import OAI.NumberTheory.DirichletL.CubicSieve.CrossFactorizationnamespace OAInoncomputable sectionopen scoped BigOperatorsopen MulChar AddCharopen scoped BigOperatorsopen Filter Asymptotics MeasureTheoryopen scoped Topologyopen MeasureTheory Realopen scoped FourierTransform SchwartzMapopen Finset Complexopen scoped Classicalopen scoped Classicalopen Filter Real Asymptoticsopen ActualEisensteinCubicopen Filteropen ActualEisensteinCubic RationalPrimeExtraction ShortDraftLatticeCountopen ActualEisensteinCubic ShortDraftLatticeCountopen Filteropen scoped Topologyopen EisensteinEmbedding ConcreteTraceCRT ActualEisensteinCubicopen MulChar AddCharopen Filter Asymptoticsopen scoped LSeries.notation ArithmeticFunction.Moebiusopen Filteropen MulChar AddCharopen MulChar AddCharopen scoped LSeries.notation ArithmeticFunction.Moebiusopen Filter Asymptotics MeasureTheoryopen scoped Topologyopen Filter Asymptoticsopen Ideal NumberField RingOfIntegers UniqueFactorizationMonoidopen Ideal NumberField RingOfIntegers UniqueFactorizationMonoidopen Ideal NumberField RingOfIntegers UniqueFactorizationMonoidopen Ideal NumberField RingOfIntegers UniqueFactorizationMonoidopen Ideal NumberField RingOfIntegers UniqueFactorizationMonoidopen Filter Asymptoticsopen Filter Asymptotics MeasureTheoryopen scoped Topologyopen Filter Asymptotics Ideal NumberFieldopen Filteropen Filter Asymptotics MeasureTheoryopen scoped Topologyopen Filter Asymptotics MeasureTheoryopen scoped Topologyopen Filter Asymptotics MeasureTheoryopen scoped Topologyopen MeasureTheory Realopen scoped ContDiff FourierTransform SchwartzMapopen scoped BigOperators Classicalopen scoped BigOperators Classicalopen scoped BigOperators Classicalopen scoped BigOperators Classical SchwartzMap ContDiffopen scoped BigOperators Classical SchwartzMap ContDiffopen scoped BigOperators Classicalopen scoped BigOperators Classical SchwartzMap ContDiffopen scoped BigOperators Classicalopen scoped BigOperators Classical SchwartzMap ContDiffopen scoped BigOperators Classical SchwartzMap ContDiffopen scoped BigOperators Classical SchwartzMap ContDiffopen scoped BigOperators Classicalopen scoped BigOperators Classical SchwartzMap ContDiffopen MeasureTheory Setopen scoped BigOperatorsopen scoped BigOperators Classicalopen scoped BigOperators Classicalopen ActualEisensteinCubic UniqueFactorizationMonoidopen scoped BigOperators Classicalnamespace CoprimeBilinearSieveopen FiniteSieveOperator CoprimeSieveOperator CoprimeMobiusExtensiontheorem disjoint_bilinear_expansion {ι n p : Type*} [DecidableEq ι] [Fintype n] [Fintype p] (B : Finset ι) (left : n → Finset ι) (right : p → Finset ι) (hl : ∀ j, left j ⊆ B) (hr : ∀ k, right k ⊆ B) (z : n → ℂ) (v : p → ℂ) : (∑ j, ∑ k, if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1 : ℂ) ^ D.card * (star (∑ j, if D ⊆ left j then z j else 0) * (∑ k, if D ⊆ right k then v k else 0)) := byι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂ⊢ (∑ j, ∑ k, if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (∑ j, if D ⊆ left j then z j else 0) * ∑ k, if D ⊆ right k then v k else 0) have hp (j : n) (k : p) : (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1 : ℂ) ^ D.card * (if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0) := by calc _ = (if Disjoint (left j) (right k) then (1 : ℂ) else 0) * (star (z j) * v k) := byι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂj:nk:p⊢ (if Disjoint (left j) (right k) then star (z j) * v k else 0) = (if Disjoint (left j) (right k) then 1 else 0) * (star (z j) * v k) split_ifsposι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂj:nk:ph✝:Disjoint (left j) (right k)⊢ star (z j) * v k = 1 * (star (z j) * v k)negι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂj:nk:ph✝:¬Disjoint (left j) (right k)⊢ 0 = 0 * (star (z j) * v k) <;>posι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂj:nk:ph✝:Disjoint (left j) (right k)⊢ star (z j) * v k = 1 * (star (z j) * v k)negι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂj:nk:ph✝:¬Disjoint (left j) (right k)⊢ 0 = 0 * (star (z j) * v k) simpAll goals completed! 🐙 _ = _ := byι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂj:nk:p⊢ (if Disjoint (left j) (right k) then 1 else 0) * (star (z j) * v k) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0 rw [disjoint_indicator_expansion B _ _ (hl j) (hr k),ι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂj:nk:p⊢ (∑ D ∈ B.powerset, if D ⊆ left j ∧ D ⊆ right k then (-1) ^ #D else 0) * (star (z j) * v k) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0 Finset.sum_mulι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂj:nk:p⊢ ∑ i ∈ B.powerset, (if i ⊆ left j ∧ i ⊆ right k then (-1) ^ #i else 0) * (star (z j) * v k) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0]ι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂj:nk:p⊢ ∑ i ∈ B.powerset, (if i ⊆ left j ∧ i ⊆ right k then (-1) ^ #i else 0) * (star (z j) * v k) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0 apply Finset.sum_congr rflι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂj:nk:p⊢ ∀ x ∈ B.powerset, (if x ⊆ left j ∧ x ⊆ right k then (-1) ^ #x else 0) * (star (z j) * v k) = (-1) ^ #x * if x ⊆ left j ∧ x ⊆ right k then star (z j) * v k else 0 intro D _ι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂj:nk:pD:Finset ιa✝:D ∈ B.powerset⊢ (if D ⊆ left j ∧ D ⊆ right k then (-1) ^ #D else 0) * (star (z j) * v k) = (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0 split_ifsposι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂj:nk:pD:Finset ιa✝:D ∈ B.powerseth✝:D ⊆ left j ∧ D ⊆ right k⊢ (-1) ^ #D * (star (z j) * v k) = (-1) ^ #D * (star (z j) * v k)negι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂj:nk:pD:Finset ιa✝:D ∈ B.powerseth✝:¬(D ⊆ left j ∧ D ⊆ right k)⊢ 0 * (star (z j) * v k) = (-1) ^ #D * 0 <;>posι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂj:nk:pD:Finset ιa✝:D ∈ B.powerseth✝:D ⊆ left j ∧ D ⊆ right k⊢ (-1) ^ #D * (star (z j) * v k) = (-1) ^ #D * (star (z j) * v k)negι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂj:nk:pD:Finset ιa✝:D ∈ B.powerseth✝:¬(D ⊆ left j ∧ D ⊆ right k)⊢ 0 * (star (z j) * v k) = (-1) ^ #D * 0 simpι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0⊢ (∑ j, ∑ k, if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (∑ j, if D ⊆ left j then z j else 0) * ∑ k, if D ⊆ right k then v k else 0) simp_rw [ι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0⊢ (∑ j, ∑ k, if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (∑ j, if D ⊆ left j then z j else 0) * ∑ k, if D ⊆ right k then v k else 0)hpι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0⊢ (∑ x, ∑ x_1, ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left x ∧ D ⊆ right x_1 then star (z x) * v x_1 else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (∑ j, if D ⊆ left j then z j else 0) * ∑ k, if D ⊆ right k then v k else 0)] calc _ = ∑ j, ∑ D ∈ B.powerset, ∑ k, (-1 : ℂ) ^ D.card * (if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0) := byι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0⊢ (∑ x, ∑ x_1, ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left x ∧ D ⊆ right x_1 then star (z x) * v x_1 else 0) = ∑ j, ∑ D ∈ B.powerset, ∑ k, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0 apply Finset.sum_congr rflι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0⊢ ∀ x ∈ Finset.univ, (∑ x_1, ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left x ∧ D ⊆ right x_1 then star (z x) * v x_1 else 0) = ∑ D ∈ B.powerset, ∑ k, (-1) ^ #D * if D ⊆ left x ∧ D ⊆ right k then star (z x) * v k else 0 intro j _ι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0j:na✝:j ∈ Finset.univ⊢ (∑ x, ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right x then star (z j) * v x else 0) = ∑ D ∈ B.powerset, ∑ k, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0 exact Finset.sum_commAll goals completed! 🐙 _ = ∑ D ∈ B.powerset, ∑ j, ∑ k, (-1 : ℂ) ^ D.card * (if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0) := Finset.sum_comm _ = _ := byι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0⊢ (∑ D ∈ B.powerset, ∑ j, ∑ k, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (∑ j, if D ⊆ left j then z j else 0) * ∑ k, if D ⊆ right k then v k else 0) apply Finset.sum_congr rflι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0⊢ ∀ x ∈ B.powerset, (∑ j, ∑ k, (-1) ^ #x * if x ⊆ left j ∧ x ⊆ right k then star (z j) * v k else 0) = (-1) ^ #x * (star (∑ j, if x ⊆ left j then z j else 0) * ∑ k, if x ⊆ right k then v k else 0) intro D _ι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0D:Finset ιa✝:D ∈ B.powerset⊢ (∑ j, ∑ k, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0) = (-1) ^ #D * (star (∑ j, if D ⊆ left j then z j else 0) * ∑ k, if D ⊆ right k then v k else 0) simp_rw [ι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0D:Finset ιa✝:D ∈ B.powerset⊢ (∑ j, ∑ k, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0) = (-1) ^ #D * (star (∑ j, if D ⊆ left j then z j else 0) * ∑ k, if D ⊆ right k then v k else 0)← Finset.mul_sumι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0D:Finset ιa✝:D ∈ B.powerset⊢ ((-1) ^ #D * ∑ i, ∑ i_1, if D ⊆ left i ∧ D ⊆ right i_1 then star (z i) * v i_1 else 0) = (-1) ^ #D * (star (∑ j, if D ⊆ left j then z j else 0) * ∑ k, if D ⊆ right k then v k else 0)] congr 1e_aι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0D:Finset ιa✝:D ∈ B.powerset⊢ (∑ i, ∑ i_1, if D ⊆ left i ∧ D ⊆ right i_1 then star (z i) * v i_1 else 0) = star (∑ j, if D ⊆ left j then z j else 0) * ∑ k, if D ⊆ right k then v k else 0 simp only [star_sum, Finset.sum_mul_sum]e_aι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0D:Finset ιa✝:D ∈ B.powerset⊢ (∑ i, ∑ i_1, if D ⊆ left i ∧ D ⊆ right i_1 then star (z i) * v i_1 else 0) = ∑ i, ∑ j, star (if D ⊆ left i then z i else 0) * if D ⊆ right j then v j else 0 apply Finset.sum_congr rfle_aι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0D:Finset ιa✝:D ∈ B.powerset⊢ ∀ x ∈ Finset.univ, (∑ i, if D ⊆ left x ∧ D ⊆ right i then star (z x) * v i else 0) = ∑ j, star (if D ⊆ left x then z x else 0) * if D ⊆ right j then v j else 0 intro j _e_aι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0D:Finset ιa✝¹:D ∈ B.powersetj:na✝:j ∈ Finset.univ⊢ (∑ i, if D ⊆ left j ∧ D ⊆ right i then star (z j) * v i else 0) = ∑ j_1, star (if D ⊆ left j then z j else 0) * if D ⊆ right j_1 then v j_1 else 0 apply Finset.sum_congr rfle_aι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0D:Finset ιa✝¹:D ∈ B.powersetj:na✝:j ∈ Finset.univ⊢ ∀ x ∈ Finset.univ, (if D ⊆ left j ∧ D ⊆ right x then star (z j) * v x else 0) = star (if D ⊆ left j then z j else 0) * if D ⊆ right x then v x else 0 intro k _e_aι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0D:Finset ιa✝²:D ∈ B.powersetj:na✝¹:j ∈ Finset.univk:pa✝:k ∈ Finset.univ⊢ (if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0) = star (if D ⊆ left j then z j else 0) * if D ⊆ right k then v k else 0 by_cases hj : D ⊆ left jposι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0D:Finset ιa✝²:D ∈ B.powersetj:na✝¹:j ∈ Finset.univk:pa✝:k ∈ Finset.univhj:D ⊆ left j⊢ (if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0) = star (if D ⊆ left j then z j else 0) * if D ⊆ right k then v k else 0negι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0D:Finset ιa✝²:D ∈ B.powersetj:na✝¹:j ∈ Finset.univk:pa✝:k ∈ Finset.univhj:¬D ⊆ left j⊢ (if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0) = star (if D ⊆ left j then z j else 0) * if D ⊆ right k then v k else 0 <;>posι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0D:Finset ιa✝²:D ∈ B.powersetj:na✝¹:j ∈ Finset.univk:pa✝:k ∈ Finset.univhj:D ⊆ left j⊢ (if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0) = star (if D ⊆ left j then z j else 0) * if D ⊆ right k then v k else 0negι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0D:Finset ιa✝²:D ∈ B.powersetj:na✝¹:j ∈ Finset.univk:pa✝:k ∈ Finset.univhj:¬D ⊆ left j⊢ (if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0) = star (if D ⊆ left j then z j else 0) * if D ⊆ right k then v k else 0 by_cases hk : D ⊆ right kposι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0D:Finset ιa✝²:D ∈ B.powersetj:na✝¹:j ∈ Finset.univk:pa✝:k ∈ Finset.univhj:¬D ⊆ left jhk:D ⊆ right k⊢ (if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0) = star (if D ⊆ left j then z j else 0) * if D ⊆ right k then v k else 0negι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0D:Finset ιa✝²:D ∈ B.powersetj:na✝¹:j ∈ Finset.univk:pa✝:k ∈ Finset.univhj:¬D ⊆ left jhk:¬D ⊆ right k⊢ (if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0) = star (if D ⊆ left j then z j else 0) * if D ⊆ right k then v k else 0 <;>posι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0D:Finset ιa✝²:D ∈ B.powersetj:na✝¹:j ∈ Finset.univk:pa✝:k ∈ Finset.univhj:D ⊆ left jhk:D ⊆ right k⊢ (if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0) = star (if D ⊆ left j then z j else 0) * if D ⊆ right k then v k else 0negι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0D:Finset ιa✝²:D ∈ B.powersetj:na✝¹:j ∈ Finset.univk:pa✝:k ∈ Finset.univhj:D ⊆ left jhk:¬D ⊆ right k⊢ (if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0) = star (if D ⊆ left j then z j else 0) * if D ⊆ right k then v k else 0posι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0D:Finset ιa✝²:D ∈ B.powersetj:na✝¹:j ∈ Finset.univk:pa✝:k ∈ Finset.univhj:¬D ⊆ left jhk:D ⊆ right k⊢ (if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0) = star (if D ⊆ left j then z j else 0) * if D ⊆ right k then v k else 0negι:Type u_1n:Type u_2p:Type u_3inst✝²:DecidableEq ιinst✝¹:Fintype ninst✝:Fintype pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ Bz:n → ℂv:p → ℂhp:∀ (j : n) (k : p), (if Disjoint (left j) (right k) then star (z j) * v k else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0D:Finset ιa✝²:D ∈ B.powersetj:na✝¹:j ∈ Finset.univk:pa✝:k ∈ Finset.univhj:¬D ⊆ left jhk:¬D ⊆ right k⊢ (if D ⊆ left j ∧ D ⊆ right k then star (z j) * v k else 0) = star (if D ⊆ left j then z j else 0) * if D ⊆ right k then v k else 0 simp [hj, hk]All goals completed! 