1. OAI
  2. NumberTheory
  3. DirichletL
  4. CubicSieve
  5. SecondPoisson
import OAI.NumberTheory.DirichletL.CubicSieve.RayCoefficientsnamespace OAInoncomputable sectionnamespace SecondPassArithmeticopen 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 UniqueFactorizationMonoidsectionopen scoped BigOperators Classical SchwartzMap ContDiffsectionopen ActualEisensteinCubicopen ConcreteTraceCRT (eisEmbedding)open EisensteinSchwartzPoisson (paperRadialFourier)open FirstCauchyArithmetic (activeGaussRowFactor)theorem masked_pair_second_poisson {ι : Type*} [DecidableEq ι] (p : ι → O) (hp : ∀ i, p i ≠ 0) [∀ i, (Ideal.span {p i}).IsMaximal] (hinj : Function.Injective (fun i => Ideal.span {p i})) (hg : ∀ i, lambda ∉ Ideal.span {p i}) (hc : ∀ i, ringChar (O ⧸ Ideal.span {p i}) ≠ 2) (M S T : Finset ι) (hd : Disjoint S T) (W : 𝓢(ℝ, ℂ)) (Y : ℝ) (hY : 0 < Y) : let n := ∏ i : activeSupport T S, p i.val (∑' y : O, rowCoprimeMask (fun i => Ideal.span {p i}) M y * (star (finiteSquarefreeRow (fun i => Ideal.span {p i}) hg T y) * finiteSquarefreeRow (fun i => Ideal.span {p i}) hg S y) * W (‖eisEmbedding y‖ ^ 2 / Y)) = ((Y : ℂ) / (‖eisEmbedding n‖ : ℂ)) * ∑ E ∈ M.powerset, let e := primeSubsetGenerator (fun i => Ideal.span {p i}) E ((UniqueFactorizationMonoid.moebius (∏ i ∈ E, Ideal.span {p i}) : ℂ) / (‖eisEmbedding e‖ ^ 2 : ℝ)) * ∑' k : O, paperRadialFourier W (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S e k := ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < Y⊢ let n := ∏ i, p ↑i; ∑' (y : O), rowCoprimeMask (fun i => span {p i}) M y * (star (finiteSquarefreeRow (fun i => span {p i}) hg T y) * finiteSquarefreeRow (fun i => span {p i}) hg S y) * W (‖eisEmbedding y‖ ^ 2 / Y) = ↑Y / ↑‖eisEmbedding n‖ * ∑ E ∈ M.powerset, let e := primeSubsetGenerator (fun i => span {p i}) E; ↑(moebius (∏ i ∈ E, span {p i})) / ↑(‖eisEmbedding e‖ ^ 2) * ∑' (k : O), paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S e k ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < Y⊢ ∑' (y : O), rowCoprimeMask (fun i => span {p i}) M y * (star (finiteSquarefreeRow (fun i => span {p i}) hg T y) * finiteSquarefreeRow (fun i => span {p i}) hg S y) * W (‖eisEmbedding y‖ ^ 2 / Y) = ↑Y / ↑‖eisEmbedding (∏ i, p ↑i)‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, span {p i})) / ↑(‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2) * ∑' (k : O), paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2 * ‖eisEmbedding (∏ i, p ↑i)‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S (primeSubsetGenerator (fun i => span {p i}) E) k ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}⊢ ∑' (y : O), rowCoprimeMask (fun i => span {p i}) M y * (star (finiteSquarefreeRow (fun i => span {p i}) hg T y) * finiteSquarefreeRow (fun i => span {p i}) hg S y) * W (‖eisEmbedding y‖ ^ 2 / Y) = ↑Y / ↑‖eisEmbedding (∏ i, p ↑i)‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, span {p i})) / ↑(‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2) * ∑' (k : O), paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2 * ‖eisEmbedding (∏ i, p ↑i)‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S (primeSubsetGenerator (fun i => span {p i}) E) k ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T S⊢ ∑' (y : O), rowCoprimeMask (fun i => span {p i}) M y * (star (finiteSquarefreeRow (fun i => span {p i}) hg T y) * finiteSquarefreeRow (fun i => span {p i}) hg S y) * W (‖eisEmbedding y‖ ^ 2 / Y) = ↑Y / ↑‖eisEmbedding (∏ i, p ↑i)‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, span {p i})) / ↑(‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2) * ∑' (k : O), paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2 * ‖eisEmbedding (∏ i, p ↑i)‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S (primeSubsetGenerator (fun i => span {p i}) E) k ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑i⊢ ∑' (y : O), rowCoprimeMask (fun i => span {p i}) M y * (star (finiteSquarefreeRow (fun i => span {p i}) hg T y) * finiteSquarefreeRow (fun i => span {p i}) hg S y) * W (‖eisEmbedding y‖ ^ 2 / Y) = ↑Y / ↑‖eisEmbedding (∏ i, p ↑i)‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, span {p i})) / ↑(‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2) * ∑' (k : O), paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2 * ‖eisEmbedding (∏ i, p ↑i)‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S (primeSubsetGenerator (fun i => span {p i}) E) k ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···⊢ ∑' (y : O), rowCoprimeMask (fun i => span {p i}) M y * (star (finiteSquarefreeRow (fun i => span {p i}) hg T y) * finiteSquarefreeRow (fun i => span {p i}) hg S y) * W (‖eisEmbedding y‖ ^ 2 / Y) = ↑Y / ↑‖eisEmbedding (∏ i, p ↑i)‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, span {p i})) / ↑(‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2) * ∑' (k : O), paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2 * ‖eisEmbedding (∏ i, p ↑i)‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S (primeSubsetGenerator (fun i => span {p i}) E) k ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···j:↥(activeSupport T S) → ℕ := activeExponent T S⊢ ∑' (y : O), rowCoprimeMask (fun i => span {p i}) M y * (star (finiteSquarefreeRow (fun i => span {p i}) hg T y) * finiteSquarefreeRow (fun i => span {p i}) hg S y) * W (‖eisEmbedding y‖ ^ 2 / Y) = ↑Y / ↑‖eisEmbedding (∏ i, p ↑i)‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, span {p i})) / ↑(‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2) * ∑' (k : O), paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2 * ‖eisEmbedding (∏ i, p ↑i)‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S (primeSubsetGenerator (fun i => span {p i}) E) k ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···j:↥(activeSupport T S) → ℕ := activeExponent T Srow:ActualEisensteinCubic.O → ℂ := finiteSexticRow Q ⋯ j⊢ ∑' (y : O), rowCoprimeMask (fun i => span {p i}) M y * (star (finiteSquarefreeRow (fun i => span {p i}) hg T y) * finiteSquarefreeRow (fun i => span {p i}) hg S y) * W (‖eisEmbedding y‖ ^ 2 / Y) = ↑Y / ↑‖eisEmbedding (∏ i, p ↑i)‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, span {p i})) / ↑(‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2) * ∑' (k : O), paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2 * ‖eisEmbedding (∏ i, p ↑i)‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S (primeSubsetGenerator (fun i => span {p i}) E) k ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···j:↥(activeSupport T S) → ℕ := activeExponent T Srow:ActualEisensteinCubic.O → ℂ := finiteSexticRow Q ⋯ jn:O := ∏ i, q i⊢ ∑' (y : O), rowCoprimeMask (fun i => span {p i}) M y * (star (finiteSquarefreeRow (fun i => span {p i}) hg T y) * finiteSquarefreeRow (fun i => span {p i}) hg S y) * W (‖eisEmbedding y‖ ^ 2 / Y) = ↑Y / ↑‖eisEmbedding (∏ i, p ↑i)‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, span {p i})) / ↑(‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2) * ∑' (k : O), paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2 * ‖eisEmbedding (∏ i, p ↑i)‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S (primeSubsetGenerator (fun i => span {p i}) E) k ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···j:↥(activeSupport T S) → ℕ := activeExponent T Srow:ActualEisensteinCubic.O → ℂ := finiteSexticRow Q ⋯ jn:O := ∏ i, q ihr:∀ (y : O), star (finiteSquarefreeRow P hg T y) * finiteSquarefreeRow P hg S y = row y⊢ ∑' (y : O), rowCoprimeMask (fun i => span {p i}) M y * (star (finiteSquarefreeRow (fun i => span {p i}) hg T y) * finiteSquarefreeRow (fun i => span {p i}) hg S y) * W (‖eisEmbedding y‖ ^ 2 / Y) = ↑Y / ↑‖eisEmbedding (∏ i, p ↑i)‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, span {p i})) / ↑(‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2) * ∑' (k : O), paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2 * ‖eisEmbedding (∏ i, p ↑i)‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S (primeSubsetGenerator (fun i => span {p i}) E) k ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···j:↥(activeSupport T S) → ℕ := activeExponent T Srow:ActualEisensteinCubic.O → ℂ := finiteSexticRow Q ⋯ jn:O := ∏ i, q ihr:∀ (y : O), star (finiteSquarefreeRow P hg T y) * finiteSquarefreeRow