/
prop819.lean
548 lines (505 loc) · 19.8 KB
/
prop819.lean
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import algebra.homology.homological_complex
import topology.category.Profinite.cofiltered_limit
import for_mathlib.Cech.split
import for_mathlib.Profinite.arrow_limit
import for_mathlib.Profinite.clopen_limit
import for_mathlib.simplicial.complex
import locally_constant.Vhat
import prop819.completion
--import prop819.locally_constant
open_locale nnreal
noncomputable theory
open category_theory opposite
open SemiNormedGroup
universes u
-- We have a surjective morphism of profinite sets.
variables (F : arrow Profinite.{u}) (surj : function.surjective F.hom)
variables (M : SemiNormedGroup.{u})
/-- The cochain complex built out of the cosimplicial object obtained by applying
`LocallyConstant.obj M` to the augmented Cech nerve of `F`. -/
abbreviation FL : cochain_complex SemiNormedGroup ℕ :=
(((cosimplicial_object.augmented.whiskering _ _).obj (LocallyConstant.{u u}.obj M)).obj
F.augmented_cech_nerve.right_op).to_cocomplex
/-- The cochain complex built out of the cosimplicial object obtained by applying
`LCC.obj M` to the augmented Cech nerve of `F`. -/
abbreviation FLC : cochain_complex SemiNormedGroup ℕ :=
(((cosimplicial_object.augmented.whiskering _ _).obj (LCC.{u u}.obj M)).obj
F.augmented_cech_nerve.right_op).to_cocomplex
--def Rop : (simplicial_object.augmented Profinite)ᵒᵖ ⥤ cosimplicial_object.augmented Profiniteᵒᵖ :=
--{ obj := λ X, X.unop.right_op,
-- map := λ X Y f,
-- { left := quiver.hom.op (comma_morphism.right f.unop),
-- right := nat_trans.right_op (comma_morphism.left f.unop),
-- w' := by { ext, exact congr_arg (λ η, (nat_trans.app η (op x)).op) f.unop.w.symm, } } }
/-- A functorial version of `FL`. -/
def FL_functor : (arrow Profinite.{u})ᵒᵖ ⥤ cochain_complex SemiNormedGroup ℕ :=
simplicial_object.augmented_cech_nerve.op ⋙
simplicial_to_cosimplicial_augmented _ ⋙
(cosimplicial_object.augmented.whiskering _ _).obj (LocallyConstant.obj M) ⋙
cosimplicial_object.augmented.cocomplex
/-- The functor sending an augmented cosimplicial object `X` to
the cochain complex associated to the composition of `X` with `LCC.obj M`. -/
@[simps obj map]
def FLC_functor' : (simplicial_object.augmented Profinite.{u})ᵒᵖ ⥤ cochain_complex SemiNormedGroup ℕ :=
simplicial_to_cosimplicial_augmented _ ⋙
(cosimplicial_object.augmented.whiskering _ _).obj (SemiNormedGroup.LCC.{u u}.obj M) ⋙
cosimplicial_object.augmented.cocomplex
/-- A functorial version of `FLC`. -/
def FLC_functor : (arrow Profinite.{u})ᵒᵖ ⥤ cochain_complex SemiNormedGroup ℕ :=
simplicial_object.augmented_cech_nerve.op ⋙ FLC_functor' M
-- Sanity checks
example : FL F M = (FL_functor M).obj (op F) := rfl
example : FLC F M = (FLC_functor M).obj (op F) := rfl
lemma _root_.cosimplicial_object.augmented.cocomplex_map_norm_noninc
{C₁ C₂ : cosimplicial_object.augmented SemiNormedGroup} (f : C₁ ⟶ C₂)
(hf1 : f.left.norm_noninc) (hf2 : ∀ n, (f.right.app n).norm_noninc) (i : ℕ) :
((cosimplicial_object.augmented.cocomplex.map f).f i).norm_noninc :=
begin
cases i,
{ exact hf1 },
{ exact hf2 _ },
end
lemma FLC_functor_map_norm_noninc {f g : (arrow Profinite.{u})ᵒᵖ} (α : f ⟶ g) (i : ℕ) :
(((FLC_functor M).map α).f i).norm_noninc :=
begin
refine cosimplicial_object.augmented.cocomplex_map_norm_noninc _ _ _ _,
{ exact SemiNormedGroup.LCC_obj_map_norm_noninc _ _ },
{ intro n,
exact SemiNormedGroup.LCC_obj_map_norm_noninc _ _ },
end
--⊢ cosimplicial_object.δ
-- (functor.right_op F.cech_nerve ⋙ (curry.obj (uncurry.obj LocallyConstant ⋙ Completion)).obj M)
-- k =
-- Completion.map (cosimplicial_object.δ (functor.right_op F.cech_nerve ⋙ LocallyConstant.obj M) k)
lemma FLC_iso_helper {x y : simplex_category} (f : x ⟶ y) :
(F.cech_nerve.right_op ⋙ LCC.obj M).map f =
Completion.map ((F.cech_nerve.right_op ⋙ LocallyConstant.obj M).map f) :=
begin
change Completion.map _ = _,
congr' 1,
dsimp [uncurry],
erw locally_constant.map_hom_id,
change 𝟙 _ ≫ _ = _,
rw category.id_comp,
end
/--
This is a strict (i.e. norm-preserving) isomorphism between `FLC F M` and
the cochain complex obtained by mapping `FL F M` along the `Completion` functor.
