| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| namespace PiExponent.NormalBasisRigidity | |
| open Set Submodule Module | |
| variable {K V ι : Type*} [Field K] [AddCommGroup V] [Module K V] | |
| def IsNormalBasis (v : ι → V) (A : Finset ι) : Prop := | |
| LinearIndepOn K v (A : Set ι) ∧ span K (v '' (A : Set ι)) = ⊤ | |
| lemma exists_normalBasis_subset [Fintype ι] (v : ι → V) (S : Set ι) | |
| (hS : span K (v '' S) = ⊤) : | |
| ∃ A : Finset ι, (A : Set ι) ⊆ S ∧ IsNormalBasis (K := K) v A := by | |
| classical | |
| let h := linearIndepOn_empty K v | |
| let A := h.extend (Set.empty_subset S) | |
| refine ⟨A.toFinset, ?_, ?_, ?_⟩ | |
| · simpa only [Set.coe_toFinset] using h.extend_subset (Set.empty_subset S) | |
| · simpa only [Set.coe_toFinset] using h.linearIndepOn_extend (Set.empty_subset S) | |
| · simpa only [Set.coe_toFinset, A] using | |
| (h.span_image_extend_eq_span_image (Set.empty_subset S)).trans hS | |
| lemma exists_normalBasis_containing [Fintype ι] (v : ι → V) | |
| (hv : span K (Set.range v) = ⊤) (i : ι) (hi : v i ≠ 0) : | |
| ∃ A : Finset ι, i ∈ A ∧ IsNormalBasis (K := K) v A := by | |
| classical | |
| have h : LinearIndepOn K v ({i} : Set ι) := by | |
| exact LinearIndepOn.singleton hi | |
| let A := h.extend (Set.subset_univ _) | |
| refine ⟨A.toFinset, ?_, ?_, ?_⟩ | |
| · exact Set.mem_toFinset.mpr (h.subset_extend (Set.subset_univ _) (Set.mem_singleton i)) | |
| · simpa only [Set.coe_toFinset] using h.linearIndepOn_extend (Set.subset_univ _) | |
| · simpa only [Set.coe_toFinset, Set.image_univ, A] using | |
| (h.span_image_extend_eq_span_image (Set.subset_univ _)).trans (by simpa using hv) | |
| lemma IsNormalBasis.ne_zero [DecidableEq ι] {v : ι → V} {A : Finset ι} | |
| (hA : IsNormalBasis (K := K) v A) {i : ι} (hi : i ∈ A) : v i ≠ 0 := | |
| hA.1.ne_zero hi | |
| lemma exists_normalBasis_omitting [Fintype ι] (v : ι → V) (i : ι) | |
| (h : span K (v '' {j | j ≠ i}) = ⊤) : | |
| ∃ A : Finset ι, i ∉ A ∧ IsNormalBasis (K := K) v A := by | |
| obtain ⟨A, hA, hB⟩ := exists_normalBasis_subset v {j | j ≠ i} h | |
| exact ⟨A, fun hi => hA hi rfl, hB⟩ | |
| lemma mandatory_not_mem_span [Fintype ι] (v : ι → V) | |
| (hv : span K (Set.range v) = ⊤) (i : ι) | |
| (hi : ∀ A, IsNormalBasis (K := K) v A → i ∈ A) : | |
| v i ∉ span K (v '' {j | j ≠ i}) := by | |
| intro hmem | |
| have hspan : span K (v '' {j | j ≠ i}) = ⊤ := by | |
| apply top_unique | |
| rw [← hv] | |
| apply span_le.mpr | |
| rintro _ ⟨j, rfl⟩ | |
| by_cases hj : j = i | |
| · subst j | |
| exact hmem | |
| · exact subset_span ⟨j, hj, rfl⟩ | |