P hg S y = row yh:have c := finitePrimeModulus Q; ∑' (u : ActualEisensteinCubic.O), rowCoprimeMask P M u * finiteSexticRow Q ⋯ j u * W (‖eisEmbedding u‖ ^ 2 / Y) = ↑Y * canonicalNormalizedGauss Q hQ ⋯ j / ↑‖eisEmbedding c‖ * ∑ E ∈ M.powerset, have d := primeSubsetGenerator P E; ↑(moebius (∏ i ∈ E, P i)) * finiteSexticRow Q ⋯ j d / ↑(‖eisEmbedding d‖ ^ 2) * ∑' (h : ActualEisensteinCubic.O), star (finiteSexticRow Q ⋯ j h) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding d‖ ^ 2 * ‖eisEmbedding c‖ ^ 2))⊢ ∑' (y : O), rowCoprimeMask (fun i => span {p i}) M y * (star (finiteSquarefreeRow (fun i => span {p i}) hg T y) * finiteSquarefreeRow (fun i => span {p i}) hg S y) * W (‖eisEmbedding y‖ ^ 2 / Y) = ↑Y / ↑‖eisEmbedding (∏ i, p ↑i)‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, span {p i})) / ↑(‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2) * ∑' (k : O), paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2 * ‖eisEmbedding (∏ i, p ↑i)‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S (primeSubsetGenerator (fun i => span {p i}) E) k ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···j:↥(activeSupport T S) → ℕ := activeExponent T Srow:ActualEisensteinCubic.O → ℂ := finiteSexticRow Q ⋯ jn:O := ∏ i, q ihr:∀ (y : O), star (finiteSquarefreeRow P hg T y) * finiteSquarefreeRow P hg S y = row yh:have c := finitePrimeModulus Q; ∑' (u : ActualEisensteinCubic.O), rowCoprimeMask P M u * finiteSexticRow Q ⋯ j u * W (‖eisEmbedding u‖ ^ 2 / Y) = ↑Y * canonicalNormalizedGauss Q hQ ⋯ j / ↑‖eisEmbedding c‖ * ∑ E ∈ M.powerset, have d := primeSubsetGenerator P E; ↑(moebius (∏ i ∈ E, P i)) * finiteSexticRow Q ⋯ j d / ↑(‖eisEmbedding d‖ ^ 2) * ∑' (h : ActualEisensteinCubic.O), star (finiteSexticRow Q ⋯ j h) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding d‖ ^ 2 * ‖eisEmbedding c‖ ^ 2))hn:‖eisEmbedding (finitePrimeModulus Q)‖ = ‖eisEmbedding n‖⊢ ∑' (y : O), rowCoprimeMask (fun i => span {p i}) M y * (star (finiteSquarefreeRow (fun i => span {p i}) hg T y) * finiteSquarefreeRow (fun i => span {p i}) hg S y) * W (‖eisEmbedding y‖ ^ 2 / Y) = ↑Y / ↑‖eisEmbedding (∏ i, p ↑i)‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, span {p i})) / ↑(‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2) * ∑' (k : O), paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2 * ‖eisEmbedding (∏ i, p ↑i)‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S (primeSubsetGenerator (fun i => span {p i}) E) k ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···j:↥(activeSupport T S) → ℕ := activeExponent T Srow:ActualEisensteinCubic.O → ℂ := finiteSexticRow Q ⋯ jn:O := ∏ i, q ihr:∀ (y : O), star (finiteSquarefreeRow P hg T y) * finiteSquarefreeRow P hg S y = row yh:∑' (u : ActualEisensteinCubic.O), rowCoprimeMask P M u * finiteSexticRow Q ⋯ j u * W (‖eisEmbedding u‖ ^ 2 / Y) = ↑Y * canonicalNormalizedGauss Q hQ ⋯ j / ↑‖eisEmbedding (finitePrimeModulus Q)‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, P i)) * finiteSexticRow Q ⋯ j (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * ∑' (h : ActualEisensteinCubic.O), star (finiteSexticRow Q ⋯ j h) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding (finitePrimeModulus Q)‖ ^ 2))hn:‖eisEmbedding (finitePrimeModulus Q)‖ = ‖eisEmbedding n‖⊢ ∑' (y : O), rowCoprimeMask (fun i => span {p i}) M y * (star (finiteSquarefreeRow (fun i => span {p i}) hg T y) * finiteSquarefreeRow (fun i => span {p i}) hg S y) * W (‖eisEmbedding y‖ ^ 2 / Y) = ↑Y / ↑‖eisEmbedding (∏ i, p ↑i)‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, span {p i})) / ↑(‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2) * ∑' (k : O), paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2 * ‖eisEmbedding (∏ i, p ↑i)‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S (primeSubsetGenerator (fun i => span {p i}) E) k ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···j:↥(activeSupport T S) → ℕ := activeExponent T Srow:ActualEisensteinCubic.O → ℂ := finiteSexticRow Q ⋯ jn:O := ∏ i, q ihr:∀ (y : O), star (finiteSquarefreeRow P hg T y) * finiteSquarefreeRow P hg S y = row yh:∑' (u : ActualEisensteinCubic.O), rowCoprimeMask P M u * finiteSexticRow Q ⋯ j u * W (‖eisEmbedding u‖ ^ 2 / Y) = ↑Y * canonicalNormalizedGauss Q hQ ⋯ j / ↑‖eisEmbedding n‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, P i)) * finiteSexticRow Q ⋯ j (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * ∑' (h : ActualEisensteinCubic.O), star (finiteSexticRow Q ⋯ j h) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))hn:‖eisEmbedding (finitePrimeModulus Q)‖ = ‖eisEmbedding n‖⊢ ∑' (y : O), rowCoprimeMask (fun i => span {p i}) M y * (star (finiteSquarefreeRow (fun i => span {p i}) hg T y) * finiteSquarefreeRow (fun i => span {p i}) hg S y) * W (‖eisEmbedding y‖ ^ 2 / Y) = ↑Y / ↑‖eisEmbedding (∏ i, p ↑i)‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, span {p i})) / ↑(‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2) * ∑' (k : O), paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2 * ‖eisEmbedding (∏ i, p ↑i)‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S (primeSubsetGenerator (fun i => span {p i}) E) k ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···j:↥(activeSupport T S) → ℕ := activeExponent T Srow:ActualEisensteinCubic.O → ℂ := finiteSexticRow Q ⋯ jn:O := ∏ i, q ihr:∀ (y : O), star (finiteSquarefreeRow P hg T y) * finiteSquarefreeRow P hg S y = row yh:∑' (u : ActualEisensteinCubic.O), rowCoprimeMask P M u * finiteSexticRow Q ⋯ j u * W (‖eisEmbedding u‖ ^ 2 / Y) = ↑Y * canonicalNormalizedGauss Q hQ ⋯ j / ↑‖eisEmbedding n‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, P i)) * finiteSexticRow Q ⋯ j (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * ∑' (h : ActualEisensteinCubic.O), star (finiteSexticRow Q ⋯ j h) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))hn:‖eisEmbedding (finitePrimeModulus Q)‖ = ‖eisEmbedding n‖ht:(canonicalNormalizedGauss (fun i => span {q i}) hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))) = FiniteGaussPhase.canonicalProductGauss q ⋯ hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))⊢ ∑' (y : O), rowCoprimeMask (fun i => span {p i}) M y * (star (finiteSquarefreeRow (fun i => span {p i}) hg T y) * finiteSquarefreeRow (fun i => span {p i}) hg S y) * W (‖eisEmbedding y‖ ^ 2 / Y) = ↑Y / ↑‖eisEmbedding (∏ i, p ↑i)‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, span {p i})) / ↑(‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2) * ∑' (k : O), paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2 * ‖eisEmbedding (∏ i, p ↑i)‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S (primeSubsetGenerator (fun i => span {p i}) E) k calc _ = ∑' y : O, rowCoprimeMask P M y * row y * W (‖eisEmbedding y‖ ^ 2 / Y) := ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···j:↥(activeSupport T S) → ℕ := activeExponent T Srow:ActualEisensteinCubic.O → ℂ := finiteSexticRow Q ⋯ jn:O := ∏ i, q ihr:∀ (y : O), star (finiteSquarefreeRow P hg T y) * finiteSquarefreeRow P hg S y = row yh:∑' (u : ActualEisensteinCubic.O), rowCoprimeMask P M u * finiteSexticRow Q ⋯ j u * W (‖eisEmbedding u‖ ^ 2 / Y) = ↑Y * canonicalNormalizedGauss Q hQ ⋯ j / ↑‖eisEmbedding n‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, P i)) * finiteSexticRow Q ⋯ j (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * ∑' (h : ActualEisensteinCubic.O), star (finiteSexticRow Q ⋯ j h) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))hn:‖eisEmbedding (finitePrimeModulus Q)‖ = ‖eisEmbedding n‖ht:(canonicalNormalizedGauss (fun i => span {q i}) hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))) = FiniteGaussPhase.canonicalProductGauss q ⋯ hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))⊢ ∑' (y : O), rowCoprimeMask (fun i => span {p i}) M y * (star (finiteSquarefreeRow (fun i => span {p i}) hg T y) * finiteSquarefreeRow (fun i => span {p i}) hg S y) * W (‖eisEmbedding y‖ ^ 2 / Y) = ∑' (y : O), rowCoprimeMask P M y * row y * W (‖eisEmbedding y‖ ^ 2 / Y) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···j:↥(activeSupport T S) → ℕ := activeExponent T Srow:ActualEisensteinCubic.O → ℂ := finiteSexticRow Q ⋯ jn:O := ∏ i, q ihr:∀ (y : O), star (finiteSquarefreeRow P