-/
def FLC_iso : strict_iso ((Completion.map_homological_complex _).obj (FL F M)) (FLC F M) :=
{ iso := homological_complex.hom.iso_of_components
(λ i, nat.rec_on i (eq_to_iso rfl) (λ _ _, eq_to_iso rfl))
begin
rintro (_|i) (_|j) (_|⟨i,w⟩); ext,
{ dsimp only [],
delta FLC FL,
dsimp only [
cosimplicial_object.augmented.whiskering,
cosimplicial_object.augmented.whiskering_obj,
cosimplicial_object.augmented.to_cocomplex,
cosimplicial_object.augmented.to_cocomplex_obj,
cochain_complex.of,
functor.map_homological_complex ],
rw dif_pos rfl,
rw dif_pos rfl,
erw [category.id_comp, category.comp_id, category.comp_id, category.comp_id],
dsimp only [cosimplicial_object.augmented.to_cocomplex_d,
cosimplicial_object.augmented.drop, comma.snd, cosimplicial_object.whiskering,
whiskering_right, cosimplicial_object.coboundary, functor.const_comp, LCC],
simp only [quiver.hom.unop_op, arrow.augmented_cech_nerve_hom_app,
whisker_right_app, nat_trans.comp_app, curry.obj_obj_map, category.id_comp,
nat_trans.right_op_app, uncurry.obj_map, nat_trans.id_app,
simplicial_object.augmented.right_op_hom,
category_theory.functor.map_id, category_theory.functor.comp_map,
SemiNormedGroup.LocallyConstant_obj_map, SemiNormedGroup.Completion_map],
erw category.id_comp },
{ dsimp only [],
delta FLC FL,
dsimp only [
cosimplicial_object.augmented.whiskering,
cosimplicial_object.augmented.whiskering_obj,
cosimplicial_object.augmented.to_cocomplex,
cosimplicial_object.augmented.to_cocomplex_obj,
cochain_complex.of,
functor.map_homological_complex ],
rw dif_pos rfl,
rw dif_pos rfl,
erw [category.id_comp, category.comp_id, category.comp_id, category.comp_id],
dsimp only [
cosimplicial_object.augmented.to_cocomplex_d,
cosimplicial_object.augmented.drop,
comma.snd,
cosimplicial_object.whiskering,
whiskering_right,
cosimplicial_object.coboundary,
LCC ],
rw [Completion.map_sum],
congr,
funext k,
rw [Completion.map_gsmul],
congr' 1,
apply FLC_iso_helper }
end,
is_strict := λ i, { strict_hom' := λ a, by { cases i; refl } } }.
open_locale simplicial
-- TODO: Move this to mathlib (also relax the has_limits condition).