| obtain ⟨A, hAi, hA⟩ := exists_normalBasis_omitting v i hspan | |
| exact hAi (hi A hA) | |
| lemma IsNormalBasis.card_eq {v : ι → V} {A : Finset ι} | |
| (hA : IsNormalBasis (K := K) v A) : A.card = Module.finrank K V := by | |
| classical | |
| have hli : LinearIndependent K (fun i : A => v i) := hA.1 | |
| let b : Basis A K V := Basis.mk hli (by | |
| have hr : Set.range (fun i : A => v i) = v '' (A : Set ι) := by | |
| ext x | |
| simp | |
| rw [hr, hA.2]) | |
| exact (Module.finrank_eq_card_finset_basis b).symm | |
| variable {W : Type*} [AddCommGroup W] [Module K W] | |
| lemma mandatory_coordinate_eq_zero [Fintype ι] (b : Basis ι K V) (q : V →ₗ[K] W) | |
| (hq : span K (Set.range (fun j => q (b j))) = ⊤) (i : ι) | |
| (hi : ∀ A, IsNormalBasis (K := K) (fun j => q (b j)) A → i ∈ A) | |
| (t : V) (ht : q t = 0) : b.repr t i = 0 := by | |
| classical | |
| by_contra hti | |
| let S : Submodule K W := span K ((fun j => q (b j)) '' {j | j ≠ i}) | |
| have hsum : (∑ j ∈ Finset.univ.erase i, b.repr t j • q (b j)) ∈ S := by | |
| apply S.sum_mem | |
| intro j hj | |
| exact S.smul_mem _ (subset_span ⟨j, (Finset.mem_erase.mp hj).1, rfl⟩) | |
| have hall : (∑ j, b.repr t j • q (b j)) = 0 := by | |
| calc | |
| _ = q (∑ j, b.repr t j • b j) := by simp only [map_sum, map_smul] | |
| _ = 0 := by rw [b.sum_repr, ht] | |
| rw [← Finset.sum_erase_add _ _ (Finset.mem_univ i)] at hall | |
| have hterm : b.repr t i • q (b i) ∈ S := by | |
| have heq : b.repr t i • q (b i) = -(∑ j ∈ Finset.univ.erase i, | |
| b.repr t j • q (b j)) := eq_neg_of_add_eq_zero_left (by simpa only [add_comm] using hall) | |
| rw [heq] | |
| exact S.neg_mem hsum | |
| exact mandatory_not_mem_span (fun j => q (b j)) hq i hi | |
| ((S.smul_mem_iff hti).mp hterm) | |
| lemma largest_nontangent_mandatory {m : ℕ} | |
| (u v : Fin (m + 1) → V) (hu : span K (Set.range u) = ⊤) | |
| (hvu : ∀ i, i ≠ 0 → v i = u i) | |
| (hexclude : ∀ A B : Finset (Fin (m + 1)), | |
| IsNormalBasis (K := K) u A → IsNormalBasis (K := K) v B → | |
| ∀ i, i ≠ 0 → i ∈ A → i ∉ B → | |
| (∀ j, i < j → (j ∈ A ↔ j ∈ B)) → False) | |
| (i : Fin (m + 1)) (hi0 : i ≠ 0) (hi : u i ≠ 0) | |
| (hlarge : ∀ j, i < j → u j = 0) | |
| (B : Finset (Fin (m + 1))) (hB : IsNormalBasis (K := K) v B) : i ∈ B := by | |
| classical | |
| obtain ⟨A, hiA, hA⟩ := exists_normalBasis_containing u hu i hi | |
| by_contra hiB | |
| apply hexclude A B hA hB i hi0 hiA hiB | |
| intro j hj | |
| have hj0 : j ≠ 0 := by | |
| intro h | |
| subst j | |
| exact (not_lt_of_ge (Fin.zero_le _)) hj | |
| have hjA : j ∉ A := fun h => hA.ne_zero h (hlarge j hj) | |
