import OAI.NumberTheory.DirichletL.Mellin.CompactWeights import OAI.NumberTheory.DirichletL.QuadraticSieve.CoefficientCancellationnamespace 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 Classical SchwartzMapnamespace QuadraticInitialBoundopen ActualEisensteinCubic ConcreteTraceCRT EisensteinSchwartzPoissondef quadraticRow {α : Type*} (P : α → Ideal O) [∀ i, (P i).IsMaximal] (hg : ∀ i, lambda ∉ P i) (S : Finset α) (z : O) : ℂ := (finiteSquarefreeRow P hg S z) ^ 3theorem quadraticRow_eq_canonical {α : Type*} (P : α → Ideal O) [∀ i, (P i).IsMaximal] (hg : ∀ i, lambda ∉ P i) (S : Finset α) (z : O) : quadraticRow P hg S z = finiteSexticRow (fun i : S => P i.val) (fun i => hg i.val) (fun _ => 3) z := byα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αz:O⊢ quadraticRow P hg S z = finiteSexticRow (fun i => P ↑i) ⋯ (fun x => 3) z simp only [quadraticRow, finiteSexticRow, finiteSquarefreeRow, MulChar.pow_apply' _ (by decide : (3 : ℕ) ≠ 0), Finset.prod_pow]α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αz:O⊢ (∏ i ∈ S, (canonicalSextic (P i) ⋯) ((Ideal.Quotient.mk (P i)) z)) ^ 3 = (∏ x, (canonicalSextic (P ↑x) ⋯) ((Ideal.Quotient.mk (P ↑x)) z)) ^ 3 rw [Finset.prod_coe_sort (s := S) (f := fun i => canonicalSextic (P i) (hg i) (Ideal.Quotient.mk (P i) z))α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αz:O⊢ (∏ i ∈ S, (canonicalSextic (P i) ⋯) ((Ideal.Quotient.mk (P i)) z)) ^ 3 = (∏ i ∈ S, (canonicalSextic (P i) ⋯) ((Ideal.Quotient.mk (P i)) z)) ^ 3]All goals completed! 🐙theorem quadraticRow_star {α : Type*} (P : α → Ideal O) [∀ i, (P i).IsMaximal] (hg : ∀ i, lambda ∉ P i) (S : Finset α) (z : O) : star (quadraticRow P hg S z) = quadraticRow P hg S z := byα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αz:O⊢ star (quadraticRow P hg S z) = quadraticRow P hg S z rw [quadraticRow_eq_canonicalα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αz:O⊢ star (finiteSexticRow (fun i => P ↑i) ⋯ (fun x => 3) z) = finiteSexticRow (fun i => P ↑i) ⋯ (fun x => 3) z]α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αz:O⊢ star (finiteSexticRow (fun i => P ↑i) ⋯ (fun x => 3) z) = finiteSexticRow (fun i => P ↑i) ⋯ (fun x => 3) z simp only [finiteSexticRow, star_prod]α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αz:O⊢ ∏ x, star ((canonicalSextic (P ↑x) ⋯ ^ 3) ((Ideal.Quotient.mk (P ↑x)) z)) = ∏ x, (canonicalSextic (P ↑x) ⋯ ^ 3) ((Ideal.Quotient.mk (P ↑x)) z) apply Finset.prod_congr rflα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αz:O⊢ ∀ x ∈ Finset.univ, star ((canonicalSextic (P ↑x) ⋯ ^ 3) ((Ideal.Quotient.mk (P ↑x)) z)) = (canonicalSextic (P ↑x) ⋯ ^ 3) ((Ideal.Quotient.mk (P ↑x)) z) intro i _α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αz:Oi:↥Sa✝:i ∈ Finset.univ⊢ star ((canonicalSextic (P ↑i) ⋯ ^ 3) ((Ideal.Quotient.mk (P ↑i)) z)) = (canonicalSextic (P ↑i) ⋯ ^ 3) ((Ideal.Quotient.mk (P ↑i)) z) rw [canonicalSextic_pow_three_quadraticα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αz:Oi:↥Sa✝:i ∈ Finset.univ⊢ star (((quadraticChar (ActualEisensteinCubic.O ⧸ P ↑i)).ringHomComp (Int.castRingHom ℂ)) ((Ideal.Quotient.mk (P ↑i)) z)) = ((quadraticChar (ActualEisensteinCubic.O ⧸ P ↑i)).ringHomComp (Int.castRingHom ℂ)) ((Ideal.Quotient.mk (P ↑i)) z)]α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αz:Oi:↥Sa✝:i ∈ Finset.univ⊢ star (((quadraticChar (ActualEisensteinCubic.O ⧸ P ↑i)).ringHomComp (Int.castRingHom ℂ)) ((Ideal.Quotient.mk (P ↑i)) z)) = ((quadraticChar (ActualEisensteinCubic.O ⧸ P ↑i)).ringHomComp (Int.castRingHom ℂ)) ((Ideal.Quotient.mk (P ↑i)) z) simp only [MulChar.ringHomComp_apply, Int.coe_castRingHom, star_intCast]All goals completed! 🐙theorem quadraticRow_disjoint_union {α : Type*} [DecidableEq α] (P : α → Ideal O) [∀ i, (P i).IsMaximal] (hg : ∀ i, lambda ∉ P i) (S T : Finset α) (hd : Disjoint S T) (z : O) : quadraticRow P hg (S ∪ T) z = quadraticRow P hg S z * quadraticRow P hg T z := byα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αhd:Disjoint S Tz:O⊢ quadraticRow P hg (S ∪ T) z = quadraticRow P hg S z * quadraticRow P hg T z simp only [quadraticRow, finiteSquarefreeRow, Finset.prod_union hd, mul_pow]All goals completed! 🐙theorem quadraticRow_pair_kernel {α : Type*} [DecidableEq α] (P : α → Ideal O) [∀ i, (P i).IsMaximal] (hg : ∀ i, lambda ∉ P i) (S T : Finset α) (z : O) : star (quadraticRow P hg S z) * quadraticRow P hg T z = rowCoprimeMask P (S ∩ T) z * finiteSexticRow (activePrimes P S T) (fun i => hg i.val) (fun _ => 3) z := byα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αz:O⊢ star (quadraticRow P hg S z) * quadraticRow P hg T z = rowCoprimeMask P (S ∩ T) z * finiteSexticRow (activePrimes P S T) ⋯ (fun x => 3) z have h := congrArg (fun v : ℂ => v ^ 3) (finiteSquarefreeRow_pair_kernel P hg S T z)α:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αz:Oh:(star (finiteSquarefreeRow P hg S z) * finiteSquarefreeRow P hg T z) ^ 3 = (rowCoprimeMask P (S ∩ T) z * star (finiteSquarefreeRow P hg (S \ T) z) * finiteSquarefreeRow P hg (T \ S) z) ^ 3⊢ star (quadraticRow P hg S z) * quadraticRow P hg T z = rowCoprimeMask P (S ∩ T) z * finiteSexticRow (activePrimes P S T) ⋯ (fun x => 3) z have hm : rowCoprimeMask P (S ∩ T) z ^ 3 = rowCoprimeMask P (S ∩ T) z := by unfold rowCoprimeMaskα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αz:Oh:(star (finiteSquarefreeRow P hg S z) * finiteSquarefreeRow P hg T z) ^ 3 = (rowCoprimeMask P (S ∩ T) z * star (finiteSquarefreeRow P hg (S \ T) z) * finiteSquarefreeRow P hg (T \ S) z) ^ 3⊢ (if ∃ i ∈ S ∩ T, z ∈ P i then 0 else 1) ^ 3 = if ∃ i ∈ S ∩ T, z ∈ P i then 0 else 1 split_ifsposα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αz:Oh:(star (finiteSquarefreeRow P hg S z) * finiteSquarefreeRow P hg T z) ^ 3 = (rowCoprimeMask P (S ∩ T) z * star (finiteSquarefreeRow P hg (S \ T) z) * finiteSquarefreeRow P hg (T \ S) z) ^ 3h✝:∃ i ∈ S ∩ T, z ∈ P i⊢ 0 ^ 3 = 0negα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αz:Oh:(star (finiteSquarefreeRow P hg S z) * finiteSquarefreeRow P hg T z) ^ 3 = (rowCoprimeMask P (S ∩ T) z * star (finiteSquarefreeRow P hg (S \ T) z) * finiteSquarefreeRow P hg (T \ S) z) ^ 3h✝:¬∃ i ∈ S ∩ T, z ∈ P i⊢ 1 ^ 3 = 1 <;>posα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αz:Oh:(star (finiteSquarefreeRow P hg S z) * finiteSquarefreeRow P hg T z) ^ 3 = (rowCoprimeMask P (S ∩ T) z * star (finiteSquarefreeRow P hg (S \ T) z) * finiteSquarefreeRow P hg (T \ S) z) ^ 3h✝:∃ i ∈ S ∩ T, z ∈ P i⊢ 0 ^ 3 = 0negα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αz:Oh:(star (finiteSquarefreeRow P hg S z) * finiteSquarefreeRow P hg T z) ^ 3 = (rowCoprimeMask P (S ∩ T) z * star (finiteSquarefreeRow P hg (S \ T) z) * finiteSquarefreeRow P hg (T \ S) z) ^ 3h✝:¬∃ i ∈ S ∩ T, z ∈ P i⊢ 1 ^ 3 = 1 simpα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αz:Oh:(star (finiteSquarefreeRow P hg S z) * finiteSquarefreeRow P hg T z) ^ 3 = (rowCoprimeMask P (S ∩ T) z * star (finiteSquarefreeRow P hg (S \ T) z) * finiteSquarefreeRow P hg (T \ S) z) ^ 3hm:rowCoprimeMask P (S ∩ T) z ^ 3 = rowCoprimeMask P (S ∩ T) z⊢ star (quadraticRow P hg S z) * quadraticRow P hg T z = rowCoprimeMask P (S ∩ T) z * finiteSexticRow (activePrimes P S T) ⋯ (fun x => 3) z have hd : Disjoint (S \ T) (T \ S) := by apply Finset.disjoint_left.mprα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αz:Oh:(star (finiteSquarefreeRow P hg S z) * finiteSquarefreeRow P hg T z) ^ 3 = (rowCoprimeMask P (S ∩ T) z * star (finiteSquarefreeRow P hg (S \ T) z) * finiteSquarefreeRow P hg (T \ S) z) ^ 3hm:rowCoprimeMask P (S ∩ T) z ^ 3 = rowCoprimeMask P (S ∩ T) z⊢ ∀ ⦃a : α⦄, a ∈ S \ T → a ∉ T \ S intro i hi hjα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αz:Oh:(star (finiteSquarefreeRow P hg S z) * finiteSquarefreeRow P hg T z) ^ 3 = (rowCoprimeMask P (S ∩ T) z * star (finiteSquarefreeRow P hg (S \ T) z) * finiteSquarefreeRow P hg (T \ S) z) ^ 3hm:rowCoprimeMask P (S ∩ T) z ^ 3 = rowCoprimeMask P (S ∩ T) zi:αhi:i ∈ S \ Thj:i ∈ T \ S⊢ False exact (Finset.mem_sdiff.mp hi).2 (Finset.mem_sdiff.mp hj).1α:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αz:Oh:(star (finiteSquarefreeRow P hg S z) * finiteSquarefreeRow P hg T z) ^ 3 = (rowCoprimeMask P (S ∩ T) z * star (finiteSquarefreeRow P hg (S \ T) z) * finiteSquarefreeRow P hg (T \ S) z) ^ 3hm:rowCoprimeMask P (S ∩ T) z ^ 3 = rowCoprimeMask P (S ∩ T) zhd:Disjoint (S \ T) (T \ S)⊢ star (quadraticRow P hg S z) * quadraticRow P hg T z = rowCoprimeMask P (S ∩ T) z * finiteSexticRow (activePrimes P S T) ⋯ (fun x => 3) z simp only [mul_pow, ← star_pow, hm] at hα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αz:Ohm:rowCoprimeMask P (S ∩ T) z ^ 3 = rowCoprimeMask P (S ∩ T) zhd:Disjoint (S \ T) (T \ S)h:star (finiteSquarefreeRow P hg S z ^ 3) * finiteSquarefreeRow P hg T z ^ 3 = rowCoprimeMask P (S ∩ T) z * star (finiteSquarefreeRow P hg (S \ T) z ^ 3) * finiteSquarefreeRow P hg (T \ S) z ^ 3⊢ star (quadraticRow P hg S z) * quadraticRow P hg T z = rowCoprimeMask P (S ∩ T) z * finiteSexticRow (activePrimes P S T) ⋯ (fun x => 3) z change star (quadraticRow P hg S z) * quadraticRow P hg T z = rowCoprimeMask P (S ∩ T) z * star (quadraticRow P hg (S \ T) z) * quadraticRow P hg (T \ S) z at hα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αz:Ohm:rowCoprimeMask P (S ∩ T) z ^ 3 = rowCoprimeMask P (S ∩ T) zhd:Disjoint (S \ T) (T \ S)h:star (quadraticRow P hg S z) * quadraticRow P hg T z = rowCoprimeMask P (S ∩ T) z * star (quadraticRow P hg (S \ T) z) * quadraticRow P hg (T \ S) z⊢ star (quadraticRow P hg S z) * quadraticRow P hg T z = rowCoprimeMask P (S ∩ T) z * finiteSexticRow (activePrimes P S T) ⋯ (fun x => 3) z simp only [quadraticRow_star] at h ⊢α:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αz:Ohm:rowCoprimeMask P (S ∩ T) z ^ 3 = rowCoprimeMask P (S ∩ T) zhd:Disjoint (S \ T) (T \ S)h:quadraticRow P hg S z * quadraticRow P hg T z = rowCoprimeMask P (S ∩ T) z * quadraticRow P hg (S \ T) z * quadraticRow P hg (T \ S) z⊢ quadraticRow P hg S z * quadraticRow P hg T z = rowCoprimeMask P (S ∩ T) z * finiteSexticRow (activePrimes P S T) ⋯ (fun x => 3) z rw [← quadraticRow_eq_canonicalα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αz:Ohm:rowCoprimeMask P (S ∩ T) z ^ 3 = rowCoprimeMask P (S ∩ T) zhd:Disjoint (S \ T) (T \ S)h:quadraticRow P hg S z * quadraticRow P hg T z = rowCoprimeMask P (S ∩ T) z * quadraticRow P hg (S \ T) z * quadraticRow P hg (T \ S) z⊢ quadraticRow P hg S z * quadraticRow P hg T z = rowCoprimeMask P (S ∩ T) z * quadraticRow P hg (activeSupport S T) z]α:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αz:Ohm:rowCoprimeMask P (S ∩ T) z ^ 3 = rowCoprimeMask P (S ∩ T) zhd:Disjoint (S \ T) (T \ S)h:quadraticRow P hg S z * quadraticRow P hg T z = rowCoprimeMask P (S ∩ T) z * quadraticRow P hg (S \ T) z * quadraticRow P hg (T \ S) z⊢ quadraticRow P hg S z * quadraticRow P hg T z = rowCoprimeMask P (S ∩ T) z * quadraticRow P hg (activeSupport S T) z change _ = rowCoprimeMask P (S ∩ T) z * quadraticRow P hg ((S \ T) ∪ (T \ S)) zα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αz:Ohm:rowCoprimeMask P (S ∩ T) z ^ 3 = rowCoprimeMask P (S ∩ T) zhd:Disjoint (S \ T) (T \ S)h:quadraticRow P hg S z * quadraticRow P hg T z = rowCoprimeMask P (S ∩ T) z * quadraticRow P hg (S \ T) z * quadraticRow P hg (T \ S) z⊢ quadraticRow P hg S z * quadraticRow P hg T z = rowCoprimeMask P (S ∩ T) z * quadraticRow P hg (S \ T ∪ T \ S) z rw [quadraticRow_disjoint_union P hg _ _ hdα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αz:Ohm:rowCoprimeMask P (S ∩ T) z ^ 3 = rowCoprimeMask P (S ∩ T) zhd:Disjoint (S \ T) (T \ S)h:quadraticRow P hg S z * quadraticRow P hg T z = rowCoprimeMask P (S ∩ T) z * quadraticRow P hg (S \ T) z * quadraticRow P hg (T \ S) z⊢ quadraticRow P hg S z * quadraticRow P hg T z = rowCoprimeMask P (S ∩ T) z * (quadraticRow P hg (S \ T) z * quadraticRow P hg (T \ S) z)]α:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αz:Ohm:rowCoprimeMask P (S ∩ T) z ^ 3 = rowCoprimeMask P (S ∩ T) zhd:Disjoint (S \ T) (T \ S)h:quadraticRow P hg S z * quadraticRow P hg T z = rowCoprimeMask P (S ∩ T) z * quadraticRow P hg (S \ T) z * quadraticRow P hg (T \ S) z⊢ quadraticRow P hg S z * quadraticRow P hg T z = rowCoprimeMask P (S ∩ T) z * (quadraticRow P hg (S \ T) z * quadraticRow P hg (T \ S) z) simpa only [mul_assoc] using hAll goals completed! 🐙theorem quadratic_gram_offdiagonal {α : Type*} [DecidableEq α] (P : α → Ideal O) [∀ i, (P i).IsMaximal] (hinj : Function.Injective P) (hg : ∀ i, lambda ∉ P i) (hc : ∀ i, ringChar (O ⧸ P i) ≠ 2) (S T : Finset α) (hne : S ≠ T) (W : 𝓢(ℝ, ℂ)) (M : ℝ) (hM : 0 < M) : ‖∑' z : O, (star (quadraticRow P hg S z) * quadraticRow P hg T z) * W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ (2 : ℝ) ^ (S ∩ T).card * (pvControl W * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖) := byα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2S:Finset αT:Finset αhne:S ≠ TW:𝓢(ℝ, ℂ)M:ℝhM:0 < M⊢ ‖∑' (z : O), star (quadraticRow P hg S z) * quadraticRow P hg T z * W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ 2 ^ #(S ∩ T) * (pvControl W * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖) let : Nonempty (activeSupport S T) := (activeSupport_nonempty_of_ne hne).to_subtypeα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2S:Finset αT:Finset αhne:S ≠ TW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mthis:Nonempty ↥(activeSupport S T) := ···⊢ ‖∑' (z : O), star (quadraticRow P hg S z) * quadraticRow P hg T z * W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ 2 ^ #(S ∩ T) * (pvControl W * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖) simp_rw [α:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2S:Finset αT:Finset αhne:S ≠ TW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mthis:Nonempty ↥(activeSupport S T) := ···⊢ ‖∑' (z : O), star (quadraticRow P hg S z) * quadraticRow P hg T z * W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ 2 ^ #(S ∩ T) * (pvControl W * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖)quadraticRow_pair_kernel P hg S Tα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2S:Finset αT:Finset αhne:S ≠ TW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mthis:Nonempty ↥(activeSupport S T) := ···⊢ ‖∑' (z : O), rowCoprimeMask P (S ∩ T) z * finiteSexticRow (activePrimes P S T) ⋯ (fun x => 3) z * W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ 2 ^ #(S ∩ T) * (pvControl W * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖)] exact canonical_masked_polya_vinogradov P hinj (S ∩ T) (activePrimes P S T) (activePrimes_pairwise_isCoprime P hinj S T) (fun i => hg i.val) (fun i => hc i.val) (fun _ => 3) (fun _ => byα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2S:Finset αT:Finset αhne:S ≠ TW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mthis:Nonempty ↥(activeSupport S T) := ···x✝:↥(activeSupport S T)⊢ 3 ≠ 0 decideAll goals completed! 