/-- the isomorphism between the 0-th term of the Cech nerve and F.left-/
@[simps]
def cech_iso_zero {C : Type*} [category C] (F : arrow C) [limits.has_limits C]
: F.cech_nerve _[0] ≅ F.left :=
{ hom := limits.wide_pullback.π _ ⟨0⟩,
inv := limits.wide_pullback.lift F.hom (λ _, 𝟙 _) (by simp),
hom_inv_id' := begin
apply limits.wide_pullback.hom_ext,
{ intro i,
simp only [limits.wide_pullback.lift_π, category.id_comp, category.comp_id, category.assoc],
congr,
tidy },
{ simp }
end }
lemma augmentation_zero {C : Type*} [category C] (F : arrow C) [limits.has_limits C] :
(cech_iso_zero F).inv ≫ F.augmented_cech_nerve.hom.app _ = F.hom := by tidy
lemma locally_constant_norm_empty (X : Profinite) (hX : ¬ nonempty X)
(g : (LocallyConstant.obj M).obj (op X)) : ∥ g ∥ = 0 :=
begin
rw locally_constant.norm_def,
dsimp [supr],
suffices : set.range (λ x : ↥X, ∥ g.to_fun x ∥) = ∅,
{ erw [this, real.Sup_empty], },
simp only [set.range_eq_empty, not_nonempty_iff],
exact not_nonempty_iff.mp hX
end
lemma Profinite.coe_comp_apply {X Y Z : Profinite} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
(f ≫ g) x = g (f x) := rfl
lemma locally_constant_to_fun_eq {X : Profinite} (f : locally_constant X M) :
f.to_fun = f := rfl
lemma locally_constant_eq {X : Profinite} (f g : locally_constant X M) :
f.to_fun = g.to_fun ↔ f = g :=
begin
split,
{ intro h, ext, change f.to_fun _ = _, rw h, refl, },
{ intro h, rw h }
end
lemma locally_constant_eq_zero {X : Profinite} (f : locally_constant X M) :
f = 0 ↔ set.range f.to_fun ⊆ {0} :=
begin
split,
{ intro h, rw h, simp, },
{ intro h, ext x,
dsimp,
apply h,
use x,
refl }
end
include surj
lemma prop819_degree_zero_helper :
function.surjective (limits.wide_pullback.base (λ i : ulift (fin 1), F.hom)) :=
begin
intro x,
obtain ⟨x,rfl⟩ := surj x,
dsimp at *,
refine ⟨(cech_iso_zero F).inv x, _⟩,
dsimp,
change (limits.wide_pullback.lift F.hom _ _ ≫ limits.wide_pullback.base _) _ = _,
simp,
end
lemma prop819_zero_norm_le (g : (LocallyConstant.obj M).obj (op F.right)) : ∥ g ∥ ≤
∥ (LocallyConstant.obj M).map (limits.wide_pullback.base (λ i : ulift (fin 1), F.hom)).op g ∥ :=
begin
by_cases hh : nonempty F.right,
{ erw real.Sup_le,
{ rintros z ⟨z,rfl⟩,
obtain ⟨z,rfl⟩ := (prop819_degree_zero_helper _ surj) z,
change ∥ g.to_fun _ ∥ ≤ _,
erw ← LocallyConstant_map_apply M _ F.right (limits.wide_pullback.base (λ i, F.hom)) g z,
apply locally_constant.norm_apply_le },
{ rcases hh with ⟨x⟩,
refine ⟨∥ g.to_fun x ∥, x, rfl⟩ },
{ use ∥ g ∥,
rintro y ⟨y,rfl⟩,
dsimp,
apply locally_constant.norm_apply_le } },
{ rw locally_constant_norm_empty _ _ hh g,
simp }
end
theorem prop819_degree_zero (f : (FLC F M).X 0) (hf : (FLC F M).d 0 1 f = 0) :
f = 0 :=
begin
apply injective_of_strict_iso _ _ (FLC_iso F M) _ _ hf,
intros f hf,
have := @controlled_exactness ((FL F M).X 0) (0 : SemiNormedGroup) ((FL F M).X 1) _ _ _ 0 1
zero_lt_one 1 ((FL F M).d _ _) _ _ 1 zero_lt_one f _,
{ rcases this with ⟨g,h1,h2⟩,
rw ← h1,
simp },
{ intros g hg,
refine ⟨0,_, by simp⟩,
change (FL F M).d 0 1 g = 0 at hg,
dsimp,
symmetry,
delta FL at hg,
dsimp only [cosimplicial_object.augmented.whiskering,
cosimplicial_object.augmented.whiskering_obj,
cosimplicial_object.augmented.to_cocomplex,
cochain_complex.of] at hg,
rw dif_pos at hg,
swap, {simp},
dsimp [cosimplicial_object.augmented.to_cocomplex_d] at hg,
ext x,
obtain ⟨x,rfl⟩ := (prop819_degree_zero_helper F surj) x,
apply_fun (λ e, e x) at hg,
rw locally_constant.coe_comap at hg,
swap, { continuity },
exact hg },
{ rintro g ⟨g,rfl⟩,
refine ⟨g,rfl,_⟩,
dsimp [cosimplicial_object.augmented.to_cocomplex_d],
simp only [locally_constant.comap_hom_apply, one_mul,
if_true, eq_self_iff_true, category.id_comp, category.comp_id],
apply prop819_zero_norm_le _ surj },
{ exact hf }
end
.