| have hjB : j ∉ B := fun h => hB.ne_zero h ((hvu j hj0).trans (hlarge j hj)) | |
| simp only [hjA, hjB] | |
| lemma weighted_normalBasis_exclusion {m : ℕ} | |
| (u v : Fin (m + 1) → V) (w cost : Fin (m + 1) → ℝ) (C : ℝ) | |
| (hcomparison : ∀ A B, IsNormalBasis (K := K) u A → | |
| IsNormalBasis (K := K) v B → (∏ j ∈ A, w j) ≤ C * ∏ j ∈ B, cost j) | |
| (hseparated : ∀ A B : Finset (Fin (m + 1)), A.card = B.card → | |
| ∀ i, i ≠ 0 → i ∈ A → i ∉ B → | |
| (∀ j, i < j → (j ∈ A ↔ j ∈ B)) → | |
| C * (∏ j ∈ B, cost j) < ∏ j ∈ A, w j) | |
| (A B : Finset (Fin (m + 1))) | |
| (hA : IsNormalBasis (K := K) u A) (hB : IsNormalBasis (K := K) v B) | |
| (i : Fin (m + 1)) (hi0 : i ≠ 0) (hiA : i ∈ A) (hiB : i ∉ B) | |
| (hlarge : ∀ j, i < j → (j ∈ A ↔ j ∈ B)) : False := by | |
| exact (not_lt_of_ge (hcomparison A B hA hB)) | |
| (hseparated A B (hA.card_eq.trans hB.card_eq.symm) i hi0 hiA hiB hlarge) | |
| lemma normalBasis_unique_of_discrepancy_excluded [LinearOrder ι] | |
| (v : ι → V) | |
| (hexclude : ∀ A B : Finset ι, | |
| IsNormalBasis (K := K) v A → IsNormalBasis (K := K) v B → | |
| ∀ i, i ∈ A → i ∉ B → | |
| (∀ j, i < j → (j ∈ A ↔ j ∈ B)) → False) | |
| (A B : Finset ι) (hA : IsNormalBasis (K := K) v A) | |
| (hB : IsNormalBasis (K := K) v B) : A = B := by | |
| classical | |
| by_contra hne | |
| let S := (A \ B) ∪ (B \ A) | |
| have hS : S.Nonempty := by | |
| by_contra he | |
| have he' : S = ∅ := Finset.not_nonempty_iff_eq_empty.mp he | |
| apply hne | |
| apply Finset.ext | |
| intro i | |
| constructor | |
| · intro hi | |
| by_contra hn | |
| have : i ∈ S := Finset.mem_union_left _ (Finset.mem_sdiff.mpr ⟨hi, hn⟩) | |
| simp [he'] at this | |
| · intro hi | |
| by_contra hn | |
| have : i ∈ S := Finset.mem_union_right _ (Finset.mem_sdiff.mpr ⟨hi, hn⟩) | |
| simp [he'] at this | |
| let i := S.max' hS | |
| have hiS : i ∈ S := Finset.max'_mem _ _ | |
| have hlarge : ∀ j, i < j → (j ∈ A ↔ j ∈ B) := by | |
| intro j hj | |
| have hjS : j ∉ S := fun h => (not_lt_of_ge (Finset.le_max' _ _ h)) hj | |
| constructor | |
| · intro hja | |
| by_contra hjb | |
| exact hjS (Finset.mem_union_left _ (Finset.mem_sdiff.mpr ⟨hja, hjb⟩)) | |
| · intro hjb | |
| by_contra hja | |
| exact hjS (Finset.mem_union_right _ (Finset.mem_sdiff.mpr ⟨hjb, hja⟩)) | |
| rcases Finset.mem_union.mp hiS with hi | hi | |
| · exact hexclude A B hA hB i (Finset.mem_sdiff.mp hi).1 | |
| (Finset.mem_sdiff.mp hi).2 hlarge | |
| · exact hexclude B A hB hA i (Finset.mem_sdiff.mp hi).1 | |
| (Finset.mem_sdiff.mp hi).2 (fun j hj => (hlarge j hj).symm) | |
| end PiExponent.NormalBasisRigidity | |
| end | |
| end OAI | |