🐙) (fun _ => byα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2S:Finset αT:Finset αhne:S ≠ TW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mthis:Nonempty ↥(activeSupport S T) := ···x✝:↥(activeSupport S T)⊢ 3 < 6 decideAll goals completed! 🐙) W M hMdef supportNorm {α : Type*} (P : α → Ideal O) (S : Finset α) : ℝ := Ideal.absNorm (∏ i ∈ S, P i)theorem supportNorm_nonneg {α : Type*} (P : α → Ideal O) (S : Finset α) : 0 ≤ supportNorm P S := Nat.cast_nonneg _theorem two_pow_card_le_supportNorm {α : Type*} (P : α → Ideal O) [∀ i, (P i).IsMaximal] (S : Finset α) : (2 : ℝ) ^ S.card ≤ supportNorm P S := byα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset α⊢ 2 ^ #S ≤ supportNorm P S have hnorm (i : α) : (2 : ℝ) ≤ Ideal.absNorm (P i) := by have h0 : Ideal.absNorm (P i) ≠ 0 := byα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset α⊢ 2 ^ #S ≤ supportNorm P S rw [ne_eq,α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αi:α⊢ ¬absNorm (P i) = 0 Ideal.absNorm_eq_zero_iffα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αi:α⊢ ¬P i = ⊥]α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αi:α⊢ ¬P i = ⊥ exact NeZero.ne (P i)α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αi:αh0:absNorm (P i) ≠ 0⊢ 2 ≤ ↑(absNorm (P i)) have h1 : Ideal.absNorm (P i) ≠ 1 := byα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset α⊢ 2 ^ #S ≤ supportNorm P S rw [ne_eq,α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αi:αh0:absNorm (P i) ≠ 0⊢ ¬absNorm (P i) = 1 Ideal.absNorm_eq_one_iffα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αi:αh0:absNorm (P i) ≠ 0⊢ ¬P i = ⊤]α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αi:αh0:absNorm (P i) ≠ 0⊢ ¬P i = ⊤ exact (inferInstance : (P i).IsMaximal).ne_topα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αi:αh0:absNorm (P i) ≠ 0h1:absNorm (P i) ≠ 1⊢ 2 ≤ ↑(absNorm (P i)) exact_mod_cast (show 2 ≤ Ideal.absNorm (P i) by omegaAll goals completed! 🐙)α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αhnorm:∀ (i : α), 2 ≤ ↑(absNorm (P i))⊢ 2 ^ #S ≤ supportNorm P S change (2 : ℝ) ^ S.card ≤ (Ideal.absNorm (∏ i ∈ S, P i) : ℝ)α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αhnorm:∀ (i : α), 2 ≤ ↑(absNorm (P i))⊢ 2 ^ #S ≤ ↑(absNorm (∏ i ∈ S, P i)) rw [map_prod,α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αhnorm:∀ (i : α), 2 ≤ ↑(absNorm (P i))⊢ 2 ^ #S ≤ ↑(∏ x ∈ S, absNorm (P x)) Nat.cast_prodα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αhnorm:∀ (i : α), 2 ≤ ↑(absNorm (P i))⊢ 2 ^ #S ≤ ∏ i ∈ S, ↑(absNorm (P i))]α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αhnorm:∀ (i : α), 2 ≤ ↑(absNorm (P i))⊢ 2 ^ #S ≤ ∏ i ∈ S, ↑(absNorm (P i)) calc _ = ∏ _i ∈ S, (2 : ℝ) := byα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αhnorm:∀ (i : α), 2 ≤ ↑(absNorm (P i))⊢ 2 ^ #S = ∏ _i ∈ S, 2 simpAll goals completed! 🐙 _ ≤ _ := Finset.prod_le_prod₀ (fun _ _ => byα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αhnorm:∀ (i : α), 2 ≤ ↑(absNorm (P i))x✝¹:αx✝:x✝¹ ∈ S⊢ 0 ≤ 2 norm_numAll goals completed! 🐙) (fun i _ => hnorm i)theorem divisor_conductor_bound {α : Type*} [DecidableEq α] (P : α → Ideal O) [∀ i, (P i).IsMaximal] (S T : Finset α) (N : ℝ) (hN : 0 ≤ N) (hS : supportNorm P S ≤ N) (hT : supportNorm P T ≤ N) : (2 : ℝ) ^ (S ∩ T).card * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ≤ N := byα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αT:Finset αN:ℝhN:0 ≤ NhS:supportNorm P S ≤ NhT:supportNorm P T ≤ N⊢ 2 ^ #(S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ≤ N have hid := congrArg (fun I : Ideal O => (Ideal.absNorm I : ℝ)) (activeSupport_product_mul_common_sq P S T)α:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αT:Finset αN:ℝhN:0 ≤ NhS:supportNorm P S ≤ NhT:supportNorm P T ≤ Nhid:↑(absNorm ((∏ i ∈ activeSupport S T, P i) * (∏ i ∈ S ∩ T, P i) ^ 2)) = ↑(absNorm ((∏ i ∈ S, P i) * ∏ i ∈ T, P i))⊢ 2 ^ #(S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ≤ N simp only [map_mul, map_pow, Nat.cast_mul, Nat.cast_pow] at hidα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αT:Finset αN:ℝhN:0 ≤ NhS:supportNorm P S ≤ NhT:supportNorm P T ≤ Nhid:↑(absNorm (∏ i ∈ activeSupport S T, P i)) * ↑(absNorm (∏ i ∈ S ∩ T, P i)) ^ 2 = ↑(absNorm (∏ i ∈ S, P i)) * ↑(absNorm (∏ i ∈ T, P i))⊢ 2 ^ #(S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ≤ N rw [← activePrimeModulus_norm_sqα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αT:Finset αN:ℝhN:0 ≤ NhS:supportNorm P S ≤ NhT:supportNorm P T ≤ Nhid:‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ^ 2 * ↑(absNorm (∏ i ∈ S ∩ T, P i)) ^ 2 = ↑(absNorm (∏ i ∈ S, P i)) * ↑(absNorm (∏ i ∈ T, P i))⊢ 2 ^ #(S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ≤ N] at hidα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αT:Finset αN:ℝhN:0 ≤ NhS:supportNorm P S ≤ NhT:supportNorm P T ≤ Nhid:‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ^ 2 * ↑(absNorm (∏ i ∈ S ∩ T, P i)) ^ 2 = ↑(absNorm (∏ i ∈ S, P i)) * ↑(absNorm (∏ i ∈ T, P i))⊢ 2 ^ #(S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ≤ N change ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ^ 2 * (supportNorm P (S ∩ T)) ^ 2 = supportNorm P S * supportNorm P T at hidα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αT:Finset αN:ℝhN:0 ≤ NhS:supportNorm P S ≤ NhT:supportNorm P T ≤ Nhid:‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ^ 2 * supportNorm P (S ∩ T) ^ 2 = supportNorm P S * supportNorm P T⊢ 2 ^ #(S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ≤ N have hpow := two_pow_card_le_supportNorm P (S ∩ T)α:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αT:Finset αN:ℝhN:0 ≤ NhS:supportNorm P S ≤ NhT:supportNorm P T ≤ Nhid:‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ^ 2 * supportNorm P (S ∩ T) ^ 2 = supportNorm P S * supportNorm P Thpow:2 ^ #(S ∩ T) ≤ supportNorm P (S ∩ T)⊢ 2 ^ #(S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ≤ N have hmul := mul_le_mul_of_nonneg_right hpow (norm_nonneg (eisEmbedding (finitePrimeModulus (activePrimes P S T))))α:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αT:Finset αN:ℝhN:0 ≤ NhS:supportNorm P S ≤ NhT:supportNorm P T ≤ Nhid:‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ^ 2 * supportNorm P (S ∩ T) ^ 2 = supportNorm P S * supportNorm P Thpow:2 ^ #(S ∩ T) ≤ supportNorm P (S ∩ T)hmul:2 ^ #(S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ≤ supportNorm P (S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖⊢ 2 ^ #(S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ≤ N have hsq := pow_le_pow_left₀ (byα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αT:Finset αN:ℝhN:0 ≤ NhS:supportNorm P S ≤ NhT:supportNorm P T ≤ Nhid:‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ^ 2 * supportNorm P (S ∩ T) ^ 2 = supportNorm P S * supportNorm P Thpow:2 ^ #(S ∩ T) ≤ supportNorm P (S ∩ T)hmul:2 ^ #(S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ≤ supportNorm P (S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖⊢ 0 ≤ 2 ^ #(S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ positivityAll goals completed! 🐙 : 0 ≤ (2 : ℝ) ^ (S ∩ T).card * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖) hmul 2α:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αT:Finset αN:ℝhN:0 ≤ NhS:supportNorm P S ≤ NhT:supportNorm P T ≤ Nhid:‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ^ 2 * supportNorm P (S ∩ T) ^ 2 = supportNorm P S * supportNorm P Thpow:2 ^ #(S ∩ T) ≤ supportNorm P (S ∩ T)hmul:2 ^ #(S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ≤ supportNorm P (S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖hsq:(2 ^ #(S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖) ^ 2 ≤ (supportNorm P (S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖) ^ 2⊢ 2 ^ #(S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ≤ N have hprod := mul_le_mul hS hT (supportNorm_nonneg P T) hNα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalS:Finset αT:Finset αN:ℝhN:0 ≤ NhS:supportNorm P S ≤ NhT:supportNorm P T ≤ Nhid:‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ^ 2 * supportNorm P (S ∩ T) ^ 2 = supportNorm P S * supportNorm P Thpow:2 ^ #(S ∩ T) ≤ supportNorm P (S ∩ T)hmul:2 ^ #(S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ≤ supportNorm P (S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖hsq:(2 ^ #(S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖) ^ 2 ≤ (supportNorm P (S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖) ^ 2hprod:supportNorm P S * supportNorm P T ≤ N * N⊢ 2 ^ #(S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖ ≤ N nlinarithAll goals completed! 🐙theorem quadratic_gram_offdiagonal_of_norm_le {α : Type*} [DecidableEq α] (P : α → Ideal O) [∀ i, (P i).IsMaximal] (hinj : Function.Injective P) (hg : ∀ i, lambda ∉ P i) (hc : ∀ i, ringChar (O ⧸ P i) ≠ 2) (S T : Finset α) (hne : S ≠ T) (N : ℝ) (hN : 0 ≤ N) (hS : supportNorm P S ≤ N) (hT : supportNorm P T ≤ N) (W : 𝓢(ℝ, ℂ)) (M : ℝ) (hM : 0 < M) : ‖∑' z : O, (star (quadraticRow P hg S z) * quadraticRow P hg T z) * W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ pvControl W * N := byα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2S:Finset αT:Finset αhne:S ≠ TN:ℝhN:0 ≤ NhS:supportNorm P S ≤ NhT:supportNorm P T ≤ NW:𝓢(ℝ, ℂ)M:ℝhM:0 < M⊢ ‖∑' (z : O), star (quadraticRow P hg S z) * quadraticRow P hg T z * W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ pvControl W * N apply (quadratic_gram_offdiagonal P hinj hg hc S T hne W M hM).transα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2S:Finset αT:Finset αhne:S ≠ TN:ℝhN:0 ≤ NhS:supportNorm P S ≤ NhT:supportNorm P T ≤ NW:𝓢(ℝ, ℂ)M:ℝhM:0 < M⊢ 2 ^ #(S ∩ T) * (pvControl W * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖) ≤ pvControl W * N rw [mul_left_commα:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2S:Finset αT:Finset αhne:S ≠ TN:ℝhN:0 ≤ NhS:supportNorm P S ≤ NhT:supportNorm P T ≤ NW:𝓢(ℝ, ℂ)M:ℝhM:0 < M⊢ pvControl W * (2 ^ #(S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖) ≤ pvControl W * N]α:Type u_1inst✝¹:DecidableEq αP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2S:Finset αT:Finset αhne:S ≠ TN:ℝhN:0 ≤ NhS:supportNorm P S ≤ NhT:supportNorm P T ≤ NW:𝓢(ℝ, ℂ)M:ℝhM:0 < M⊢ pvControl W * (2 ^ #(S ∩ T) * ‖eisEmbedding (finitePrimeModulus (activePrimes P S T))‖) ≤ pvControl W * N exact mul_le_mul_of_nonneg_left (divisor_conductor_bound P S T N hN hS hT) (pvControl_nonneg W)All goals completed! 🐙def diagonalControl (W : 𝓢(ℝ, ℂ)) : ℝ := 16 * (1 + Real.pi) ^ 2 * (Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ) Wtheorem diagonalControl_nonneg (W : 𝓢(ℝ, ℂ)) : 0 ≤ diagonalControl W := byW:𝓢(ℝ, ℂ)⊢ 0 ≤ diagonalControl W unfold diagonalControlW:𝓢(ℝ, ℂ)⊢ 0 ≤ 16 * (1 + π) ^ 2 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) W positivityAll goals completed! 🐙theorem radial_weight_lattice_bound (W : 𝓢(ℝ, ℂ)) (M : ℝ) (hM : 1 ≤ M) : (∑' z : O, ‖W (‖eisEmbedding z‖ ^ 2 / M)‖) ≤ diagonalControl W * M := byW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ M⊢ ∑' (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W * M have hMp : 0 < M := by linarithW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < M⊢ ∑' (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W * M let t := M⁻¹W:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹⊢ ∑' (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W * M have ht : 0 < t := inv_pos.mpr hMpW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < t⊢ ∑' (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W * M have ht1 : t ≤ 1 := (inv_le_one₀ hMp).mpr hMW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1⊢ ∑' (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W * M let B := 4 * (Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ) WW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1B:ℝ := 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) W⊢ ∑' (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W * M have hB : 0 ≤ B := by dsimp [B]W:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1B:ℝ := 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) W⊢ 0 ≤ 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) W; positivityW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1B:ℝ := 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) WhB:0 ≤ B⊢ ∑' (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W * M have hp (z : O) : ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B * ((1 + t * ‖eisEmbedding z‖ ^ 2) ^ 2)⁻¹ := by have hq : 0 ≤ ‖eisEmbedding z‖ ^ 2 / M := div_nonneg (sq_nonneg _) hMp.leW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1B:ℝ := 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) WhB:0 ≤ Bz:Ohq:0 ≤ ‖eisEmbedding z‖ ^ 2 / M⊢ ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B * ((1 + t * ‖eisEmbedding z‖ ^ 2) ^ 2)⁻¹ have hh := SchwartzMap.one_add_le_sup_seminorm_apply (𝕜 := ℝ) (m := (2, 0)) (k := 2) (n := 0) le_rfl le_rfl W (‖eisEmbedding z‖ ^ 2 / M)W:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1B:ℝ := 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) WhB:0 ≤ Bz:Ohq:0 ≤ ‖eisEmbedding z‖ ^ 2 / Mhh:(1 + ‖‖eisEmbedding z‖ ^ 2 / M‖) ^ 2 * ‖iteratedFDeriv ℝ 0 (⇑W) (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ 2 ^ (2, 0).1 * ((Finset.Iic (2, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) W⊢ ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B * ((1 + t * ‖eisEmbedding z‖ ^ 2) ^ 2)⁻¹ simp only [norm_iteratedFDeriv_zero, Real.norm_of_nonneg hq] at hhW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1B:ℝ := 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) WhB:0 ≤ Bz:Ohq:0 ≤ ‖eisEmbedding z‖ ^ 2 / Mhh:(1 + ‖eisEmbedding z‖ ^ 2 / M) ^ 2 * ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ 2 ^ 2 * ((Finset.Iic (2, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) W⊢ ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B * ((1 + t * ‖eisEmbedding z‖ ^ 2) ^ 2)⁻¹ norm_num only [show (2 : ℝ) ^ 2 = 4 by norm_numAll goals completed! 