/-- Any discrete quotient `S` of `F.left` yields a cochain complex, as follows.
First, let `T` be the maximal quotient of `F.right` such that `F.hom : F.left ⟶ F.right`
descends to `S → T`. Next construct the augmented Cech nerve of `S → T`, and finally
apply `FL_functor M` to this augmented Cech nerve.
-/
def FLF : (discrete_quotient F.left)ᵒᵖ ⥤ cochain_complex SemiNormedGroup ℕ :=
(Profinite.arrow_diagram F surj).op ⋙ FL_functor M
/--
The diagram of cochain complexes given by `FLF F surj` fits together into a cocone
whose cocone point is defeq to `FL F M`. This is precisely this cocone.
-/
def FLF_cocone : limits.cocone (FLF F surj M) :=
(FL_functor M).map_cocone $ (Profinite.arrow_cone F surj).op
open Profinite
lemma exists_locally_constant_FLF (n : ℕ) (f : (FL F M).X (n+1)) :
∃ (S : discrete_quotient F.left) (g : ((FLF F surj M).obj (op S)).X (n+1)),
((FLF_cocone F surj M).ι.app (op S)).f _ g = f :=
begin
have hC := Cech_cone_is_limit F surj n,
obtain ⟨i,g,hg⟩ := Profinite.exists_locally_constant _ hC f,
use [i, g],
exact hg.symm,
end
lemma FLF_cocone_app_coe_eq (n : ℕ) (S : discrete_quotient F.left)
(g : ((FLF F surj M).obj (op S)).X (n+1)) :
(((FLF_cocone F surj M).ι.app (op S)).f _ g).to_fun =
g.to_fun ∘ ((Cech_cone F surj n).π.app _) :=
begin
ext x,
change locally_constant.comap _ _ _ = _,
rw locally_constant.coe_comap,
swap, { continuity },
refl,
end
lemma FLF_map_coe_eq (n : ℕ) (S T : discrete_quotient F.left) (hh : T ≤ S)
(g : ((FLF F surj M).obj (op S)).X (n+1)) :
(((FLF F surj M).map (hom_of_le hh).op).f _ g).to_fun =
g.to_fun ∘ ((Cech_cone_diagram F surj n).map (hom_of_le hh)) :=
begin
ext x,
change locally_constant.comap _ _ _ = _,
rw locally_constant.coe_comap,
swap, { continuity },
refl,
end
lemma eq_zero_FLF (n : ℕ) (S : discrete_quotient F.left)
(g : ((FLF F surj M).obj (op S)).X (n+1))
(hg : ((FLF_cocone F surj M).ι.app (op S)).f _ g = 0) :
∃ (T : discrete_quotient F.left) (hT : T ≤ S),
((FLF F surj M).map (hom_of_le hT).op).f _ g = 0 :=
begin
have := exists_image (Cech_cone_diagram F surj n)
(Cech_cone F surj n) (Cech_cone_is_limit F surj n) S,
obtain ⟨T,hT,hh⟩ := this,
use T, use hT,
rw [locally_constant_eq_zero, FLF_map_coe_eq],
rw [locally_constant_eq_zero, FLF_cocone_app_coe_eq] at hg,
rintro x ⟨x,rfl⟩,
apply hg,
let P : (Cech_cone F surj n).X ⟶ (Cech_cone_diagram F surj n).obj S :=
(Cech_cone F surj n).π.app _,
let p : (Cech_cone_diagram F surj n).obj T ⟶ (Cech_cone_diagram F surj n).obj S :=
(Cech_cone_diagram F surj n).map (hom_of_le hT),
change ↥((Cech_cone_diagram F surj n).obj T) at x,
have : p x ∈ set.range p, use x,
erw ← hh at this,
obtain ⟨y,hy⟩ := this,
dsimp only [p] at hy,
use y,
dsimp only [function.comp_apply],
erw hy,
end
.