🐙theorem coprime_bilinear_operator_bound {ι m n p : Type*} [DecidableEq ι] [Fintype m] [Fintype n] [Fintype p] [DecidableEq m] [DecidableEq n] [DecidableEq p] (B : Finset ι) (left : n → Finset ι) (right : p → Finset ι) (hl : ∀ j, left j ⊆ B) (hr : ∀ k, right k ⊆ B) (A : Matrix m n ℂ) (C : Matrix m p ℂ) (a : n → ℂ) (b : p → ℂ) : (∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, (2 : ℝ) ^ (left j).card * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, (2 : ℝ) ^ (right k).card * ‖b k‖ ^ 2) := byι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂ⊢ (∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(right k) * ‖b k‖ ^ 2) let X (D : Finset ι) (i : m) : ℂ := ∑ j, A i j * (if D ⊆ left j then a j else 0)ι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0⊢ (∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(right k) * ‖b k‖ ^ 2) let Y (D : Finset ι) (i : m) : ℂ := ∑ k, C i k * (if D ⊆ right k then b k else 0)ι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0⊢ (∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(right k) * ‖b k‖ ^ 2) have heq (i : m) : (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1 : ℂ) ^ D.card * (star (X D i) * Y D i) := by simpa only [X, Y, mul_ite, mul_zero] using disjoint_bilinear_expansion B left right hl hr (fun j => A i j * a j) (fun k => C i k * b k)ι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)⊢ (∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(right k) * ‖b k‖ ^ 2) have hnorm : (∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖) ≤ ∑ D ∈ B.powerset, ∑ i, ‖X D i‖ * ‖Y D i‖ := by rw [Finset.sum_commι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)⊢ ∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ y, ∑ x ∈ B.powerset, ‖X x y‖ * ‖Y x y‖]ι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)⊢ ∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ y, ∑ x ∈ B.powerset, ‖X x y‖ * ‖Y x y‖ apply Finset.sum_le_sumι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)⊢ ∀ i ∈ Finset.univ, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ x ∈ B.powerset, ‖X x i‖ * ‖Y x i‖ intro i _ι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)i:ma✝:i ∈ Finset.univ⊢ ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ x ∈ B.powerset, ‖X x i‖ * ‖Y x i‖ rw [heqι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)i:ma✝:i ∈ Finset.univ⊢ ‖∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)‖ ≤ ∑ x ∈ B.powerset, ‖X x i‖ * ‖Y x i‖]ι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)i:ma✝:i ∈ Finset.univ⊢ ‖∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)‖ ≤ ∑ x ∈ B.powerset, ‖X x i‖ * ‖Y x i‖ apply (norm_sum_le _ _).transι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)i:ma✝:i ∈ Finset.univ⊢ ∑ i_1 ∈ B.powerset, ‖(-1) ^ #i_1 * (star (X i_1 i) * Y i_1 i)‖ ≤ ∑ x ∈ B.powerset, ‖X x i‖ * ‖Y x i‖ apply Finset.sum_le_sumι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)i:ma✝:i ∈ Finset.univ⊢ ∀ i_1 ∈ B.powerset, ‖(-1) ^ #i_1 * (star (X i_1 i) * Y i_1 i)‖ ≤ ‖X i_1 i‖ * ‖Y i_1 i‖ intro D _ι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)i:ma✝¹:i ∈ Finset.univD:Finset ιa✝:D ∈ B.powerset⊢ ‖(-1) ^ #D * (star (X D i) * Y D i)‖ ≤ ‖X D i‖ * ‖Y D i‖ simp only [norm_mul, norm_pow, norm_neg, norm_one, one_pow, one_mul, norm_star]ι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)i:ma✝¹:i ∈ Finset.univD:Finset ιa✝:D ∈ B.powerset⊢ ‖X D i‖ * ‖Y D i‖ ≤ ‖X D i‖ * ‖Y D i‖ exact le_rflι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)hnorm:∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ D ∈ B.powerset, ∑ i, ‖X D i‖ * ‖Y D i‖⊢ (∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(right k) * ‖b k‖ ^ 2) have hCS := Finset.sum_mul_sq_le_sq_mul_sq (B.powerset ×ˢ (Finset.univ : Finset m)) (fun di => ‖X di.1 di.2‖) (fun di => ‖Y di.1 di.2‖)ι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)hnorm:∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ D ∈ B.powerset, ∑ i, ‖X D i‖ * ‖Y D i‖hCS:(∑ i ∈ B.powerset ×ˢ Finset.univ, ‖X i.1 i.2‖ * ‖Y i.1 i.2‖) ^ 2 ≤ (∑ i ∈ B.powerset ×ˢ Finset.univ, ‖X i.1 i.2‖ ^ 2) * ∑ i ∈ B.powerset ×ˢ Finset.univ, ‖Y i.1 i.2‖ ^ 2⊢ (∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(right k) * ‖b k‖ ^ 2) simp only [Finset.sum_product] at hCSι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)hnorm:∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ D ∈ B.powerset, ∑ i, ‖X D i‖ * ‖Y D i‖hCS:(∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ * ‖Y x x_1‖) ^ 2 ≤ (∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ ^ 2) * ∑ x ∈ B.powerset, ∑ x_1, ‖Y x x_1‖ ^ 2⊢ (∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(right k) * ‖b k‖ ^ 2) have hX : (∑ D ∈ B.powerset, ∑ i, ‖X D i‖ ^ 2) ≤ ‖operator A‖ ^ 2 * ∑ j, (2 : ℝ) ^ (left j).card * ‖a j‖ ^ 2 := by calc _ ≤ ∑ D ∈ B.powerset, ‖operator A‖ ^ 2 * ∑ j, ‖if D ⊆ left j then a j else 0‖ ^ 2 := byι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)hnorm:∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ D ∈ B.powerset, ∑ i, ‖X D i‖ * ‖Y D i‖hCS:(∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ * ‖Y x x_1‖) ^ 2 ≤ (∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ ^ 2) * ∑ x ∈ B.powerset, ∑ x_1, ‖Y x x_1‖ ^ 2⊢ ∑ D ∈ B.powerset, ∑ i, ‖X D i‖ ^ 2 ≤ ∑ D ∈ B.powerset, ‖operator A‖ ^ 2 * ∑ j, ‖if D ⊆ left j then a j else 0‖ ^ 2 apply Finset.sum_le_sumι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)hnorm:∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ D ∈ B.powerset, ∑ i, ‖X D i‖ * ‖Y D i‖hCS:(∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ * ‖Y x x_1‖) ^ 2 ≤ (∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ ^ 2) * ∑ x ∈ B.powerset, ∑ x_1, ‖Y x x_1‖ ^ 2⊢ ∀ i ∈ B.powerset, ∑ i_1, ‖X i i_1‖ ^ 2 ≤ ‖operator A‖ ^ 2 * ∑ j, ‖if i ⊆ left j then a j else 0‖ ^ 2 intro D _ι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)hnorm:∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ D ∈ B.powerset, ∑ i, ‖X D i‖ * ‖Y D i‖hCS:(∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ * ‖Y x x_1‖) ^ 2 ≤ (∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ ^ 2) * ∑ x ∈ B.powerset, ∑ x_1, ‖Y x x_1‖ ^ 2D:Finset ιa✝:D ∈ B.powerset⊢ ∑ i, ‖X D i‖ ^ 2 ≤ ‖operator A‖ ^ 2 * ∑ j, ‖if D ⊆ left j then a j else 0‖ ^ 2 exact energy_bound A _All goals completed! 🐙 _ = _ := byι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)hnorm:∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ D ∈ B.powerset, ∑ i, ‖X D i‖ * ‖Y D i‖hCS:(∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ * ‖Y x x_1‖) ^ 2 ≤ (∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ ^ 2) * ∑ x ∈ B.powerset, ∑ x_1, ‖Y x x_1‖ ^ 2⊢ ∑ D ∈ B.powerset, ‖operator A‖ ^ 2 * ∑ j, ‖if D ⊆ left j then a j else 0‖ ^ 2 = ‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2 rw [← Finset.mul_sum,ι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)hnorm:∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ D ∈ B.powerset, ∑ i, ‖X D i‖ * ‖Y D i‖hCS:(∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ * ‖Y x x_1‖) ^ 2 ≤ (∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ ^ 2) * ∑ x ∈ B.powerset, ∑ x_1, ‖Y x x_1‖ ^ 2⊢ ‖operator A‖ ^ 2 * ∑ i ∈ B.powerset, ∑ j, ‖if i ⊆ left j then a j else 0‖ ^ 2 = ‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2 masked_coefficient_energy B left hl aι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)hnorm:∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ D ∈ B.powerset, ∑ i, ‖X D i‖ * ‖Y D i‖hCS:(∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ * ‖Y x x_1‖) ^ 2 ≤ (∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ ^ 2) * ∑ x ∈ B.powerset, ∑ x_1, ‖Y x x_1‖ ^ 2⊢ ‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2 = ‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2]ι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)hnorm:∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ D ∈ B.powerset, ∑ i, ‖X D i‖ * ‖Y D i‖hCS:(∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ * ‖Y x x_1‖) ^ 2 ≤ (∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ ^ 2) * ∑ x ∈ B.powerset, ∑ x_1, ‖Y x x_1‖ ^ 2hX:∑ D ∈ B.powerset, ∑ i, ‖X D i‖ ^ 2 ≤ ‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2⊢ (∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(right k) * ‖b k‖ ^ 2) have hY : (∑ D ∈ B.powerset, ∑ i, ‖Y D i‖ ^ 2) ≤ ‖operator C‖ ^ 2 * ∑ k, (2 : ℝ) ^ (right k).card * ‖b k‖ ^ 2 := by calc _ ≤ ∑ D ∈ B.powerset, ‖operator C‖ ^ 2 * ∑ k, ‖if D ⊆ right k then b k else 0‖ ^ 2 := byι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)hnorm:∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ D ∈ B.powerset, ∑ i, ‖X D i‖ * ‖Y D i‖hCS:(∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ * ‖Y x x_1‖) ^ 2 ≤ (∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ ^ 2) * ∑ x ∈ B.powerset, ∑ x_1, ‖Y x x_1‖ ^ 2hX:∑ D ∈ B.powerset, ∑ i, ‖X D i‖ ^ 2 ≤ ‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2⊢ ∑ D ∈ B.powerset, ∑ i, ‖Y D i‖ ^ 2 ≤ ∑ D ∈ B.powerset, ‖operator C‖ ^ 2 * ∑ k, ‖if D ⊆ right k then b k else 0‖ ^ 2 apply Finset.sum_le_sumι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)hnorm:∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ D ∈ B.powerset, ∑ i, ‖X D i‖ * ‖Y D i‖hCS:(∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ * ‖Y x x_1‖) ^ 2 ≤ (∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ ^ 2) * ∑ x ∈ B.powerset, ∑ x_1, ‖Y x x_1‖ ^ 2hX:∑ D ∈ B.powerset, ∑ i, ‖X D i‖ ^ 2 ≤ ‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2⊢ ∀ i ∈ B.powerset, ∑ i_1, ‖Y i i_1‖ ^ 2 ≤ ‖operator C‖ ^ 2 * ∑ k, ‖if i ⊆ right k then b k else 0‖ ^ 2 intro D _ι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)hnorm:∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ D ∈ B.powerset, ∑ i, ‖X D i‖ * ‖Y D i‖hCS:(∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ * ‖Y x x_1‖) ^ 2 ≤ (∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ ^ 2) * ∑ x ∈ B.powerset, ∑ x_1, ‖Y x x_1‖ ^ 2hX:∑ D ∈ B.powerset, ∑ i, ‖X D i‖ ^ 2 ≤ ‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2D:Finset ιa✝:D ∈ B.powerset⊢ ∑ i, ‖Y D i‖ ^ 2 ≤ ‖operator C‖ ^ 2 * ∑ k, ‖if D ⊆ right k then b k else 0‖ ^ 2 exact energy_bound C _All goals completed! 🐙 _ = _ := byι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)hnorm:∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ D ∈ B.powerset, ∑ i, ‖X D i‖ * ‖Y D i‖hCS:(∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ * ‖Y x x_1‖) ^ 2 ≤ (∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ ^ 2) * ∑ x ∈ B.powerset, ∑ x_1, ‖Y x x_1‖ ^ 2hX:∑ D ∈ B.powerset, ∑ i, ‖X D i‖ ^ 2 ≤ ‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2⊢ ∑ D ∈ B.powerset, ‖operator C‖ ^ 2 * ∑ k, ‖if D ⊆ right k then b k else 0‖ ^ 2 = ‖operator C‖ ^ 2 * ∑ k, 2 ^ #(right k) * ‖b k‖ ^ 2 rw [← Finset.mul_sum,ι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)hnorm:∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ D ∈ B.powerset, ∑ i, ‖X D i‖ * ‖Y D i‖hCS:(∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ * ‖Y x x_1‖) ^ 2 ≤ (∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ ^ 2) * ∑ x ∈ B.powerset, ∑ x_1, ‖Y x x_1‖ ^ 2hX:∑ D ∈ B.powerset, ∑ i, ‖X D i‖ ^ 2 ≤ ‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2⊢ ‖operator C‖ ^ 2 * ∑ i ∈ B.powerset, ∑ k, ‖if i ⊆ right k then b k else 0‖ ^ 2 = ‖operator C‖ ^ 2 * ∑ k, 2 ^ #(right k) * ‖b k‖ ^ 2 masked_coefficient_energy B right hr bι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)hnorm:∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ D ∈ B.powerset, ∑ i, ‖X D i‖ * ‖Y D i‖hCS:(∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ * ‖Y x x_1‖) ^ 2 ≤ (∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ ^ 2) * ∑ x ∈ B.powerset, ∑ x_1, ‖Y x x_1‖ ^ 2hX:∑ D ∈ B.powerset, ∑ i, ‖X D i‖ ^ 2 ≤ ‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2⊢ ‖operator C‖ ^ 2 * ∑ j, 2 ^ #(right j) * ‖b j‖ ^ 2 = ‖operator C‖ ^ 2 * ∑ k, 2 ^ #(right k) * ‖b k‖ ^ 2]ι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)hnorm:∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ D ∈ B.powerset, ∑ i, ‖X D i‖ * ‖Y D i‖hCS:(∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ * ‖Y x x_1‖) ^ 2 ≤ (∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ ^ 2) * ∑ x ∈ B.powerset, ∑ x_1, ‖Y x x_1‖ ^ 2hX:∑ D ∈ B.powerset, ∑ i, ‖X D i‖ ^ 2 ≤ ‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2hY:∑ D ∈ B.powerset, ∑ i, ‖Y D i‖ ^ 2 ≤ ‖operator C‖ ^ 2 * ∑ k, 2 ^ #(right k) * ‖b k‖ ^ 2⊢ (∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(right k) * ‖b k‖ ^ 2) exact (pow_le_pow_left₀ (Finset.sum_nonneg (fun _ _ => norm_nonneg _)) hnorm 2).trans (hCS.trans (mul_le_mul hX hY (byι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)hnorm:∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ D ∈ B.powerset, ∑ i, ‖X D i‖ * ‖Y D i‖hCS:(∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ * ‖Y x x_1‖) ^ 2 ≤ (∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ ^ 2) * ∑ x ∈ B.powerset, ∑ x_1, ‖Y x x_1‖ ^ 2hX:∑ D ∈ B.powerset, ∑ i, ‖X D i‖ ^ 2 ≤ ‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2hY:∑ D ∈ B.powerset, ∑ i, ‖Y D i‖ ^ 2 ≤ ‖operator C‖ ^ 2 * ∑ k, 2 ^ #(right k) * ‖b k‖ ^ 2⊢ 0 ≤ ∑ x ∈ B.powerset, ∑ x_1, ‖Y x x_1‖ ^ 2 positivityAll goals completed! 🐙) (byι:Type u_1m:Type u_2n:Type u_3p:Type u_4inst✝⁶:DecidableEq ιinst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pB:Finset ιleft:n → Finset ιright:p → Finset ιhl:∀ (j : n), left j ⊆ Bhr:∀ (k : p), right k ⊆ BA:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂX:Finset ι → m → ℂ := fun D i => ∑ j, A i j * if D ⊆ left j then a j else 0Y:Finset ι → m → ℂ := fun D i => ∑ k, C i k * if D ⊆ right k then b k else 0heq:∀ (i : m), (∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0) = ∑ D ∈ B.powerset, (-1) ^ #D * (star (X D i) * Y D i)hnorm:∑ i, ‖∑ j, ∑ k, if Disjoint (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖ ≤ ∑ D ∈ B.powerset, ∑ i, ‖X D i‖ * ‖Y D i‖hCS:(∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ * ‖Y x x_1‖) ^ 2 ≤ (∑ x ∈ B.powerset, ∑ x_1, ‖X x x_1‖ ^ 2) * ∑ x ∈ B.powerset, ∑ x_1, ‖Y x x_1‖ ^ 2hX:∑ D ∈ B.powerset, ∑ i, ‖X D i‖ ^ 2 ≤ ‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2hY:∑ D ∈ B.powerset, ∑ i, ‖Y D i‖ ^ 2 ≤ ‖operator C‖ ^ 2 * ∑ k, 2 ^ #(right k) * ‖b k‖ ^ 2⊢ 0 ≤ ‖operator A‖ ^ 2 * ∑ j, 2 ^ #(left j) * ‖a j‖ ^ 2 positivityAll goals completed! 