🐙] at hh change (1 + ‖eisEmbedding z‖ ^ 2 / M) ^ 2 * ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B at hhW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1B:ℝ := 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) WhB:0 ≤ Bz:Ohq:0 ≤ ‖eisEmbedding z‖ ^ 2 / Mhh:(1 + ‖eisEmbedding z‖ ^ 2 / M) ^ 2 * ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B⊢ ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B * ((1 + t * ‖eisEmbedding z‖ ^ 2) ^ 2)⁻¹ rw [← div_eq_mul_invW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1B:ℝ := 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) WhB:0 ≤ Bz:Ohq:0 ≤ ‖eisEmbedding z‖ ^ 2 / Mhh:(1 + ‖eisEmbedding z‖ ^ 2 / M) ^ 2 * ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B⊢ ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B / (1 + t * ‖eisEmbedding z‖ ^ 2) ^ 2]W:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1B:ℝ := 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) WhB:0 ≤ Bz:Ohq:0 ≤ ‖eisEmbedding z‖ ^ 2 / Mhh:(1 + ‖eisEmbedding z‖ ^ 2 / M) ^ 2 * ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B⊢ ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B / (1 + t * ‖eisEmbedding z‖ ^ 2) ^ 2 apply (le_div_iff₀ (byW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1B:ℝ := 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) WhB:0 ≤ Bz:Ohq:0 ≤ ‖eisEmbedding z‖ ^ 2 / Mhh:(1 + ‖eisEmbedding z‖ ^ 2 / M) ^ 2 * ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B⊢ 0 < (1 + t * ‖eisEmbedding z‖ ^ 2) ^ 2 positivityAll goals completed! 🐙)).mpr simpa only [t, div_eq_mul_inv, mul_comm, mul_left_comm, mul_assoc] using hhW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1B:ℝ := 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) WhB:0 ≤ Bhp:∀ (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B * ((1 + t * ‖eisEmbedding z‖ ^ 2) ^ 2)⁻¹⊢ ∑' (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W * M have hs : Summable (fun z : O => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖) := by simpa only [scaledRadialTest_apply] using actual_eisenstein_summable_norm (scaledRadialTest W M hMp)W:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1B:ℝ := 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) WhB:0 ≤ Bhp:∀ (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B * ((1 + t * ‖eisEmbedding z‖ ^ 2) ^ 2)⁻¹hs:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖⊢ ∑' (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W * M have hb := hs.tsum_le_tsum hp ((scaled_eisenstein_cauchy_summable t ht).mul_left B)W:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1B:ℝ := 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) WhB:0 ≤ Bhp:∀ (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B * ((1 + t * ‖eisEmbedding z‖ ^ 2) ^ 2)⁻¹hs:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖hb:∑' (i : O), ‖W (‖eisEmbedding i‖ ^ 2 / M)‖ ≤ ∑' (i : O), B * ((1 + t * ‖eisEmbedding i‖ ^ 2) ^ 2)⁻¹⊢ ∑' (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W * M rw [tsum_mul_leftW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1B:ℝ := 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) WhB:0 ≤ Bhp:∀ (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B * ((1 + t * ‖eisEmbedding z‖ ^ 2) ^ 2)⁻¹hs:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖hb:∑' (i : O), ‖W (‖eisEmbedding i‖ ^ 2 / M)‖ ≤ B * ∑' (x : O), ((1 + t * ‖eisEmbedding x‖ ^ 2) ^ 2)⁻¹⊢ ∑' (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W * M] at hbW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1B:ℝ := 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) WhB:0 ≤ Bhp:∀ (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B * ((1 + t * ‖eisEmbedding z‖ ^ 2) ^ 2)⁻¹hs:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖hb:∑' (i : O), ‖W (‖eisEmbedding i‖ ^ 2 / M)‖ ≤ B * ∑' (x : O), ((1 + t * ‖eisEmbedding x‖ ^ 2) ^ 2)⁻¹⊢ ∑' (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W * M have hsmall := scaled_eisenstein_cauchy_small t ht ht1W:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1B:ℝ := 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) WhB:0 ≤ Bhp:∀ (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B * ((1 + t * ‖eisEmbedding z‖ ^ 2) ^ 2)⁻¹hs:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖hb:∑' (i : O), ‖W (‖eisEmbedding i‖ ^ 2 / M)‖ ≤ B * ∑' (x : O), ((1 + t * ‖eisEmbedding x‖ ^ 2) ^ 2)⁻¹hsmall:t * ∑' (h : EisensteinSchwartzPoisson.O), ((1 + t * ‖eisEmbedding h‖ ^ 2) ^ 2)⁻¹ ≤ 4 * (1 + π) ^ 2⊢ ∑' (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W * M have hmul := mul_le_mul_of_nonneg_left hb ht.leW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1B:ℝ := 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) WhB:0 ≤ Bhp:∀ (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B * ((1 + t * ‖eisEmbedding z‖ ^ 2) ^ 2)⁻¹hs:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖hb:∑' (i : O), ‖W (‖eisEmbedding i‖ ^ 2 / M)‖ ≤ B * ∑' (x : O), ((1 + t * ‖eisEmbedding x‖ ^ 2) ^ 2)⁻¹hsmall:t * ∑' (h : EisensteinSchwartzPoisson.O), ((1 + t * ‖eisEmbedding h‖ ^ 2) ^ 2)⁻¹ ≤ 4 * (1 + π) ^ 2hmul:t * ∑' (i : O), ‖W (‖eisEmbedding i‖ ^ 2 / M)‖ ≤ t * (B * ∑' (x : O), ((1 + t * ‖eisEmbedding x‖ ^ 2) ^ 2)⁻¹)⊢ ∑' (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W * M have hcontrol := mul_le_mul_of_nonneg_left hsmall hBW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1B:ℝ := 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) WhB:0 ≤ Bhp:∀ (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B * ((1 + t * ‖eisEmbedding z‖ ^ 2) ^ 2)⁻¹hs:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖hb:∑' (i : O), ‖W (‖eisEmbedding i‖ ^ 2 / M)‖ ≤ B * ∑' (x : O), ((1 + t * ‖eisEmbedding x‖ ^ 2) ^ 2)⁻¹hsmall:t * ∑' (h : EisensteinSchwartzPoisson.O), ((1 + t * ‖eisEmbedding h‖ ^ 2) ^ 2)⁻¹ ≤ 4 * (1 + π) ^ 2hmul:t * ∑' (i : O), ‖W (‖eisEmbedding i‖ ^ 2 / M)‖ ≤ t * (B * ∑' (x : O), ((1 + t * ‖eisEmbedding x‖ ^ 2) ^ 2)⁻¹)hcontrol:B * (t * ∑' (h : EisensteinSchwartzPoisson.O), ((1 + t * ‖eisEmbedding h‖ ^ 2) ^ 2)⁻¹) ≤ B * (4 * (1 + π) ^ 2)⊢ ∑' (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W * M have hfinal : t * (∑' z : O, ‖W (‖eisEmbedding z‖ ^ 2 / M)‖) ≤ diagonalControl W := by unfold diagonalControlW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1B:ℝ := 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) WhB:0 ≤ Bhp:∀ (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B * ((1 + t * ‖eisEmbedding z‖ ^ 2) ^ 2)⁻¹hs:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖hb:∑' (i : O), ‖W (‖eisEmbedding i‖ ^ 2 / M)‖ ≤ B * ∑' (x : O), ((1 + t * ‖eisEmbedding x‖ ^ 2) ^ 2)⁻¹hsmall:t * ∑' (h : EisensteinSchwartzPoisson.O), ((1 + t * ‖eisEmbedding h‖ ^ 2) ^ 2)⁻¹ ≤ 4 * (1 + π) ^ 2hmul:t * ∑' (i : O), ‖W (‖eisEmbedding i‖ ^ 2 / M)‖ ≤ t * (B * ∑' (x : O), ((1 + t * ‖eisEmbedding x‖ ^ 2) ^ 2)⁻¹)hcontrol:B * (t * ∑' (h : EisensteinSchwartzPoisson.O), ((1 + t * ‖eisEmbedding h‖ ^ 2) ^ 2)⁻¹) ≤ B * (4 * (1 + π) ^ 2)⊢ t * ∑' (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ 16 * (1 + π) ^ 2 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) W dsimp [B] at hcontrol hmulW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1B:ℝ := 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) WhB:0 ≤ Bhp:∀ (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B * ((1 + t * ‖eisEmbedding z‖ ^ 2) ^ 2)⁻¹hs:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖hb:∑' (i : O), ‖W (‖eisEmbedding i‖ ^ 2 / M)‖ ≤ B * ∑' (x : O), ((1 + t * ‖eisEmbedding x‖ ^ 2) ^ 2)⁻¹hsmall:t * ∑' (h : EisensteinSchwartzPoisson.O), ((1 + t * ‖eisEmbedding h‖ ^ 2) ^ 2)⁻¹ ≤ 4 * (1 + π) ^ 2hmul:t * ∑' (i : O), ‖W (‖eisEmbedding i‖ ^ 2 / M)‖ ≤ t * (4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) W * ∑' (x : O), ((1 + t * ‖eisEmbedding x‖ ^ 2) ^ 2)⁻¹)hcontrol:4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) W * (t * ∑' (h : EisensteinSchwartzPoisson.O), ((1 + t * ‖eisEmbedding h‖ ^ 2) ^ 2)⁻¹) ≤ 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) W * (4 * (1 + π) ^ 2)⊢ t * ∑' (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ 16 * (1 + π) ^ 2 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) W nlinarithW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1B:ℝ := 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) WhB:0 ≤ Bhp:∀ (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B * ((1 + t * ‖eisEmbedding z‖ ^ 2) ^ 2)⁻¹hs:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖hb:∑' (i : O), ‖W (‖eisEmbedding i‖ ^ 2 / M)‖ ≤ B * ∑' (x : O), ((1 + t * ‖eisEmbedding x‖ ^ 2) ^ 2)⁻¹hsmall:t * ∑' (h : EisensteinSchwartzPoisson.O), ((1 + t * ‖eisEmbedding h‖ ^ 2) ^ 2)⁻¹ ≤ 4 * (1 + π) ^ 2hmul:t * ∑' (i : O), ‖W (‖eisEmbedding i‖ ^ 2 / M)‖ ≤ t * (B * ∑' (x : O), ((1 + t * ‖eisEmbedding x‖ ^ 2) ^ 2)⁻¹)hcontrol:B * (t * ∑' (h : EisensteinSchwartzPoisson.O), ((1 + t * ‖eisEmbedding h‖ ^ 2) ^ 2)⁻¹) ≤ B * (4 * (1 + π) ^ 2)hfinal:t * ∑' (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W⊢ ∑' (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W * M have hh := mul_le_mul_of_nonneg_left hfinal hMp.leW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mt:ℝ := M⁻¹ht:0 < tht1:t ≤ 1B:ℝ := 4 * ((Finset.Iic (2, 0)).sup (schwartzSeminormFamily ℝ ℝ ℂ)) WhB:0 ≤ Bhp:∀ (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ B * ((1 + t * ‖eisEmbedding z‖ ^ 2) ^ 2)⁻¹hs:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖hb:∑' (i : O), ‖W (‖eisEmbedding i‖ ^ 2 / M)‖ ≤ B * ∑' (x : O), ((1 + t * ‖eisEmbedding x‖ ^ 2) ^ 2)⁻¹hsmall:t * ∑' (h : EisensteinSchwartzPoisson.O), ((1 + t * ‖eisEmbedding h‖ ^ 2) ^ 2)⁻¹ ≤ 4 * (1 + π) ^ 2hmul:t * ∑' (i : O), ‖W (‖eisEmbedding i‖ ^ 2 / M)‖ ≤ t * (B * ∑' (x : O), ((1 + t * ‖eisEmbedding x‖ ^ 2) ^ 2)⁻¹)hcontrol:B * (t * ∑' (h : EisensteinSchwartzPoisson.O), ((1 + t * ‖eisEmbedding h‖ ^ 2) ^ 2)⁻¹) ≤ B * (4 * (1 + π) ^ 2)hfinal:t * ∑' (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl Whh:M * (t * ∑' (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖) ≤ M * diagonalControl W⊢ ∑' (z : O), ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W * M simpa only [t, ← mul_assoc, mul_inv_cancel₀ hMp.ne', one_mul, mul_comm] using hhAll goals completed! 🐙theorem quadraticRow_norm_le_one {α : Type*} (P : α → Ideal O) [∀ i, (P i).IsMaximal] (hg : ∀ i, lambda ∉ P i) (S : Finset α) (z : O) : ‖quadraticRow P hg S z‖ ≤ 1 := byα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αz:O⊢ ‖quadraticRow P hg S z‖ ≤ 1 rw [quadraticRow_eq_canonicalα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αz:O⊢ ‖finiteSexticRow (fun i => P ↑i) ⋯ (fun x => 3) z‖ ≤ 1]α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αz:O⊢ ‖finiteSexticRow (fun i => P ↑i) ⋯ (fun x => 3) z‖ ≤ 1 exact finiteSexticRow_norm_le_one _ _ _ _All goals completed! 🐙theorem quadratic_pair_summable {α : Type*} (P : α → Ideal O) [∀ i, (P i).IsMaximal] (hg : ∀ i, lambda ∉ P i) (S T : Finset α) (W : 𝓢(ℝ, ℂ)) (M : ℝ) (hM : 0 < M) : Summable (fun z : O => (star (quadraticRow P hg S z) * quadraticRow P hg T z) * W (‖eisEmbedding z‖ ^ 2 / M)) := byα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αW:𝓢(ℝ, ℂ)M:ℝhM:0 < M⊢ Summable fun z => star (quadraticRow P hg S z) * quadraticRow P hg T z * W (‖eisEmbedding z‖ ^ 2 / M) have hW : Summable (fun z : O => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖) := by simpa only [scaledRadialTest_apply] using actual_eisenstein_summable_norm (scaledRadialTest W M hM)α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αW:𝓢(ℝ, ℂ)M:ℝhM:0 < MhW:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖⊢ Summable fun z => star (quadraticRow P hg S z) * quadraticRow P hg T z * W (‖eisEmbedding z‖ ^ 2 / M) apply Summable.of_normα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αW:𝓢(ℝ, ℂ)M:ℝhM:0 < MhW:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖⊢ Summable fun a => ‖star (quadraticRow P hg S a) * quadraticRow P hg T a * W (‖eisEmbedding a‖ ^ 2 / M)‖ apply Summable.of_nonneg_of_le (fun _ => norm_nonneg _) _ hWα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αW:𝓢(ℝ, ℂ)M:ℝhM:0 < MhW:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖⊢ ∀ (b : O), ‖star (quadraticRow P hg S b) * quadraticRow P hg T b * W (‖eisEmbedding b‖ ^ 2 / M)‖ ≤ ‖W (‖eisEmbedding b‖ ^ 2 / M)‖ intro zα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αW:𝓢(ℝ, ℂ)M:ℝhM:0 < MhW:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖z:O⊢ ‖star (quadraticRow P hg S z) * quadraticRow P hg T z * W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ rw [norm_mul,α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αW:𝓢(ℝ, ℂ)M:ℝhM:0 < MhW:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖z:O⊢ ‖star (quadraticRow P hg S z) * quadraticRow P hg T z‖ * ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ norm_mul,α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αW:𝓢(ℝ, ℂ)M:ℝhM:0 < MhW:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖z:O⊢ ‖star (quadraticRow P hg S z)‖ * ‖quadraticRow P hg T z‖ * ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ norm_starα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αW:𝓢(ℝ, ℂ)M:ℝhM:0 < MhW:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖z:O⊢ ‖quadraticRow P hg S z‖ * ‖quadraticRow P hg T z‖ * ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ ‖W (‖eisEmbedding z‖ ^ 2 / M)‖]α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αT:Finset αW:𝓢(ℝ, ℂ)M:ℝhM:0 < MhW:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖z:O⊢ ‖quadraticRow P hg S z‖ * ‖quadraticRow P hg T z‖ * ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ exact mul_le_of_le_one_left (norm_nonneg _) ((mul_le_mul (quadraticRow_norm_le_one P hg S z) (quadraticRow_norm_le_one P hg T z) (norm_nonneg _) zero_le_one).trans_eq (one_mul 1))All goals completed! 