hg T y) * finiteSquarefreeRow P hg S y = row yh:∑' (u : ActualEisensteinCubic.O), rowCoprimeMask P M u * finiteSexticRow Q ⋯ j u * W (‖eisEmbedding u‖ ^ 2 / Y) = ↑Y * canonicalNormalizedGauss Q hQ ⋯ j / ↑‖eisEmbedding n‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, P i)) * finiteSexticRow Q ⋯ j (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * ∑' (h : ActualEisensteinCubic.O), star (finiteSexticRow Q ⋯ j h) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))hn:‖eisEmbedding (finitePrimeModulus Q)‖ = ‖eisEmbedding n‖ht:(canonicalNormalizedGauss (fun i => span {q i}) hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))) = FiniteGaussPhase.canonicalProductGauss q ⋯ hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))⊢ ∀ (b : O), rowCoprimeMask (fun i => span {p i}) M b * (star (finiteSquarefreeRow (fun i => span {p i}) hg T b) * finiteSquarefreeRow (fun i => span {p i}) hg S b) * W (‖eisEmbedding b‖ ^ 2 / Y) = rowCoprimeMask P M b * row b * W (‖eisEmbedding b‖ ^ 2 / Y) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···j:↥(activeSupport T S) → ℕ := activeExponent T Srow:ActualEisensteinCubic.O → ℂ := finiteSexticRow Q ⋯ jn:O := ∏ i, q ihr:∀ (y : O), star (finiteSquarefreeRow P hg T y) * finiteSquarefreeRow P hg S y = row yh:∑' (u : ActualEisensteinCubic.O), rowCoprimeMask P M u * finiteSexticRow Q ⋯ j u * W (‖eisEmbedding u‖ ^ 2 / Y) = ↑Y * canonicalNormalizedGauss Q hQ ⋯ j / ↑‖eisEmbedding n‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, P i)) * finiteSexticRow Q ⋯ j (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * ∑' (h : ActualEisensteinCubic.O), star (finiteSexticRow Q ⋯ j h) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))hn:‖eisEmbedding (finitePrimeModulus Q)‖ = ‖eisEmbedding n‖ht:(canonicalNormalizedGauss (fun i => span {q i}) hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))) = FiniteGaussPhase.canonicalProductGauss q ⋯ hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))y:O⊢ rowCoprimeMask (fun i => span {p i}) M y * (star (finiteSquarefreeRow (fun i => span {p i}) hg T y) * finiteSquarefreeRow (fun i => span {p i}) hg S y) * W (‖eisEmbedding y‖ ^ 2 / Y) = rowCoprimeMask P M y * row y * W (‖eisEmbedding y‖ ^ 2 / Y) All goals completed! 🐙 _ = ((Y : ℂ) / (‖eisEmbedding n‖ : ℂ)) * (canonicalNormalizedGauss Q hQ (fun i => hg i.val) j * ∑ E ∈ M.powerset, ((UniqueFactorizationMonoid.moebius (∏ i ∈ E, P i) : ℂ) * row (primeSubsetGenerator P E) / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 : ℝ)) * ∑' k : O, star (row k) * paperRadialFourier W (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))) := ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···j:↥(activeSupport T S) → ℕ := activeExponent T Srow:ActualEisensteinCubic.O → ℂ := finiteSexticRow Q ⋯ jn:O := ∏ i, q ihr:∀ (y : O), star (finiteSquarefreeRow P hg T y) * finiteSquarefreeRow P hg S y = row yh:∑' (u : ActualEisensteinCubic.O), rowCoprimeMask P M u * finiteSexticRow Q ⋯ j u * W (‖eisEmbedding u‖ ^ 2 / Y) = ↑Y * canonicalNormalizedGauss Q hQ ⋯ j / ↑‖eisEmbedding n‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, P i)) * finiteSexticRow Q ⋯ j (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * ∑' (h : ActualEisensteinCubic.O), star (finiteSexticRow Q ⋯ j h) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))hn:‖eisEmbedding (finitePrimeModulus Q)‖ = ‖eisEmbedding n‖ht:(canonicalNormalizedGauss (fun i => span {q i}) hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))) = FiniteGaussPhase.canonicalProductGauss q ⋯ hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))⊢ ∑' (y : O), rowCoprimeMask P M y * row y * W (‖eisEmbedding y‖ ^ 2 / Y) = ↑Y / ↑‖eisEmbedding n‖ * (canonicalNormalizedGauss Q hQ ⋯ j * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, P i)) * row (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * ∑' (k : O), star (row k) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···j:↥(activeSupport T S) → ℕ := activeExponent T Srow:ActualEisensteinCubic.O → ℂ := finiteSexticRow Q ⋯ jn:O := ∏ i, q ihr:∀ (y : O), star (finiteSquarefreeRow P hg T y) * finiteSquarefreeRow