🐙)))end CoprimeBilinearSievenamespace IdealCoprimeBilinearSieveabbrev O := ActualEisensteinCubic.Oopen IdealMobiusDivisorSum IdealCoprimeSieveOperator FiniteSieveOperatortheorem support_weighted_energy_bound {n : Type*} [Fintype n] (ε : ℝ) (hε : 0 < ε) (column : n → Ideal O) (hcol : ∀ j, column j ≠ 0) (N : ℝ) (hnorm : ∀ j, (Ideal.absNorm (column j) : ℝ) ≤ N) (a : n → ℂ) : (∑ j, (2 : ℝ) ^ (primeSupport (column j)).card * ‖a j‖ ^ 2) ≤ (supportConstant ε hε * N ^ ε) * ∑ j, ‖a j‖ ^ 2 := byn:Type u_1inst✝:Fintype nε:ℝhε:0 < εcolumn:n → Ideal Ohcol:∀ (j : n), column j ≠ 0N:ℝhnorm:∀ (j : n), ↑(absNorm (column j)) ≤ Na:n → ℂ⊢ ∑ j, 2 ^ #(primeSupport (column j)) * ‖a j‖ ^ 2 ≤ supportConstant ε hε * N ^ ε * ∑ j, ‖a j‖ ^ 2 rw [Finset.mul_sumn:Type u_1inst✝:Fintype nε:ℝhε:0 < εcolumn:n → Ideal Ohcol:∀ (j : n), column j ≠ 0N:ℝhnorm:∀ (j : n), ↑(absNorm (column j)) ≤ Na:n → ℂ⊢ ∑ j, 2 ^ #(primeSupport (column j)) * ‖a j‖ ^ 2 ≤ ∑ i, supportConstant ε hε * N ^ ε * ‖a i‖ ^ 2]n:Type u_1inst✝:Fintype nε:ℝhε:0 < εcolumn:n → Ideal Ohcol:∀ (j : n), column j ≠ 0N:ℝhnorm:∀ (j : n), ↑(absNorm (column j)) ≤ Na:n → ℂ⊢ ∑ j, 2 ^ #(primeSupport (column j)) * ‖a j‖ ^ 2 ≤ ∑ i, supportConstant ε hε * N ^ ε * ‖a i‖ ^ 2 apply Finset.sum_le_sumn:Type u_1inst✝:Fintype nε:ℝhε:0 < εcolumn:n → Ideal Ohcol:∀ (j : n), column j ≠ 0N:ℝhnorm:∀ (j : n), ↑(absNorm (column j)) ≤ Na:n → ℂ⊢ ∀ i ∈ Finset.univ, 2 ^ #(primeSupport (column i)) * ‖a i‖ ^ 2 ≤ supportConstant ε hε * N ^ ε * ‖a i‖ ^ 2 intro j _n:Type u_1inst✝:Fintype nε:ℝhε:0 < εcolumn:n → Ideal Ohcol:∀ (j : n), column j ≠ 0N:ℝhnorm:∀ (j : n), ↑(absNorm (column j)) ≤ Na:n → ℂj:na✝:j ∈ Finset.univ⊢ 2 ^ #(primeSupport (column j)) * ‖a j‖ ^ 2 ≤ supportConstant ε hε * N ^ ε * ‖a j‖ ^ 2 apply mul_le_mul_of_nonneg_right _ (sq_nonneg _)n:Type u_1inst✝:Fintype nε:ℝhε:0 < εcolumn:n → Ideal Ohcol:∀ (j : n), column j ≠ 0N:ℝhnorm:∀ (j : n), ↑(absNorm (column j)) ≤ Na:n → ℂj:na✝:j ∈ Finset.univ⊢ 2 ^ #(primeSupport (column j)) ≤ supportConstant ε hε * N ^ ε exact (support_card_bound ε hε (column j) (hcol j)).trans (mul_le_mul_of_nonneg_left (Real.rpow_le_rpow (Nat.cast_nonneg _) (hnorm j) hε.le) (supportConstant_pos ε hε).le)All goals completed! 🐙theorem ideal_coprime_bilinear_operator_bound {m n p : Type*} [Fintype m] [Fintype n] [Fintype p] [DecidableEq m] [DecidableEq n] [DecidableEq p] (ε : ℝ) (hε : 0 < ε) (left : n → Ideal O) (right : p → Ideal O) (hl0 : ∀ j, left j ≠ 0) (hr0 : ∀ k, right k ≠ 0) (N₁ N₂ : ℝ) (hN₁ : 0 ≤ N₁) (_hN₂ : 0 ≤ N₂) (hlN : ∀ j, (Ideal.absNorm (left j) : ℝ) ≤ N₁) (hrN : ∀ k, (Ideal.absNorm (right k) : ℝ) ≤ N₂) (A : Matrix m n ℂ) (C : Matrix m p ℂ) (a : n → ℂ) (b : p → ℂ) : (∑ i, ‖∑ j, ∑ k, if IsCoprime (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * ∑ j, ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ∑ k, ‖b k‖ ^ 2) := bym:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εleft:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁_hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂ⊢ (∑ i, ‖∑ j, ∑ k, if IsCoprime (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * ∑ j, ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ∑ k, ‖b k‖ ^ 2) let B : Finset (Ideal O) := (Finset.univ.biUnion (fun j => primeSupport (left j))) ∪ (Finset.univ.biUnion (fun k => primeSupport (right k)))m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εleft:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁_hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂB:Finset (Ideal O) := (Finset.univ.biUnion fun j => primeSupport (left j)) ∪ Finset.univ.biUnion fun k => primeSupport (right k)⊢ (∑ i, ‖∑ j, ∑ k, if IsCoprime (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * ∑ j, ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ∑ k, ‖b k‖ ^ 2) have hl (j : n) : primeSupport (left j) ⊆ B := by intro P hPm:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εleft:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁_hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂB:Finset (Ideal O) := (Finset.univ.biUnion fun j => primeSupport (left j)) ∪ Finset.univ.biUnion fun k => primeSupport (right k)j:nP:Ideal IdealMobiusDivisorSum.OhP:P ∈ primeSupport (left j)⊢ P ∈ B exact Finset.mem_union.mpr (Or.inl (Finset.mem_biUnion.mpr ⟨j, Finset.mem_univ j, hP⟩))m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εleft:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁_hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂB:Finset (Ideal O) := (Finset.univ.biUnion fun j => primeSupport (left j)) ∪ Finset.univ.biUnion fun k => primeSupport (right k)hl:∀ (j : n), primeSupport (left j) ⊆ B⊢ (∑ i, ‖∑ j, ∑ k, if IsCoprime (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * ∑ j, ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ∑ k, ‖b k‖ ^ 2) have hr (k : p) : primeSupport (right k) ⊆ B := by intro P hPm:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εleft:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁_hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂB:Finset (Ideal O) := (Finset.univ.biUnion fun j => primeSupport (left j)) ∪ Finset.univ.biUnion fun k => primeSupport (right k)hl:∀ (j : n), primeSupport (left j) ⊆ Bk:pP:Ideal IdealMobiusDivisorSum.OhP:P ∈ primeSupport (right k)⊢ P ∈ B exact Finset.mem_union.mpr (Or.inr (Finset.mem_biUnion.mpr ⟨k, Finset.mem_univ k, hP⟩))m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εleft:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁_hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂB:Finset (Ideal O) := (Finset.univ.biUnion fun j => primeSupport (left j)) ∪ Finset.univ.biUnion fun k => primeSupport (right k)hl:∀ (j : n), primeSupport (left j) ⊆ Bhr:∀ (k : p), primeSupport (right k) ⊆ B⊢ (∑ i, ‖∑ j, ∑ k, if IsCoprime (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * ∑ j, ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ∑ k, ‖b k‖ ^ 2) have h := CoprimeBilinearSieve.coprime_bilinear_operator_bound B (fun j => primeSupport (left j)) (fun k => primeSupport (right k)) hl hr A C a bm:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εleft:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁_hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂB:Finset (Ideal O) := (Finset.univ.biUnion fun j => primeSupport (left j)) ∪ Finset.univ.biUnion fun k => primeSupport (right k)hl:∀ (j : n), primeSupport (left j) ⊆ Bhr:∀ (k : p), primeSupport (right k) ⊆ Bh:(∑ i, ‖∑ j, ∑ k, if Disjoint (primeSupport (left j)) (primeSupport (right k)) then star (A i j * a j) * (C i k * b k) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(primeSupport (left j)) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(primeSupport (right k)) * ‖b k‖ ^ 2)⊢ (∑ i, ‖∑ j, ∑ k, if IsCoprime (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * ∑ j, ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ∑ k, ‖b k‖ ^ 2) simp_rw [m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εleft:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁_hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂB:Finset (Ideal O) := (Finset.univ.biUnion fun j => primeSupport (left j)) ∪ Finset.univ.biUnion fun k => primeSupport (right k)hl:∀ (j : n), primeSupport (left j) ⊆ Bhr:∀ (k : p), primeSupport (right k) ⊆ Bh:(∑ i, ‖∑ j, ∑ k, if Disjoint (primeSupport (left j)) (primeSupport (right k)) then star (A i j * a j) * (C i k * b k) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(primeSupport (left j)) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(primeSupport (right k)) * ‖b k‖ ^ 2)⊢ (∑ i, ‖∑ j, ∑ k, if IsCoprime (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * ∑ j, ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ∑ k, ‖b k‖ ^ 2)primeSupport_disjoint_iff (hl0 _) (hr0 _)m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εleft:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁_hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂB:Finset (Ideal O) := (Finset.univ.biUnion fun j => primeSupport (left j)) ∪ Finset.univ.biUnion fun k => primeSupport (right k)hl:∀ (j : n), primeSupport (left j) ⊆ Bhr:∀ (k : p), primeSupport (right k) ⊆ Bh:(∑ x, ‖∑ x_1, ∑ x_2, if IsCoprime (left x_1) (right x_2) then star (A x x_1 * a x_1) * (C x x_2 * b x_2) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(primeSupport (left j)) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(primeSupport (right k)) * ‖b k‖ ^ 2)⊢ (∑ i, ‖∑ j, ∑ k, if IsCoprime (left j) (right k) then star (A i j * a j) * (C i k * b k) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * ∑ j, ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ∑ k, ‖b k‖ ^ 2)] at h apply h.transm:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εleft:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁_hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂB:Finset (Ideal O) := (Finset.univ.biUnion fun j => primeSupport (left j)) ∪ Finset.univ.biUnion fun k => primeSupport (right k)hl:∀ (j : n), primeSupport (left j) ⊆ Bhr:∀ (k : p), primeSupport (right k) ⊆ Bh:(∑ x, ‖∑ x_1, ∑ x_2, if IsCoprime (left x_1) (right x_2) then star (A x x_1 * a x_1) * (C x x_2 * b x_2) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(primeSupport (left j)) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(primeSupport (right k)) * ‖b k‖ ^ 2)⊢ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(primeSupport (left j)) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(primeSupport (right k)) * ‖b k‖ ^ 2) ≤ (‖operator A‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * ∑ j, ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ∑ k, ‖b k‖ ^ 2) apply mul_le_mulh₁m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εleft:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁_hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂB:Finset (Ideal O) := (Finset.univ.biUnion fun j => primeSupport (left j)) ∪ Finset.univ.biUnion fun k => primeSupport (right k)hl:∀ (j : n), primeSupport (left j) ⊆ Bhr:∀ (k : p), primeSupport (right k) ⊆ Bh:(∑ x, ‖∑ x_1, ∑ x_2, if IsCoprime (left x_1) (right x_2) then star (A x x_1 * a x_1) * (C x x_2 * b x_2) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(primeSupport (left j)) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(primeSupport (right k)) * ‖b k‖ ^ 2)⊢ ‖operator A‖ ^ 2 * ∑ j, 2 ^ #(primeSupport (left j)) * ‖a j‖ ^ 2 ≤ ‖operator A‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * ∑ j, ‖a j‖ ^ 2h₂m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εleft:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁_hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂB:Finset (Ideal O) := (Finset.univ.biUnion fun j => primeSupport (left j)) ∪ Finset.univ.biUnion fun k => primeSupport (right k)hl:∀ (j : n), primeSupport (left j) ⊆ Bhr:∀ (k : p), primeSupport (right k) ⊆ Bh:(∑ x, ‖∑ x_1, ∑ x_2, if IsCoprime (left x_1) (right x_2) then star (A x x_1 * a x_1) * (C x x_2 * b x_2) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(primeSupport (left j)) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(primeSupport (right k)) * ‖b k‖ ^ 2)⊢ ‖operator C‖ ^ 2 * ∑ k, 2 ^ #(primeSupport (right k)) * ‖b k‖ ^ 2 ≤ ‖operator C‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ∑ k, ‖b k‖ ^ 2c0m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εleft:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁_hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂB:Finset (Ideal O) := (Finset.univ.biUnion fun j => primeSupport (left j)) ∪ Finset.univ.biUnion fun k => primeSupport (right k)hl:∀ (j : n), primeSupport (left j) ⊆ Bhr:∀ (k : p), primeSupport (right k) ⊆ Bh:(∑ x, ‖∑ x_1, ∑ x_2, if IsCoprime (left x_1) (right x_2) then star (A x x_1 * a x_1) * (C x x_2 * b x_2) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(primeSupport (left j)) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(primeSupport (right k)) * ‖b k‖ ^ 2)⊢ 0 ≤ ‖operator C‖ ^ 2 * ∑ k, 2 ^ #(primeSupport (right k)) * ‖b k‖ ^ 2b0m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εleft:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁_hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂB:Finset (Ideal O) := (Finset.univ.biUnion fun j => primeSupport (left j)) ∪ Finset.univ.biUnion fun k => primeSupport (right k)hl:∀ (j : n), primeSupport (left j) ⊆ Bhr:∀ (k : p), primeSupport (right k) ⊆ Bh:(∑ x, ‖∑ x_1, ∑ x_2, if IsCoprime (left x_1) (right x_2) then star (A x x_1 * a x_1) * (C x x_2 * b x_2) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(primeSupport (left j)) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(primeSupport (right k)) * ‖b k‖ ^ 2)⊢ 0 ≤ ‖operator A‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * ∑ j, ‖a j‖ ^ 2 ·h₁m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εleft:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁_hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂB:Finset (Ideal O) := (Finset.univ.biUnion fun j => primeSupport (left j)) ∪ Finset.univ.biUnion fun k => primeSupport (right k)hl:∀ (j : n), primeSupport (left j) ⊆ Bhr:∀ (k : p), primeSupport (right k) ⊆ Bh:(∑ x, ‖∑ x_1, ∑ x_2, if IsCoprime (left x_1) (right x_2) then star (A x x_1 * a x_1) * (C x x_2 * b x_2) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(primeSupport (left j)) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(primeSupport (right k)) * ‖b k‖ ^ 2)⊢ ‖operator A‖ ^ 2 * ∑ j, 2 ^ #(primeSupport (left j)) * ‖a j‖ ^ 2 ≤ ‖operator A‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * ∑ j, ‖a j‖ ^ 2 simpa only [mul_assoc] using mul_le_mul_of_nonneg_left (support_weighted_energy_bound ε hε left hl0 N₁ hlN a) (sq_nonneg ‖operator A‖)All goals completed! 🐙 ·h₂m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εleft:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁_hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂB:Finset (Ideal O) := (Finset.univ.biUnion fun j => primeSupport (left j)) ∪ Finset.univ.biUnion fun k => primeSupport (right k)hl:∀ (j : n), primeSupport (left j) ⊆ Bhr:∀ (k : p), primeSupport (right k) ⊆ Bh:(∑ x, ‖∑ x_1, ∑ x_2, if IsCoprime (left x_1) (right x_2) then star (A x x_1 * a x_1) * (C x x_2 * b x_2) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(primeSupport (left j)) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(primeSupport (right k)) * ‖b k‖ ^ 2)⊢ ‖operator C‖ ^ 2 * ∑ k, 2 ^ #(primeSupport (right k)) * ‖b k‖ ^ 2 ≤ ‖operator C‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ∑ k, ‖b k‖ ^ 2 simpa only [mul_assoc] using mul_le_mul_of_nonneg_left (support_weighted_energy_bound ε hε right hr0 N₂ hrN b) (sq_nonneg ‖operator C‖)All goals completed! 