| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| open CategoryTheory |
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| namespace Problem340.IteratedSubdivision |
|
|
| section Linear |
|
|
| variable {R M : Type*} [Semiring R] [AddCommMonoid M] [Module R M] |
|
|
| def eventuallyIn (s : Module.End R M) (P : Submodule R M) : Submodule R M where |
| carrier := {x | ∃ N : ℕ, ∀ n, N ≤ n → (s ^ n) x ∈ P} |
| zero_mem' := ⟨0, fun n _ => by simp⟩ |
| add_mem' := by |
| rintro x y ⟨N, hN⟩ ⟨K, hK⟩ |
| refine ⟨max N K, fun n hn => ?_⟩ |
| simpa only [map_add] using P.add_mem |
| (hN n ((le_max_left N K).trans hn)) |
| (hK n ((le_max_right N K).trans hn)) |
| smul_mem' := by |
| rintro a x ⟨N, hN⟩ |
| exact ⟨N, fun n hn => by simpa only [map_smul] using P.smul_mem a (hN n hn)⟩ |
|
|
| theorem pow_mem_of_mem (s : Module.End R M) (P : Submodule R M) |
| (hP : ∀ x ∈ P, s x ∈ P) {x : M} (hx : x ∈ P) (n : ℕ) : |
| (s ^ n) x ∈ P := by |
| induction n with |
| | zero => simpa using hx |
| | succ n ih => |
| simpa only [pow_succ', Module.End.mul_apply] using hP ((s ^ n) x) ih |
|
|
| theorem mem_eventuallyIn_of_pow_mem (s : Module.End R M) (P : Submodule R M) |
| (hP : ∀ x ∈ P, s x ∈ P) {x : M} {N : ℕ} (hx : (s ^ N) x ∈ P) : |
| x ∈ eventuallyIn s P := by |
| refine ⟨N, fun n hn => ?_⟩ |
| obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le hn |
| rw [Nat.add_comm N k] |
| simpa only [pow_add, Module.End.mul_apply] using |
| pow_mem_of_mem s P hP hx k |
|
|
| theorem finsupp_eventuallyIn {ι : Type*} |
| (s : Module.End R (ι →₀ R)) (P : Submodule R (ι →₀ R)) |
| (hgen : ∀ i, Finsupp.single i 1 ∈ eventuallyIn s P) |
| (c : ι →₀ R) : c ∈ eventuallyIn s P := by |
| classical |
| induction c using Finsupp.induction_linear with |
| | zero => exact (eventuallyIn s P).zero_mem |
| | add f g hf hg => exact (eventuallyIn s P).add_mem hf hg |
| | single i a => |
| simpa using (eventuallyIn s P).smul_mem a (hgen i) |
|
|
| theorem finsupp_exists_uniform_iteration {ι : Type*} |
| (s : Module.End R (ι →₀ R)) (P : Submodule R (ι →₀ R)) |
| (hP : ∀ x ∈ P, s x ∈ P) |
| (hgen : ∀ i, ∃ N : ℕ, (s ^ N) (Finsupp.single i 1) ∈ P) |
| (c : ι →₀ R) : ∃ N : ℕ, ∀ n, N ≤ n → (s ^ n) c ∈ P := by |
| apply finsupp_eventuallyIn s P _ c |
| intro i |
| obtain ⟨N, hN⟩ := hgen i |
| exact mem_eventuallyIn_of_pow_mem s P hP hN |
|
|
| theorem finite_family_uniform_iteration {ι : Type*} (t : Finset ι) |
| (s : Module.End R M) (P : Submodule R M) (z : ι → M) |
| (hz : ∀ i ∈ t, z i ∈ eventuallyIn s P) : |
| ∃ N : ℕ, ∀ n, N ≤ n → ∀ i ∈ t, (s ^ n) (z i) ∈ P := by |
| classical |
| induction t using Finset.induction_on with |
| | empty => exact ⟨0, by simp⟩ |
| | @insert i t hi ih => |
| obtain ⟨N, hN⟩ := hz i (Finset.mem_insert_self i t) |
| obtain ⟨K, hK⟩ := ih (fun j hj => hz j (Finset.mem_insert_of_mem hj)) |
| refine ⟨max N K, fun n hn j hj => ?_⟩ |
| rcases Finset.mem_insert.mp hj with rfl | hj |
| · exact hN n ((le_max_left N K).trans hn) |
| · exact hK n ((le_max_right N K).trans hn) j hj |
|