🐙theorem quadratic_gram_diagonal {α : Type*} (P : α → Ideal O) [∀ i, (P i).IsMaximal] (hg : ∀ i, lambda ∉ P i) (S : Finset α) (W : 𝓢(ℝ, ℂ)) (M : ℝ) (hM : 1 ≤ M) : ‖∑' z : O, (star (quadraticRow P hg S z) * quadraticRow P hg S z) * W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W * M := byα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ M⊢ ‖∑' (z : O), star (quadraticRow P hg S z) * quadraticRow P hg S z * W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W * M have hMp : 0 < M := by linarithα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < M⊢ ‖∑' (z : O), star (quadraticRow P hg S z) * quadraticRow P hg S z * W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W * M have hs := (quadratic_pair_summable P hg S S W M hMp).normα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mhs:Summable fun x => ‖star (quadraticRow P hg S x) * quadraticRow P hg S x * W (‖eisEmbedding x‖ ^ 2 / M)‖⊢ ‖∑' (z : O), star (quadraticRow P hg S z) * quadraticRow P hg S z * W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W * M have hW : Summable (fun z : O => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖) := by simpa only [scaledRadialTest_apply] using actual_eisenstein_summable_norm (scaledRadialTest W M hMp)α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mhs:Summable fun x => ‖star (quadraticRow P hg S x) * quadraticRow P hg S x * W (‖eisEmbedding x‖ ^ 2 / M)‖hW:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖⊢ ‖∑' (z : O), star (quadraticRow P hg S z) * quadraticRow P hg S z * W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ diagonalControl W * M refine (norm_tsum_le_tsum_norm hs).trans ((hs.tsum_le_tsum ?_ hW).trans (radial_weight_lattice_bound W M hM))α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mhs:Summable fun x => ‖star (quadraticRow P hg S x) * quadraticRow P hg S x * W (‖eisEmbedding x‖ ^ 2 / M)‖hW:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖⊢ ∀ (i : O), ‖star (quadraticRow P hg S i) * quadraticRow P hg S i * W (‖eisEmbedding i‖ ^ 2 / M)‖ ≤ ‖W (‖eisEmbedding i‖ ^ 2 / M)‖ intro zα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mhs:Summable fun x => ‖star (quadraticRow P hg S x) * quadraticRow P hg S x * W (‖eisEmbedding x‖ ^ 2 / M)‖hW:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖z:O⊢ ‖star (quadraticRow P hg S z) * quadraticRow P hg S z * W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ rw [norm_mul,α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mhs:Summable fun x => ‖star (quadraticRow P hg S x) * quadraticRow P hg S x * W (‖eisEmbedding x‖ ^ 2 / M)‖hW:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖z:O⊢ ‖star (quadraticRow P hg S z) * quadraticRow P hg S z‖ * ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ norm_mul,α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mhs:Summable fun x => ‖star (quadraticRow P hg S x) * quadraticRow P hg S x * W (‖eisEmbedding x‖ ^ 2 / M)‖hW:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖z:O⊢ ‖star (quadraticRow P hg S z)‖ * ‖quadraticRow P hg S z‖ * ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ norm_starα:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mhs:Summable fun x => ‖star (quadraticRow P hg S x) * quadraticRow P hg S x * W (‖eisEmbedding x‖ ^ 2 / M)‖hW:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖z:O⊢ ‖quadraticRow P hg S z‖ * ‖quadraticRow P hg S z‖ * ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ ‖W (‖eisEmbedding z‖ ^ 2 / M)‖]α:Type u_1P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iS:Finset αW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < Mhs:Summable fun x => ‖star (quadraticRow P hg S x) * quadraticRow P hg S x * W (‖eisEmbedding x‖ ^ 2 / M)‖hW:Summable fun z => ‖W (‖eisEmbedding z‖ ^ 2 / M)‖z:O⊢ ‖quadraticRow P hg S z‖ * ‖quadraticRow P hg S z‖ * ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ ‖W (‖eisEmbedding z‖ ^ 2 / M)‖ exact mul_le_of_le_one_left (norm_nonneg _) ((mul_le_mul (quadraticRow_norm_le_one P hg S z) (quadraticRow_norm_le_one P hg S z) (norm_nonneg _) zero_le_one).trans_eq (one_mul 1))All goals completed! 🐙private theorem norm_sq_finite_sum {β : Type*} (C : Finset β) (a r : β → ℂ) : (↑(‖∑ n ∈ C, a n * r n‖ ^ 2) : ℂ) = ∑ n ∈ C, ∑ m ∈ C, (star (a n) * a m) * (star (r n) * r m) := byβ:Type u_1C:Finset βa:β → ℂr:β → ℂ⊢ ↑(‖∑ n ∈ C, a n * r n‖ ^ 2) = ∑ n ∈ C, ∑ m ∈ C, star (a n) * a m * (star (r n) * r m) rw [Complex.sq_norm,β:Type u_1C:Finset βa:β → ℂr:β → ℂ⊢ ↑(normSq (∑ n ∈ C, a n * r n)) = ∑ n ∈ C, ∑ m ∈ C, star (a n) * a m * (star (r n) * r m) Complex.normSq_eq_conj_mul_selfβ:Type u_1C:Finset βa:β → ℂr:β → ℂ⊢ (starRingEnd ℂ) (∑ n ∈ C, a n * r n) * ∑ n ∈ C, a n * r n = ∑ n ∈ C, ∑ m ∈ C, star (a n) * a m * (star (r n) * r m)]β:Type u_1C:Finset βa:β → ℂr:β → ℂ⊢ (starRingEnd ℂ) (∑ n ∈ C, a n * r n) * ∑ n ∈ C, a n * r n = ∑ n ∈ C, ∑ m ∈ C, star (a n) * a m * (star (r n) * r m) simp only [map_sum, map_mul, starRingEnd_apply]β:Type u_1C:Finset βa:β → ℂr:β → ℂ⊢ (∑ x ∈ C, star (a x) * star (r x)) * ∑ n ∈ C, a n * r n = ∑ n ∈ C, ∑ m ∈ C, star (a n) * a m * (star (r n) * r m) rw [Finset.sum_mul_sumβ:Type u_1C:Finset βa:β → ℂr:β → ℂ⊢ ∑ i ∈ C, ∑ j ∈ C, star (a i) * star (r i) * (a j * r j) = ∑ n ∈ C, ∑ m ∈ C, star (a n) * a m * (star (r n) * r m)]β:Type u_1C:Finset βa:β → ℂr:β → ℂ⊢ ∑ i ∈ C, ∑ j ∈ C, star (a i) * star (r i) * (a j * r j) = ∑ n ∈ C, ∑ m ∈ C, star (a n) * a m * (star (r n) * r m) apply Finset.sum_congr rflβ:Type u_1C:Finset βa:β → ℂr:β → ℂ⊢ ∀ x ∈ C, ∑ j ∈ C, star (a x) * star (r x) * (a j * r j) = ∑ m ∈ C, star (a x) * a m * (star (r x) * r m) intro n hnβ:Type u_1C:Finset βa:β → ℂr:β → ℂn:βhn:n ∈ C⊢ ∑ j ∈ C, star (a n) * star (r n) * (a j * r j) = ∑ m ∈ C, star (a n) * a m * (star (r n) * r m) apply Finset.sum_congr rflβ:Type u_1C:Finset βa:β → ℂr:β → ℂn:βhn:n ∈ C⊢ ∀ x ∈ C, star (a n) * star (r n) * (a x * r x) = star (a n) * a x * (star (r n) * r x) intro m hmβ:Type u_1C:Finset βa:β → ℂr:β → ℂn:βhn:n ∈ Cm:βhm:m ∈ C⊢ star (a n) * star (r n) * (a m * r m) = star (a n) * a m * (star (r n) * r m) ringAll goals completed! 🐙theorem quadratic_mean_summable {α β : Type*} (P : α → Ideal O) [∀ i, (P i).IsMaximal] (hg : ∀ i, lambda ∉ P i) (C : Finset β) (support : β → Finset α) (a : β → ℂ) (W : 𝓢(ℝ, ℂ)) (M : ℝ) (hM : 0 < M) : Summable (fun z : O => W (‖eisEmbedding z‖ ^ 2 / M) * (↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2) : ℂ)) := byα:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < M⊢ Summable fun z => W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2) have hs := summable_sum (s := C) (fun n hn => summable_sum (s := C) (fun m hm => (quadratic_pair_summable P hg (support n) (support m) W M hM).mul_left (star (a n) * a m)))α:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mhs:Summable fun b => ∑ i ∈ C, ∑ i_1 ∈ C, star (a i) * a i_1 * (star (quadraticRow P hg (support i) b) * quadraticRow P hg (support i_1) b * W (‖eisEmbedding b‖ ^ 2 / M))⊢ Summable fun z => W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2) apply hs.congrα:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mhs:Summable fun b => ∑ i ∈ C, ∑ i_1 ∈ C, star (a i) * a i_1 * (star (quadraticRow P hg (support i) b) * quadraticRow P hg (support i_1) b * W (‖eisEmbedding b‖ ^ 2 / M))⊢ ∀ (b : O), ∑ i ∈ C, ∑ i_1 ∈ C, star (a i) * a i_1 * (star (quadraticRow P hg (support i) b) * quadraticRow P hg (support i_1) b * W (‖eisEmbedding b‖ ^ 2 / M)) = W (‖eisEmbedding b‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) b‖ ^ 2) intro zα:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mhs:Summable fun b => ∑ i ∈ C, ∑ i_1 ∈ C, star (a i) * a i_1 * (star (quadraticRow P hg (support i) b) * quadraticRow P hg (support i_1) b * W (‖eisEmbedding b‖ ^ 2 / M))z:O⊢ ∑ i ∈ C, ∑ i_1 ∈ C, star (a i) * a i_1 * (star (quadraticRow P hg (support i) z) * quadraticRow P hg (support i_1) z * W (‖eisEmbedding z‖ ^ 2 / M)) = W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2) rw [norm_sq_finite_sumα:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mhs:Summable fun b => ∑ i ∈ C, ∑ i_1 ∈ C, star (a i) * a i_1 * (star (quadraticRow P hg (support i) b) * quadraticRow P hg (support i_1) b * W (‖eisEmbedding b‖ ^ 2 / M))z:O⊢ ∑ i ∈ C, ∑ i_1 ∈ C, star (a i) * a i_1 * (star (quadraticRow P hg (support i) z) * quadraticRow P hg (support i_1) z * W (‖eisEmbedding z‖ ^ 2 / M)) = W (‖eisEmbedding z‖ ^ 2 / M) * ∑ n ∈ C, ∑ m ∈ C, star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z)]α:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mhs:Summable fun b => ∑ i ∈ C, ∑ i_1 ∈ C, star (a i) * a i_1 * (star (quadraticRow P hg (support i) b) * quadraticRow P hg (support i_1) b * W (‖eisEmbedding b‖ ^ 2 / M))z:O⊢ ∑ i ∈ C, ∑ i_1 ∈ C, star (a i) * a i_1 * (star (quadraticRow P hg (support i) z) * quadraticRow P hg (support i_1) z * W (‖eisEmbedding z‖ ^ 2 / M)) = W (‖eisEmbedding z‖ ^ 2 / M) * ∑ n ∈ C, ∑ m ∈ C, star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z) simp only [Finset.mul_sum]α:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mhs:Summable fun b => ∑ i ∈ C, ∑ i_1 ∈ C, star (a i) * a i_1 * (star (quadraticRow P hg (support i) b) * quadraticRow P hg (support i_1) b * W (‖eisEmbedding b‖ ^ 2 / M))z:O⊢ ∑ i ∈ C, ∑ i_1 ∈ C, star (a i) * a i_1 * (star (quadraticRow P hg (support i) z) * quadraticRow P hg (support i_1) z * W (‖eisEmbedding z‖ ^ 2 / M)) = ∑ x ∈ C, ∑ i ∈ C, W (‖eisEmbedding z‖ ^ 2 / M) * (star (a x) * a i * (star (quadraticRow P hg (support x) z) * quadraticRow P hg (support i) z)) apply Finset.sum_congr rflα:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mhs:Summable fun b => ∑ i ∈ C, ∑ i_1 ∈ C, star (a i) * a i_1 * (star (quadraticRow P hg (support i) b) * quadraticRow P hg (support i_1) b * W (‖eisEmbedding b‖ ^ 2 / M))z:O⊢ ∀ x ∈ C, ∑ i ∈ C, star (a x) * a i * (star (quadraticRow P hg (support x) z) * quadraticRow P hg (support i) z * W (‖eisEmbedding z‖ ^ 2 / M)) = ∑ i ∈ C, W (‖eisEmbedding z‖ ^ 2 / M) * (star (a x) * a i * (star (quadraticRow P hg (support x) z) * quadraticRow P hg (support i) z)) intro n hnα:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mhs:Summable fun b => ∑ i ∈ C, ∑ i_1 ∈ C, star (a i) * a i_1 * (star (quadraticRow P hg (support i) b) * quadraticRow P hg (support i_1) b * W (‖eisEmbedding b‖ ^ 2 / M))z:On:βhn:n ∈ C⊢ ∑ i ∈ C, star (a n) * a i * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support i) z * W (‖eisEmbedding z‖ ^ 2 / M)) = ∑ i ∈ C, W (‖eisEmbedding z‖ ^ 2 / M) * (star (a n) * a i * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support i) z)) apply Finset.sum_congr rflα:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mhs:Summable fun b => ∑ i ∈ C, ∑ i_1 ∈ C, star (a i) * a i_1 * (star (quadraticRow P hg (support i) b) * quadraticRow P hg (support i_1) b * W (‖eisEmbedding b‖ ^ 2 / M))z:On:βhn:n ∈ C⊢ ∀ x ∈ C, star (a n) * a x * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support x) z * W (‖eisEmbedding z‖ ^ 2 / M)) = W (‖eisEmbedding z‖ ^ 2 / M) * (star (a n) * a x * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support x) z)) intro m hmα:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mhs:Summable fun b => ∑ i ∈ C, ∑ i_1 ∈ C, star (a i) * a i_1 * (star (quadraticRow P hg (support i) b) * quadraticRow P hg (support i_1) b * W (‖eisEmbedding b‖ ^ 2 / M))z:On:βhn:n ∈ Cm:βhm:m ∈ C⊢ star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)) = W (‖eisEmbedding z‖ ^ 2 / M) * (star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z)) ringAll goals completed! 🐙theorem quadratic_mean_expand {α β : Type*} (P : α → Ideal O) [∀ i, (P i).IsMaximal] (hg : ∀ i, lambda ∉ P i) (C : Finset β) (support : β → Finset α) (a : β → ℂ) (W : 𝓢(ℝ, ℂ)) (M : ℝ) (hM : 0 < M) : (∑' z : O, W (‖eisEmbedding z‖ ^ 2 / M) * (↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2) : ℂ)) = ∑ n ∈ C, ∑ m ∈ C, (star (a n) * a m) * ∑' z : O, (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z) * W (‖eisEmbedding z‖ ^ 2 / M) := byα:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < M⊢ ∑' (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2) = ∑ n ∈ C, ∑ m ∈ C, star (a n) * a m * ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M) let f : β → β → O → ℂ := fun n m z => (star (a n) * a m) * ((star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z) * W (‖eisEmbedding z‖ ^ 2 / M))α:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mf:β → β → O → ℂ := fun n m z => star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M))⊢ ∑' (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2) = ∑ n ∈ C, ∑ m ∈ C, star (a n) * a m * ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M) have hf (n m : β) : Summable (f n m) := (quadratic_pair_summable P hg (support n) (support m) W M hM).mul_left _α:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mf:β → β → O → ℂ := fun n m z => star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M))hf:∀ (n m : β), Summable (f n m)⊢ ∑' (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2) = ∑ n ∈ C, ∑ m ∈ C, star (a n) * a m * ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M) have heq (z : O) : W (‖eisEmbedding z‖ ^ 2 / M) * (↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2) : ℂ) = ∑ n ∈ C, ∑ m ∈ C, f n m z := by rw [norm_sq_finite_sumα:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mf:β → β → O → ℂ := fun n m z => star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M))hf:∀ (n m : β), Summable (f n m)z:O⊢ W (‖eisEmbedding z‖ ^ 2 / M) * ∑ n ∈ C, ∑ m ∈ C, star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z) = ∑ n ∈ C, ∑ m ∈ C, f n m z]α:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mf:β → β → O → ℂ := fun n m z => star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M))hf:∀ (n m : β), Summable (f n m)z:O⊢ W (‖eisEmbedding z‖ ^ 2 / M) * ∑ n ∈ C, ∑ m ∈ C, star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z) = ∑ n ∈ C, ∑ m ∈ C, f n m z simp only [Finset.mul_sum]α:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mf:β → β → O → ℂ := fun n m z => star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M))hf:∀ (n m : β), Summable (f n m)z:O⊢ ∑ x ∈ C, ∑ i ∈ C, W (‖eisEmbedding z‖ ^ 2 / M) * (star (a x) * a i * (star (quadraticRow P hg (support x) z) * quadraticRow P hg (support i) z)) = ∑ n ∈ C, ∑ m ∈ C, f n m z apply Finset.sum_congr rflα:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mf:β → β → O → ℂ := fun n m z => star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M))hf:∀ (n m : β), Summable (f n m)z:O⊢ ∀ x ∈ C, ∑ i ∈ C, W (‖eisEmbedding z‖ ^ 2 / M) * (star (a x) * a i * (star (quadraticRow P hg (support x) z) * quadraticRow P hg (support i) z)) = ∑ m ∈ C, f x m z intro n hnα:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mf:β → β → O → ℂ := fun n m z => star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M))hf:∀ (n m : β), Summable (f n m)z:On:βhn:n ∈ C⊢ ∑ i ∈ C, W (‖eisEmbedding z‖ ^ 2 / M) * (star (a n) * a i * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support i) z)) = ∑ m ∈ C, f n m z apply Finset.sum_congr rflα:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mf:β → β → O → ℂ := fun n m z => star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M))hf:∀ (n m : β), Summable (f n m)z:On:βhn:n ∈ C⊢ ∀ x ∈ C, W (‖eisEmbedding z‖ ^ 2 / M) * (star (a n) * a x * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support x) z)) = f n x z intro m hmα:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mf:β → β → O → ℂ := fun n m z => star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M))hf:∀ (n m : β), Summable (f n m)z:On:βhn:n ∈ Cm:βhm:m ∈ C⊢ W (‖eisEmbedding z‖ ^ 2 / M) * (star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z)) = f n m z dsimp [f]α:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mf:β → β → O → ℂ := fun n m z => star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M))hf:∀ (n m : β), Summable (f n m)z:On:βhn:n ∈ Cm:βhm:m ∈ C⊢ W (‖eisEmbedding z‖ ^ 2 / M) * ((starRingEnd ℂ) (a n) * a m * ((starRingEnd ℂ) (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z)) = (starRingEnd ℂ) (a n) * a m * ((starRingEnd ℂ) (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)) ringα:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mf:β → β → O → ℂ := fun n m z => star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M))hf:∀ (n m : β), Summable (f n m)heq:∀ (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2) = ∑ n ∈ C, ∑ m ∈ C, f n m z⊢ ∑' (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2) = ∑ n ∈ C, ∑ m ∈ C, star (a n) * a m * ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M) rw [tsum_congr heq,α:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mf:β → β → O → ℂ := fun n m z => star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M))hf:∀ (n m : β), Summable (f n m)heq:∀ (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2) = ∑ n ∈ C, ∑ m ∈ C, f n m z⊢ ∑' (b : O), ∑ n ∈ C, ∑ m ∈ C, f n m b = ∑ n ∈ C, ∑ m ∈ C, star (a n) * a m * ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M) Summable.tsum_finsetSum (fun n hn => summable_sum (fun m hm => hf n m))α:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mf:β → β → O → ℂ := fun n m z => star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M))hf:∀ (n m : β), Summable (f n m)heq:∀ (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2) = ∑ n ∈ C, ∑ m ∈ C, f n m z⊢ ∑ i ∈ C, ∑' (b : O), ∑ i_1 ∈ C, f i i_1 b = ∑ n ∈ C, ∑ m ∈ C, star (a n) * a m * ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)]α:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mf:β → β → O → ℂ := fun n m z => star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M))hf:∀ (n m : β), Summable (f n m)heq:∀ (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2) = ∑ n ∈ C, ∑ m ∈ C, f n m z⊢ ∑ i ∈ C, ∑' (b : O), ∑ i_1 ∈ C, f i i_1 b = ∑ n ∈ C, ∑ m ∈ C, star (a n) * a m * ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M) apply Finset.sum_congr rflα:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mf:β → β → O → ℂ := fun n m z => star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M))hf:∀ (n m : β), Summable (f n m)heq:∀ (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2) = ∑ n ∈ C, ∑ m ∈ C, f n m z⊢ ∀ x ∈ C, ∑' (b : O), ∑ i ∈ C, f x i b = ∑ m ∈ C, star (a x) * a m * ∑' (z : O), star (quadraticRow P hg (support x) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M) intro n hnα:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mf:β → β → O → ℂ := fun n m z => star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M))hf:∀ (n m : β), Summable (f n m)heq:∀ (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2) = ∑ n ∈ C, ∑ m ∈ C, f n m zn:βhn:n ∈ C⊢ ∑' (b : O), ∑ i ∈ C, f n i b = ∑ m ∈ C, star (a n) * a m * ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M) rw [Summable.tsum_finsetSum (fun m hm => hf n m)α:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mf:β → β → O → ℂ := fun n m z => star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M))hf:∀ (n m : β), Summable (f n m)heq:∀ (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2) = ∑ n ∈ C, ∑ m ∈ C, f n m zn:βhn:n ∈ C⊢ ∑ i ∈ C, ∑' (b : O), f n i b = ∑ m ∈ C, star (a n) * a m * ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)]α:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mf:β → β → O → ℂ := fun n m z => star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M))hf:∀ (n m : β), Summable (f n m)heq:∀ (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2) = ∑ n ∈ C, ∑ m ∈ C, f n m zn:βhn:n ∈ C⊢ ∑ i ∈ C, ∑' (b : O), f n i b = ∑ m ∈ C, star (a n) * a m * ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M) apply Finset.sum_congr rflα:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mf:β → β → O → ℂ := fun n m z => star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M))hf:∀ (n m : β), Summable (f n m)heq:∀ (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2) = ∑ n ∈ C, ∑ m ∈ C, f n m zn:βhn:n ∈ C⊢ ∀ x ∈ C, ∑' (b : O), f n x b = star (a n) * a x * ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support x) z * W (‖eisEmbedding z‖ ^ 2 / M) intro m hmα:Type u_1β:Type u_2P:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhg:∀ (i : α), lambda ∉ P iC:Finset βsupport:β → Finset αa:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:0 < Mf:β → β → O → ℂ := fun n m z => star (a n) * a m * (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M))hf:∀ (n m : β), Summable (f n m)heq:∀ (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2) = ∑ n ∈ C, ∑ m ∈ C, f n m zn:βhn:n ∈ Cm:βhm:m ∈ C⊢ ∑' (b : O), f n m b = star (a n) * a m * ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M) exact tsum_mul_leftAll goals completed! 🐙theorem quadratic_initial_mean_bound {α β : Type*} [DecidableEq α] [DecidableEq β] (P : α → Ideal O) [∀ i, (P i).IsMaximal] (hinj : Function.Injective P) (hg : ∀ i, lambda ∉ P i) (hc : ∀ i, ringChar (O ⧸ P i) ≠ 2) (C : Finset β) (support : β → Finset α) (hsupp : Set.InjOn support (↑C : Set β)) (N : ℝ) (hN : 0 ≤ N) (hNorm : ∀ n ∈ C, supportNorm P (support n) ≤ N) (a : β → ℂ) (W : 𝓢(ℝ, ℂ)) (M : ℝ) (hM : 1 ≤ M) : ‖∑' z : O, W (‖eisEmbedding z‖ ^ 2 / M) * (↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2) : ℂ)‖ ≤ (diagonalControl W * M + pvControl W * N * C.card) * ∑ n ∈ C, ‖a n‖ ^ 2 := byα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ M⊢ ‖∑' (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2)‖ ≤ (diagonalControl W * M + pvControl W * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2 have hMp : 0 < M := by linarithα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < M⊢ ‖∑' (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2)‖ ≤ (diagonalControl W * M + pvControl W * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2 let D := diagonalControl W * Mα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * M⊢ ‖∑' (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2)‖ ≤ (diagonalControl W * M + pvControl W * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2 let B := pvControl W * Nα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * N⊢ ‖∑' (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2)‖ ≤ (diagonalControl W * M + pvControl W * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2 have hB : 0 ≤ B := mul_nonneg (pvControl_nonneg W) hNα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ B⊢ ‖∑' (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2)‖ ≤ (diagonalControl W * M + pvControl W * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2 let K : β → β → ℂ := fun n m => ∑' z : O, (star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z) * W (‖eisEmbedding z‖ ^ 2 / M)α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)⊢ ‖∑' (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2)‖ ≤ (diagonalControl W * M + pvControl W * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2 have hK (n : β) (hn : n ∈ C) (m : β) (hm : m ∈ C) : ‖K n m‖ ≤ (if n = m then D else 0) + B := by by_cases hnm : n = mposα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)n:βhn:n ∈ Cm:βhm:m ∈ Chnm:n = m⊢ ‖K n m‖ ≤ (if n = m then D else 0) + Bnegα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)n:βhn:n ∈ Cm:βhm:m ∈ Chnm:¬n = m⊢ ‖K n m‖ ≤ (if n = m then D else 0) + B ·posα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)n:βhn:n ∈ Cm:βhm:m ∈ Chnm:n = m⊢ ‖K n m‖ ≤ (if n = m then D else 0) + B subst mposα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)n:βhn:n ∈ Chm:n ∈ C⊢ ‖K n n‖ ≤ (if n = n then D else 0) + B simp only [ite_true]posα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)n:βhn:n ∈ Chm:n ∈ C⊢ ‖K n n‖ ≤ D + B exact (quadratic_gram_diagonal P hg (support n) W M hM).trans (le_add_of_nonneg_right hB)All goals completed! 🐙 ·negα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)n:βhn:n ∈ Cm:βhm:m ∈ Chnm:¬n = m⊢ ‖K n m‖ ≤ (if n = m then D else 0) + B simp only [hnm, ite_false, zero_add]negα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)n:βhn:n ∈ Cm:βhm:m ∈ Chnm:¬n = m⊢ ‖K n m‖ ≤ B exact quadratic_gram_offdiagonal_of_norm_le P hinj hg hc _ _ (fun h => hnm (hsupp hn hm h)) N hN (hNorm n hn) (hNorm m hm) W M hMpα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + B⊢ ‖∑' (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2)‖ ≤ (diagonalControl W * M + pvControl W * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2 have heq (n : β) (hn : n ∈ C) : (∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B)) = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖ := by simp only [mul_add, Finset.sum_add_distrib, mul_ite, mul_zero]α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bn:βhn:n ∈ C⊢ (∑ x ∈ C, if n = x then ‖a n‖ * ‖a x‖ * D else 0) + ∑ x ∈ C, ‖a n‖ * ‖a x‖ * B = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖ simp only [Finset.sum_ite_eq, hn, ite_true]α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bn:βhn:n ∈ C⊢ ‖a n‖ * ‖a n‖ * D + ∑ x ∈ C, ‖a n‖ * ‖a x‖ * B = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖ rw [← Finset.sum_mul,α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bn:βhn:n ∈ C⊢ ‖a n‖ * ‖a n‖ * D + (∑ i ∈ C, ‖a n‖ * ‖a i‖) * B = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖ ← Finset.mul_sumα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bn:βhn:n ∈ C⊢ ‖a n‖ * ‖a n‖ * D + (‖a n‖ * ∑ i ∈ C, ‖a i‖) * B = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖]α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bn:βhn:n ∈ C⊢ ‖a n‖ * ‖a n‖ * D + (‖a n‖ * ∑ i ∈ C, ‖a i‖) * B = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖ ringα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bheq:∀ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖⊢ ‖∑' (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2)‖ ≤ (diagonalControl W * M + pvControl W * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2 have hcalc : (∑ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B)) = D * (∑ n ∈ C, ‖a n‖ ^ 2) + B * (∑ n ∈ C, ‖a n‖) ^ 2 := by rw [Finset.sum_congr rfl heqα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bheq:∀ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖⊢ ∑ x ∈ C, (D * ‖a x‖ ^ 2 + B * ‖a x‖ * ∑ m ∈ C, ‖a m‖) = D * ∑ n ∈ C, ‖a n‖ ^ 2 + B * (∑ n ∈ C, ‖a n‖) ^ 2]α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bheq:∀ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖⊢ ∑ x ∈ C, (D * ‖a x‖ ^ 2 + B * ‖a x‖ * ∑ m ∈ C, ‖a m‖) = D * ∑ n ∈ C, ‖a n‖ ^ 2 + B * (∑ n ∈ C, ‖a n‖) ^ 2 simp only [Finset.sum_add_distrib, ← Finset.mul_sum, ← Finset.sum_mul]α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bheq:∀ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖⊢ D * ∑ i ∈ C, ‖a i‖ ^ 2 + (B * ∑ m ∈ C, ‖a m‖) * ∑ m ∈ C, ‖a m‖ = D * ∑ i ∈ C, ‖a i‖ ^ 2 + B * (∑ m ∈ C, ‖a m‖) ^ 2 ringα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bheq:∀ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖hcalc:∑ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ∑ n ∈ C, ‖a n‖ ^ 2 + B * (∑ n ∈ C, ‖a n‖) ^ 2⊢ ‖∑' (z : O), W (‖eisEmbedding z‖ ^ 2 / M) * ↑(‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2)‖ ≤ (diagonalControl W * M + pvControl W * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2 rw [quadratic_mean_expand P hg C support a W M hMpα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bheq:∀ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖hcalc:∑ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ∑ n ∈ C, ‖a n‖ ^ 2 + B * (∑ n ∈ C, ‖a n‖) ^ 2⊢ ‖∑ n ∈ C, ∑ m ∈ C, star (a n) * a m * ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ (diagonalControl W * M + pvControl W * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2]α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bheq:∀ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖hcalc:∑ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ∑ n ∈ C, ‖a n‖ ^ 2 + B * (∑ n ∈ C, ‖a n‖) ^ 2⊢ ‖∑ n ∈ C, ∑ m ∈ C, star (a n) * a m * ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ (diagonalControl W * M + pvControl W * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2 calc _ ≤ ∑ n ∈ C, ∑ m ∈ C, ‖(star (a n) * a m) * K n m‖ := byα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bheq:∀ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖hcalc:∑ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ∑ n ∈ C, ‖a n‖ ^ 2 + B * (∑ n ∈ C, ‖a n‖) ^ 2⊢ ‖∑ n ∈ C, ∑ m ∈ C, star (a n) * a m * ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)‖ ≤ ∑ n ∈ C, ∑ m ∈ C, ‖star (a n) * a m * K n m‖ exact (norm_sum_le _ _).trans (Finset.sum_le_sum (fun _ _ => norm_sum_le _ _))All goals completed! 