🐙 ·c0m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εleft:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁_hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂB:Finset (Ideal O) := (Finset.univ.biUnion fun j => primeSupport (left j)) ∪ Finset.univ.biUnion fun k => primeSupport (right k)hl:∀ (j : n), primeSupport (left j) ⊆ Bhr:∀ (k : p), primeSupport (right k) ⊆ Bh:(∑ x, ‖∑ x_1, ∑ x_2, if IsCoprime (left x_1) (right x_2) then star (A x x_1 * a x_1) * (C x x_2 * b x_2) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(primeSupport (left j)) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(primeSupport (right k)) * ‖b k‖ ^ 2)⊢ 0 ≤ ‖operator C‖ ^ 2 * ∑ k, 2 ^ #(primeSupport (right k)) * ‖b k‖ ^ 2 positivityAll goals completed! 🐙 ·b0m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εleft:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁_hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂB:Finset (Ideal O) := (Finset.univ.biUnion fun j => primeSupport (left j)) ∪ Finset.univ.biUnion fun k => primeSupport (right k)hl:∀ (j : n), primeSupport (left j) ⊆ Bhr:∀ (k : p), primeSupport (right k) ⊆ Bh:(∑ x, ‖∑ x_1, ∑ x_2, if IsCoprime (left x_1) (right x_2) then star (A x x_1 * a x_1) * (C x x_2 * b x_2) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(primeSupport (left j)) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(primeSupport (right k)) * ‖b k‖ ^ 2)⊢ 0 ≤ ‖operator A‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * ∑ j, ‖a j‖ ^ 2 have hpos := (supportConstant_pos ε hε).leb0m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εleft:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁_hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Matrix m n ℂC:Matrix m p ℂa:n → ℂb:p → ℂB:Finset (Ideal O) := (Finset.univ.biUnion fun j => primeSupport (left j)) ∪ Finset.univ.biUnion fun k => primeSupport (right k)hl:∀ (j : n), primeSupport (left j) ⊆ Bhr:∀ (k : p), primeSupport (right k) ⊆ Bh:(∑ x, ‖∑ x_1, ∑ x_2, if IsCoprime (left x_1) (right x_2) then star (A x x_1 * a x_1) * (C x x_2 * b x_2) else 0‖) ^ 2 ≤ (‖operator A‖ ^ 2 * ∑ j, 2 ^ #(primeSupport (left j)) * ‖a j‖ ^ 2) * (‖operator C‖ ^ 2 * ∑ k, 2 ^ #(primeSupport (right k)) * ‖b k‖ ^ 2)hpos:0 ≤ supportConstant ε hε⊢ 0 ≤ ‖operator A‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * ∑ j, ‖a j‖ ^ 2 positivityAll goals completed! 🐙end IdealCoprimeBilinearSievenamespace DivisorBlockCauchyabbrev O := ActualEisensteinCubic.Oopen IdealMobiusDivisorSum FiniteSieveOperator IdealCoprimeSieveOperatortheorem divisor_mask_energy {n : Type*} [Fintype n] (S : Finset (Ideal O)) (column : n → Ideal O) (hcol : ∀ j, column j ≠ 0) (a : n → ℂ) : (∑ D ∈ S, ∑ j, ‖if D ∣ column j then a j else 0‖ ^ 2) ≤ ∑ j, ((idealDivisors (column j)).card : ℝ) * ‖a j‖ ^ 2 := byn:Type u_1inst✝:Fintype nS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0a:n → ℂ⊢ ∑ D ∈ S, ∑ j, ‖if D ∣ column j then a j else 0‖ ^ 2 ≤ ∑ j, ↑(#(idealDivisors (column j))) * ‖a j‖ ^ 2 rw [Finset.sum_commn:Type u_1inst✝:Fintype nS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0a:n → ℂ⊢ ∑ y, ∑ x ∈ S, ‖if x ∣ column y then a y else 0‖ ^ 2 ≤ ∑ j, ↑(#(idealDivisors (column j))) * ‖a j‖ ^ 2]n:Type u_1inst✝:Fintype nS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0a:n → ℂ⊢ ∑ y, ∑ x ∈ S, ‖if x ∣ column y then a y else 0‖ ^ 2 ≤ ∑ j, ↑(#(idealDivisors (column j))) * ‖a j‖ ^ 2 apply Finset.sum_le_sumn:Type u_1inst✝:Fintype nS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0a:n → ℂ⊢ ∀ i ∈ Finset.univ, ∑ x ∈ S, ‖if x ∣ column i then a i else 0‖ ^ 2 ≤ ↑(#(idealDivisors (column i))) * ‖a i‖ ^ 2 intro j _n:Type u_1inst✝:Fintype nS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0a:n → ℂj:na✝:j ∈ Finset.univ⊢ ∑ x ∈ S, ‖if x ∣ column j then a j else 0‖ ^ 2 ≤ ↑(#(idealDivisors (column j))) * ‖a j‖ ^ 2 have hsub : S.filter (fun D => D ∣ column j) ⊆ idealDivisors (column j) := byn:Type u_1inst✝:Fintype nS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0a:n → ℂ⊢ ∑ D ∈ S, ∑ j, ‖if D ∣ column j then a j else 0‖ ^ 2 ≤ ∑ j, ↑(#(idealDivisors (column j))) * ‖a j‖ ^ 2 intro D hDn:Type u_1inst✝:Fintype nS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0a:n → ℂj:na✝:j ∈ Finset.univD:Ideal OhD:D ∈ {D ∈ S | D ∣ column j}⊢ D ∈ idealDivisors (column j) exact (mem_idealDivisors (hcol j)).mpr (Finset.mem_filter.mp hD).2n:Type u_1inst✝:Fintype nS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0a:n → ℂj:na✝:j ∈ Finset.univhsub:{D ∈ S | D ∣ column j} ⊆ idealDivisors (column j)⊢ ∑ x ∈ S, ‖if x ∣ column j then a j else 0‖ ^ 2 ≤ ↑(#(idealDivisors (column j))) * ‖a j‖ ^ 2 have hite (D : Ideal O) : ‖if D ∣ column j then a j else 0‖ ^ 2 = if D ∣ column j then ‖a j‖ ^ 2 else 0 := byn:Type u_1inst✝:Fintype nS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0a:n → ℂ⊢ ∑ D ∈ S, ∑ j, ‖if D ∣ column j then a j else 0‖ ^ 2 ≤ ∑ j, ↑(#(idealDivisors (column j))) * ‖a j‖ ^ 2 split_ifsposn:Type u_1inst✝:Fintype nS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0a:n → ℂj:na✝:j ∈ Finset.univhsub:{D ∈ S | D ∣ column j} ⊆ idealDivisors (column j)D:Ideal Oh✝:D ∣ column j⊢ ‖a j‖ ^ 2 = ‖a j‖ ^ 2negn:Type u_1inst✝:Fintype nS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0a:n → ℂj:na✝:j ∈ Finset.univhsub:{D ∈ S | D ∣ column j} ⊆ idealDivisors (column j)D:Ideal Oh✝:¬D ∣ column j⊢ ‖0‖ ^ 2 = 0 <;>posn:Type u_1inst✝:Fintype nS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0a:n → ℂj:na✝:j ∈ Finset.univhsub:{D ∈ S | D ∣ column j} ⊆ idealDivisors (column j)D:Ideal Oh✝:D ∣ column j⊢ ‖a j‖ ^ 2 = ‖a j‖ ^ 2negn:Type u_1inst✝:Fintype nS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0a:n → ℂj:na✝:j ∈ Finset.univhsub:{D ∈ S | D ∣ column j} ⊆ idealDivisors (column j)D:Ideal Oh✝:¬D ∣ column j⊢ ‖0‖ ^ 2 = 0 simpn:Type u_1inst✝:Fintype nS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0a:n → ℂj:na✝:j ∈ Finset.univhsub:{D ∈ S | D ∣ column j} ⊆ idealDivisors (column j)hite:∀ (D : Ideal O), ‖if D ∣ column j then a j else 0‖ ^ 2 = if D ∣ column j then ‖a j‖ ^ 2 else 0⊢ ∑ x ∈ S, ‖if x ∣ column j then a j else 0‖ ^ 2 ≤ ↑(#(idealDivisors (column j))) * ‖a j‖ ^ 2 simp_rw [n:Type u_1inst✝:Fintype nS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0a:n → ℂj:na✝:j ∈ Finset.univhsub:{D ∈ S | D ∣ column j} ⊆ idealDivisors (column j)hite:∀ (D : Ideal O), ‖if D ∣ column j then a j else 0‖ ^ 2 = if D ∣ column j then ‖a j‖ ^ 2 else 0⊢ ∑ x ∈ S, ‖if x ∣ column j then a j else 0‖ ^ 2 ≤ ↑(#(idealDivisors (column j))) * ‖a j‖ ^ 2hiten:Type u_1inst✝:Fintype nS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0a:n → ℂj:na✝:j ∈ Finset.univhsub:{D ∈ S | D ∣ column j} ⊆ idealDivisors (column j)hite:∀ (D : Ideal O), ‖if D ∣ column j then a j else 0‖ ^ 2 = if D ∣ column j then ‖a j‖ ^ 2 else 0⊢ (∑ x ∈ S, if x ∣ column j then ‖a j‖ ^ 2 else 0) ≤ ↑(#(idealDivisors (column j))) * ‖a j‖ ^ 2] rw [← Finset.sum_filter,n:Type u_1inst✝:Fintype nS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0a:n → ℂj:na✝:j ∈ Finset.univhsub:{D ∈ S | D ∣ column j} ⊆ idealDivisors (column j)hite:∀ (D : Ideal O), ‖if D ∣ column j then a j else 0‖ ^ 2 = if D ∣ column j then ‖a j‖ ^ 2 else 0⊢ ∑ a_1 ∈ S with a_1 ∣ column j, ‖a j‖ ^ 2 ≤ ↑(#(idealDivisors (column j))) * ‖a j‖ ^ 2 Finset.sum_const,n:Type u_1inst✝:Fintype nS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0a:n → ℂj:na✝:j ∈ Finset.univhsub:{D ∈ S | D ∣ column j} ⊆ idealDivisors (column j)hite:∀ (D : Ideal O), ‖if D ∣ column j then a j else 0‖ ^ 2 = if D ∣ column j then ‖a j‖ ^ 2 else 0⊢ #({a ∈ S | a ∣ column j}) • ‖a j‖ ^ 2 ≤ ↑(#(idealDivisors (column j))) * ‖a j‖ ^ 2 nsmul_eq_muln:Type u_1inst✝:Fintype nS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0a:n → ℂj:na✝:j ∈ Finset.univhsub:{D ∈ S | D ∣ column j} ⊆ idealDivisors (column j)hite:∀ (D : Ideal O), ‖if D ∣ column j then a j else 0‖ ^ 2 = if D ∣ column j then ‖a j‖ ^ 2 else 0⊢ ↑(#({a ∈ S | a ∣ column j})) * ‖a j‖ ^ 2 ≤ ↑(#(idealDivisors (column j))) * ‖a j‖ ^ 2]n:Type u_1inst✝:Fintype nS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0a:n → ℂj:na✝:j ∈ Finset.univhsub:{D ∈ S | D ∣ column j} ⊆ idealDivisors (column j)hite:∀ (D : Ideal O), ‖if D ∣ column j then a j else 0‖ ^ 2 = if D ∣ column j then ‖a j‖ ^ 2 else 0⊢ ↑(#({a ∈ S | a ∣ column j})) * ‖a j‖ ^ 2 ≤ ↑(#(idealDivisors (column j))) * ‖a j‖ ^ 2 exact mul_le_mul_of_nonneg_right (byn:Type u_1inst✝:Fintype nS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0a:n → ℂj:na✝:j ∈ Finset.univhsub:{D ∈ S | D ∣ column j} ⊆ idealDivisors (column j)hite:∀ (D : Ideal O), ‖if D ∣ column j then a j else 0‖ ^ 2 = if D ∣ column j then ‖a j‖ ^ 2 else 0⊢ ↑(#({a ∈ S | a ∣ column j})) ≤ ↑(#(idealDivisors (column j))) exact_mod_cast Finset.card_le_card hsubAll goals completed! 🐙) (sq_nonneg _)def divisorConstant (ε : ℝ) (hε : 0 < ε) : ℝ := (IdealDivisorBound.ideal_divisor_small_power ε hε).choosetheorem divisorConstant_pos (ε : ℝ) (hε : 0 < ε) : 0 < divisorConstant ε hε := (IdealDivisorBound.ideal_divisor_small_power ε hε).choose_spec.1theorem divisor_mask_energy_small_power {n : Type*} [Fintype n] (ε : ℝ) (hε : 0 < ε) (S : Finset (Ideal O)) (column : n → Ideal O) (hcol : ∀ j, column j ≠ 0) (N : ℝ) (hnorm : ∀ j, (Ideal.absNorm (column j) : ℝ) ≤ N) (a : n → ℂ) : (∑ D ∈ S, ∑ j, ‖if D ∣ column j then a j else 0‖ ^ 2) ≤ (divisorConstant ε hε * N ^ ε) * ∑ j, ‖a j‖ ^ 2 := byn:Type u_1inst✝:Fintype nε:ℝhε:0 < εS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0N:ℝhnorm:∀ (j : n), ↑(absNorm (column j)) ≤ Na:n → ℂ⊢ ∑ D ∈ S, ∑ j, ‖if D ∣ column j then a j else 0‖ ^ 2 ≤ divisorConstant ε hε * N ^ ε * ∑ j, ‖a j‖ ^ 2 apply (divisor_mask_energy S column hcol a).transn:Type u_1inst✝:Fintype nε:ℝhε:0 < εS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0N:ℝhnorm:∀ (j : n), ↑(absNorm (column j)) ≤ Na:n → ℂ⊢ ∑ j, ↑(#(idealDivisors (column j))) * ‖a j‖ ^ 2 ≤ divisorConstant ε hε * N ^ ε * ∑ j, ‖a j‖ ^ 2 rw [Finset.mul_sumn:Type u_1inst✝:Fintype nε:ℝhε:0 < εS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0N:ℝhnorm:∀ (j : n), ↑(absNorm (column j)) ≤ Na:n → ℂ⊢ ∑ j, ↑(#(idealDivisors (column j))) * ‖a j‖ ^ 2 ≤ ∑ i, divisorConstant ε hε * N ^ ε * ‖a i‖ ^ 2]n:Type u_1inst✝:Fintype nε:ℝhε:0 < εS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0N:ℝhnorm:∀ (j : n), ↑(absNorm (column j)) ≤ Na:n → ℂ⊢ ∑ j, ↑(#(idealDivisors (column j))) * ‖a j‖ ^ 2 ≤ ∑ i, divisorConstant ε hε * N ^ ε * ‖a i‖ ^ 2 apply Finset.sum_le_sumn:Type u_1inst✝:Fintype nε:ℝhε:0 < εS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0N:ℝhnorm:∀ (j : n), ↑(absNorm (column j)) ≤ Na:n → ℂ⊢ ∀ i ∈ Finset.univ, ↑(#(idealDivisors (column i))) * ‖a i‖ ^ 2 ≤ divisorConstant ε hε * N ^ ε * ‖a i‖ ^ 2 intro j _n:Type u_1inst✝:Fintype nε:ℝhε:0 < εS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0N:ℝhnorm:∀ (j : n), ↑(absNorm (column j)) ≤ Na:n → ℂj:na✝:j ∈ Finset.univ⊢ ↑(#(idealDivisors (column j))) * ‖a j‖ ^ 2 ≤ divisorConstant ε hε * N ^ ε * ‖a j‖ ^ 2 apply mul_le_mul_of_nonneg_right _ (sq_nonneg _)n:Type u_1inst✝:Fintype nε:ℝhε:0 < εS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0N:ℝhnorm:∀ (j : n), ↑(absNorm (column j)) ≤ Na:n → ℂj:na✝:j ∈ Finset.univ⊢ ↑(#(idealDivisors (column j))) ≤ divisorConstant ε hε * N ^ ε apply ((IdealDivisorBound.ideal_divisor_small_power ε hε).choose_spec.2 (column j) (hcol j)).transn:Type u_1inst✝:Fintype nε:ℝhε:0 < εS:Finset (Ideal O)column:n → Ideal Ohcol:∀ (j : n), column j ≠ 0N:ℝhnorm:∀ (j : n), ↑(absNorm (column j)) ≤ Na:n → ℂj:na✝:j ∈ Finset.univ⊢ ⋯.choose * ↑(absNorm (column j)) ^ ε ≤ divisorConstant ε hε * N ^ ε exact mul_le_mul_of_nonneg_left (Real.rpow_le_rpow (Nat.cast_nonneg _) (hnorm j) hε.le) (divisorConstant_pos ε hε).leAll goals completed! 🐙theorem rectangular_cauchy {α β : Type*} (S : Finset α) (T : Finset β) (H : α → β → ℝ) (E : α → ℝ) (F : β → ℝ) (C : ℝ) (_hC : 0 ≤ C) (hH : ∀ d ∈ S, ∀ e ∈ T, (H d e) ^ 2 ≤ C * E d * F e) : (∑ d ∈ S, ∑ e ∈ T, H d e) ^ 2 ≤ (S.card : ℝ) * T.card * C * (∑ d ∈ S, E d) * (∑ e ∈ T, F e) := byα:Type u_1β:Type u_2S:Finset αT:Finset βH:α → β → ℝE:α → ℝF:β → ℝC:ℝ_hC:0 ≤ ChH:∀ d ∈ S, ∀ e ∈ T, H d e ^ 2 ≤ C * E d * F e⊢ (∑ d ∈ S, ∑ e ∈ T, H d e) ^ 2 ≤ (↑(#S) * ↑(#T) * C * ∑ d ∈ S, E d) * ∑ e ∈ T, F e classical have hCS := Finset.sum_mul_sq_le_sq_mul_sq (S ×ˢ T) (fun _ => (1 : ℝ)) (fun x => H x.1 x.2)α:Type u_1β:Type u_2S:Finset αT:Finset βH:α → β → ℝE:α → ℝF:β → ℝC:ℝ_hC:0 ≤ ChH:∀ d ∈ S, ∀ e ∈ T, H d e ^ 2 ≤ C * E d * F ehCS:(∑ i ∈ S ×ˢ T, 1 * H i.1 i.2) ^ 2 ≤ (∑ i ∈ S ×ˢ T, 1 ^ 2) * ∑ i ∈ S ×ˢ T, H i.1 i.2 ^ 2⊢ (∑ d ∈ S, ∑ e ∈ T, H d e) ^ 2 ≤ (↑(#S) * ↑(#T) * C * ∑ d ∈ S, E d) * ∑ e ∈ T, F e simp only [one_mul, one_pow, Finset.sum_const, nsmul_eq_mul, mul_one, Finset.card_product, Nat.cast_mul, Finset.sum_product] at hCSα:Type u_1β:Type u_2S:Finset αT:Finset βH:α → β → ℝE:α → ℝF:β → ℝC:ℝ_hC:0 ≤ ChH:∀ d ∈ S, ∀ e ∈ T, H d e ^ 2 ≤ C * E d * F ehCS:(∑ x ∈ S, ∑ x_1 ∈ T, H x x_1) ^ 2 ≤ ↑(#S) * ↑(#T) * ∑ x ∈ S, ∑ x_1 ∈ T, H x x_1 ^ 2⊢ (∑ d ∈ S, ∑ e ∈ T, H d e) ^ 2 ≤ (↑(#S) * ↑(#T) * C * ∑ d ∈ S, E d) * ∑ e ∈ T, F e have hsum : (∑ d ∈ S, ∑ e ∈ T, (H d e) ^ 2) ≤ C * (∑ d ∈ S, E d) * (∑ e ∈ T, F e) := byα:Type u_1β:Type u_2S:Finset αT:Finset βH:α → β → ℝE:α → ℝF:β → ℝC:ℝ_hC:0 ≤ ChH:∀ d ∈ S, ∀ e ∈ T, H d e ^ 2 ≤ C * E d * F e⊢ (∑ d ∈ S, ∑ e ∈ T, H d e) ^ 2 ≤ (↑(#S) * ↑(#T) * C * ∑ d ∈ S, E d) * ∑ e ∈ T, F e calc _ ≤ ∑ d ∈ S, ∑ e ∈ T, C * E d * F e := byα:Type u_1β:Type u_2S:Finset αT:Finset βH:α → β → ℝE:α → ℝF:β → ℝC:ℝ_hC:0 ≤ ChH:∀ d ∈ S, ∀ e ∈ T, H d e ^ 2 ≤ C * E d * F ehCS:(∑ x ∈ S, ∑ x_1 ∈ T, H x x_1) ^ 2 ≤ ↑(#S) * ↑(#T) * ∑ x ∈ S, ∑ x_1 ∈ T, H x x_1 ^ 2⊢ ∑ d ∈ S, ∑ e ∈ T, H d e ^ 2 ≤ ∑ d ∈ S, ∑ e ∈ T, C * E d * F e apply Finset.sum_le_sumα:Type u_1β:Type u_2S:Finset αT:Finset βH:α → β → ℝE:α → ℝF:β → ℝC:ℝ_hC:0 ≤ ChH:∀ d ∈ S, ∀ e ∈ T, H d e ^ 2 ≤ C * E d * F ehCS:(∑ x ∈ S, ∑ x_1 ∈ T, H x x_1) ^ 2 ≤ ↑(#S) * ↑(#T) * ∑ x ∈ S, ∑ x_1 ∈ T, H x x_1 ^ 2⊢ ∀ i ∈ S, ∑ e ∈ T, H i e ^ 2 ≤ ∑ e ∈ T, C * E i * F e intro d hdα:Type u_1β:Type u_2S:Finset αT:Finset βH:α → β → ℝE:α → ℝF:β → ℝC:ℝ_hC:0 ≤ ChH:∀ d ∈ S, ∀ e ∈ T, H d e ^ 2 ≤ C * E d * F ehCS:(∑ x ∈ S, ∑ x_1 ∈ T, H x x_1) ^ 2 ≤ ↑(#S) * ↑(#T) * ∑ x ∈ S, ∑ x_1 ∈ T, H x x_1 ^ 2d:αhd:d ∈ S⊢ ∑ e ∈ T, H d e ^ 2 ≤ ∑ e ∈ T, C * E d * F e exact Finset.sum_le_sum (fun e he => hH d hd e he)All goals completed! 