lemma d_eq_zero_FLF (n : ℕ) (S : discrete_quotient F.left)
(g : ((FLF F surj M).obj (op S)).X (n+1))
(hg : (FL F M).d (n+1) (n+2)
(((FLF_cocone F surj M).ι.app (op S)).f _ g) = 0) :
∃ (T : discrete_quotient F.left) (hT : T ≤ S),
((FLF F surj M).obj (op T)).d (n+1) (n+2)
(((FLF F surj M).map $ (hom_of_le hT).op).f _ g) = 0 :=
begin
have := ((FLF_cocone F surj M).ι.app (op S)).comm (n+1) (n+2),
apply_fun (λ e, e g) at this,
erw this at hg,
dsimp only [SemiNormedGroup.coe_comp] at hg,
have := eq_zero_FLF F surj M (n+1) S _ hg,
obtain ⟨T,hT,h⟩ := this,
use T, use hT,
have hh := ((FLF F surj M).map (hom_of_le hT).op).comm (n+1) (n+2),
apply_fun (λ e, e g) at hh,
erw ← hh at h,
exact h,
end
lemma norm_eq_FLF (n : ℕ) (S : discrete_quotient F.left)
(g : ((FLF F surj M).obj (op S)).X (n+1)) :
∃ (T : discrete_quotient F.left) (hT : T ≤ S),
∥((FLF_cocone F surj M).ι.app (op S)).f _ g∥₊ =
∥(((FLF F surj M)).map (hom_of_le hT).op).f _ g∥₊ :=
begin
have := exists_image (Cech_cone_diagram F surj n)
(Cech_cone F surj n) (Cech_cone_is_limit F surj n) S,
obtain ⟨T,hT,hh⟩ := this,
use T, use hT,
ext,
dsimp,
change Sup _ = Sup _,
congr' 1,
ext r,
split,
{ rintros ⟨x,rfl⟩,
dsimp only,
change ↥(Cech_cone F surj n).X at x,
use (Cech_cone F surj n).π.app T x,
dsimp only,
rw [← locally_constant_to_fun_eq, FLF_map_coe_eq,
function.comp_apply, ← Profinite.coe_comp_apply,
(Cech_cone F surj n).w, ← locally_constant_to_fun_eq,
FLF_cocone_app_coe_eq],
refl },
{ rintros ⟨x,rfl⟩,
dsimp only,
change ↥((Cech_cone_diagram F surj n).obj T) at x,
have : (Cech_cone_diagram F surj n).map (hom_of_le hT) x ∈
set.range ((Cech_cone_diagram F surj n).map (hom_of_le hT)), use x,
rw ← hh at this,
obtain ⟨y,hy⟩ := this,
change ↥(Cech_cone F surj n).X at y,
use y,
dsimp only,
simp_rw ← locally_constant_to_fun_eq,
rw [FLF_map_coe_eq, FLF_cocone_app_coe_eq],
dsimp only [function.comp_apply],
erw hy }
end
lemma exists_locally_constant (n : ℕ) (f : (FL F M).X (n+1))
(hf : (FL F M).d _ (n+2) f = 0) :
-- TODO: ∃ ..., true looks a bit fuuny
∃ (S : discrete_quotient F.left)
(g : ((FLF F surj M).obj (op S)).X (n+1))
(hgf : ((FLF_cocone F surj M).ι.app (op S)).f _ g = f)
(hgd : (((FLF F surj M).obj (op S)).d _ (n+2) g = 0))
(hgnorm : ∥f∥₊ = ∥g∥₊), true :=
begin
obtain ⟨S,f,rfl⟩ := exists_locally_constant_FLF F surj M n f,
obtain ⟨T1,hT1,h1⟩ := d_eq_zero_FLF F surj M n S f hf,
obtain ⟨T2,hT2,h2⟩ := norm_eq_FLF F surj M n S f,
let T := T1 ⊓ T2,
have hT : T ≤ S := le_trans inf_le_left hT1,
have hhT1 : T ≤ T1 := inf_le_left,
have hhT2 : T ≤ T2 := inf_le_right,
let g := ((FLF F surj M).map (hom_of_le hT).op).f _ f,
let g1 := ((FLF F surj M).map (hom_of_le hT1).op).f _ f,
let g2 := ((FLF F surj M).map (hom_of_le hT2).op).f _ f,
have hg1 : ((FLF F surj M).map (hom_of_le hhT1).op).f _ g1 = g,
{ dsimp only [g, g1],