🐙 _ ≤ ∑ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) := byα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bheq:∀ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖hcalc:∑ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ∑ n ∈ C, ‖a n‖ ^ 2 + B * (∑ n ∈ C, ‖a n‖) ^ 2⊢ ∑ n ∈ C, ∑ m ∈ C, ‖star (a n) * a m * K n m‖ ≤ ∑ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) apply Finset.sum_le_sumα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bheq:∀ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖hcalc:∑ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ∑ n ∈ C, ‖a n‖ ^ 2 + B * (∑ n ∈ C, ‖a n‖) ^ 2⊢ ∀ i ∈ C, ∑ m ∈ C, ‖star (a i) * a m * K i m‖ ≤ ∑ m ∈ C, ‖a i‖ * ‖a m‖ * ((if i = m then D else 0) + B) intro n hnα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bheq:∀ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖hcalc:∑ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ∑ n ∈ C, ‖a n‖ ^ 2 + B * (∑ n ∈ C, ‖a n‖) ^ 2n:βhn:n ∈ C⊢ ∑ m ∈ C, ‖star (a n) * a m * K n m‖ ≤ ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) apply Finset.sum_le_sumα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bheq:∀ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖hcalc:∑ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ∑ n ∈ C, ‖a n‖ ^ 2 + B * (∑ n ∈ C, ‖a n‖) ^ 2n:βhn:n ∈ C⊢ ∀ i ∈ C, ‖star (a n) * a i * K n i‖ ≤ ‖a n‖ * ‖a i‖ * ((if n = i then D else 0) + B) intro m hmα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bheq:∀ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖hcalc:∑ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ∑ n ∈ C, ‖a n‖ ^ 2 + B * (∑ n ∈ C, ‖a n‖) ^ 2n:βhn:n ∈ Cm:βhm:m ∈ C⊢ ‖star (a n) * a m * K n m‖ ≤ ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) simp only [norm_mul, norm_star]α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bheq:∀ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖hcalc:∑ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ∑ n ∈ C, ‖a n‖ ^ 2 + B * (∑ n ∈ C, ‖a n‖) ^ 2n:βhn:n ∈ Cm:βhm:m ∈ C⊢ ‖a n‖ * ‖a m‖ * ‖K n m‖ ≤ ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) exact mul_le_mul_of_nonneg_left (hK n hn m hm) (mul_nonneg (norm_nonneg _) (norm_nonneg _))All goals completed! 🐙 _ = D * (∑ n ∈ C, ‖a n‖ ^ 2) + B * (∑ n ∈ C, ‖a n‖) ^ 2 := hcalc _ ≤ D * (∑ n ∈ C, ‖a n‖ ^ 2) + B * ((C.card : ℝ) * ∑ n ∈ C, ‖a n‖ ^ 2) := byα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bheq:∀ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖hcalc:∑ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ∑ n ∈ C, ‖a n‖ ^ 2 + B * (∑ n ∈ C, ‖a n‖) ^ 2⊢ D * ∑ n ∈ C, ‖a n‖ ^ 2 + B * (∑ n ∈ C, ‖a n‖) ^ 2 ≤ D * ∑ n ∈ C, ‖a n‖ ^ 2 + B * (↑(#C) * ∑ n ∈ C, ‖a n‖ ^ 2) apply add_le_add le_rflα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bheq:∀ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖hcalc:∑ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ∑ n ∈ C, ‖a n‖ ^ 2 + B * (∑ n ∈ C, ‖a n‖) ^ 2⊢ B * (∑ n ∈ C, ‖a n‖) ^ 2 ≤ B * (↑(#C) * ∑ n ∈ C, ‖a n‖ ^ 2) apply mul_le_mul_of_nonneg_left _ hBα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bheq:∀ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖hcalc:∑ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ∑ n ∈ C, ‖a n‖ ^ 2 + B * (∑ n ∈ C, ‖a n‖) ^ 2⊢ (∑ n ∈ C, ‖a n‖) ^ 2 ≤ ↑(#C) * ∑ n ∈ C, ‖a n‖ ^ 2 simpa only [mul_one, one_pow, Finset.sum_const, nsmul_eq_mul, mul_one, mul_comm] using Finset.sum_mul_sq_le_sq_mul_sq C (fun n => ‖a n‖) (fun _ => (1 : ℝ))All goals completed! 🐙 _ = _ := byα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bheq:∀ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖hcalc:∑ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ∑ n ∈ C, ‖a n‖ ^ 2 + B * (∑ n ∈ C, ‖a n‖) ^ 2⊢ D * ∑ n ∈ C, ‖a n‖ ^ 2 + B * (↑(#C) * ∑ n ∈ C, ‖a n‖ ^ 2) = (diagonalControl W * M + pvControl W * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2 dsimp [D, B]α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂW:𝓢(ℝ, ℂ)M:ℝhM:1 ≤ MhMp:0 < MD:ℝ := diagonalControl W * MB:ℝ := pvControl W * NhB:0 ≤ BK:β → β → ℂ := fun n m => ∑' (z : O), star (quadraticRow P hg (support n) z) * quadraticRow P hg (support m) z * W (‖eisEmbedding z‖ ^ 2 / M)hK:∀ n ∈ C, ∀ m ∈ C, ‖K n m‖ ≤ (if n = m then D else 0) + Bheq:∀ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ‖a n‖ ^ 2 + B * ‖a n‖ * ∑ m ∈ C, ‖a m‖hcalc:∑ n ∈ C, ∑ m ∈ C, ‖a n‖ * ‖a m‖ * ((if n = m then D else 0) + B) = D * ∑ n ∈ C, ‖a n‖ ^ 2 + B * (∑ n ∈ C, ‖a n‖) ^ 2⊢ diagonalControl W * M * ∑ n ∈ C, ‖a n‖ ^ 2 + pvControl W * N * (↑(#C) * ∑ n ∈ C, ‖a n‖ ^ 2) = (diagonalControl W * M + pvControl W * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2; ringAll goals completed! 🐙def sieveBump : ContDiffBump (0 : ℝ) where rIn := 1 rOut := 2 rIn_pos := by⊢ 0 < 1 norm_numAll goals completed! 🐙 rIn_lt_rOut := by⊢ 1 < 2 norm_numAll goals completed! 🐙def sieveCutoff : 𝓢(ℝ, ℂ) := by⊢ 𝓢(ℝ, ℂ) have hc : HasCompactSupport (fun t : ℝ => (sieveBump t : ℂ)) := sieveBump.hasCompactSupport.comp_left Complex.ofReal_zerohc:HasCompactSupport fun t => ↑(↑sieveBump t)⊢ 𝓢(ℝ, ℂ) have hs : ContDiff ℝ ∞ (fun t : ℝ => (sieveBump t : ℂ)) := Complex.ofRealCLM.contDiff.comp sieveBump.contDiffhc:HasCompactSupport fun t => ↑(↑sieveBump t)hs:ContDiff ℝ ∞ fun t => ↑(↑sieveBump t)⊢ 𝓢(ℝ, ℂ) exact hc.toSchwartzMap hsAll goals completed! 🐙@[simp] theorem sieveCutoff_apply (t : ℝ) : sieveCutoff t = (sieveBump t : ℂ) := rfltheorem sieveBump_eq_one {t : ℝ} (ht0 : 0 ≤ t) (ht1 : t ≤ 1) : sieveBump t = 1 := byt:ℝht0:0 ≤ tht1:t ≤ 1⊢ ↑sieveBump t = 1 apply sieveBump.one_of_mem_closedBallt:ℝht0:0 ≤ tht1:t ≤ 1⊢ t ∈ Metric.closedBall 0 sieveBump.rIn change dist t 0 ≤ 1t:ℝht0:0 ≤ tht1:t ≤ 1⊢ dist t 0 ≤ 1 simpa only [dist_zero_right, Real.norm_eq_abs, abs_of_nonneg ht0] using ht1All goals completed! 🐙theorem quadratic_initial_finite_bound {α β : Type*} [DecidableEq α] [DecidableEq β] (P : α → Ideal O) [∀ i, (P i).IsMaximal] (hinj : Function.Injective P) (hg : ∀ i, lambda ∉ P i) (hc : ∀ i, ringChar (O ⧸ P i) ≠ 2) (C : Finset β) (support : β → Finset α) (hsupp : Set.InjOn support (↑C : Set β)) (N : ℝ) (hN : 0 ≤ N) (hNorm : ∀ n ∈ C, supportNorm P (support n) ≤ N) (a : β → ℂ) (M : ℝ) (hM : 1 ≤ M) (R : Finset O) (hR : ∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ M) : (∑ z ∈ R, ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2) ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * N * C.card) * ∑ n ∈ C, ‖a n‖ ^ 2 := byα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ M⊢ ∑ z ∈ R, ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2 have hMp : 0 < M := by linarithα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < M⊢ ∑ z ∈ R, ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2 let E : O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2⊢ ∑ z ∈ R, ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2 let F : O → ℝ := fun z => sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E zα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2F:O → ℝ := fun z => ↑sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E z⊢ ∑ z ∈ R, ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2 let H : O → ℂ := fun z => sieveCutoff (‖eisEmbedding z‖ ^ 2 / M) * (E z : ℂ)α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2F:O → ℝ := fun z => ↑sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E zH:O → ℂ := fun z => sieveCutoff (‖eisEmbedding z‖ ^ 2 / M) * ↑(E z)⊢ ∑ z ∈ R, ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2 have hF (z : O) : 0 ≤ F z := mul_nonneg sieveBump.nonneg (sq_nonneg _)α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2F:O → ℝ := fun z => ↑sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E zH:O → ℂ := fun z => sieveCutoff (‖eisEmbedding z‖ ^ 2 / M) * ↑(E z)hF:∀ (z : O), 0 ≤ F z⊢ ∑ z ∈ R, ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2 have hH (z : O) : H z = (F z : ℂ) := by simp only [H, F, sieveCutoff_apply, Complex.ofReal_mul]α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2F:O → ℝ := fun z => ↑sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E zH:O → ℂ := fun z => sieveCutoff (‖eisEmbedding z‖ ^ 2 / M) * ↑(E z)hF:∀ (z : O), 0 ≤ F zhH:∀ (z : O), H z = ↑(F z)⊢ ∑ z ∈ R, ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2 have hHs : Summable H := quadratic_mean_summable P hg C support a sieveCutoff M hMpα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2F:O → ℝ := fun z => ↑sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E zH:O → ℂ := fun z => sieveCutoff (‖eisEmbedding z‖ ^ 2 / M) * ↑(E z)hF:∀ (z : O), 0 ≤ F zhH:∀ (z : O), H z = ↑(F z)hHs:Summable H⊢ ∑ z ∈ R, ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2 have hFs : Summable F := hHs.norm.congr (fun z => byα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2F:O → ℝ := fun z => ↑sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E zH:O → ℂ := fun z => sieveCutoff (‖eisEmbedding z‖ ^ 2 / M) * ↑(E z)hF:∀ (z : O), 0 ≤ F zhH:∀ (z : O), H z = ↑(F z)hHs:Summable Hz:O⊢ ‖H z‖ = F z rw [hH,α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2F:O → ℝ := fun z => ↑sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E zH:O → ℂ := fun z => sieveCutoff (‖eisEmbedding z‖ ^ 2 / M) * ↑(E z)hF:∀ (z : O), 0 ≤ F zhH:∀ (z : O), H z = ↑(F z)hHs:Summable Hz:O⊢ ‖↑(F z)‖ = F z Complex.norm_real,α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2F:O → ℝ := fun z => ↑sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E zH:O → ℂ := fun z => sieveCutoff (‖eisEmbedding z‖ ^ 2 / M) * ↑(E z)hF:∀ (z : O), 0 ≤ F zhH:∀ (z : O), H z = ↑(F z)hHs:Summable Hz:O⊢ ‖F z‖ = F z Real.norm_eq_abs,α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2F:O → ℝ := fun z => ↑sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E zH:O → ℂ := fun z => sieveCutoff (‖eisEmbedding z‖ ^ 2 / M) * ↑(E z)hF:∀ (z : O), 0 ≤ F zhH:∀ (z : O), H z = ↑(F z)hHs:Summable Hz:O⊢ |F z| = F z abs_of_nonneg (hF z)α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2F:O → ℝ := fun z => ↑sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E zH:O → ℂ := fun z => sieveCutoff (‖eisEmbedding z‖ ^ 2 / M) * ↑(E z)hF:∀ (z : O), 0 ≤ F zhH:∀ (z : O), H z = ↑(F z)hHs:Summable Hz:O⊢ F z = F z]All goals completed! 🐙)α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2F:O → ℝ := fun z => ↑sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E zH:O → ℂ := fun z => sieveCutoff (‖eisEmbedding z‖ ^ 2 / M) * ↑(E z)hF:∀ (z : O), 0 ≤ F zhH:∀ (z : O), H z = ↑(F z)hHs:Summable HhFs:Summable F⊢ ∑ z ∈ R, ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2 have hsum : ‖∑' z : O, H z‖ = ∑' z : O, F z := by rw [tsum_congr hH,α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2F:O → ℝ := fun z => ↑sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E zH:O → ℂ := fun z => sieveCutoff (‖eisEmbedding z‖ ^ 2 / M) * ↑(E z)hF:∀ (z : O), 0 ≤ F zhH:∀ (z : O), H z = ↑(F z)hHs:Summable HhFs:Summable F⊢ ‖∑' (b : O), ↑(F b)‖ = ∑' (z : O), F z ← Complex.ofReal_tsum,α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2F:O → ℝ := fun z => ↑sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E zH:O → ℂ := fun z => sieveCutoff (‖eisEmbedding z‖ ^ 2 / M) * ↑(E z)hF:∀ (z : O), 0 ≤ F zhH:∀ (z : O), H z = ↑(F z)hHs:Summable HhFs:Summable F⊢ ‖↑(∑' (a : O), F a)‖ = ∑' (z : O), F z Complex.norm_real,α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2F:O → ℝ := fun z => ↑sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E zH:O → ℂ := fun z => sieveCutoff (‖eisEmbedding z‖ ^ 2 / M) * ↑(E z)hF:∀ (z : O), 0 ≤ F zhH:∀ (z : O), H z = ↑(F z)hHs:Summable HhFs:Summable F⊢ ‖∑' (a : O), F a‖ = ∑' (z : O), F z Real.norm_eq_abs,α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2F:O → ℝ := fun z => ↑sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E zH:O → ℂ := fun z => sieveCutoff (‖eisEmbedding z‖ ^ 2 / M) * ↑(E z)hF:∀ (z : O), 0 ≤ F zhH:∀ (z : O), H z = ↑(F z)hHs:Summable HhFs:Summable F⊢ |∑' (a : O), F a| = ∑' (z : O), F z abs_of_nonneg (tsum_nonneg hF)α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2F:O → ℝ := fun z => ↑sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E zH:O → ℂ := fun z => sieveCutoff (‖eisEmbedding z‖ ^ 2 / M) * ↑(E z)hF:∀ (z : O), 0 ≤ F zhH:∀ (z : O), H z = ↑(F z)hHs:Summable HhFs:Summable F⊢ ∑' (i : O), F i = ∑' (z : O), F z]α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2F:O → ℝ := fun z => ↑sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E zH:O → ℂ := fun z => sieveCutoff (‖eisEmbedding z‖ ^ 2 / M) * ↑(E z)hF:∀ (z : O), 0 ≤ F zhH:∀ (z : O), H z = ↑(F z)hHs:Summable HhFs:Summable Fhsum:‖∑' (z : O), H z‖ = ∑' (z : O), F z⊢ ∑ z ∈ R, ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * N * ↑(#C)) * ∑ n ∈ C, ‖a n‖ ^ 2 calc _ = ∑ z ∈ R, F z := byα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2F:O → ℝ := fun z => ↑sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E zH:O → ℂ := fun z => sieveCutoff (‖eisEmbedding z‖ ^ 2 / M) * ↑(E z)hF:∀ (z : O), 0 ≤ F zhH:∀ (z : O), H z = ↑(F z)hHs:Summable HhFs:Summable Fhsum:‖∑' (z : O), H z‖ = ∑' (z : O), F z⊢ ∑ z ∈ R, ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2 = ∑ z ∈ R, F z apply Finset.sum_congr rflα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2F:O → ℝ := fun z => ↑sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E zH:O → ℂ := fun z => sieveCutoff (‖eisEmbedding z‖ ^ 2 / M) * ↑(E z)hF:∀ (z : O), 0 ≤ F zhH:∀ (z : O), H z = ↑(F z)hHs:Summable HhFs:Summable Fhsum:‖∑' (z : O), H z‖ = ∑' (z : O), F z⊢ ∀ x ∈ R, ‖∑ n ∈ C, a n * quadraticRow P hg (support n) x‖ ^ 2 = F x intro z hzα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2F:O → ℝ := fun z => ↑sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E zH:O → ℂ := fun z => sieveCutoff (‖eisEmbedding z‖ ^ 2 / M) * ↑(E z)hF:∀ (z : O), 0 ≤ F zhH:∀ (z : O), H z = ↑(F z)hHs:Summable HhFs:Summable Fhsum:‖∑' (z : O), H z‖ = ∑' (z : O), F zz:Ohz:z ∈ R⊢ ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2 = F z have ht0 : 0 ≤ ‖eisEmbedding z‖ ^ 2 / M := div_nonneg (sq_nonneg _) hMp.leα:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2F:O → ℝ := fun z => ↑sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E zH:O → ℂ := fun z => sieveCutoff (‖eisEmbedding z‖ ^ 2 / M) * ↑(E z)hF:∀ (z : O), 0 ≤ F zhH:∀ (z : O), H z = ↑(F z)hHs:Summable HhFs:Summable Fhsum:‖∑' (z : O), H z‖ = ∑' (z : O), F zz:Ohz:z ∈ Rht0:0 ≤ ‖eisEmbedding z‖ ^ 2 / M⊢ ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2 = F z have ht1 : ‖eisEmbedding z‖ ^ 2 / M ≤ 1 := (div_le_one hMp).mpr (hR z hz)α:Type u_1β:Type u_2inst✝²:DecidableEq αinst✝¹:DecidableEq βP:α → Ideal Oinst✝:∀ (i : α), (P i).IsMaximalhinj:Function.Injective Phg:∀ (i : α), lambda ∉ P ihc:∀ (i : α), ringChar (O ⧸ P i) ≠ 2C:Finset βsupport:β → Finset αhsupp:InjOn support ↑CN:ℝhN:0 ≤ NhNorm:∀ n ∈ C, supportNorm P (support n) ≤ Na:β → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ MhMp:0 < ME:O → ℝ := fun z => ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2F:O → ℝ := fun z => ↑sieveBump (‖eisEmbedding z‖ ^ 2 / M) * E zH:O → ℂ := fun z => sieveCutoff (‖eisEmbedding z‖ ^ 2 / M) * ↑(E z)hF:∀ (z : O), 0 ≤ F zhH:∀ (z : O), H z = ↑(F z)hHs:Summable HhFs:Summable Fhsum:‖∑' (z : O), H z‖ = ∑' (z : O), F zz:Ohz:z ∈ Rht0:0 ≤ ‖eisEmbedding z‖ ^ 2 / Mht1:‖eisEmbedding z‖ ^ 2 / M ≤ 1⊢ ‖∑ n ∈ C, a n * quadraticRow P hg (support n) z‖ ^ 2 = F z simp only [F, sieveBump_eq_one ht0 ht1, one_mul, E]All goals completed! 