🐙 _ = _ := byα:Type u_1β:Type u_2S:Finset αT:Finset βH:α → β → ℝE:α → ℝF:β → ℝC:ℝ_hC:0 ≤ ChH:∀ d ∈ S, ∀ e ∈ T, H d e ^ 2 ≤ C * E d * F ehCS:(∑ x ∈ S, ∑ x_1 ∈ T, H x x_1) ^ 2 ≤ ↑(#S) * ↑(#T) * ∑ x ∈ S, ∑ x_1 ∈ T, H x x_1 ^ 2⊢ ∑ d ∈ S, ∑ e ∈ T, C * E d * F e = (C * ∑ d ∈ S, E d) * ∑ e ∈ T, F e simp only [← Finset.mul_sum, ← Finset.sum_mul]α:Type u_1β:Type u_2S:Finset αT:Finset βH:α → β → ℝE:α → ℝF:β → ℝC:ℝ_hC:0 ≤ ChH:∀ d ∈ S, ∀ e ∈ T, H d e ^ 2 ≤ C * E d * F ehCS:(∑ x ∈ S, ∑ x_1 ∈ T, H x x_1) ^ 2 ≤ ↑(#S) * ↑(#T) * ∑ x ∈ S, ∑ x_1 ∈ T, H x x_1 ^ 2hsum:∑ d ∈ S, ∑ e ∈ T, H d e ^ 2 ≤ (C * ∑ d ∈ S, E d) * ∑ e ∈ T, F e⊢ (∑ d ∈ S, ∑ e ∈ T, H d e) ^ 2 ≤ (↑(#S) * ↑(#T) * C * ∑ d ∈ S, E d) * ∑ e ∈ T, F e apply hCS.transα:Type u_1β:Type u_2S:Finset αT:Finset βH:α → β → ℝE:α → ℝF:β → ℝC:ℝ_hC:0 ≤ ChH:∀ d ∈ S, ∀ e ∈ T, H d e ^ 2 ≤ C * E d * F ehCS:(∑ x ∈ S, ∑ x_1 ∈ T, H x x_1) ^ 2 ≤ ↑(#S) * ↑(#T) * ∑ x ∈ S, ∑ x_1 ∈ T, H x x_1 ^ 2hsum:∑ d ∈ S, ∑ e ∈ T, H d e ^ 2 ≤ (C * ∑ d ∈ S, E d) * ∑ e ∈ T, F e⊢ ↑(#S) * ↑(#T) * ∑ x ∈ S, ∑ x_1 ∈ T, H x x_1 ^ 2 ≤ (↑(#S) * ↑(#T) * C * ∑ d ∈ S, E d) * ∑ e ∈ T, F e simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hsum (show (0 : ℝ) ≤ (S.card : ℝ) * T.card byα:Type u_1β:Type u_2S:Finset αT:Finset βH:α → β → ℝE:α → ℝF:β → ℝC:ℝ_hC:0 ≤ ChH:∀ d ∈ S, ∀ e ∈ T, H d e ^ 2 ≤ C * E d * F e⊢ (∑ d ∈ S, ∑ e ∈ T, H d e) ^ 2 ≤ (↑(#S) * ↑(#T) * C * ∑ d ∈ S, E d) * ∑ e ∈ T, F e positivityAll goals completed! 🐙)end DivisorBlockCauchynamespace IdealDivisorBilinearBlocksabbrev O := ActualEisensteinCubic.Oopen FiniteSieveOperator IdealCoprimeSieveOperator DivisorBlockCauchydef blockValue {m n p : Type*} [Fintype m] [Fintype n] [Fintype p] (left : n → Ideal O) (right : p → Ideal O) (A : Ideal O → Matrix m n ℂ) (B : Ideal O → Matrix m p ℂ) (a : n → ℂ) (b : p → ℂ) (D E : Ideal O) : ℝ := ∑ i, ‖∑ j, ∑ k, if IsCoprime (left j) (right k) then star (A D i j * (if D ∣ left j then a j else 0)) * (B E i k * (if E ∣ right k then b k else 0)) else 0‖theorem blockValue_nonneg {m n p : Type*} [Fintype m] [Fintype n] [Fintype p] (left : n → Ideal O) (right : p → Ideal O) (A : Ideal O → Matrix m n ℂ) (B : Ideal O → Matrix m p ℂ) (a : n → ℂ) (b : p → ℂ) (D E : Ideal O) : 0 ≤ blockValue left right A B a b D E := Finset.sum_nonneg (fun _ _ => norm_nonneg _)theorem divisor_block_bound {m n p : Type*} [Fintype m] [Fintype n] [Fintype p] [DecidableEq m] [DecidableEq n] [DecidableEq p] (ε : ℝ) (hε : 0 < ε) (S T : Finset (Ideal O)) (left : n → Ideal O) (right : p → Ideal O) (hl0 : ∀ j, left j ≠ 0) (hr0 : ∀ k, right k ≠ 0) (N₁ N₂ : ℝ) (hN₁ : 0 ≤ N₁) (hN₂ : 0 ≤ N₂) (hlN : ∀ j, (Ideal.absNorm (left j) : ℝ) ≤ N₁) (hrN : ∀ k, (Ideal.absNorm (right k) : ℝ) ≤ N₂) (A : Ideal O → Matrix m n ℂ) (B : Ideal O → Matrix m p ℂ) (a : n → ℂ) (b : p → ℂ) (UA UB : ℝ) (hUA : 0 ≤ UA) (hUB : 0 ≤ UB) (hA : ∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UA) (hB : ∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UB) : (∑ D ∈ S, ∑ E ∈ T, blockValue left right A B a b D E) ^ 2 ≤ (S.card : ℝ) * T.card * (UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)) * ((divisorConstant ε hε * N₁ ^ ε) * ∑ j, ‖a j‖ ^ 2) * ((divisorConstant ε hε * N₂ ^ ε) * ∑ k, ‖b k‖ ^ 2) := bym:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UB⊢ (∑ D ∈ S, ∑ E ∈ T, blockValue left right A B a b D E) ^ 2 ≤ ↑(#S) * ↑(#T) * (UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)) * (divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2) * (divisorConstant ε hε * N₂ ^ ε * ∑ k, ‖b k‖ ^ 2) let EL (D : Ideal O) : ℝ := ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2⊢ (∑ D ∈ S, ∑ E ∈ T, blockValue left right A B a b D E) ^ 2 ≤ ↑(#S) * ↑(#T) * (UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)) * (divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2) * (divisorConstant ε hε * N₂ ^ ε * ∑ k, ‖b k‖ ^ 2) let ER (E : Ideal O) : ℝ := ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2⊢ (∑ D ∈ S, ∑ E ∈ T, blockValue left right A B a b D E) ^ 2 ≤ ↑(#S) * ↑(#T) * (UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)) * (divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2) * (divisorConstant ε hε * N₂ ^ ε * ∑ k, ‖b k‖ ^ 2) let C : ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)⊢ (∑ D ∈ S, ∑ E ∈ T, blockValue left right A B a b D E) ^ 2 ≤ ↑(#S) * ↑(#T) * (UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)) * (divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2) * (divisorConstant ε hε * N₂ ^ ε * ∑ k, ‖b k‖ ^ 2) have hs := (supportConstant_pos ε hε).lem:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hε⊢ (∑ D ∈ S, ∑ E ∈ T, blockValue left right A B a b D E) ^ 2 ≤ ↑(#S) * ↑(#T) * (UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)) * (divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2) * (divisorConstant ε hε * N₂ ^ ε * ∑ k, ‖b k‖ ^ 2) have hd := (divisorConstant_pos ε hε).lem:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hε⊢ (∑ D ∈ S, ∑ E ∈ T, blockValue left right A B a b D E) ^ 2 ≤ ↑(#S) * ↑(#T) * (UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)) * (divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2) * (divisorConstant ε hε * N₂ ^ ε * ∑ k, ‖b k‖ ^ 2) have hC : 0 ≤ C := by dsimp [C]m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hε⊢ 0 ≤ UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε); positivitym:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ C⊢ (∑ D ∈ S, ∑ E ∈ T, blockValue left right A B a b D E) ^ 2 ≤ ↑(#S) * ↑(#T) * (UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)) * (divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2) * (divisorConstant ε hε * N₂ ^ ε * ∑ k, ‖b k‖ ^ 2) have hEL (D) : 0 ≤ EL D := by dsimp [EL]m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ CD:Ideal O⊢ 0 ≤ ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2; positivitym:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ ChEL:∀ (D : Ideal O), 0 ≤ EL D⊢ (∑ D ∈ S, ∑ E ∈ T, blockValue left right A B a b D E) ^ 2 ≤ ↑(#S) * ↑(#T) * (UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)) * (divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2) * (divisorConstant ε hε * N₂ ^ ε * ∑ k, ‖b k‖ ^ 2) have hER (E) : 0 ≤ ER E := by dsimp [ER]m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ ChEL:∀ (D : Ideal O), 0 ≤ EL DE:Ideal O⊢ 0 ≤ ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2; positivitym:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ ChEL:∀ (D : Ideal O), 0 ≤ EL DhER:∀ (E : Ideal O), 0 ≤ ER E⊢ (∑ D ∈ S, ∑ E ∈ T, blockValue left right A B a b D E) ^ 2 ≤ ↑(#S) * ↑(#T) * (UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)) * (divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2) * (divisorConstant ε hε * N₂ ^ ε * ∑ k, ‖b k‖ ^ 2) have hp (D : Ideal O) (hD : D ∈ S) (E : Ideal O) (hE : E ∈ T) : (blockValue left right A B a b D E) ^ 2 ≤ C * EL D * ER E := by have hh := IdealCoprimeBilinearSieve.ideal_coprime_bilinear_operator_bound ε hε left right hl0 hr0 N₁ N₂ hN₁ hN₂ hlN hrN (A D) (B E) (fun j => if D ∣ left j then a j else 0) (fun k => if E ∣ right k then b k else 0)m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ ChEL:∀ (D : Ideal O), 0 ≤ EL DhER:∀ (E : Ideal O), 0 ≤ ER ED:Ideal OhD:D ∈ SE:Ideal OhE:E ∈ Thh:(∑ i, ‖∑ j, ∑ k, if IsCoprime (left j) (right k) then star (A D i j * if D ∣ left j then a j else 0) * (B E i k * if E ∣ right k then b k else 0) else 0‖) ^ 2 ≤ (‖operator (A D)‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2) * (‖operator (B E)‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2)⊢ blockValue left right A B a b D E ^ 2 ≤ C * EL D * ER E change (blockValue left right A B a b D E) ^ 2 ≤ (‖operator (A D)‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * EL D) * (‖operator (B E)‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ER E) at hhm:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ ChEL:∀ (D : Ideal O), 0 ≤ EL DhER:∀ (E : Ideal O), 0 ≤ ER ED:Ideal OhD:D ∈ SE:Ideal OhE:E ∈ Thh:blockValue left right A B a b D E ^ 2 ≤ ‖operator (A D)‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * EL D * (‖operator (B E)‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ER E)⊢ blockValue left right A B a b D E ^ 2 ≤ C * EL D * ER E apply hh.transm:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ ChEL:∀ (D : Ideal O), 0 ≤ EL DhER:∀ (E : Ideal O), 0 ≤ ER ED:Ideal OhD:D ∈ SE:Ideal OhE:E ∈ Thh:blockValue left right A B a b D E ^ 2 ≤ ‖operator (A D)‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * EL D * (‖operator (B E)‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ER E)⊢ ‖operator (A D)‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * EL D * (‖operator (B E)‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ER E) ≤ C * EL D * ER E calc _ ≤ (UA * (supportConstant ε hε * N₁ ^ ε) * EL D) * (UB * (supportConstant ε hε * N₂ ^ ε) * ER E) := bym:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ ChEL:∀ (D : Ideal O), 0 ≤ EL DhER:∀ (E : Ideal O), 0 ≤ ER ED:Ideal OhD:D ∈ SE:Ideal OhE:E ∈ Thh:blockValue left right A B a b D E ^ 2 ≤ ‖operator (A D)‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * EL D * (‖operator (B E)‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ER E)⊢ ‖operator (A D)‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * EL D * (‖operator (B E)‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ER E) ≤ UA * (supportConstant ε hε * N₁ ^ ε) * EL D * (UB * (supportConstant ε hε * N₂ ^ ε) * ER E) gcongrhbcm:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ ChEL:∀ (D : Ideal O), 0 ≤ EL DhER:∀ (E : Ideal O), 0 ≤ ER ED:Ideal OhD:D ∈ SE:Ideal OhE:E ∈ Thh:blockValue left right A B a b D E ^ 2 ≤ ‖operator (A D)‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * EL D * (‖operator (B E)‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ER E)⊢ ‖operator (A D)‖ ^ 2 ≤ UAhbcm:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ ChEL:∀ (D : Ideal O), 0 ≤ EL DhER:∀ (E : Ideal O), 0 ≤ ER ED:Ideal OhD:D ∈ SE:Ideal OhE:E ∈ Thh:blockValue left right A B a b D E ^ 2 ≤ ‖operator (A D)‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * EL D * (‖operator (B E)‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ER E)⊢ ‖operator (B E)‖ ^ 2 ≤ UB ·hbcm:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ ChEL:∀ (D : Ideal O), 0 ≤ EL DhER:∀ (E : Ideal O), 0 ≤ ER ED:Ideal OhD:D ∈ SE:Ideal OhE:E ∈ Thh:blockValue left right A B a b D E ^ 2 ≤ ‖operator (A D)‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * EL D * (‖operator (B E)‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ER E)⊢ ‖operator (A D)‖ ^ 2 ≤ UA exact hA D hDAll goals completed! 🐙 ·hbcm:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ ChEL:∀ (D : Ideal O), 0 ≤ EL DhER:∀ (E : Ideal O), 0 ≤ ER ED:Ideal OhD:D ∈ SE:Ideal OhE:E ∈ Thh:blockValue left right A B a b D E ^ 2 ≤ ‖operator (A D)‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * EL D * (‖operator (B E)‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ER E)⊢ ‖operator (B E)‖ ^ 2 ≤ UB exact hB E hEAll goals completed! 🐙 _ = _ := bym:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ ChEL:∀ (D : Ideal O), 0 ≤ EL DhER:∀ (E : Ideal O), 0 ≤ ER ED:Ideal OhD:D ∈ SE:Ideal OhE:E ∈ Thh:blockValue left right A B a b D E ^ 2 ≤ ‖operator (A D)‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * EL D * (‖operator (B E)‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ER E)⊢ UA * (supportConstant ε hε * N₁ ^ ε) * EL D * (UB * (supportConstant ε hε * N₂ ^ ε) * ER E) = C * EL D * ER E dsimp [C]m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ ChEL:∀ (D : Ideal O), 0 ≤ EL DhER:∀ (E : Ideal O), 0 ≤ ER ED:Ideal OhD:D ∈ SE:Ideal OhE:E ∈ Thh:blockValue left right A B a b D E ^ 2 ≤ ‖operator (A D)‖ ^ 2 * (supportConstant ε hε * N₁ ^ ε) * EL D * (‖operator (B E)‖ ^ 2 * (supportConstant ε hε * N₂ ^ ε) * ER E)⊢ UA * (supportConstant ε hε * N₁ ^ ε) * EL D * (UB * (supportConstant ε hε * N₂ ^ ε) * ER E) = UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε) * EL D * ER E; ringm:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ ChEL:∀ (D : Ideal O), 0 ≤ EL DhER:∀ (E : Ideal O), 0 ≤ ER Ehp:∀ D ∈ S, ∀ E ∈ T, blockValue left right A B a b D E ^ 2 ≤ C * EL D * ER E⊢ (∑ D ∈ S, ∑ E ∈ T, blockValue left right A B a b D E) ^ 2 ≤ ↑(#S) * ↑(#T) * (UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)) * (divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2) * (divisorConstant ε hε * N₂ ^ ε * ∑ k, ‖b k‖ ^ 2) have hc := rectangular_cauchy S T (blockValue left right A B a b) EL ER C hC hpm:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ ChEL:∀ (D : Ideal O), 0 ≤ EL DhER:∀ (E : Ideal O), 0 ≤ ER Ehp:∀ D ∈ S, ∀ E ∈ T, blockValue left right A B a b D E ^ 2 ≤ C * EL D * ER Ehc:(∑ d ∈ S, ∑ e ∈ T, blockValue left right A B a b d e) ^ 2 ≤ (↑(#S) * ↑(#T) * C * ∑ d ∈ S, EL d) * ∑ e ∈ T, ER e⊢ (∑ D ∈ S, ∑ E ∈ T, blockValue left right A B a b D E) ^ 2 ≤ ↑(#S) * ↑(#T) * (UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)) * (divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2) * (divisorConstant ε hε * N₂ ^ ε * ∑ k, ‖b k‖ ^ 2) have he1 := divisor_mask_energy_small_power ε hε S left hl0 N₁ hlN am:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ ChEL:∀ (D : Ideal O), 0 ≤ EL DhER:∀ (E : Ideal O), 0 ≤ ER Ehp:∀ D ∈ S, ∀ E ∈ T, blockValue left right A B a b D E ^ 2 ≤ C * EL D * ER Ehc:(∑ d ∈ S, ∑ e ∈ T, blockValue left right A B a b d e) ^ 2 ≤ (↑(#S) * ↑(#T) * C * ∑ d ∈ S, EL d) * ∑ e ∈ T, ER ehe1:∑ D ∈ S, ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2 ≤ divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2⊢ (∑ D ∈ S, ∑ E ∈ T, blockValue left right A B a b D E) ^ 2 ≤ ↑(#S) * ↑(#T) * (UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)) * (divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2) * (divisorConstant ε hε * N₂ ^ ε * ∑ k, ‖b k‖ ^ 2) have he2 := divisor_mask_energy_small_power ε hε T right hr0 N₂ hrN bm:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ ChEL:∀ (D : Ideal O), 0 ≤ EL DhER:∀ (E : Ideal O), 0 ≤ ER Ehp:∀ D ∈ S, ∀ E ∈ T, blockValue left right A B a b D E ^ 2 ≤ C * EL D * ER Ehc:(∑ d ∈ S, ∑ e ∈ T, blockValue left right A B a b d e) ^ 2 ≤ (↑(#S) * ↑(#T) * C * ∑ d ∈ S, EL d) * ∑ e ∈ T, ER ehe1:∑ D ∈ S, ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2 ≤ divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2he2:∑ D ∈ T, ∑ j, ‖if D ∣ right j then b j else 0‖ ^ 2 ≤ divisorConstant ε hε * N₂ ^ ε * ∑ j, ‖b j‖ ^ 2⊢ (∑ D ∈ S, ∑ E ∈ T, blockValue left right A B a b D E) ^ 2 ≤ ↑(#S) * ↑(#T) * (UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)) * (divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2) * (divisorConstant ε hε * N₂ ^ ε * ∑ k, ‖b k‖ ^ 2) apply hc.transm:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ ChEL:∀ (D : Ideal O), 0 ≤ EL DhER:∀ (E : Ideal O), 0 ≤ ER Ehp:∀ D ∈ S, ∀ E ∈ T, blockValue left right A B a b D E ^ 2 ≤ C * EL D * ER Ehc:(∑ d ∈ S, ∑ e ∈ T, blockValue left right A B a b d e) ^ 2 ≤ (↑(#S) * ↑(#T) * C * ∑ d ∈ S, EL d) * ∑ e ∈ T, ER ehe1:∑ D ∈ S, ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2 ≤ divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2he2:∑ D ∈ T, ∑ j, ‖if D ∣ right j then b j else 0‖ ^ 2 ≤ divisorConstant ε hε * N₂ ^ ε * ∑ j, ‖b j‖ ^ 2⊢ (↑(#S) * ↑(#T) * C * ∑ d ∈ S, EL d) * ∑ e ∈ T, ER e ≤ ↑(#S) * ↑(#T) * (UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)) * (divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2) * (divisorConstant ε hε * N₂ ^ ε * ∑ k, ‖b k‖ ^ 2) change (S.card : ℝ) * T.card * C * (∑ D ∈ S, EL D) * (∑ E ∈ T, ER E) ≤ _m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ ChEL:∀ (D : Ideal O), 0 ≤ EL DhER:∀ (E : Ideal O), 0 ≤ ER Ehp:∀ D ∈ S, ∀ E ∈ T, blockValue left right A B a b D E ^ 2 ≤ C * EL D * ER Ehc:(∑ d ∈ S, ∑ e ∈ T, blockValue left right A B a b d e) ^ 2 ≤ (↑(#S) * ↑(#T) * C * ∑ d ∈ S, EL d) * ∑ e ∈ T, ER ehe1:∑ D ∈ S, ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2 ≤ divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2he2:∑ D ∈ T, ∑ j, ‖if D ∣ right j then b j else 0‖ ^ 2 ≤ divisorConstant ε hε * N₂ ^ ε * ∑ j, ‖b j‖ ^ 2⊢ (↑(#S) * ↑(#T) * C * ∑ D ∈ S, EL D) * ∑ E ∈ T, ER E ≤ ↑(#S) * ↑(#T) * (UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)) * (divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2) * (divisorConstant ε hε * N₂ ^ ε * ∑ k, ‖b k‖ ^ 2) change (∑ D ∈ S, EL D) ≤ _ at he1m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ ChEL:∀ (D : Ideal O), 0 ≤ EL DhER:∀ (E : Ideal O), 0 ≤ ER Ehp:∀ D ∈ S, ∀ E ∈ T, blockValue left right A B a b D E ^ 2 ≤ C * EL D * ER Ehc:(∑ d ∈ S, ∑ e ∈ T, blockValue left right A B a b d e) ^ 2 ≤ (↑(#S) * ↑(#T) * C * ∑ d ∈ S, EL d) * ∑ e ∈ T, ER ehe2:∑ D ∈ T, ∑ j, ‖if D ∣ right j then b j else 0‖ ^ 2 ≤ divisorConstant ε hε * N₂ ^ ε * ∑ j, ‖b j‖ ^ 2he1:∑ D ∈ S, EL D ≤ divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2⊢ (↑(#S) * ↑(#T) * C * ∑ D ∈ S, EL D) * ∑ E ∈ T, ER E ≤ ↑(#S) * ↑(#T) * (UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)) * (divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2) * (divisorConstant ε hε * N₂ ^ ε * ∑ k, ‖b k‖ ^ 2) change (∑ E ∈ T, ER E) ≤ _ at he2m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ ChEL:∀ (D : Ideal O), 0 ≤ EL DhER:∀ (E : Ideal O), 0 ≤ ER Ehp:∀ D ∈ S, ∀ E ∈ T, blockValue left right A B a b D E ^ 2 ≤ C * EL D * ER Ehc:(∑ d ∈ S, ∑ e ∈ T, blockValue left right A B a b d e) ^ 2 ≤ (↑(#S) * ↑(#T) * C * ∑ d ∈ S, EL d) * ∑ e ∈ T, ER ehe1:∑ D ∈ S, EL D ≤ divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2he2:∑ E ∈ T, ER E ≤ divisorConstant ε hε * N₂ ^ ε * ∑ j, ‖b j‖ ^ 2⊢ (↑(#S) * ↑(#T) * C * ∑ D ∈ S, EL D) * ∑ E ∈ T, ER E ≤ ↑(#S) * ↑(#T) * (UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)) * (divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2) * (divisorConstant ε hε * N₂ ^ ε * ∑ k, ‖b k‖ ^ 2) dsimp only [C]m:Type u_1n:Type u_2p:Type u_3inst✝⁵:Fintype minst✝⁴:Fintype ninst✝³:Fintype pinst✝²:DecidableEq minst✝¹:DecidableEq ninst✝:DecidableEq pε:ℝhε:0 < εS:Finset (Ideal O)T:Finset (Ideal O)left:n → Ideal Oright:p → Ideal Ohl0:∀ (j : n), left j ≠ 0hr0:∀ (k : p), right k ≠ 0N₁:ℝN₂:ℝhN₁:0 ≤ N₁hN₂:0 ≤ N₂hlN:∀ (j : n), ↑(absNorm (left j)) ≤ N₁hrN:∀ (k : p), ↑(absNorm (right k)) ≤ N₂A:Ideal O → Matrix m n ℂB:Ideal O → Matrix m p ℂa:n → ℂb:p → ℂUA:ℝUB:ℝhUA:0 ≤ UAhUB:0 ≤ UBhA:∀ D ∈ S, ‖operator (A D)‖ ^ 2 ≤ UAhB:∀ E ∈ T, ‖operator (B E)‖ ^ 2 ≤ UBEL:Ideal O → ℝ := fun D => ∑ j, ‖if D ∣ left j then a j else 0‖ ^ 2ER:Ideal O → ℝ := fun E => ∑ k, ‖if E ∣ right k then b k else 0‖ ^ 2C:ℝ := UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)hs:0 ≤ supportConstant ε hεhd:0 ≤ divisorConstant ε hεhC:0 ≤ ChEL:∀ (D : Ideal O), 0 ≤ EL DhER:∀ (E : Ideal O), 0 ≤ ER Ehp:∀ D ∈ S, ∀ E ∈ T, blockValue left right A B a b D E ^ 2 ≤ C * EL D * ER Ehc:(∑ d ∈ S, ∑ e ∈ T, blockValue left right A B a b d e) ^ 2 ≤ (↑(#S) * ↑(#T) * C * ∑ d ∈ S, EL d) * ∑ e ∈ T, ER ehe1:∑ D ∈ S, EL D ≤ divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2he2:∑ E ∈ T, ER E ≤ divisorConstant ε hε * N₂ ^ ε * ∑ j, ‖b j‖ ^ 2⊢ (↑(#S) * ↑(#T) * (UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)) * ∑ D ∈ S, EL D) * ∑ E ∈ T, ER E ≤ ↑(#S) * ↑(#T) * (UA * UB * (supportConstant ε hε * N₁ ^ ε) * (supportConstant ε hε * N₂ ^ ε)) * (divisorConstant ε hε * N₁ ^ ε * ∑ j, ‖a j‖ ^ 2) * (divisorConstant ε hε * N₂ ^ ε * ∑ k, ‖b k‖ ^ 2) gcongrAll goals completed! 🐙end IdealDivisorBilinearBlocksnamespace FiniteSieveRestrictionopen FiniteSieveOperatorvariable {n p : Type*} [Fintype n] [Fintype p] [DecidableEq n] [DecidableEq p]def extendByZero (S : Finset n) (f : n → p) (u : n → ℂ) (k : p) : ℂ := ∑ j ∈ S, if f j = k then u j else 0omit [Fintype n] [Fintype p] [DecidableEq n] in theorem extendByZero_apply_image (S : Finset n) (f : n → p) (u : n → ℂ) (hf : Set.InjOn f S) (j : n) (hj : j ∈ S) : extendByZero S f u (f j) = u j := byn:Type u_1p:Type u_2inst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑Sj:nhj:j ∈ S⊢ extendByZero S f u (f j) = u j unfold extendByZeron:Type u_1p:Type u_2inst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑Sj:nhj:j ∈ S⊢ (∑ j_1 ∈ S, if f j_1 = f j then u j_1 else 0) = u j rw [Finset.sum_eq_single jn:Type u_1p:Type u_2inst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑Sj:nhj:j ∈ S⊢ (if f j = f j then u j else 0) = u jh₀n:Type u_1p:Type u_2inst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑Sj:nhj:j ∈ S⊢ ∀ b ∈ S, b ≠ j → (if f b = f j then u b else 0) = 0h₁n:Type u_1p:Type u_2inst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑Sj:nhj:j ∈ S⊢ j ∉ S → (if f j = f j then u j else 0) = 0]n:Type u_1p:Type u_2inst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑Sj:nhj:j ∈ S⊢ (if f j = f j then u j else 0) = u jh₀n:Type u_1p:Type u_2inst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑Sj:nhj:j ∈ S⊢ ∀ b ∈ S, b ≠ j → (if f b = f j then u b else 0) = 0h₁n:Type u_1p:Type u_2inst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑Sj:nhj:j ∈ S⊢ j ∉ S → (if f j = f j then u j else 0) = 0 ·n:Type u_1p:Type u_2inst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑Sj:nhj:j ∈ S⊢ (if f j = f j then u j else 0) = u j simpAll goals completed! 🐙 ·h₀n:Type u_1p:Type u_2inst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑Sj:nhj:j ∈ S⊢ ∀ b ∈ S, b ≠ j → (if f b = f j then u b else 0) = 0 intro k hk hkjh₀n:Type u_1p:Type u_2inst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑Sj:nhj:j ∈ Sk:nhk:k ∈ Shkj:k ≠ j⊢ (if f k = f j then u k else 0) = 0 simp only [ite_eq_right (fun h => hkj (hf hk hj h))]All goals completed! 🐙 ·h₁n:Type u_1p:Type u_2inst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑Sj:nhj:j ∈ S⊢ j ∉ S → (if f j = f j then u j else 0) = 0 exact fun h => (h hj).elimAll goals completed! 🐙omit [Fintype n] [Fintype p] [DecidableEq n] in theorem extendByZero_zero (S : Finset n) (f : n → p) (u : n → ℂ) (k : p) (hk : k ∉ S.image f) : extendByZero S f u k = 0 := byn:Type u_1p:Type u_2inst✝:DecidableEq pS:Finset nf:n → pu:n → ℂk:phk:k ∉ Finset.image f S⊢ extendByZero S f u k = 0 apply Finset.sum_eq_zeron:Type u_1p:Type u_2inst✝:DecidableEq pS:Finset nf:n → pu:n → ℂk:phk:k ∉ Finset.image f S⊢ ∀ x ∈ S, (if f x = k then u x else 0) = 0 intro j hjn:Type u_1p:Type u_2inst✝:DecidableEq pS:Finset nf:n → pu:n → ℂk:phk:k ∉ Finset.image f Sj:nhj:j ∈ S⊢ (if f j = k then u j else 0) = 0 exact ite_eq_right (fun h => hk (Finset.mem_image.mpr ⟨j, hj, h⟩))All goals completed! 🐙omit [Fintype n] [DecidableEq n] in theorem extendByZero_energy (S : Finset n) (f : n → p) (u : n → ℂ) (hf : Set.InjOn f S) : (∑ k, ‖extendByZero S f u k‖ ^ 2) = ∑ j ∈ S, ‖u j‖ ^ 2 := byn:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑S⊢ ∑ k, ‖extendByZero S f u k‖ ^ 2 = ∑ j ∈ S, ‖u j‖ ^ 2 calc _ = ∑ k ∈ S.image f, ‖extendByZero S f u k‖ ^ 2 := byn:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑S⊢ ∑ k, ‖extendByZero S f u k‖ ^ 2 = ∑ k ∈ Finset.image f S, ‖extendByZero S f u k‖ ^ 2 symmn:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑S⊢ ∑ k ∈ Finset.image f S, ‖extendByZero S f u k‖ ^ 2 = ∑ k, ‖extendByZero S f u k‖ ^ 2 apply Finset.sum_subset (Finset.subset_univ _)n:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑S⊢ ∀ x ∈ Finset.univ, x ∉ Finset.image f S → ‖extendByZero S f u x‖ ^ 2 = 0 intro k _ hkn:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑Sk:pa✝:k ∈ Finset.univhk:k ∉ Finset.image f S⊢ ‖extendByZero S f u k‖ ^ 2 = 0 simp [extendByZero_zero S f u k hk]All goals completed! 🐙 _ = ∑ j ∈ S, ‖extendByZero S f u (f j)‖ ^ 2 := byn:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑S⊢ ∑ k ∈ Finset.image f S, ‖extendByZero S f u k‖ ^ 2 = ∑ j ∈ S, ‖extendByZero S f u (f j)‖ ^ 2 rw [Finset.sum_imagen:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑S⊢ ∑ x ∈ S, ‖extendByZero S f u (f x)‖ ^ 2 = ∑ j ∈ S, ‖extendByZero S f u (f j)‖ ^ 2n:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑S⊢ InjOn f ↑S]n:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑S⊢ InjOn f ↑S exact fun j hj k hk h => hf hj hk hAll goals completed! 🐙 _ = _ := Finset.sum_congr rfl (fun j hj => byn:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑Sj:nhj:j ∈ S⊢ ‖extendByZero S f u (f j)‖ ^ 2 = ‖u j‖ ^ 2 rw [extendByZero_apply_image S f u hf j hjn:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pS:Finset nf:n → pu:n → ℂhf:InjOn f ↑Sj:nhj:j ∈ S⊢ ‖u j‖ ^ 2 = ‖u j‖ ^ 2]All goals completed! 🐙)omit [Fintype n] [DecidableEq n] in theorem extendByZero_matrix {m : Type*} (S : Finset n) (f : n → p) (u : n → ℂ) (A : Matrix m p ℂ) (i : m) : (∑ k, A i k * extendByZero S f u k) = ∑ j ∈ S, A i (f j) * u j := byn:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pm:Type u_3S:Finset nf:n → pu:n → ℂA:Matrix m p ℂi:m⊢ ∑ k, A i k * extendByZero S f u k = ∑ j ∈ S, A i (f j) * u j simp only [extendByZero, Finset.mul_sum]n:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pm:Type u_3S:Finset nf:n → pu:n → ℂA:Matrix m p ℂi:m⊢ (∑ k, ∑ i_1 ∈ S, A i k * if f i_1 = k then u i_1 else 0) = ∑ j ∈ S, A i (f j) * u j rw [Finset.sum_commn:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pm:Type u_3S:Finset nf:n → pu:n → ℂA:Matrix m p ℂi:m⊢ (∑ y ∈ S, ∑ x, A i x * if f y = x then u y else 0) = ∑ j ∈ S, A i (f j) * u j]n:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pm:Type u_3S:Finset nf:n → pu:n → ℂA:Matrix m p ℂi:m⊢ (∑ y ∈ S, ∑ x, A i x * if f y = x then u y else 0) = ∑ j ∈ S, A i (f j) * u j apply Finset.sum_congr rfln:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pm:Type u_3S:Finset nf:n → pu:n → ℂA:Matrix m p ℂi:m⊢ ∀ x ∈ S, (∑ x_1, A i x_1 * if f x = x_1 then u x else 0) = A i (f x) * u x intro j _n:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pm:Type u_3S:Finset nf:n → pu:n → ℂA:Matrix m p ℂi:mj:na✝:j ∈ S⊢ (∑ x, A i x * if f j = x then u j else 0) = A i (f j) * u j rw [Finset.sum_eq_single (f j)n:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pm:Type u_3S:Finset nf:n → pu:n → ℂA:Matrix m p ℂi:mj:na✝:j ∈ S⊢ (A i (f j) * if f j = f j then u j else 0) = A i (f j) * u jh₀n:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pm:Type u_3S:Finset nf:n → pu:n → ℂA:Matrix m p ℂi:mj:na✝:j ∈ S⊢ ∀ b ∈ Finset.univ, b ≠ f j → (A i b * if f j = b then u j else 0) = 0h₁n:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pm:Type u_3S:Finset nf:n → pu:n → ℂA:Matrix m p ℂi:mj:na✝:j ∈ S⊢ f j ∉ Finset.univ → (A i (f j) * if f j = f j then u j else 0) = 0]n:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pm:Type u_3S:Finset nf:n → pu:n → ℂA:Matrix m p ℂi:mj:na✝:j ∈ S⊢ (A i (f j) * if f j = f j then u j else 0) = A i (f j) * u jh₀n:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pm:Type u_3S:Finset nf:n → pu:n → ℂA:Matrix m p ℂi:mj:na✝:j ∈ S⊢ ∀ b ∈ Finset.univ, b ≠ f j → (A i b * if f j = b then u j else 0) = 0h₁n:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pm:Type u_3S:Finset nf:n → pu:n → ℂA:Matrix m p ℂi:mj:na✝:j ∈ S⊢ f j ∉ Finset.univ → (A i (f j) * if f j = f j then u j else 0) = 0 ·n:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pm:Type u_3S:Finset nf:n → pu:n → ℂA:Matrix m p ℂi:mj:na✝:j ∈ S⊢ (A i (f j) * if f j = f j then u j else 0) = A i (f j) * u j simpAll goals completed! 🐙 ·h₀n:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pm:Type u_3S:Finset nf:n → pu:n → ℂA:Matrix m p ℂi:mj:na✝:j ∈ S⊢ ∀ b ∈ Finset.univ, b ≠ f j → (A i b * if f j = b then u j else 0) = 0 intro k _ hkjh₀n:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pm:Type u_3S:Finset nf:n → pu:n → ℂA:Matrix m p ℂi:mj:na✝¹:j ∈ Sk:pa✝:k ∈ Finset.univhkj:k ≠ f j⊢ (A i k * if f j = k then u j else 0) = 0 simp [Ne.symm hkj]All goals completed! 🐙 ·h₁n:Type u_1p:Type u_2inst✝¹:Fintype pinst✝:DecidableEq pm:Type u_3S:Finset nf:n → pu:n → ℂA:Matrix m p ℂi:mj:na✝:j ∈ S⊢ f j ∉ Finset.univ → (A i (f j) * if f j = f j then u j else 0) = 0 simpAll goals completed! 