openAI-math-dataset / lean /OAI /GroupTheory /PolycyclicRecognition /Finiteness /BicomplexDiagonalComponents.lean
| import OAI.GroupTheory.PolycyclicRecognition.Finiteness.BicomplexDiagonalDifferential | |
| namespace OAI | |
| /-! # Component formula for the first-quadrant total differential -/ | |
| noncomputable section | |
| namespace PolycyclicRecognition | |
| open CategoryTheory | |
| open scoped BigOperators | |
| attribute [local instance 2000] ModuleCat.isModule | |
| attribute [local instance 2300] Pi.module | |
| open Finset.HasAntidiagonal | |
| variable {R : Type} [CommRing R] (K : NatBicomplex R) | |
| abbrev diagonalSucc₁ {n : ℕ} (p : ↥(antidiagonal n)) : ↥(antidiagonal (n + 1)) := | |
| ⟨(p.1.1 + 1, p.1.2), mem_antidiagonal.mpr (by | |
| have := mem_antidiagonal.mp p.2; omega)⟩ | |
| abbrev diagonalSucc₂ {n : ℕ} (p : ↥(antidiagonal n)) : ↥(antidiagonal (n + 1)) := | |
| ⟨(p.1.1, p.1.2 + 1), mem_antidiagonal.mpr (by | |
| have := mem_antidiagonal.mp p.2; omega)⟩ | |
| theorem bicomplex_horizontal_component (n : ℕ) (p : ↥(antidiagonal n)) | |
| (x : bicomplexDiagonal K (n + 1)) : | |
| (∑ q : ↥(antidiagonal (n + 1)), | |
| bicomplexToDiagonal K n ((K.d₁ (ComplexShape.down ℕ) q.1.1 q.1.2 n).hom (x q))) p = | |
| ((K.d (p.1.1 + 1) p.1.1).f p.1.2).hom (x (diagonalSucc₁ p)) := by | |
| classical | |
| rw [Finset.sum_apply, Finset.sum_eq_single (diagonalSucc₁ p)] | |
| · rw [bicomplexToDiagonal_d₁_succ K n p.1.1 p.1.2 (mem_antidiagonal.mp p.2)] | |
| exact Pi.single_eq_same _ _ | |
| · intro q _ hq | |
| rcases q with ⟨⟨i, j⟩, hqmem⟩ | |
| cases i with | |
| | zero => | |
| rw [bicomplexToDiagonal_d₁_zero] | |
| rfl | |
| | succ i => | |
| have hij : i + j = n := by | |
| have := mem_antidiagonal.mp hqmem; omega | |
| rw [bicomplexToDiagonal_d₁_succ K n i j hij] | |
| apply Pi.single_eq_of_ne | |
| intro he | |
| apply hq | |
| apply Subtype.ext | |
| have hv := congrArg Subtype.val he | |
| have h₁ := congrArg Prod.fst hv | |
| have h₂ := congrArg Prod.snd hv | |
| apply Prod.ext <;> dsimp [diagonalSucc₁] at * <;> omega | |
| · simp | |
| theorem bicomplex_vertical_component (n : ℕ) (p : ↥(antidiagonal n)) | |
| (x : bicomplexDiagonal K (n + 1)) : | |
| (∑ q : ↥(antidiagonal (n + 1)), | |
| bicomplexToDiagonal K n ((K.d₂ (ComplexShape.down ℕ) q.1.1 q.1.2 n).hom (x q))) p = | |
| (-1 : ℤˣ) ^ p.1.1 • | |
| ((K.X p.1.1).d (p.1.2 + 1) p.1.2).hom (x (diagonalSucc₂ p)) := by | |
| classical | |
| rw [Finset.sum_apply, Finset.sum_eq_single (diagonalSucc₂ p)] | |
| · rw [bicomplexToDiagonal_d₂_succ K n p.1.1 p.1.2 (mem_antidiagonal.mp p.2)] | |
| simp only [Pi.smul_apply, Pi.single_eq_same] | |
| · intro q _ hq | |
| rcases q with ⟨⟨i, j⟩, hqmem⟩ | |
| cases j with | |
| | zero => | |
| rw [bicomplexToDiagonal_d₂_zero] | |
| rfl | |
| | succ j => | |
| have hij : i + j = n := by | |
| have := mem_antidiagonal.mp hqmem; omega | |
| rw [bicomplexToDiagonal_d₂_succ K n i j hij, Pi.smul_apply] | |
| have he : p ≠ ⟨(i, j), mem_antidiagonal.mpr hij⟩ := by | |
| intro he | |
| apply hq | |
| apply Subtype.ext | |
| have hv := congrArg Subtype.val he | |
| have h₁ := congrArg Prod.fst hv | |
| have h₂ := congrArg Prod.snd hv | |
| apply Prod.ext <;> dsimp [diagonalSucc₂] at * <;> omega | |
| rw [Pi.single_eq_of_ne he] | |
| simp only [Units.smul_def, zsmul_zero] | |
| · simp | |
| theorem bicomplexDiagonalDifferential_apply (n : ℕ) (p : ↥(antidiagonal n)) | |
| (x : bicomplexDiagonal K (n + 1)) : | |
| bicomplexDiagonalDifferential K n x p = | |
| ((K.d (p.1.1 + 1) p.1.1).f p.1.2).hom (x (diagonalSucc₁ p)) + | |
| (-1 : ℤˣ) ^ p.1.1 • | |
| ((K.X p.1.1).d (p.1.2 + 1) p.1.2).hom (x (diagonalSucc₂ p)) := by | |
| classical | |
| conv_lhs => rw [← LinearMap.sum_single_apply _ x] | |
| simp only [map_sum, bicomplexDiagonalDifferential_single, Finset.sum_add_distrib, | |
| Pi.add_apply] | |
| rw [bicomplex_horizontal_component, bicomplex_vertical_component] | |
| end PolycyclicRecognition | |
| end | |
| end OAI | |