|
| end Linear |
|
|
| section ChainHomotopy |
|
|
| variable {V : Type*} [Category V] [Preadditive V] |
| variable {ι : Type*} {c : ComplexShape ι} {C : HomologicalComplex V c} |
|
|
| def chainIterate (s : C ⟶ C) : ℕ → (C ⟶ C) |
| | 0 => 𝟙 C |
| | n + 1 => chainIterate s n ≫ s |
|
|
| def iterateHomotopy {s : C ⟶ C} (h : Homotopy s (𝟙 C)) : |
| (n : ℕ) → Homotopy (chainIterate s n) (𝟙 C) |
| | 0 => Homotopy.refl _ |
| | n + 1 => (h.compLeftId (chainIterate s n)).trans (iterateHomotopy h n) |
|
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| theorem iterateHomotopy_hom {s : C ⟶ C} (h : Homotopy s (𝟙 C)) |
| (n : ℕ) (i j : ι) : |
| (iterateHomotopy h n).hom i j = |
| ∑ k ∈ Finset.range n, (chainIterate s k).f i ≫ h.hom i j := by |
| induction n with |
| | zero => simp [iterateHomotopy] |
| | succ n ih => |
| simp [iterateHomotopy, Homotopy.compLeftId, ih, Finset.sum_range_succ, |
| add_comm] |
|
|
| theorem chainIterate_eq_pow (s : C ⟶ C) (n : ℕ) : |
| chainIterate s n = End.asHom ((End.of s) ^ n) := by |
| induction n with |
| | zero => rfl |
| | succ n ih => simp only [chainIterate, ih, pow_succ', End.mul_def] |
|
|
| theorem homologyMap_chainIterate {s : C ⟶ C} (h : Homotopy s (𝟙 C)) |
| (n : ℕ) (i : ι) [C.HasHomology i] : |
| HomologicalComplex.homologyMap (chainIterate s n) i = 𝟙 (C.homology i) := by |
| rw [(iterateHomotopy h n).homologyMap_eq, HomologicalComplex.homologyMap_id] |
|
|
| end ChainHomotopy |
|
|
| section Support |
|
|
| variable {R : Type*} [Ring R] {ι : Type*} {c : ComplexShape ι} |
| variable {C : HomologicalComplex (ModuleCat R) c} |
|
|
| theorem chainIterate_f_hom (s : C ⟶ C) (n : ℕ) (i : ι) : |
| ((chainIterate s n).f i).hom = ((s.f i).hom ^ n) := by |
| induction n with |
| | zero => rfl |
| | succ n ih => |
| ext x |
| simp only [chainIterate, HomologicalComplex.comp_f, ModuleCat.hom_comp, |
| LinearMap.comp_apply, ih, pow_succ', Module.End.mul_apply] |
|
|
| theorem chainIterate_preserves (s : C ⟶ C) (P : ∀ i, Submodule R (C.X i)) |
| (hs : ∀ i x, x ∈ P i → (s.f i).hom x ∈ P i) |
| (n : ℕ) (i : ι) {x : C.X i} (hx : x ∈ P i) : |
| ((chainIterate s n).f i).hom x ∈ P i := by |
| rw [chainIterate_f_hom] |
| exact pow_mem_of_mem _ _ (hs i) hx n |
|
|
| theorem iterateHomotopy_preserves {s : C ⟶ C} (h : Homotopy s (𝟙 C)) |
| (P : ∀ i, Submodule R (C.X i)) |
| (hs : ∀ i x, x ∈ P i → (s.f i).hom x ∈ P i) |
| (hh : ∀ i j x, x ∈ P i → (h.hom i j).hom x ∈ P j) |
| (n : ℕ) (i j : ι) {x : C.X i} (hx : x ∈ P i) : |
| ((iterateHomotopy h n).hom i j).hom x ∈ P j := by |
| rw [iterateHomotopy_hom, ModuleCat.hom_sum, LinearMap.sum_apply] |
| apply (P j).sum_mem |
| intro k _ |
| change (h.hom i j).hom (((chainIterate s k).f i).hom x) ∈ P j |
| exact hh i j _ (chainIterate_preserves s P hs k i hx) |
|
|
| end Support |
|
|
| end Problem340.IteratedSubdivision |
|
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| end |
|
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| end OAI |
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|