🐙theorem masked_column_restriction_norm_le {m : Type*} [Fintype m] [DecidableEq m] (S : Finset n) (f : n → p) (hf : Set.InjOn f S) (A : Matrix m p ℂ) : ‖operator (fun i j => if j ∈ S then A i (f j) else 0)‖ ≤ ‖operator A‖ := byn:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mS:Finset nf:n → phf:InjOn f ↑SA:Matrix m p ℂ⊢ ‖operator fun i j => if j ∈ S then A i (f j) else 0‖ ≤ ‖operator A‖ apply ContinuousLinearMap.opNorm_le_bound _ (norm_nonneg _)n:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mS:Finset nf:n → phf:InjOn f ↑SA:Matrix m p ℂ⊢ ∀ (x : EuclideanSpace ℂ n), ‖(operator fun i j => if j ∈ S then A i (f j) else 0) x‖ ≤ ‖operator A‖ * ‖x‖ intro un:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mS:Finset nf:n → phf:InjOn f ↑SA:Matrix m p ℂu:EuclideanSpace ℂ n⊢ ‖(operator fun i j => if j ∈ S then A i (f j) else 0) u‖ ≤ ‖operator A‖ * ‖u‖ let v : EuclideanSpace ℂ p := WithLp.toLp 2 (extendByZero S f u)n:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mS:Finset nf:n → phf:InjOn f ↑SA:Matrix m p ℂu:EuclideanSpace ℂ nv:EuclideanSpace ℂ p := WithLp.toLp 2 (extendByZero S f u.ofLp)⊢ ‖(operator fun i j => if j ∈ S then A i (f j) else 0) u‖ ≤ ‖operator A‖ * ‖u‖ have hv : ‖v‖ ≤ ‖u‖ := byn:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mS:Finset nf:n → phf:InjOn f ↑SA:Matrix m p ℂ⊢ ‖operator fun i j => if j ∈ S then A i (f j) else 0‖ ≤ ‖operator A‖ apply (sq_le_sq₀ (norm_nonneg _) (norm_nonneg _)).mpn:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mS:Finset nf:n → phf:InjOn f ↑SA:Matrix m p ℂu:EuclideanSpace ℂ nv:EuclideanSpace ℂ p := WithLp.toLp 2 (extendByZero S f u.ofLp)⊢ ‖v‖ ^ 2 ≤ ‖u‖ ^ 2 rw [EuclideanSpace.norm_sq_eq,n:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mS:Finset nf:n → phf:InjOn f ↑SA:Matrix m p ℂu:EuclideanSpace ℂ nv:EuclideanSpace ℂ p := WithLp.toLp 2 (extendByZero S f u.ofLp)⊢ ∑ i, ‖v.ofLp i‖ ^ 2 ≤ ‖u‖ ^ 2 EuclideanSpace.norm_sq_eqn:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mS:Finset nf:n → phf:InjOn f ↑SA:Matrix m p ℂu:EuclideanSpace ℂ nv:EuclideanSpace ℂ p := WithLp.toLp 2 (extendByZero S f u.ofLp)⊢ ∑ i, ‖v.ofLp i‖ ^ 2 ≤ ∑ i, ‖u.ofLp i‖ ^ 2]n:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mS:Finset nf:n → phf:InjOn f ↑SA:Matrix m p ℂu:EuclideanSpace ℂ nv:EuclideanSpace ℂ p := WithLp.toLp 2 (extendByZero S f u.ofLp)⊢ ∑ i, ‖v.ofLp i‖ ^ 2 ≤ ∑ i, ‖u.ofLp i‖ ^ 2 change (∑ k, ‖extendByZero S f u k‖ ^ 2) ≤ ∑ j, ‖u j‖ ^ 2n:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mS:Finset nf:n → phf:InjOn f ↑SA:Matrix m p ℂu:EuclideanSpace ℂ nv:EuclideanSpace ℂ p := WithLp.toLp 2 (extendByZero S f u.ofLp)⊢ ∑ k, ‖extendByZero S f u.ofLp k‖ ^ 2 ≤ ∑ j, ‖u.ofLp j‖ ^ 2 rw [extendByZero_energy S f u hfn:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mS:Finset nf:n → phf:InjOn f ↑SA:Matrix m p ℂu:EuclideanSpace ℂ nv:EuclideanSpace ℂ p := WithLp.toLp 2 (extendByZero S f u.ofLp)⊢ ∑ j ∈ S, ‖u.ofLp j‖ ^ 2 ≤ ∑ j, ‖u.ofLp j‖ ^ 2]n:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mS:Finset nf:n → phf:InjOn f ↑SA:Matrix m p ℂu:EuclideanSpace ℂ nv:EuclideanSpace ℂ p := WithLp.toLp 2 (extendByZero S f u.ofLp)⊢ ∑ j ∈ S, ‖u.ofLp j‖ ^ 2 ≤ ∑ j, ‖u.ofLp j‖ ^ 2 exact Finset.sum_le_univ_sum_of_nonneg (fun j => sq_nonneg ‖u j‖)n:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mS:Finset nf:n → phf:InjOn f ↑SA:Matrix m p ℂu:EuclideanSpace ℂ nv:EuclideanSpace ℂ p := WithLp.toLp 2 (extendByZero S f u.ofLp)hv:‖v‖ ≤ ‖u‖⊢ ‖(operator fun i j => if j ∈ S then A i (f j) else 0) u‖ ≤ ‖operator A‖ * ‖u‖ have heq : operator (fun i j => if j ∈ S then A i (f j) else 0) u = operator A v := byn:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mS:Finset nf:n → phf:InjOn f ↑SA:Matrix m p ℂ⊢ ‖operator fun i j => if j ∈ S then A i (f j) else 0‖ ≤ ‖operator A‖ ext in:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mS:Finset nf:n → phf:InjOn f ↑SA:Matrix m p ℂu:EuclideanSpace ℂ nv:EuclideanSpace ℂ p := WithLp.toLp 2 (extendByZero S f u.ofLp)hv:‖v‖ ≤ ‖u‖i:m⊢ ((operator fun i j => if j ∈ S then A i (f j) else 0) u).ofLp i = ((operator A) v).ofLp i change (∑ j, (if j ∈ S then A i (f j) else 0) * u j) = ∑ k, A i k * extendByZero S f u kn:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mS:Finset nf:n → phf:InjOn f ↑SA:Matrix m p ℂu:EuclideanSpace ℂ nv:EuclideanSpace ℂ p := WithLp.toLp 2 (extendByZero S f u.ofLp)hv:‖v‖ ≤ ‖u‖i:m⊢ ∑ j, (if j ∈ S then A i (f j) else 0) * u.ofLp j = ∑ k, A i k * extendByZero S f u.ofLp k rw [extendByZero_matrixn:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mS:Finset nf:n → phf:InjOn f ↑SA:Matrix m p ℂu:EuclideanSpace ℂ nv:EuclideanSpace ℂ p := WithLp.toLp 2 (extendByZero S f u.ofLp)hv:‖v‖ ≤ ‖u‖i:m⊢ ∑ j, (if j ∈ S then A i (f j) else 0) * u.ofLp j = ∑ j ∈ S, A i (f j) * u.ofLp j]n:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mS:Finset nf:n → phf:InjOn f ↑SA:Matrix m p ℂu:EuclideanSpace ℂ nv:EuclideanSpace ℂ p := WithLp.toLp 2 (extendByZero S f u.ofLp)hv:‖v‖ ≤ ‖u‖i:m⊢ ∑ j, (if j ∈ S then A i (f j) else 0) * u.ofLp j = ∑ j ∈ S, A i (f j) * u.ofLp j simp only [ite_mul, zero_mul, ← Finset.sum_filter]n:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mS:Finset nf:n → phf:InjOn f ↑SA:Matrix m p ℂu:EuclideanSpace ℂ nv:EuclideanSpace ℂ p := WithLp.toLp 2 (extendByZero S f u.ofLp)hv:‖v‖ ≤ ‖u‖i:m⊢ ∑ a with a ∈ S, A i (f a) * u.ofLp a = ∑ a ∈ S, A i (f a) * u.ofLp a simpn:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mS:Finset nf:n → phf:InjOn f ↑SA:Matrix m p ℂu:EuclideanSpace ℂ nv:EuclideanSpace ℂ p := WithLp.toLp 2 (extendByZero S f u.ofLp)hv:‖v‖ ≤ ‖u‖heq:(operator fun i j => if j ∈ S then A i (f j) else 0) u = (operator A) v⊢ ‖(operator fun i j => if j ∈ S then A i (f j) else 0) u‖ ≤ ‖operator A‖ * ‖u‖ rw [heqn:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mS:Finset nf:n → phf:InjOn f ↑SA:Matrix m p ℂu:EuclideanSpace ℂ nv:EuclideanSpace ℂ p := WithLp.toLp 2 (extendByZero S f u.ofLp)hv:‖v‖ ≤ ‖u‖heq:(operator fun i j => if j ∈ S then A i (f j) else 0) u = (operator A) v⊢ ‖(operator A) v‖ ≤ ‖operator A‖ * ‖u‖]n:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mS:Finset nf:n → phf:InjOn f ↑SA:Matrix m p ℂu:EuclideanSpace ℂ nv:EuclideanSpace ℂ p := WithLp.toLp 2 (extendByZero S f u.ofLp)hv:‖v‖ ≤ ‖u‖heq:(operator fun i j => if j ∈ S then A i (f j) else 0) u = (operator A) v⊢ ‖(operator A) v‖ ≤ ‖operator A‖ * ‖u‖ exact ((operator A).le_opNorm v).trans (mul_le_mul_of_nonneg_left hv (norm_nonneg _))All goals completed! 🐙theorem column_restriction_norm_le {m : Type*} [Fintype m] [DecidableEq m] (f : n → p) (hf : Function.Injective f) (A : Matrix m p ℂ) : ‖operator (fun i j => A i (f j))‖ ≤ ‖operator A‖ := byn:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mf:n → phf:Function.Injective fA:Matrix m p ℂ⊢ ‖operator fun i j => A i (f j)‖ ≤ ‖operator A‖ simpa only [Finset.mem_univ, ite_true] using masked_column_restriction_norm_le Finset.univ f hf.injOn AAll goals completed! 🐙theorem row_restriction_norm_le {m : Type*} [Fintype m] [DecidableEq m] (f : n → p) (hf : Function.Injective f) (A : Matrix p m ℂ) : ‖operator (fun i j => A (f i) j)‖ ≤ ‖operator A‖ := byn:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mf:n → phf:Function.Injective fA:Matrix p m ℂ⊢ ‖operator fun i j => A (f i) j‖ ≤ ‖operator A‖ let B : Matrix n m ℂ := fun i j => A (f i) jn:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mf:n → phf:Function.Injective fA:Matrix p m ℂB:Matrix n m ℂ := fun i j => A (f i) j⊢ ‖operator fun i j => A (f i) j‖ ≤ ‖operator A‖ have h := column_restriction_norm_le f hf A.conjTransposen:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mf:n → phf:Function.Injective fA:Matrix p m ℂB:Matrix n m ℂ := fun i j => A (f i) jh:‖operator fun i j => A.conjTranspose i (f j)‖ ≤ ‖operator A.conjTranspose‖⊢ ‖operator fun i j => A (f i) j‖ ≤ ‖operator A‖ change ‖operator B.conjTranspose‖ ≤ ‖operator A.conjTranspose‖ at hn:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mf:n → phf:Function.Injective fA:Matrix p m ℂB:Matrix n m ℂ := fun i j => A (f i) jh:‖operator B.conjTranspose‖ ≤ ‖operator A.conjTranspose‖⊢ ‖operator fun i j => A (f i) j‖ ≤ ‖operator A‖ rw [operator_conjTranspose_norm B,n:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mf:n → phf:Function.Injective fA:Matrix p m ℂB:Matrix n m ℂ := fun i j => A (f i) jh:‖operator B‖ ≤ ‖operator A.conjTranspose‖⊢ ‖operator fun i j => A (f i) j‖ ≤ ‖operator A‖ operator_conjTranspose_norm An:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mf:n → phf:Function.Injective fA:Matrix p m ℂB:Matrix n m ℂ := fun i j => A (f i) jh:‖operator B‖ ≤ ‖operator A‖⊢ ‖operator fun i j => A (f i) j‖ ≤ ‖operator A‖] at hn:Type u_1p:Type u_2inst✝⁵:Fintype ninst✝⁴:Fintype pinst✝³:DecidableEq ninst✝²:DecidableEq pm:Type u_3inst✝¹:Fintype minst✝:DecidableEq mf:n → phf:Function.Injective fA:Matrix p m ℂB:Matrix n m ℂ := fun i j => A (f i) jh:‖operator B‖ ≤ ‖operator A‖⊢ ‖operator fun i j => A (f i) j‖ ≤ ‖operator A‖ exact hAll goals completed! 🐙theorem submatrix_norm_le {m q : Type*} [Fintype m] [Fintype q] [DecidableEq m] [DecidableEq q] (r : m → q) (hr : Function.Injective r) (c : n → p) (hc : Function.Injective c) (A : Matrix q p ℂ) : ‖operator (fun i j => A (r i) (c j))‖ ≤ ‖operator A‖ := (row_restriction_norm_le r hr (fun i j => A i (c j))).trans (column_restriction_norm_le c hc A)end FiniteSieveRestrictionopen scoped BigOperatorsnamespace FiniteSieveOperatortheorem squared_norm_le_of_energy {m n : Type*} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] (A : Matrix m n ℂ) (C : ℝ) (hC : 0 ≤ C) (h : ∀ a : n → ℂ, (∑ i, ‖∑ j, A i j * a j‖ ^ 2) ≤ C * ∑ j, ‖a j‖ ^ 2) : ‖operator A‖ ^ 2 ≤ C := bym:Type u_1n:Type u_2inst✝³:Fintype minst✝²:Fintype ninst✝¹:DecidableEq minst✝:DecidableEq nA:Matrix m n ℂC:ℝhC:0 ≤ Ch:∀ (a : n → ℂ), ∑ i, ‖∑ j, A i j * a j‖ ^ 2 ≤ C * ∑ j, ‖a j‖ ^ 2⊢ ‖operator A‖ ^ 2 ≤ C have hop : ‖operator A‖ ≤ Real.sqrt C := by apply ContinuousLinearMap.opNorm_le_bound _ (Real.sqrt_nonneg C)m:Type u_1n:Type u_2inst✝³:Fintype minst✝²:Fintype ninst✝¹:DecidableEq minst✝:DecidableEq nA:Matrix m n ℂC:ℝhC:0 ≤ Ch:∀ (a : n → ℂ), ∑ i, ‖∑ j, A i j * a j‖ ^ 2 ≤ C * ∑ j, ‖a j‖ ^ 2⊢ ∀ (x : EuclideanSpace ℂ n), ‖(operator A) x‖ ≤ √C * ‖x‖ intro am:Type u_1n:Type u_2inst✝³:Fintype minst✝²:Fintype ninst✝¹:DecidableEq minst✝:DecidableEq nA:Matrix m n ℂC:ℝhC:0 ≤ Ch:∀ (a : n → ℂ), ∑ i, ‖∑ j, A i j * a j‖ ^ 2 ≤ C * ∑ j, ‖a j‖ ^ 2a:EuclideanSpace ℂ n⊢ ‖(operator A) a‖ ≤ √C * ‖a‖ apply (sq_le_sq₀ (norm_nonneg _) (mul_nonneg (Real.sqrt_nonneg C) (norm_nonneg a))).mpm:Type u_1n:Type u_2inst✝³:Fintype minst✝²:Fintype ninst✝¹:DecidableEq minst✝:DecidableEq nA:Matrix m n ℂC:ℝhC:0 ≤ Ch:∀ (a : n → ℂ), ∑ i, ‖∑ j, A i j * a j‖ ^ 2 ≤ C * ∑ j, ‖a j‖ ^ 2a:EuclideanSpace ℂ n⊢ ‖(operator A) a‖ ^ 2 ≤ (√C * ‖a‖) ^ 2 simpa only [mul_pow, Real.sq_sqrt hC, EuclideanSpace.norm_sq_eq, operator_apply] using h am:Type u_1n:Type u_2inst✝³:Fintype minst✝²:Fintype ninst✝¹:DecidableEq minst✝:DecidableEq nA:Matrix m n ℂC:ℝhC:0 ≤ Ch:∀ (a : n → ℂ), ∑ i, ‖∑ j, A i j * a j‖ ^ 2 ≤ C * ∑ j, ‖a j‖ ^ 2hop:‖operator A‖ ≤ √C⊢ ‖operator A‖ ^ 2 ≤ C simpa only [Real.sq_sqrt hC] using pow_le_pow_left₀ (norm_nonneg _) hop 2All goals completed! 🐙end FiniteSieveOperatoropen scoped BigOperators Classicalnamespace CanonicalQuadraticSieveabbrev O := ActualEisensteinCubic.Oopen ConcretePrimeRowBridge ActualEisensteinCubic CompletedGausstheorem idealSexticRow_ambient (F G : Finset (Ideal O)) (hFp : ∀ I ∈ F, I ≠ ⊥) (hGp : ∀ I ∈ G, I ≠ ⊥) (hFg : ∀ I ∈ F, ∀ P ∈ UniqueFactorizationMonoid.normalizedFactors I, goodLambda ∉ P) (hGg : ∀ I ∈ G, ∀ P ∈ UniqueFactorizationMonoid.normalizedFactors I, goodLambda ∉ P) (I : Ideal O) (hIF : I ∈ F) (hIG : I ∈ G) (z : O) : idealSexticRow F hFp hFg I z = idealSexticRow G hGp hGg I z := byF:Finset (Ideal O)G:Finset (Ideal O)hFp:∀ I ∈ F, I ≠ ⊥hGp:∀ I ∈ G, I ≠ ⊥hFg:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ PhGg:∀ I ∈ G, ∀ P ∈ normalizedFactors I, goodLambda ∉ PI:Ideal OhIF:I ∈ FhIG:I ∈ Gz:O⊢ idealSexticRow F hFp hFg I z = idealSexticRow G hGp hGg I z let idx (i : primePool F) (hi : i ∈ idealSupport F I) : primePool G := ⟨i.val, mem_primePool_iff.mpr ⟨I, hIG, (mem_idealSupport_iff F I i).mp hi⟩⟩F:Finset (Ideal O)G:Finset (Ideal O)hFp:∀ I ∈ F, I ≠ ⊥hGp:∀ I ∈ G, I ≠ ⊥hFg:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ PhGg:∀ I ∈ G, ∀ P ∈ normalizedFactors I, goodLambda ∉ PI:Ideal OhIF:I ∈ FhIG:I ∈ Gz:Oidx:(i : ↥(primePool F)) → i ∈ idealSupport F I → ↥(primePool G) := fun i hi => ⟨↑i, ⋯⟩⊢ idealSexticRow F hFp hFg I z = idealSexticRow G hGp hGg I z unfold idealSexticRow finiteSquarefreeRowF:Finset (Ideal O)G:Finset (Ideal O)hFp:∀ I ∈ F, I ≠ ⊥hGp:∀ I ∈ G, I ≠ ⊥hFg:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ PhGg:∀ I ∈ G, ∀ P ∈ normalizedFactors I, goodLambda ∉ PI:Ideal OhIF:I ∈ FhIG:I ∈ Gz:Oidx:(i : ↥(primePool F)) → i ∈ idealSupport F I → ↥(primePool G) := fun i hi => ⟨↑i, ⋯⟩⊢ ∏ i ∈ idealSupport F I, (canonicalSextic ((fun i => ↑i) i) ⋯) ((Ideal.Quotient.mk ((fun i => ↑i) i)) z) = ∏ i ∈ idealSupport G I, (canonicalSextic ((fun i => ↑i) i) ⋯) ((Ideal.Quotient.mk ((fun i => ↑i) i)) z) apply Finset.prod_bij idxhiF:Finset (Ideal O)G:Finset (Ideal O)hFp:∀ I ∈ F, I ≠ ⊥hGp:∀ I ∈ G, I ≠ ⊥hFg:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ PhGg:∀ I ∈ G, ∀ P ∈ normalizedFactors I, goodLambda ∉ PI:Ideal OhIF:I ∈ FhIG:I ∈ Gz:Oidx:(i : ↥(primePool F)) → i ∈ idealSupport F I → ↥(primePool G) := fun i hi => ⟨↑i, ⋯⟩⊢ ∀ (a : ↥(primePool F)) (ha : a ∈ idealSupport F I), idx a ha ∈ idealSupport G Ii_injF:Finset (Ideal O)G:Finset (Ideal O)hFp:∀ I ∈ F, I ≠ ⊥hGp:∀ I ∈ G, I ≠ ⊥hFg:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ PhGg:∀ I ∈ G, ∀ P ∈ normalizedFactors I, goodLambda ∉ PI:Ideal OhIF:I ∈ FhIG:I ∈ Gz:Oidx:(i : ↥(primePool F)) → i ∈ idealSupport F I → ↥(primePool G) := fun i hi => ⟨↑i, ⋯⟩⊢ ∀ (a₁ : ↥(primePool F)) (ha₁ : a₁ ∈ idealSupport F I) (a₂ : ↥(primePool F)) (ha₂ : a₂ ∈ idealSupport F I), idx a₁ ha₁ = idx a₂ ha₂ → a₁ = a₂i_surjF:Finset (Ideal O)G:Finset (Ideal O)hFp:∀ I ∈ F, I ≠ ⊥hGp:∀ I ∈ G, I ≠ ⊥hFg:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ PhGg:∀ I ∈ G, ∀ P ∈ normalizedFactors I, goodLambda ∉ PI:Ideal OhIF:I ∈ FhIG:I ∈ Gz:Oidx:(i : ↥(primePool F)) → i ∈ idealSupport F I → ↥(primePool G) := fun i hi => ⟨↑i, ⋯⟩⊢ ∀ b ∈ idealSupport G I, ∃ a, ∃ (ha : a ∈ idealSupport F I), idx a ha = bhF:Finset (Ideal O)G:Finset (Ideal O)hFp:∀ I ∈ F, I ≠ ⊥hGp:∀ I ∈ G, I ≠ ⊥hFg:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ PhGg:∀ I ∈ G, ∀ P ∈ normalizedFactors I, goodLambda ∉ PI:Ideal OhIF:I ∈ FhIG:I ∈ Gz:Oidx:(i : ↥(primePool F)) → i ∈ idealSupport F I → ↥(primePool G) := fun i hi => ⟨↑i, ⋯⟩⊢ ∀ (a : ↥(primePool F)) (ha : a ∈ idealSupport F I), (canonicalSextic ((fun i => ↑i) a) ⋯) ((Ideal.Quotient.mk ((fun i => ↑i) a)) z) = (canonicalSextic ((fun i => ↑i) (idx a ha)) ⋯) ((Ideal.Quotient.mk ((fun i => ↑i) (idx a ha))) z) ·hiF:Finset (Ideal O)G:Finset (Ideal O)hFp:∀ I ∈ F, I ≠ ⊥hGp:∀ I ∈ G, I ≠ ⊥hFg:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ PhGg:∀ I ∈ G, ∀ P ∈ normalizedFactors I, goodLambda ∉ PI:Ideal OhIF:I ∈ FhIG:I ∈ Gz:Oidx:(i : ↥(primePool F)) → i ∈ idealSupport F I → ↥(primePool G) := fun i hi => ⟨↑i, ⋯⟩⊢ ∀ (a : ↥(primePool F)) (ha : a ∈ idealSupport F I), idx a ha ∈ idealSupport G I intro i hihiF:Finset (Ideal O)G:Finset (Ideal O)hFp:∀ I ∈ F, I ≠ ⊥hGp:∀ I ∈ G, I ≠ ⊥hFg:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ PhGg:∀ I ∈ G, ∀ P ∈ normalizedFactors I, goodLambda ∉ PI:Ideal OhIF:I ∈ FhIG:I ∈ Gz:Oidx:(i : ↥(primePool F)) → i ∈ idealSupport F I → ↥(primePool G) := fun i hi => ⟨↑i, ⋯⟩i:↥(primePool F)hi:i ∈ idealSupport F I⊢ idx i hi ∈ idealSupport G I exact (mem_idealSupport_iff G I (idx i hi)).mpr ((mem_idealSupport_iff F I i).mp hi)All goals completed! 