🐙 _ ≤ ∑' z : O, F z := hFs.sum_le_tsum R (fun z _ => hF z) _ = ‖∑' z : O, H z‖ := hsum.symm _ ≤ _ := quadratic_initial_mean_bound P hinj hg hc C support hsupp N hN hNorm a sieveCutoff M hMopen ActualEisensteinCubic ConcreteTraceCRT EisensteinSchwartzPoissonopen ConcretePrimeRowBridge ShortDraftHeckeBridgedef initialSieveConstant : ℝ := diagonalControl sieveCutoff + 128 * pvControl sieveCutoff + 1theorem initialSieveConstant_pos : 0 < initialSieveConstant := by⊢ 0 < initialSieveConstant have := diagonalControl_nonneg sieveCutoffthis:0 ≤ diagonalControl sieveCutoff⊢ 0 < initialSieveConstant have := pvControl_nonneg sieveCutoffthis✝:0 ≤ diagonalControl sieveCutoffthis:0 ≤ pvControl sieveCutoff⊢ 0 < initialSieveConstant unfold initialSieveConstantthis✝:0 ≤ diagonalControl sieveCutoffthis:0 ≤ pvControl sieveCutoff⊢ 0 < diagonalControl sieveCutoff + 128 * pvControl sieveCutoff + 1 positivityAll goals completed! 🐙theorem ideal_initial_quadratic_sieve (F : Finset (Ideal O)) (hFpos : ∀ I ∈ F, I ≠ ⊥) (hFgood : ∀ I ∈ F, ∀ P ∈ UniqueFactorizationMonoid.normalizedFactors I, goodLambda ∉ P) (hsq : ∀ I ∈ F, Squarefree I) (hchar : ∀ i : primePool F, ringChar (O ⧸ i.val) ≠ 2) (N : ℕ) (hN : 1 ≤ N) (hnorm : ∀ I ∈ F, Ideal.absNorm I ≤ N) (a : Ideal O → ℂ) (M : ℝ) (hM : 1 ≤ M) (R : Finset O) (hR : ∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ M) : (∑ z ∈ R, ‖∑ I ∈ F, a I * (idealSexticRow F hFpos hFgood I z) ^ 3‖ ^ 2) ≤ initialSieveConstant * (M + (N : ℝ) ^ 2) * ∑ I ∈ F, ‖a I‖ ^ 2 := byF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ M⊢ ∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ initialSieveConstant * (M + ↑N ^ 2) * ∑ I ∈ F, ‖a I‖ ^ 2 let : ∀ i : primePool F, (i.val).IsMaximal := primePool_maximal F hFposF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···⊢ ∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ initialSieveConstant * (M + ↑N ^ 2) * ∑ I ∈ F, ‖a I‖ ^ 2 have hNr : (1 : ℝ) ≤ N := by exact_mod_cast hNF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑N⊢ ∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ initialSieveConstant * (M + ↑N ^ 2) * ∑ I ∈ F, ‖a I‖ ^ 2 have hsupp : Set.InjOn (idealSupport F) (↑F : Set (Ideal O)) := by intro I hI J hJ hF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑NI:Ideal ConcretePrimeRowBridge.OhI:I ∈ ↑FJ:Ideal ConcretePrimeRowBridge.OhJ:J ∈ ↑Fh:idealSupport F I = idealSupport F J⊢ I = J exact idealSupport_injective_on_squarefree F hI hJ (hsq I hI) (hsq J hJ) hF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑F⊢ ∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ initialSieveConstant * (M + ↑N ^ 2) * ∑ I ∈ F, ‖a I‖ ^ 2 have hNorm (I : Ideal O) (hI : I ∈ F) : supportNorm (fun i : primePool F => i.val) (idealSupport F I) ≤ (N : ℝ) := by unfold supportNormF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FI:Ideal OhI:I ∈ F⊢ ↑(absNorm (∏ i ∈ idealSupport F I, (fun i => ↑i) i)) ≤ ↑N rw [idealSupport_product_eq F hI (hsq I hI)F:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FI:Ideal OhI:I ∈ F⊢ ↑(absNorm I) ≤ ↑N]F:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FI:Ideal OhI:I ∈ F⊢ ↑(absNorm I) ≤ ↑N exact_mod_cast hnorm I hIF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑N⊢ ∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ initialSieveConstant * (M + ↑N ^ 2) * ∑ I ∈ F, ‖a I‖ ^ 2 have hb := quadratic_initial_finite_bound (fun i : primePool F => i.val) Subtype.val_injective (primePool_good F hFgood) hchar F (idealSupport F) hsupp (N : ℝ) (byF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑N⊢ 0 ≤ ↑N positivityAll goals completed! 🐙) hNorm a M hM R hRF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑Nhb:∑ z ∈ R, ‖∑ n ∈ F, a n * quadraticRow (fun i => ↑i) ⋯ (idealSupport F n) z‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F)) * ∑ n ∈ F, ‖a n‖ ^ 2⊢ ∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ initialSieveConstant * (M + ↑N ^ 2) * ∑ I ∈ F, ‖a I‖ ^ 2 change (∑ z ∈ R, ‖∑ I ∈ F, a I * (idealSexticRow F hFpos hFgood I z) ^ 3‖ ^ 2) ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * (N : ℝ) * F.card) * ∑ I ∈ F, ‖a I‖ ^ 2 at hbF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑Nhb:∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F)) * ∑ I ∈ F, ‖a I‖ ^ 2⊢ ∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ initialSieveConstant * (M + ↑N ^ 2) * ∑ I ∈ F, ‖a I‖ ^ 2 have hsub : F ⊆ idealsUpTo N := by intro I hIF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑Nhb:∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F)) * ∑ I ∈ F, ‖a I‖ ^ 2I:Ideal OhI:I ∈ F⊢ I ∈ idealsUpTo N apply mem_idealsUpTo.mprF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑Nhb:∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F)) * ∑ I ∈ F, ‖a I‖ ^ 2I:Ideal OhI:I ∈ F⊢ 1 ≤ absNorm I ∧ absNorm I ≤ N constructorleftF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑Nhb:∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F)) * ∑ I ∈ F, ‖a I‖ ^ 2I:Ideal OhI:I ∈ F⊢ 1 ≤ absNorm IrightF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑Nhb:∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F)) * ∑ I ∈ F, ‖a I‖ ^ 2I:Ideal OhI:I ∈ F⊢ absNorm I ≤ N ·leftF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑Nhb:∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F)) * ∑ I ∈ F, ‖a I‖ ^ 2I:Ideal OhI:I ∈ F⊢ 1 ≤ absNorm I apply Nat.one_le_iff_ne_zero.mprleftF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑Nhb:∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F)) * ∑ I ∈ F, ‖a I‖ ^ 2I:Ideal OhI:I ∈ F⊢ absNorm I ≠ 0 rw [ne_eq,leftF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑Nhb:∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F)) * ∑ I ∈ F, ‖a I‖ ^ 2I:Ideal OhI:I ∈ F⊢ ¬absNorm I = 0 Ideal.absNorm_eq_zero_iffleftF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑Nhb:∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F)) * ∑ I ∈ F, ‖a I‖ ^ 2I:Ideal OhI:I ∈ F⊢ ¬I = ⊥]leftF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑Nhb:∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F)) * ∑ I ∈ F, ‖a I‖ ^ 2I:Ideal OhI:I ∈ F⊢ ¬I = ⊥ exact hFpos I hIAll goals completed! 🐙 ·rightF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑Nhb:∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F)) * ∑ I ∈ F, ‖a I‖ ^ 2I:Ideal OhI:I ∈ F⊢ absNorm I ≤ N exact hnorm I hIF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑Nhb:∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F)) * ∑ I ∈ F, ‖a I‖ ^ 2hsub:F ⊆ idealsUpTo N⊢ ∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ initialSieveConstant * (M + ↑N ^ 2) * ∑ I ∈ F, ‖a I‖ ^ 2 have hcardNat : F.card ≤ 64 * (N + 1) := (Finset.card_le_card hsub).trans (CompactScaleBridge.idealsUpTo_card_le N)F:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑Nhb:∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F)) * ∑ I ∈ F, ‖a I‖ ^ 2hsub:F ⊆ idealsUpTo NhcardNat:#F ≤ 64 * (N + 1)⊢ ∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ initialSieveConstant * (M + ↑N ^ 2) * ∑ I ∈ F, ‖a I‖ ^ 2 have hcard : (F.card : ℝ) ≤ 128 * (N : ℝ) := by have hh : (F.card : ℝ) ≤ 64 * ((N : ℝ) + 1) := byF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ M⊢ ∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ initialSieveConstant * (M + ↑N ^ 2) * ∑ I ∈ F, ‖a I‖ ^ 2 exact_mod_cast hcardNatF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑Nhb:∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F)) * ∑ I ∈ F, ‖a I‖ ^ 2hsub:F ⊆ idealsUpTo NhcardNat:#F ≤ 64 * (N + 1)hh:↑(#F) ≤ 64 * (↑N + 1)⊢ ↑(#F) ≤ 128 * ↑N linarithF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑Nhb:∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F)) * ∑ I ∈ F, ‖a I‖ ^ 2hsub:F ⊆ idealsUpTo NhcardNat:#F ≤ 64 * (N + 1)hcard:↑(#F) ≤ 128 * ↑N⊢ ∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ initialSieveConstant * (M + ↑N ^ 2) * ∑ I ∈ F, ‖a I‖ ^ 2 refine hb.trans (mul_le_mul_of_nonneg_right ?_ (Finset.sum_nonneg fun _ _ => sq_nonneg _))F:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑Nhb:∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F)) * ∑ I ∈ F, ‖a I‖ ^ 2hsub:F ⊆ idealsUpTo NhcardNat:#F ≤ 64 * (N + 1)hcard:↑(#F) ≤ 128 * ↑N⊢ diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F) ≤ initialSieveConstant * (M + ↑N ^ 2) have hdiag := diagonalControl_nonneg sieveCutoffF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑Nhb:∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F)) * ∑ I ∈ F, ‖a I‖ ^ 2hsub:F ⊆ idealsUpTo NhcardNat:#F ≤ 64 * (N + 1)hcard:↑(#F) ≤ 128 * ↑Nhdiag:0 ≤ diagonalControl sieveCutoff⊢ diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F) ≤ initialSieveConstant * (M + ↑N ^ 2) have hpv := pvControl_nonneg sieveCutoffF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑Nhb:∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F)) * ∑ I ∈ F, ‖a I‖ ^ 2hsub:F ⊆ idealsUpTo NhcardNat:#F ≤ 64 * (N + 1)hcard:↑(#F) ≤ 128 * ↑Nhdiag:0 ≤ diagonalControl sieveCutoffhpv:0 ≤ pvControl sieveCutoff⊢ diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F) ≤ initialSieveConstant * (M + ↑N ^ 2) have hmul := mul_le_mul_of_nonneg_left hcard (mul_nonneg hpv (byF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑Nhb:∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F)) * ∑ I ∈ F, ‖a I‖ ^ 2hsub:F ⊆ idealsUpTo NhcardNat:#F ≤ 64 * (N + 1)hcard:↑(#F) ≤ 128 * ↑Nhdiag:0 ≤ diagonalControl sieveCutoffhpv:0 ≤ pvControl sieveCutoff⊢ 0 ≤ ↑N positivityAll goals completed! 🐙 : (0 : ℝ) ≤ N))F:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑Nhb:∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F)) * ∑ I ∈ F, ‖a I‖ ^ 2hsub:F ⊆ idealsUpTo NhcardNat:#F ≤ 64 * (N + 1)hcard:↑(#F) ≤ 128 * ↑Nhdiag:0 ≤ diagonalControl sieveCutoffhpv:0 ≤ pvControl sieveCutoffhmul:pvControl sieveCutoff * ↑N * ↑(#F) ≤ pvControl sieveCutoff * ↑N * (128 * ↑N)⊢ diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F) ≤ initialSieveConstant * (M + ↑N ^ 2) unfold initialSieveConstantF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ Mthis:∀ (i : ↥(primePool F)), (↑i).IsMaximal := ···hNr:1 ≤ ↑Nhsupp:InjOn (idealSupport F) ↑FhNorm:∀ I ∈ F, supportNorm (fun i => ↑i) (idealSupport F I) ≤ ↑Nhb:∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ (diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F)) * ∑ I ∈ F, ‖a I‖ ^ 2hsub:F ⊆ idealsUpTo NhcardNat:#F ≤ 64 * (N + 1)hcard:↑(#F) ≤ 128 * ↑Nhdiag:0 ≤ diagonalControl sieveCutoffhpv:0 ≤ pvControl sieveCutoffhmul:pvControl sieveCutoff * ↑N * ↑(#F) ≤ pvControl sieveCutoff * ↑N * (128 * ↑N)⊢ diagonalControl sieveCutoff * M + pvControl sieveCutoff * ↑N * ↑(#F) ≤ (diagonalControl sieveCutoff + 128 * pvControl sieveCutoff + 1) * (M + ↑N ^ 2) nlinarith [mul_nonneg hdiag (sq_nonneg (N : ℝ)), mul_nonneg hpv (show 0 ≤ M byF:Finset (Ideal O)hFpos:∀ I ∈ F, I ≠ ⊥hFgood:∀ I ∈ F, ∀ P ∈ normalizedFactors I, goodLambda ∉ Phsq:∀ I ∈ F, Squarefree Ihchar:∀ (i : ↥(primePool F)), ringChar (O ⧸ ↑i) ≠ 2N:ℕhN:1 ≤ Nhnorm:∀ I ∈ F, absNorm I ≤ Na:Ideal O → ℂM:ℝhM:1 ≤ MR:Finset OhR:∀ z ∈ R, ‖eisEmbedding z‖ ^ 2 ≤ M⊢ ∑ z ∈ R, ‖∑ I ∈ F, a I * idealSexticRow F hFpos hFgood I z ^ 3‖ ^ 2 ≤ initialSieveConstant * (M + ↑N ^ 2) * ∑ I ∈ F, ‖a I‖ ^ 2 linarithAll goals completed! 🐙)]end QuadraticInitialBoundopen scoped Classical FourierTransform SchwartzMap ContDiffopen MeasureTheorynamespace CubicReflectionKerneldef gammaMultiplier (s : ℂ) : ℂ := Complex.Gamma ((7 / 6 : ℂ) + s) * Complex.Gamma ((5 / 6 : ℂ) + s) / (Complex.Gamma ((7 / 6 : ℂ) - s) * Complex.Gamma ((5 / 6 : ℂ) - s))theorem gammaMultiplier_neg (s : ℂ) : gammaMultiplier (-s) = (gammaMultiplier s)⁻¹ := bys:ℂ⊢ gammaMultiplier (-s) = (gammaMultiplier s)⁻¹ simp only [gammaMultiplier, sub_neg_eq_add, ← sub_eq_add_neg, inv_div]All goals completed! 🐙theorem Gamma_vertical_ne_zero (a u : ℝ) (ha : 0 < a) : Complex.Gamma ((a : ℂ) + u * Complex.I) ≠ 0 := bya:ℝu:ℝha:0 < a⊢ Complex.Gamma (↑a + ↑u * I) ≠ 0 apply Complex.Gamma_ne_zero_of_re_posa:ℝu:ℝha:0 < a⊢ 0 < (↑a + ↑u * I).re simpa using haAll goals completed! 🐙theorem Gamma_vertical_conj (a u : ℝ) : Complex.Gamma ((a : ℂ) - u * Complex.I) = star (Complex.Gamma ((a : ℂ) + u * Complex.I)) := bya:ℝu:ℝ⊢ Complex.Gamma (↑a - ↑u * I) = star (Complex.Gamma (↑a + ↑u * I)) change _ = (starRingEnd ℂ) (Complex.Gamma ((a : ℂ) + u * Complex.I))a:ℝu:ℝ⊢ Complex.Gamma (↑a - ↑u * I) = (starRingEnd ℂ) (Complex.Gamma (↑a + ↑u * I)) rw [← Complex.Gamma_conja:ℝu:ℝ⊢ Complex.Gamma (↑a - ↑u * I) = Complex.Gamma ((starRingEnd ℂ) (↑a + ↑u * I))]a:ℝu:ℝ⊢ Complex.Gamma (↑a - ↑u * I) = Complex.Gamma ((starRingEnd ℂ) (↑a + ↑u * I)) congr 1a:ℝu:ℝ⊢ ↑a - ↑u * I = (starRingEnd ℂ) (↑a + ↑u * I) simp [sub_eq_add_neg]All goals completed! 🐙theorem gammaMultiplier_norm_imaginary (u : ℝ) : ‖gammaMultiplier (u * Complex.I)‖ = 1 := byu:ℝ⊢ ‖gammaMultiplier (↑u * I)‖ = 1 have ha := Gamma_vertical_ne_zero (7 / 6) u (byu:ℝ⊢ 0 < 7 / 6 norm_numAll goals completed! 🐙)u:ℝha:Complex.Gamma (↑(7 / 6) + ↑u * I) ≠ 0⊢ ‖gammaMultiplier (↑u * I)‖ = 1 have hb := Gamma_vertical_ne_zero (5 / 6) u (byu:ℝha:Complex.Gamma (↑(7 / 6) + ↑u * I) ≠ 0⊢ 0 < 5 / 6 norm_numAll goals completed! 