P hg S y = row yh:∑' (u : ActualEisensteinCubic.O), rowCoprimeMask P M u * finiteSexticRow Q ⋯ j u * W (‖eisEmbedding u‖ ^ 2 / Y) = ↑Y * canonicalNormalizedGauss Q hQ ⋯ j / ↑‖eisEmbedding n‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, P i)) * finiteSexticRow Q ⋯ j (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * ∑' (h : ActualEisensteinCubic.O), star (finiteSexticRow Q ⋯ j h) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))hn:‖eisEmbedding (finitePrimeModulus Q)‖ = ‖eisEmbedding n‖ht:(canonicalNormalizedGauss (fun i => span {q i}) hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))) = FiniteGaussPhase.canonicalProductGauss q ⋯ hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))⊢ ↑Y * canonicalNormalizedGauss Q hQ ⋯ j / ↑‖eisEmbedding n‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, P i)) * finiteSexticRow Q ⋯ j (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * ∑' (h : ActualEisensteinCubic.O), star (finiteSexticRow Q ⋯ j h) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2)) = ↑Y / ↑‖eisEmbedding n‖ * (canonicalNormalizedGauss Q hQ ⋯ j * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, P i)) * row (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * ∑' (k : O), star (row k) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))) All goals completed! 🐙 _ = _ := ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···j:↥(activeSupport T S) → ℕ := activeExponent T Srow:ActualEisensteinCubic.O → ℂ := finiteSexticRow Q ⋯ jn:O := ∏ i, q ihr:∀ (y : O), star (finiteSquarefreeRow P hg T y) * finiteSquarefreeRow P hg S y = row yh:∑' (u : ActualEisensteinCubic.O), rowCoprimeMask P M u * finiteSexticRow Q ⋯ j u * W (‖eisEmbedding u‖ ^ 2 / Y) = ↑Y * canonicalNormalizedGauss Q hQ ⋯ j / ↑‖eisEmbedding n‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, P i)) * finiteSexticRow Q ⋯ j (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * ∑' (h : ActualEisensteinCubic.O), star (finiteSexticRow Q ⋯ j h) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))hn:‖eisEmbedding (finitePrimeModulus Q)‖ = ‖eisEmbedding n‖ht:(canonicalNormalizedGauss (fun i => span {q i}) hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))) = FiniteGaussPhase.canonicalProductGauss q ⋯ hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))⊢ ↑Y / ↑‖eisEmbedding n‖ * (canonicalNormalizedGauss Q hQ ⋯ j * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, P i)) * row (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * ∑' (k : O), star (row k) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))) = ↑Y / ↑‖eisEmbedding (∏ i, p ↑i)‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, span {p i})) / ↑(‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2) * ∑' (k : O), paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2 * ‖eisEmbedding (∏ i, p ↑i)‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S (primeSubsetGenerator (fun i => span {p i}) E) k ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···j:↥(activeSupport T S) → ℕ := activeExponent T Srow:ActualEisensteinCubic.O → ℂ := finiteSexticRow Q ⋯ jn:O := ∏ i, q ihr:∀ (y : O), star (finiteSquarefreeRow P hg T y) * finiteSquarefreeRow P hg S y = row yh:∑' (u : ActualEisensteinCubic.O), rowCoprimeMask P M u * finiteSexticRow Q ⋯ j u * W (‖eisEmbedding u‖ ^ 2 / Y) = ↑Y * canonicalNormalizedGauss Q hQ ⋯ j / ↑‖eisEmbedding n‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, P i)) * finiteSexticRow Q ⋯ j (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * ∑' (h : ActualEisensteinCubic.O), star (finiteSexticRow Q ⋯ j h) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))hn:‖eisEmbedding (finitePrimeModulus Q)‖ = ‖eisEmbedding n‖ht:(canonicalNormalizedGauss (fun i => span {q i}) hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))) = FiniteGaussPhase.canonicalProductGauss q ⋯ hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))⊢ ↑Y / ↑‖eisEmbedding n‖ * (FiniteGaussPhase.canonicalProductGauss q ⋯ hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))) = ↑Y / ↑‖eisEmbedding (∏ i, p ↑i)‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, span {p i})) / ↑(‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2) * ∑' (k : O), paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2 * ‖eisEmbedding (∏ i, p ↑i)‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S (primeSubsetGenerator (fun i => span {p i}) E) k ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···j:↥(activeSupport T S) → ℕ := activeExponent T Srow:ActualEisensteinCubic.O → ℂ := finiteSexticRow Q ⋯ jn:O := ∏ i, q ihr:∀ (y : O), star (finiteSquarefreeRow P hg T y) * finiteSquarefreeRow P hg S y = row yh:∑' (u : ActualEisensteinCubic.O), rowCoprimeMask P M u * finiteSexticRow Q ⋯ j u * W (‖eisEmbedding u‖ ^ 2 / Y) = ↑Y * canonicalNormalizedGauss Q hQ ⋯ j / ↑‖eisEmbedding n‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, P i)) * finiteSexticRow Q ⋯ j (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * ∑' (h : ActualEisensteinCubic.O), star (finiteSexticRow Q ⋯ j h) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))hn:‖eisEmbedding (finitePrimeModulus Q)‖ = ‖eisEmbedding n‖ht:(canonicalNormalizedGauss (fun i => span {q i}) hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))) = FiniteGaussPhase.canonicalProductGauss q ⋯ hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))⊢ ∑ x ∈ M.powerset, ∑' (x_1 : GaussGeneratorTransport.O), ↑Y / ↑‖eisEmbedding n‖ * (FiniteGaussPhase.canonicalProductGauss q ⋯ hQ ⋯ j * (↑(moebius (∏ i ∈ x, P i)) * row (primeSubsetGenerator P x) / ↑(‖eisEmbedding (primeSubsetGenerator P x)‖ ^ 2) * (star (finiteSexticRow (fun i => span {q i}) ⋯ j x_1) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding x_1‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P x)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))))) = ∑ x ∈ M.powerset, ∑' (x_1 : O), ↑Y / ↑‖eisEmbedding (∏ i, p ↑i)‖ * (↑(moebius (∏ i ∈ x, span {p i})) / ↑(‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) x)‖ ^ 2) * (paperRadialFourier (⇑W) (Y * ‖eisEmbedding x_1‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) x)‖ ^ 2 * ‖eisEmbedding (∏ i, p ↑i)‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S (primeSubsetGenerator (fun i => span {p i}) x) x_1)) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···j:↥(activeSupport T S) → ℕ := activeExponent T Srow:ActualEisensteinCubic.O → ℂ := finiteSexticRow Q ⋯ jn:O := ∏ i, q ihr:∀ (y : O), star (finiteSquarefreeRow P hg T y) * finiteSquarefreeRow P hg S y = row yh:∑' (u : ActualEisensteinCubic.O), rowCoprimeMask P M u * finiteSexticRow Q ⋯ j u * W (‖eisEmbedding u‖ ^ 2 / Y) = ↑Y * canonicalNormalizedGauss Q hQ ⋯ j / ↑‖eisEmbedding n‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, P i)) * finiteSexticRow Q ⋯ j (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * ∑' (h : ActualEisensteinCubic.O), star (finiteSexticRow Q ⋯ j h) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))hn:‖eisEmbedding (finitePrimeModulus Q)‖ = ‖eisEmbedding n‖ht:(canonicalNormalizedGauss (fun i => span {q i}) hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))) = FiniteGaussPhase.canonicalProductGauss q ⋯ hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))⊢ ∀ x ∈ M.powerset, ∑' (x_1 : GaussGeneratorTransport.O), ↑Y / ↑‖eisEmbedding n‖ * (FiniteGaussPhase.canonicalProductGauss q ⋯ hQ ⋯ j * (↑(moebius (∏ i ∈ x, P i)) * row (primeSubsetGenerator P x) / ↑(‖eisEmbedding (primeSubsetGenerator P x)‖ ^ 2) * (star (finiteSexticRow (fun i => span {q i}) ⋯ j x_1) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding x_1‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P x)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))))) = ∑' (x_1 : O), ↑Y / ↑‖eisEmbedding (∏ i, p ↑i)‖ * (↑(moebius (∏ i ∈ x, span {p i})) / ↑(‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) x)‖ ^ 2) * (paperRadialFourier (⇑W) (Y * ‖eisEmbedding x_1‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) x)‖ ^ 2 * ‖eisEmbedding (∏ i, p ↑i)‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S (primeSubsetGenerator (fun i => span {p i}) x) x_1)) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···j:↥(activeSupport T S) → ℕ := activeExponent T