := exact_couple.k EC
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local abbreviation d := differential EC
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local abbreviation H := differential_is_differential EC
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definition derived_couple_A : AbGroup :=
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ab_subgroup (image_subgroup (exact_couple.i EC))
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ab_subgroup (image_subgroup i)
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definition derived_couple_B : AbGroup :=
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homology (differential EC) (differential_is_differential EC)
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definition derived_couple_i : derived_couple_A EC →g derived_couple_A EC :=
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definition derived_couple_i : derived_couple_A →g derived_couple_A :=
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(image_lift (exact_couple.i EC)) ∘g (image_incl (exact_couple.i EC))
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definition SES_of_exact_couple_at_i : SES (ab_kernel i) A (ab_image i) :=
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begin
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fapply SES_iso_C,
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fapply SES_of_subgroup (kernel_subgroup i),
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fapply ab_group_first_iso_thm i,
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end
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definition j_factor : A →g (ab_kernel d) :=
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begin
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fapply ab_hom_lift j,
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intro a,
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unfold kernel_subgroup,
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exact sorry --refine ap (group_fun j) _,
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end
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definition derived_couple_j : derived_couple_A EC →g derived_couple_B EC :=
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begin
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exact sorry,
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@ -319,6 +319,16 @@ definition hom_lift [constructor] {G H : Group} (f : G →g H) (K : subgroup_rel
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intro g h, apply subtype_eq, esimp, apply respect_mul
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end
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definition ab_hom_lift [constructor] {G H : AbGroup} (f : G →g H) (K : subgroup_rel H) (Hyp : Π (g : G), K (f g)) : G →g ab_subgroup K :=
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begin
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fapply homomorphism.mk,
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intro g,
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fapply sigma.mk,
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exact f g,
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fapply Hyp,
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intro g h, apply subtype_eq, apply respect_mul,
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end
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definition image_lift [constructor] {G H : Group} (f : G →g H) : G →g image f :=
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begin
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fapply hom_lift f,
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@ -328,6 +338,15 @@ definition hom_lift [constructor] {G H : Group} (f : G →g H) (K : subgroup_rel
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exact g, reflexivity
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end
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definition ab_image_lift [constructor] {G H : AbGroup} (f : G →g H) : G →g image f :=
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begin
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fapply hom_lift f,
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intro g,
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apply tr,
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fapply fiber.mk,
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exact g, reflexivity
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end
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definition is_surjective_image_lift {G H : Group} (f : G →g H) : is_surjective (image_lift f) :=
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begin
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intro h,
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