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b797a5b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 | 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
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