have : (hom_of_le hT).op = (hom_of_le hT1).op ≫ (hom_of_le hhT1).op, refl,
rw [this, functor.map_comp],
refl },
have hg2 : ((FLF F surj M).map (hom_of_le hhT2).op).f _ g2 = g,
{ dsimp only [g, g2],
have : (hom_of_le hT).op = (hom_of_le hT2).op ≫ (hom_of_le hhT2).op, refl,
rw [this, functor.map_comp],
refl },
refine ⟨T, g, _, _, _, trivial⟩,
{ rw ← (FLF_cocone F surj M).w (hom_of_le hT).op,
refl },
{ rw ← hg1,
have := ((FLF F surj M).map (hom_of_le hhT1).op).comm (n+1) (n+2),
apply_fun (λ e, e g1) at this,
erw this, clear this,
dsimp only [g1, SemiNormedGroup.coe_comp, function.comp_app],
rw [h1, normed_group_hom.map_zero] },
{ apply le_antisymm,
{ dsimp only [g],
have := (FLF_cocone F surj M).w (hom_of_le hT).op,
rw ← this, clear this,
apply LocallyConstant_obj_map_norm_noninc },
{ rw [← hg2, h2],
apply LocallyConstant_obj_map_norm_noninc } }
end
lemma FLF_norm_noninc (n : ℕ) (S : discrete_quotient F.left)
(f : ((FLF F surj M).obj (op S)).X n) :
∥((FLF_cocone F surj M).ι.app (op S)).f _ f∥₊ ≤ ∥f∥₊ :=
begin
cases n,
apply LocallyConstant_obj_map_norm_noninc,
apply LocallyConstant_obj_map_norm_noninc,
end
theorem prop819 {m : ℕ} (ε : ℝ≥0) (hε : 0 < ε)
(f : (FLC F M).X (m+1)) (hf : (FLC F M).d (m+1) (m+2) f = 0) :
∃ g : (FLC F M).X m, (FLC F M).d m (m+1) g = f ∧ ∥g∥₊ ≤ (1 + ε) * ∥f∥₊ :=
begin
apply exact_of_strict_iso _ _ (FLC_iso F M) ε hε _ _ _ hf,
apply cmpl_exact_of_exact _ _ hε,
clear hf f m hε ε,
intros n f hf,
-- We've reduced to the non-completed case.
have := exists_locally_constant F surj M _ f hf,
rcases this with ⟨S,g,rfl,h2,h3,-⟩,
--let gg := ((FLF_cocone F surj M).ι.app (op S)).f _ g,
let CC : Π (n : ℕ), ((FLF F surj M).obj (op S)).X (n+1) ⟶
((FLF F surj M).obj (op S)).X n :=
((Profinite.arrow_diagram F surj).obj S).contracting_homotopy
(LocallyConstant.{u u}.obj M),
let gc := CC _ g,
let GG := ((FLF_cocone F surj M).ι.app (op S)).f _ gc,
refine ⟨GG,_,_⟩,
{ dsimp only [GG],
have := ((FLF_cocone F surj M).ι.app (op S)).comm n (n+1),
apply_fun (λ e, e gc) at this,
erw this, clear this,
change ((FLF_cocone F surj M).ι.app (op S)).f (n + 1) _ = _,
congr' 1,
change (CC n ≫ _) g = g,
cases n,
{ have hh := arrow.is_contracting_homotopy_one (LocallyConstant.{u u}.obj M)
((Profinite.arrow_diagram F surj).obj S),
apply_fun (λ e, e g) at hh,
change CC 1 (_) + _ = g at hh,
conv at hh {
congr,
congr,
erw h2 },
rw [normed_group_hom.map_zero, zero_add] at hh,
exact hh },
{ have hh := arrow.is_contracting_homotopy (LocallyConstant.{u u}.obj M)
((Profinite.arrow_diagram F surj).obj S) _,
apply_fun (λ e, e g) at hh,
change CC _ (_) + _ = g at hh,
conv at hh {
congr,
congr,
erw h2 },
rw [normed_group_hom.map_zero, zero_add] at hh,
exact hh } },
{ rw h3,
suffices : ∥GG∥₊ ≤ ∥gc∥₊,
{ apply le_trans this _,
cases n,
apply LocallyConstant_obj_map_norm_noninc,
apply LocallyConstant_obj_map_norm_noninc },
apply FLF_norm_noninc }
end