🐙)u:ℝha:Complex.Gamma (↑(7 / 6) + ↑u * I) ≠ 0hb:Complex.Gamma (↑(5 / 6) + ↑u * I) ≠ 0⊢ ‖gammaMultiplier (↑u * I)‖ = 1 have hca := Gamma_vertical_conj (7 / 6) uu:ℝha:Complex.Gamma (↑(7 / 6) + ↑u * I) ≠ 0hb:Complex.Gamma (↑(5 / 6) + ↑u * I) ≠ 0hca:Complex.Gamma (↑(7 / 6) - ↑u * I) = star (Complex.Gamma (↑(7 / 6) + ↑u * I))⊢ ‖gammaMultiplier (↑u * I)‖ = 1 have hcb := Gamma_vertical_conj (5 / 6) uu:ℝha:Complex.Gamma (↑(7 / 6) + ↑u * I) ≠ 0hb:Complex.Gamma (↑(5 / 6) + ↑u * I) ≠ 0hca:Complex.Gamma (↑(7 / 6) - ↑u * I) = star (Complex.Gamma (↑(7 / 6) + ↑u * I))hcb:Complex.Gamma (↑(5 / 6) - ↑u * I) = star (Complex.Gamma (↑(5 / 6) + ↑u * I))⊢ ‖gammaMultiplier (↑u * I)‖ = 1 push_cast at ha hb hca hcbu:ℝha:Complex.Gamma (7 / 6 + ↑u * I) ≠ 0hb:Complex.Gamma (5 / 6 + ↑u * I) ≠ 0hca:Complex.Gamma (7 / 6 - ↑u * I) = star (Complex.Gamma (7 / 6 + ↑u * I))hcb:Complex.Gamma (5 / 6 - ↑u * I) = star (Complex.Gamma (5 / 6 + ↑u * I))⊢ ‖gammaMultiplier (↑u * I)‖ = 1 rw [gammaMultiplier,u:ℝha:Complex.Gamma (7 / 6 + ↑u * I) ≠ 0hb:Complex.Gamma (5 / 6 + ↑u * I) ≠ 0hca:Complex.Gamma (7 / 6 - ↑u * I) = star (Complex.Gamma (7 / 6 + ↑u * I))hcb:Complex.Gamma (5 / 6 - ↑u * I) = star (Complex.Gamma (5 / 6 + ↑u * I))⊢ ‖Complex.Gamma (7 / 6 + ↑u * I) * Complex.Gamma (5 / 6 + ↑u * I) / (Complex.Gamma (7 / 6 - ↑u * I) * Complex.Gamma (5 / 6 - ↑u * I))‖ = 1 hca,u:ℝha:Complex.Gamma (7 / 6 + ↑u * I) ≠ 0hb:Complex.Gamma (5 / 6 + ↑u * I) ≠ 0hca:Complex.Gamma (7 / 6 - ↑u * I) = star (Complex.Gamma (7 / 6 + ↑u * I))hcb:Complex.Gamma (5 / 6 - ↑u * I) = star (Complex.Gamma (5 / 6 + ↑u * I))⊢ ‖Complex.Gamma (7 / 6 + ↑u * I) * Complex.Gamma (5 / 6 + ↑u * I) / (star (Complex.Gamma (7 / 6 + ↑u * I)) * Complex.Gamma (5 / 6 - ↑u * I))‖ = 1 hcb,u:ℝha:Complex.Gamma (7 / 6 + ↑u * I) ≠ 0hb:Complex.Gamma (5 / 6 + ↑u * I) ≠ 0hca:Complex.Gamma (7 / 6 - ↑u * I) = star (Complex.Gamma (7 / 6 + ↑u * I))hcb:Complex.Gamma (5 / 6 - ↑u * I) = star (Complex.Gamma (5 / 6 + ↑u * I))⊢ ‖Complex.Gamma (7 / 6 + ↑u * I) * Complex.Gamma (5 / 6 + ↑u * I) / (star (Complex.Gamma (7 / 6 + ↑u * I)) * star (Complex.Gamma (5 / 6 + ↑u * I)))‖ = 1 norm_div,u:ℝha:Complex.Gamma (7 / 6 + ↑u * I) ≠ 0hb:Complex.Gamma (5 / 6 + ↑u * I) ≠ 0hca:Complex.Gamma (7 / 6 - ↑u * I) = star (Complex.Gamma (7 / 6 + ↑u * I))hcb:Complex.Gamma (5 / 6 - ↑u * I) = star (Complex.Gamma (5 / 6 + ↑u * I))⊢ ‖Complex.Gamma (7 / 6 + ↑u * I) * Complex.Gamma (5 / 6 + ↑u * I)‖ / ‖star (Complex.Gamma (7 / 6 + ↑u * I)) * star (Complex.Gamma (5 / 6 + ↑u * I))‖ = 1 norm_mul,u:ℝha:Complex.Gamma (7 / 6 + ↑u * I) ≠ 0hb:Complex.Gamma (5 / 6 + ↑u * I) ≠ 0hca:Complex.Gamma (7 / 6 - ↑u * I) = star (Complex.Gamma (7 / 6 + ↑u * I))hcb:Complex.Gamma (5 / 6 - ↑u * I) = star (Complex.Gamma (5 / 6 + ↑u * I))⊢ ‖Complex.Gamma (7 / 6 + ↑u * I)‖ * ‖Complex.Gamma (5 / 6 + ↑u * I)‖ / ‖star (Complex.Gamma (7 / 6 + ↑u * I)) * star (Complex.Gamma (5 / 6 + ↑u * I))‖ = 1 norm_mul,u:ℝha:Complex.Gamma (7 / 6 + ↑u * I) ≠ 0hb:Complex.Gamma (5 / 6 + ↑u * I) ≠ 0hca:Complex.Gamma (7 / 6 - ↑u * I) = star (Complex.Gamma (7 / 6 + ↑u * I))hcb:Complex.Gamma (5 / 6 - ↑u * I) = star (Complex.Gamma (5 / 6 + ↑u * I))⊢ ‖Complex.Gamma (7 / 6 + ↑u * I)‖ * ‖Complex.Gamma (5 / 6 + ↑u * I)‖ / (‖star (Complex.Gamma (7 / 6 + ↑u * I))‖ * ‖star (Complex.Gamma (5 / 6 + ↑u * I))‖) = 1 norm_star,u:ℝha:Complex.Gamma (7 / 6 + ↑u * I) ≠ 0hb:Complex.Gamma (5 / 6 + ↑u * I) ≠ 0hca:Complex.Gamma (7 / 6 - ↑u * I) = star (Complex.Gamma (7 / 6 + ↑u * I))hcb:Complex.Gamma (5 / 6 - ↑u * I) = star (Complex.Gamma (5 / 6 + ↑u * I))⊢ ‖Complex.Gamma (7 / 6 + ↑u * I)‖ * ‖Complex.Gamma (5 / 6 + ↑u * I)‖ / (‖Complex.Gamma (7 / 6 + ↑u * I)‖ * ‖star (Complex.Gamma (5 / 6 + ↑u * I))‖) = 1 norm_staru:ℝha:Complex.Gamma (7 / 6 + ↑u * I) ≠ 0hb:Complex.Gamma (5 / 6 + ↑u * I) ≠ 0hca:Complex.Gamma (7 / 6 - ↑u * I) = star (Complex.Gamma (7 / 6 + ↑u * I))hcb:Complex.Gamma (5 / 6 - ↑u * I) = star (Complex.Gamma (5 / 6 + ↑u * I))⊢ ‖Complex.Gamma (7 / 6 + ↑u * I)‖ * ‖Complex.Gamma (5 / 6 + ↑u * I)‖ / (‖Complex.Gamma (7 / 6 + ↑u * I)‖ * ‖Complex.Gamma (5 / 6 + ↑u * I)‖) = 1]u:ℝha:Complex.Gamma (7 / 6 + ↑u * I) ≠ 0hb:Complex.Gamma (5 / 6 + ↑u * I) ≠ 0hca:Complex.Gamma (7 / 6 - ↑u * I) = star (Complex.Gamma (7 / 6 + ↑u * I))hcb:Complex.Gamma (5 / 6 - ↑u * I) = star (Complex.Gamma (5 / 6 + ↑u * I))⊢ ‖Complex.Gamma (7 / 6 + ↑u * I)‖ * ‖Complex.Gamma (5 / 6 + ↑u * I)‖ / (‖Complex.Gamma (7 / 6 + ↑u * I)‖ * ‖Complex.Gamma (5 / 6 + ↑u * I)‖) = 1 exact div_self (mul_ne_zero (norm_ne_zero_iff.mpr ha) (norm_ne_zero_iff.mpr hb))All goals completed! 🐙theorem Gamma_vertical_continuous (a : ℝ) (ha : 0 < a) : Continuous (fun u : ℝ => Complex.Gamma ((a : ℂ) + u * Complex.I)) := bya:ℝha:0 < a⊢ Continuous fun u => Complex.Gamma (↑a + ↑u * I) apply continuous_iff_continuousAt.mpra:ℝha:0 < a⊢ ∀ (x : ℝ), ContinuousAt (fun u => Complex.Gamma (↑a + ↑u * I)) x intro ua:ℝha:0 < au:ℝ⊢ ContinuousAt (fun u => Complex.Gamma (↑a + ↑u * I)) u apply (Complex.continuousAt_Gamma _ ?_).compa:ℝha:0 < au:ℝ⊢ ContinuousAt (fun u => ↑a + ↑u * I) ua:ℝha:0 < au:ℝ⊢ ∀ (m : ℕ), ↑a + ↑u * I ≠ -↑m ·a:ℝha:0 < au:ℝ⊢ ContinuousAt (fun u => ↑a + ↑u * I) u fun_propAll goals completed! 🐙 ·a:ℝha:0 < au:ℝ⊢ ∀ (m : ℕ), ↑a + ↑u * I ≠ -↑m intro n hna:ℝha:0 < au:ℝn:ℕhn:↑a + ↑u * I = -↑n⊢ False have heq := congrArg Complex.re hna:ℝha:0 < au:ℝn:ℕhn:↑a + ↑u * I = -↑nheq:(↑a + ↑u * I).re = (-↑n).re⊢ False simp only [Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, mul_zero, zero_mul, sub_zero, add_zero, Complex.neg_re, Complex.natCast_re] at heqa:ℝha:0 < au:ℝn:ℕhn:↑a + ↑u * I = -↑nheq:a = -↑n⊢ False have : (0 : ℝ) ≤ n := Nat.cast_nonneg na:ℝha:0 < au:ℝn:ℕhn:↑a + ↑u * I = -↑nheq:a = -↑nthis:0 ≤ ↑n⊢ False linarithAll goals completed! 🐙def axisMultiplier (u : ℝ) : ℂ := gammaMultiplier ((2 * Real.pi * u : ℝ) * Complex.I)theorem axisMultiplier_norm (u : ℝ) : ‖axisMultiplier u‖ = 1 := gammaMultiplier_norm_imaginary _theorem axisMultiplier_continuous : Continuous axisMultiplier := by⊢ Continuous axisMultiplier have h1 := (Gamma_vertical_continuous (7 / 6) (by⊢ 0 < 7 / 6 norm_numAll goals completed! 🐙)).comp (show Continuous (fun u : ℝ => 2 * Real.pi * u) by fun_propAll goals completed! 🐙)h1:Continuous ((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)⊢ Continuous axisMultiplier have h2 := (Gamma_vertical_continuous (5 / 6) (byh1:Continuous ((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)⊢ 0 < 5 / 6 norm_numAll goals completed! 🐙)).comp (show Continuous (fun u : ℝ => 2 * Real.pi * u) by fun_propAll goals completed! 🐙)h1:Continuous ((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)h2:Continuous ((fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)⊢ Continuous axisMultiplier have h3 := (Gamma_vertical_continuous (7 / 6) (byh1:Continuous ((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)h2:Continuous ((fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)⊢ 0 < 7 / 6 norm_numAll goals completed! 🐙)).comp (show Continuous (fun u : ℝ => -(2 * Real.pi * u)) by fun_propAll goals completed! 🐙)h1:Continuous ((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)h2:Continuous ((fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)h3:Continuous ((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))⊢ Continuous axisMultiplier have h4 := (Gamma_vertical_continuous (5 / 6) (byh1:Continuous ((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)h2:Continuous ((fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)h3:Continuous ((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))⊢ 0 < 5 / 6 norm_numAll goals completed! 🐙)).comp (show Continuous (fun u : ℝ => -(2 * Real.pi * u)) by fun_propAll goals completed! 🐙)h1:Continuous ((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)h2:Continuous ((fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)h3:Continuous ((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))h4:Continuous ((fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))⊢ Continuous axisMultiplier have hd := h3.mul h4h1:Continuous ((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)h2:Continuous ((fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)h3:Continuous ((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))h4:Continuous ((fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))hd:Continuous (((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u)) * (fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))⊢ Continuous axisMultiplier have hne (u : ℝ) : Complex.Gamma ((7 / 6 : ℝ) + (-(2 * Real.pi * u) : ℝ) * Complex.I) * Complex.Gamma ((5 / 6 : ℝ) + (-(2 * Real.pi * u) : ℝ) * Complex.I) ≠ 0 := mul_ne_zero (Gamma_vertical_ne_zero _ _ (byh1:Continuous ((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)h2:Continuous ((fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)h3:Continuous ((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))h4:Continuous ((fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))hd:Continuous (((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u)) * (fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))u:ℝ⊢ 0 < 7 / 6 norm_numAll goals completed! 🐙)) (Gamma_vertical_ne_zero _ _ (byh1:Continuous ((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)h2:Continuous ((fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)h3:Continuous ((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))h4:Continuous ((fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))hd:Continuous (((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u)) * (fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))u:ℝ⊢ 0 < 5 / 6 norm_numAll goals completed! 🐙))h1:Continuous ((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)h2:Continuous ((fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)h3:Continuous ((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))h4:Continuous ((fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))hd:Continuous (((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u)) * (fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))hne:∀ (u : ℝ), Complex.Gamma (↑(7 / 6) + ↑(-(2 * π * u)) * I) * Complex.Gamma (↑(5 / 6) + ↑(-(2 * π * u)) * I) ≠ 0⊢ Continuous axisMultiplier convert (h1.mul h2).div hd hne using 1h1:Continuous ((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)h2:Continuous ((fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)h3:Continuous ((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))h4:Continuous ((fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))hd:Continuous (((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u)) * (fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))hne:∀ (u : ℝ), Complex.Gamma (↑(7 / 6) + ↑(-(2 * π * u)) * I) * Complex.Gamma (↑(5 / 6) + ↑(-(2 * π * u)) * I) ≠ 0⊢ axisMultiplier = (((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => 2 * π * u) * (fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => 2 * π * u) / (((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u)) * (fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u)) ext uh1:Continuous ((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)h2:Continuous ((fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => 2 * π * u)h3:Continuous ((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))h4:Continuous ((fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))hd:Continuous (((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u)) * (fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))hne:∀ (u : ℝ), Complex.Gamma (↑(7 / 6) + ↑(-(2 * π * u)) * I) * Complex.Gamma (↑(5 / 6) + ↑(-(2 * π * u)) * I) ≠ 0u:ℝ⊢ axisMultiplier u = ((((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => 2 * π * u) * (fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => 2 * π * u) / (((fun u => Complex.Gamma (↑(7 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u)) * (fun u => Complex.Gamma (↑(5 / 6) + ↑u * I)) ∘ fun u => -(2 * π * u))) u simp only [axisMultiplier, gammaMultiplier, Function.comp_def, Complex.ofReal_div, Complex.ofReal_ofNat, Complex.ofReal_neg, neg_mul, ← sub_eq_add_neg, Pi.div_apply, Pi.mul_apply]All goals completed! 🐙def spectralProfile (logSource : SchwartzMap ℝ ℂ) (u : ℝ) : ℂ := axisMultiplier u * (𝓕 logSource) utheorem spectralProfile_norm (logSource : SchwartzMap ℝ ℂ) (u : ℝ) : ‖spectralProfile logSource u‖ = ‖(𝓕 logSource) u‖ := bylogSource:𝓢(ℝ, ℂ)u:ℝ⊢ ‖spectralProfile logSource u‖ = ‖(𝓕 logSource) u‖ rw [spectralProfile,logSource:𝓢(ℝ, ℂ)u:ℝ⊢ ‖axisMultiplier u * (𝓕 logSource) u‖ = ‖(𝓕 logSource) u‖ norm_mul,logSource:𝓢(ℝ, ℂ)u:ℝ⊢ ‖axisMultiplier u‖ * ‖(𝓕 logSource) u‖ = ‖(𝓕 logSource) u‖ axisMultiplier_norm,logSource:𝓢(ℝ, ℂ)u:ℝ⊢ 1 * ‖(𝓕 logSource) u‖ = ‖(𝓕 logSource) u‖ one_mullogSource:𝓢(ℝ, ℂ)u:ℝ⊢ ‖(𝓕 logSource) u‖ = ‖(𝓕 logSource) u‖]All goals completed! 🐙theorem spectralProfile_continuous (logSource : SchwartzMap ℝ ℂ) : Continuous (spectralProfile logSource) := axisMultiplier_continuous.mul (𝓕 logSource).continuoustheorem spectralProfile_moment_integrable (logSource : SchwartzMap ℝ ℂ) (j : ℕ) : Integrable (fun u : ℝ => ‖u‖ ^ j * ‖spectralProfile logSource u‖) := bylogSource:𝓢(ℝ, ℂ)j:ℕ⊢ Integrable (fun u => ‖u‖ ^ j * ‖spectralProfile logSource u‖) volume simpa only [spectralProfile_norm] using (𝓕 logSource).integrable_pow_mul volume jAll goals completed! 🐙theorem spectralProfile_pow_integrable (logSource : SchwartzMap ℝ ℂ) (j : ℕ) : Integrable (fun u : ℝ => u ^ j • spectralProfile logSource u) := bylogSource:𝓢(ℝ, ℂ)j:ℕ⊢ Integrable (fun u => u ^ j • spectralProfile logSource u) volume apply (spectralProfile_moment_integrable logSource j).mono' ((show Continuous (fun u : ℝ => u ^ j • spectralProfile logSource u) bylogSource:𝓢(ℝ, ℂ)j:ℕ⊢ Integrable (fun u => u ^ j • spectralProfile logSource u) volume exact (continuous_id.pow j).smul (spectralProfile_continuous logSource)All goals completed! 🐙).aestronglyMeasurable) exact Filter.Eventually.of_forall (fun u => bylogSource:𝓢(ℝ, ℂ)j:ℕu:ℝ⊢ ‖u ^ j • spectralProfile logSource u‖ ≤ ‖u‖ ^ j * ‖spectralProfile logSource u‖ simp only [norm_smul, norm_pow, le_rfl]All goals completed! 🐙)def reflectedLogProfile (logSource : SchwartzMap ℝ ℂ) : ℝ → ℂ := 𝓕 (spectralProfile logSource)theorem reflectedLogProfile_smooth (logSource : SchwartzMap ℝ ℂ) : ContDiff ℝ ∞ (reflectedLogProfile logSource) := Real.contDiff_fourier (fun j _ => spectralProfile_moment_integrable logSource j)theorem reflectedLogProfile_deriv (logSource : SchwartzMap ℝ ℂ) (j : ℕ) : iteratedDeriv j (reflectedLogProfile logSource) = 𝓕 (fun u : ℝ => (-2 * Real.pi * Complex.I * u) ^ j * spectralProfile logSource u) := bylogSource:𝓢(ℝ, ℂ)j:ℕ⊢ iteratedDeriv j (reflectedLogProfile logSource) = 𝓕 fun u => (-2 * ↑π * I * ↑u) ^ j * spectralProfile logSource u simpa only [reflectedLogProfile, smul_eq_mul] using (Real.iteratedDeriv_fourier (N := (⊤ : ℕ∞)) (fun n _ => spectralProfile_pow_integrable logSource n) (bylogSource:𝓢(ℝ, ℂ)j:ℕ⊢ ↑j ≤ ⊤ simpAll goals completed! 🐙 : (j : ℕ∞) ≤ ⊤))end CubicReflectionKernelendend OAI