🐙 ·i_injF:Finset (Ideal O)G:Finset (Ideal O)hFp:∀ I ∈ F, I ≠ ⊥hGp:∀ I ∈ G, I ≠ ⊥hFg:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ PhGg:∀ I ∈ G, ∀ P ∈ normalizedFactors I, goodLambda ∉ PI:Ideal OhIF:I ∈ FhIG:I ∈ Gz:Oidx:(i : ↥(primePool F)) → i ∈ idealSupport F I → ↥(primePool G) := fun i hi => ⟨↑i, ⋯⟩⊢ ∀ (a₁ : ↥(primePool F)) (ha₁ : a₁ ∈ idealSupport F I) (a₂ : ↥(primePool F)) (ha₂ : a₂ ∈ idealSupport F I), idx a₁ ha₁ = idx a₂ ha₂ → a₁ = a₂ intro i hi j hj hiji_injF:Finset (Ideal O)G:Finset (Ideal O)hFp:∀ I ∈ F, I ≠ ⊥hGp:∀ I ∈ G, I ≠ ⊥hFg:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ PhGg:∀ I ∈ G, ∀ P ∈ normalizedFactors I, goodLambda ∉ PI:Ideal OhIF:I ∈ FhIG:I ∈ Gz:Oidx:(i : ↥(primePool F)) → i ∈ idealSupport F I → ↥(primePool G) := fun i hi => ⟨↑i, ⋯⟩i:↥(primePool F)hi:i ∈ idealSupport F Ij:↥(primePool F)hj:j ∈ idealSupport F Ihij:idx i hi = idx j hj⊢ i = j exact Subtype.ext (congrArg (fun k : primePool G => k.val) hij)All goals completed! 🐙 ·i_surjF:Finset (Ideal O)G:Finset (Ideal O)hFp:∀ I ∈ F, I ≠ ⊥hGp:∀ I ∈ G, I ≠ ⊥hFg:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ PhGg:∀ I ∈ G, ∀ P ∈ normalizedFactors I, goodLambda ∉ PI:Ideal OhIF:I ∈ FhIG:I ∈ Gz:Oidx:(i : ↥(primePool F)) → i ∈ idealSupport F I → ↥(primePool G) := fun i hi => ⟨↑i, ⋯⟩⊢ ∀ b ∈ idealSupport G I, ∃ a, ∃ (ha : a ∈ idealSupport F I), idx a ha = b intro j hji_surjF:Finset (Ideal O)G:Finset (Ideal O)hFp:∀ I ∈ F, I ≠ ⊥hGp:∀ I ∈ G, I ≠ ⊥hFg:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ PhGg:∀ I ∈ G, ∀ P ∈ normalizedFactors I, goodLambda ∉ PI:Ideal OhIF:I ∈ FhIG:I ∈ Gz:Oidx:(i : ↥(primePool F)) → i ∈ idealSupport F I → ↥(primePool G) := fun i hi => ⟨↑i, ⋯⟩j:↥(primePool G)hj:j ∈ idealSupport G I⊢ ∃ a, ∃ (ha : a ∈ idealSupport F I), idx a ha = j let i : primePool F := ⟨j.val, mem_primePool_iff.mpr ⟨I, hIF, (mem_idealSupport_iff G I j).mp hj⟩⟩i_surjF:Finset (Ideal O)G:Finset (Ideal O)hFp:∀ I ∈ F, I ≠ ⊥hGp:∀ I ∈ G, I ≠ ⊥hFg:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ PhGg:∀ I ∈ G, ∀ P ∈ normalizedFactors I, goodLambda ∉ PI:Ideal OhIF:I ∈ FhIG:I ∈ Gz:Oidx:(i : ↥(primePool F)) → i ∈ idealSupport F I → ↥(primePool G) := fun i hi => ⟨↑i, ⋯⟩j:↥(primePool G)hj:j ∈ idealSupport G Ii:↥(primePool F) := ⟨↑j, ⋯⟩⊢ ∃ a, ∃ (ha : a ∈ idealSupport F I), idx a ha = j have hi : i ∈ idealSupport F I := (mem_idealSupport_iff F I i).mpr ((mem_idealSupport_iff G I j).mp hj)i_surjF:Finset (Ideal O)G:Finset (Ideal O)hFp:∀ I ∈ F, I ≠ ⊥hGp:∀ I ∈ G, I ≠ ⊥hFg:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ PhGg:∀ I ∈ G, ∀ P ∈ normalizedFactors I, goodLambda ∉ PI:Ideal OhIF:I ∈ FhIG:I ∈ Gz:Oidx:(i : ↥(primePool F)) → i ∈ idealSupport F I → ↥(primePool G) := fun i hi => ⟨↑i, ⋯⟩j:↥(primePool G)hj:j ∈ idealSupport G Ii:↥(primePool F) := ⟨↑j, ⋯⟩hi:i ∈ idealSupport F I⊢ ∃ a, ∃ (ha : a ∈ idealSupport F I), idx a ha = j exact ⟨i, hi, Subtype.ext rfl⟩All goals completed! 🐙 ·hF:Finset (Ideal O)G:Finset (Ideal O)hFp:∀ I ∈ F, I ≠ ⊥hGp:∀ I ∈ G, I ≠ ⊥hFg:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ PhGg:∀ I ∈ G, ∀ P ∈ normalizedFactors I, goodLambda ∉ PI:Ideal OhIF:I ∈ FhIG:I ∈ Gz:Oidx:(i : ↥(primePool F)) → i ∈ idealSupport F I → ↥(primePool G) := fun i hi => ⟨↑i, ⋯⟩⊢ ∀ (a : ↥(primePool F)) (ha : a ∈ idealSupport F I), (canonicalSextic ((fun i => ↑i) a) ⋯) ((Ideal.Quotient.mk ((fun i => ↑i) a)) z) = (canonicalSextic ((fun i => ↑i) (idx a ha)) ⋯) ((Ideal.Quotient.mk ((fun i => ↑i) (idx a ha))) z) intro i hihF:Finset (Ideal O)G:Finset (Ideal O)hFp:∀ I ∈ F, I ≠ ⊥hGp:∀ I ∈ G, I ≠ ⊥hFg:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ PhGg:∀ I ∈ G, ∀ P ∈ normalizedFactors I, goodLambda ∉ PI:Ideal OhIF:I ∈ FhIG:I ∈ Gz:Oidx:(i : ↥(primePool F)) → i ∈ idealSupport F I → ↥(primePool G) := fun i hi => ⟨↑i, ⋯⟩i:↥(primePool F)hi:i ∈ idealSupport F I⊢ (canonicalSextic ((fun i => ↑i) i) ⋯) ((Ideal.Quotient.mk ((fun i => ↑i) i)) z) = (canonicalSextic ((fun i => ↑i) (idx i hi)) ⋯) ((Ideal.Quotient.mk ((fun i => ↑i) (idx i hi))) z) rflAll goals completed! 🐙def Admissible (I : Ideal O) : Prop := I ≠ 0 ∧ Squarefree I ∧ ∀ P ∈ UniqueFactorizationMonoid.normalizedFactors I, lambda ∉ P ∧ ringChar (O ⧸ P) ≠ 2theorem singletonPositive (I : Ideal O) (h : Admissible I) : ∀ J ∈ ({I} : Finset (Ideal O)), J ≠ ⊥ := byI:Ideal Oh:Admissible I⊢ ∀ J ∈ {I}, J ≠ ⊥ intro J hJI:Ideal Oh:Admissible IJ:Ideal OhJ:J ∈ {I}⊢ J ≠ ⊥ rcases Finset.mem_singleton.mp hJ with rflJ:Ideal Oh:Admissible JhJ:J ∈ {J}⊢ J ≠ ⊥ exact h.1All goals completed! 🐙theorem singletonGood (I : Ideal O) (h : Admissible I) : ∀ J ∈ ({I} : Finset (Ideal O)), ∀ P ∈ UniqueFactorizationMonoid.normalizedFactors J, goodLambda ∉ P := byI:Ideal Oh:Admissible I⊢ ∀ J ∈ {I}, ∀ P ∈ normalizedFactors J, goodLambda ∉ P intro J hJ P hPI:Ideal Oh:Admissible IJ:Ideal OhJ:J ∈ {I}P:Ideal OhP:P ∈ normalizedFactors J⊢ goodLambda ∉ P rcases Finset.mem_singleton.mp hJ with rflJ:Ideal OP:Ideal OhP:P ∈ normalizedFactors Jh:Admissible JhJ:J ∈ {J}⊢ goodLambda ∉ P exact (h.2.2 P hP).1All goals completed! 🐙def quadraticRow (I : Ideal O) (z : O) : ℂ := if h : Admissible I then (idealSexticRow {I} (singletonPositive I h) (singletonGood I h) I z) ^ 3 else 0theorem quadraticRow_eq (F : Finset (Ideal O)) (hF : ∀ I ∈ F, Admissible I) (I : Ideal O) (hI : I ∈ F) (z : O) : quadraticRow I z = (idealSexticRow F (fun J hJ => (hF J hJ).1) (fun J hJ P hP => (hF J hJ).2.2 P hP |>.1) I z) ^ 3 := byF:Finset (Ideal O)hF:∀ I ∈ F, Admissible II:Ideal OhI:I ∈ Fz:O⊢ quadraticRow I z = idealSexticRow F ⋯ ⋯ I z ^ 3 rw [quadraticRow,F:Finset (Ideal O)hF:∀ I ∈ F, Admissible II:Ideal OhI:I ∈ Fz:O⊢ (if h : Admissible I then idealSexticRow {I} ⋯ ⋯ I z ^ 3 else 0) = idealSexticRow F ⋯ ⋯ I z ^ 3 dite_eq_left (hF I hI)F:Finset (Ideal O)hF:∀ I ∈ F, Admissible II:Ideal OhI:I ∈ Fz:O⊢ idealSexticRow {I} ⋯ ⋯ I z ^ 3 = idealSexticRow F ⋯ ⋯ I z ^ 3]F:Finset (Ideal O)hF:∀ I ∈ F, Admissible II:Ideal OhI:I ∈ Fz:O⊢ idealSexticRow {I} ⋯ ⋯ I z ^ 3 = idealSexticRow F ⋯ ⋯ I z ^ 3 congr 1e_aF:Finset (Ideal O)hF:∀ I ∈ F, Admissible II:Ideal OhI:I ∈ Fz:O⊢ idealSexticRow {I} ⋯ ⋯ I z = idealSexticRow F ⋯ ⋯ I z exact idealSexticRow_ambient {I} F (singletonPositive I (hF I hI)) (fun J hJ => (hF J hJ).1) (singletonGood I (hF I hI)) (fun J hJ P hP => (hF J hJ).2.2 P hP |>.1) I (Finset.mem_singleton_self I) hI zAll goals completed! 🐙def idealRange (X : ℝ) : Finset (Ideal O) := (idealsUpTo ⌊X⌋₊).filter Admissibletheorem mem_idealRange {X : ℝ} {I : Ideal O} : I ∈ idealRange X ↔ Admissible I ∧ (Ideal.absNorm I : ℝ) ≤ X := byX:ℝI:Ideal O⊢ I ∈ idealRange X ↔ Admissible I ∧ ↑(absNorm I) ≤ X rw [idealRange,X:ℝI:Ideal O⊢ I ∈ filter Admissible (idealsUpTo ⌊X⌋₊) ↔ Admissible I ∧ ↑(absNorm I) ≤ X Finset.mem_filter,X:ℝI:Ideal O⊢ I ∈ idealsUpTo ⌊X⌋₊ ∧ Admissible I ↔ Admissible I ∧ ↑(absNorm I) ≤ X mem_idealsUpToX:ℝI:Ideal O⊢ (1 ≤ absNorm I ∧ absNorm I ≤ ⌊X⌋₊) ∧ Admissible I ↔ Admissible I ∧ ↑(absNorm I) ≤ X]X:ℝI:Ideal O⊢ (1 ≤ absNorm I ∧ absNorm I ≤ ⌊X⌋₊) ∧ Admissible I ↔ Admissible I ∧ ↑(absNorm I) ≤ X constructormpX:ℝI:Ideal O⊢ (1 ≤ absNorm I ∧ absNorm I ≤ ⌊X⌋₊) ∧ Admissible I → Admissible I ∧ ↑(absNorm I) ≤ XmprX:ℝI:Ideal O⊢ Admissible I ∧ ↑(absNorm I) ≤ X → (1 ≤ absNorm I ∧ absNorm I ≤ ⌊X⌋₊) ∧ Admissible I ·mpX:ℝI:Ideal O⊢ (1 ≤ absNorm I ∧ absNorm I ≤ ⌊X⌋₊) ∧ Admissible I → Admissible I ∧ ↑(absNorm I) ≤ X rintro ⟨⟨hp, hN⟩, ha⟩mpX:ℝI:Ideal Oha:Admissible Ihp:1 ≤ absNorm IhN:absNorm I ≤ ⌊X⌋₊⊢ Admissible I ∧ ↑(absNorm I) ≤ X refine ⟨ha, ?_⟩mpX:ℝI:Ideal Oha:Admissible Ihp:1 ≤ absNorm IhN:absNorm I ≤ ⌊X⌋₊⊢ ↑(absNorm I) ≤ X have hX : 0 ≤ X := byX:ℝI:Ideal O⊢ I ∈ idealRange X ↔ Admissible I ∧ ↑(absNorm I) ≤ X by_contra hnX:ℝI:Ideal Oha:Admissible Ihp:1 ≤ absNorm IhN:absNorm I ≤ ⌊X⌋₊hn:¬0 ≤ X⊢ False have hz : ⌊X⌋₊ = 0 := Nat.floor_eq_zero.mpr (byX:ℝI:Ideal Oha:Admissible Ihp:1 ≤ absNorm IhN:absNorm I ≤ ⌊X⌋₊hn:¬0 ≤ X⊢ X < 1 linarithAll goals completed! 🐙)X:ℝI:Ideal Oha:Admissible Ihp:1 ≤ absNorm IhN:absNorm I ≤ ⌊X⌋₊hn:¬0 ≤ Xhz:⌊X⌋₊ = 0⊢ False omegampX:ℝI:Ideal Oha:Admissible Ihp:1 ≤ absNorm IhN:absNorm I ≤ ⌊X⌋₊hX:0 ≤ X⊢ ↑(absNorm I) ≤ X exact (Nat.cast_le.mpr hN).trans (Nat.floor_le hX)All goals completed! 🐙 ·mprX:ℝI:Ideal O⊢ Admissible I ∧ ↑(absNorm I) ≤ X → (1 ≤ absNorm I ∧ absNorm I ≤ ⌊X⌋₊) ∧ Admissible I rintro ⟨ha, hN⟩mprX:ℝI:Ideal Oha:Admissible IhN:↑(absNorm I) ≤ X⊢ (1 ≤ absNorm I ∧ absNorm I ≤ ⌊X⌋₊) ∧ Admissible I exact ⟨⟨Nat.one_le_iff_ne_zero.mpr (fun h => ha.1 (Ideal.absNorm_eq_zero_iff.mp h)), Nat.le_floor hN⟩, ha⟩All goals completed! 🐙theorem idealRange_mono {X Y : ℝ} (h : X ≤ Y) : idealRange X ⊆ idealRange Y := byX:ℝY:ℝh:X ≤ Y⊢ idealRange X ⊆ idealRange Y intro I hIX:ℝY:ℝh:X ≤ YI:Ideal OhI:I ∈ idealRange X⊢ I ∈ idealRange Y rcases mem_idealRange.mp hI with ⟨ha, hN⟩X:ℝY:ℝh:X ≤ YI:Ideal OhI:I ∈ idealRange Xha:Admissible IhN:↑(absNorm I) ≤ X⊢ I ∈ idealRange Y exact mem_idealRange.mpr ⟨ha, hN.trans h⟩All goals completed! 🐙def matrix (M N : ℝ) : Matrix (idealRange M) (idealRange N) ℂ := fun I J => quadraticRow J.val (primaryGenerator I.val)def sieveNorm (M N : ℝ) : ℝ := ‖FiniteSieveOperator.operator (matrix M N)‖ ^ 2theorem sieveNorm_nonneg (M N : ℝ) : 0 ≤ sieveNorm M N := sq_nonneg _def rangeInclusion {X Y : ℝ} (h : X ≤ Y) (I : idealRange X) : idealRange Y := ⟨I.val, idealRange_mono h I.property⟩theorem rangeInclusion_injective {X Y : ℝ} (h : X ≤ Y) : Function.Injective (rangeInclusion h) := byX:ℝY:ℝh:X ≤ Y⊢ Function.Injective (rangeInclusion h) intro I J heqX:ℝY:ℝh:X ≤ YI:↥(idealRange X)J:↥(idealRange X)heq:rangeInclusion h I = rangeInclusion h J⊢ I = J exact Subtype.ext (congrArg (fun x : idealRange Y => x.val) heq)All goals completed! 🐙theorem sieveNorm_mono {M N M' N' : ℝ} (hM : M ≤ M') (hN : N ≤ N') : sieveNorm M N ≤ sieveNorm M' N' := byM:ℝN:ℝM':ℝN':ℝhM:M ≤ M'hN:N ≤ N'⊢ sieveNorm M N ≤ sieveNorm M' N' have h := FiniteSieveRestriction.submatrix_norm_le (rangeInclusion hM) (rangeInclusion_injective hM) (rangeInclusion hN) (rangeInclusion_injective hN) (matrix M' N')M:ℝN:ℝM':ℝN':ℝhM:M ≤ M'hN:N ≤ N'h:‖FiniteSieveOperator.operator fun i j => matrix M' N' (rangeInclusion hM i) (rangeInclusion hN j)‖ ≤ ‖FiniteSieveOperator.operator (matrix M' N')‖⊢ sieveNorm M N ≤ sieveNorm M' N' change ‖FiniteSieveOperator.operator (matrix M N)‖ ≤ ‖FiniteSieveOperator.operator (matrix M' N')‖ at hM:ℝN:ℝM':ℝN':ℝhM:M ≤ M'hN:N ≤ N'h:‖FiniteSieveOperator.operator (matrix M N)‖ ≤ ‖FiniteSieveOperator.operator (matrix M' N')‖⊢ sieveNorm M N ≤ sieveNorm M' N' exact pow_le_pow_left₀ (norm_nonneg _) h 2All goals completed! 🐙theorem family_squared_norm_le {m n : Type*} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] (rows : m → Ideal O) (cols : n → Ideal O) (hr : Function.Injective rows) (hc : Function.Injective cols) (M N : ℝ) (hrows : ∀ i, Admissible (rows i) ∧ (Ideal.absNorm (rows i) : ℝ) ≤ M) (hcols : ∀ j, Admissible (cols j) ∧ (Ideal.absNorm (cols j) : ℝ) ≤ N) : ‖FiniteSieveOperator.operator (fun i j => quadraticRow (cols j) (primaryGenerator (rows i)))‖ ^ 2 ≤ sieveNorm M N := bym:Type u_1n:Type u_2inst✝³:Fintype minst✝²:Fintype ninst✝¹:DecidableEq minst✝:DecidableEq nrows:m → Ideal Ocols:n → Ideal Ohr:Function.Injective rowshc:Function.Injective colsM:ℝN:ℝhrows:∀ (i : m), Admissible (rows i) ∧ ↑(absNorm (rows i)) ≤ Mhcols:∀ (j : n), Admissible (cols j) ∧ ↑(absNorm (cols j)) ≤ N⊢ ‖FiniteSieveOperator.operator fun i j => quadraticRow (cols j) (primaryGenerator (rows i))‖ ^ 2 ≤ sieveNorm M N let r : m → idealRange M := fun i => ⟨rows i, mem_idealRange.mpr (hrows i)⟩m:Type u_1n:Type u_2inst✝³:Fintype minst✝²:Fintype ninst✝¹:DecidableEq minst✝:DecidableEq nrows:m → Ideal Ocols:n → Ideal Ohr:Function.Injective rowshc:Function.Injective colsM:ℝN:ℝhrows:∀ (i : m), Admissible (rows i) ∧ ↑(absNorm (rows i)) ≤ Mhcols:∀ (j : n), Admissible (cols j) ∧ ↑(absNorm (cols j)) ≤ Nr:m → ↥(idealRange M) := fun i => ⟨rows i, ⋯⟩⊢ ‖FiniteSieveOperator.operator fun i j => quadraticRow (cols j) (primaryGenerator (rows i))‖ ^ 2 ≤ sieveNorm M N let c : n → idealRange N := fun j => ⟨cols j, mem_idealRange.mpr (hcols j)⟩m:Type u_1n:Type u_2inst✝³:Fintype minst✝²:Fintype ninst✝¹:DecidableEq minst✝:DecidableEq nrows:m → Ideal Ocols:n → Ideal Ohr:Function.Injective rowshc:Function.Injective colsM:ℝN:ℝhrows:∀ (i : m), Admissible (rows i) ∧ ↑(absNorm (rows i)) ≤ Mhcols:∀ (j : n), Admissible (cols j) ∧ ↑(absNorm (cols j)) ≤ Nr:m → ↥(idealRange M) := fun i => ⟨rows i, ⋯⟩c:n → ↥(idealRange N) := fun j => ⟨cols j, ⋯⟩⊢ ‖FiniteSieveOperator.operator fun i j => quadraticRow (cols j) (primaryGenerator (rows i))‖ ^ 2 ≤ sieveNorm M N have hri : Function.Injective r := fun i j h => hr (congrArg Subtype.val h)m:Type u_1n:Type u_2inst✝³:Fintype minst✝²:Fintype ninst✝¹:DecidableEq minst✝:DecidableEq nrows:m → Ideal Ocols:n → Ideal Ohr:Function.Injective rowshc:Function.Injective colsM:ℝN:ℝhrows:∀ (i : m), Admissible (rows i) ∧ ↑(absNorm (rows i)) ≤ Mhcols:∀ (j : n), Admissible (cols j) ∧ ↑(absNorm (cols j)) ≤ Nr:m → ↥(idealRange M) := fun i => ⟨rows i, ⋯⟩c:n → ↥(idealRange N) := fun j => ⟨cols j, ⋯⟩hri:Function.Injective r⊢ ‖FiniteSieveOperator.operator fun i j => quadraticRow (cols j) (primaryGenerator (rows i))‖ ^ 2 ≤ sieveNorm M N have hci : Function.Injective c := fun i j h => hc (congrArg Subtype.val h)m:Type u_1n:Type u_2inst✝³:Fintype minst✝²:Fintype ninst✝¹:DecidableEq minst✝:DecidableEq nrows:m → Ideal Ocols:n → Ideal Ohr:Function.Injective rowshc:Function.Injective colsM:ℝN:ℝhrows:∀ (i : m), Admissible (rows i) ∧ ↑(absNorm (rows i)) ≤ Mhcols:∀ (j : n), Admissible (cols j) ∧ ↑(absNorm (cols j)) ≤ Nr:m → ↥(idealRange M) := fun i => ⟨rows i, ⋯⟩c:n → ↥(idealRange N) := fun j => ⟨cols j, ⋯⟩hri:Function.Injective rhci:Function.Injective c⊢ ‖FiniteSieveOperator.operator fun i j => quadraticRow (cols j) (primaryGenerator (rows i))‖ ^ 2 ≤ sieveNorm M N have h := FiniteSieveRestriction.submatrix_norm_le r hri c hci (matrix M N)m:Type u_1n:Type u_2inst✝³:Fintype minst✝²:Fintype ninst✝¹:DecidableEq minst✝:DecidableEq nrows:m → Ideal Ocols:n → Ideal Ohr:Function.Injective rowshc:Function.Injective colsM:ℝN:ℝhrows:∀ (i : m), Admissible (rows i) ∧ ↑(absNorm (rows i)) ≤ Mhcols:∀ (j : n), Admissible (cols j) ∧ ↑(absNorm (cols j)) ≤ Nr:m → ↥(idealRange M) := fun i => ⟨rows i, ⋯⟩c:n → ↥(idealRange N) := fun j => ⟨cols j, ⋯⟩hri:Function.Injective rhci:Function.Injective ch:‖FiniteSieveOperator.operator fun i j => matrix M N (r i) (c j)‖ ≤ ‖FiniteSieveOperator.operator (matrix M N)‖⊢ ‖FiniteSieveOperator.operator fun i j => quadraticRow (cols j) (primaryGenerator (rows i))‖ ^ 2 ≤ sieveNorm M N exact pow_le_pow_left₀ (norm_nonneg _) h 2All goals completed! 🐙theorem energy_bound (M N : ℝ) (a : idealRange N → ℂ) : (∑ I : idealRange M, ‖∑ J : idealRange N, quadraticRow J.val (primaryGenerator I.val) * a J‖ ^ 2) ≤ sieveNorm M N * ∑ J, ‖a J‖ ^ 2 := FiniteSieveOperator.energy_bound (matrix M N) aend CanonicalQuadraticSieveendend OAI