Srow:ActualEisensteinCubic.O → ℂ := finiteSexticRow Q ⋯ jn:O := ∏ i, q ihr:∀ (y : O), star (finiteSquarefreeRow P hg T y) * finiteSquarefreeRow P hg S y = row yh:∑' (u : ActualEisensteinCubic.O), rowCoprimeMask P M u * finiteSexticRow Q ⋯ j u * W (‖eisEmbedding u‖ ^ 2 / Y) = ↑Y * canonicalNormalizedGauss Q hQ ⋯ j / ↑‖eisEmbedding n‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, P i)) * finiteSexticRow Q ⋯ j (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * ∑' (h : ActualEisensteinCubic.O), star (finiteSexticRow Q ⋯ j h) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))hn:‖eisEmbedding (finitePrimeModulus Q)‖ = ‖eisEmbedding n‖ht:(canonicalNormalizedGauss (fun i => span {q i}) hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))) = FiniteGaussPhase.canonicalProductGauss q ⋯ hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))E:Finset ιhE:E ∈ M.powerset⊢ ∑' (x : GaussGeneratorTransport.O), ↑Y / ↑‖eisEmbedding n‖ * (FiniteGaussPhase.canonicalProductGauss q ⋯ hQ ⋯ j * (↑(moebius (∏ i ∈ E, P i)) * row (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * (star (finiteSexticRow (fun i => span {q i}) ⋯ j x) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding x‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))))) = ∑' (x : O), ↑Y / ↑‖eisEmbedding (∏ i, p ↑i)‖ * (↑(moebius (∏ i ∈ E, span {p i})) / ↑(‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2) * (paperRadialFourier (⇑W) (Y * ‖eisEmbedding x‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2 * ‖eisEmbedding (∏ i, p ↑i)‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S (primeSubsetGenerator (fun i => span {p i}) E) x)) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···j:↥(activeSupport T S) → ℕ := activeExponent T Srow:ActualEisensteinCubic.O → ℂ := finiteSexticRow Q ⋯ jn:O := ∏ i, q ihr:∀ (y : O), star (finiteSquarefreeRow P hg T y) * finiteSquarefreeRow P hg S y = row yh:∑' (u : ActualEisensteinCubic.O), rowCoprimeMask P M u * finiteSexticRow Q ⋯ j u * W (‖eisEmbedding u‖ ^ 2 / Y) = ↑Y * canonicalNormalizedGauss Q hQ ⋯ j / ↑‖eisEmbedding n‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, P i)) * finiteSexticRow Q ⋯ j (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * ∑' (h : ActualEisensteinCubic.O), star (finiteSexticRow Q ⋯ j h) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))hn:‖eisEmbedding (finitePrimeModulus Q)‖ = ‖eisEmbedding n‖ht:(canonicalNormalizedGauss (fun i => span {q i}) hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))) = FiniteGaussPhase.canonicalProductGauss q ⋯ hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))E:Finset ιhE:E ∈ M.powerset⊢ ∀ (b : GaussGeneratorTransport.O), ↑Y / ↑‖eisEmbedding n‖ * (FiniteGaussPhase.canonicalProductGauss q ⋯ hQ ⋯ j * (↑(moebius (∏ i ∈ E, P i)) * row (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * (star (finiteSexticRow (fun i => span {q i}) ⋯ j b) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding b‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))))) = ↑Y / ↑‖eisEmbedding (∏ i, p ↑i)‖ * (↑(moebius (∏ i ∈ E, span {p i})) / ↑(‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2) * (paperRadialFourier (⇑W) (Y * ‖eisEmbedding b‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2 * ‖eisEmbedding (∏ i, p ↑i)‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S (primeSubsetGenerator (fun i => span {p i}) E) b)) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···j:↥(activeSupport T S) → ℕ := activeExponent T Srow:ActualEisensteinCubic.O → ℂ := finiteSexticRow Q ⋯ jn:O := ∏ i, q ihr:∀ (y : O), star (finiteSquarefreeRow P hg T y) * finiteSquarefreeRow P hg S y = row yh:∑' (u : ActualEisensteinCubic.O), rowCoprimeMask P M u * finiteSexticRow Q ⋯ j u * W (‖eisEmbedding u‖ ^ 2 / Y) = ↑Y * canonicalNormalizedGauss Q hQ ⋯ j / ↑‖eisEmbedding n‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, P i)) * finiteSexticRow Q ⋯ j (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * ∑' (h : ActualEisensteinCubic.O), star (finiteSexticRow Q ⋯ j h) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))hn:‖eisEmbedding (finitePrimeModulus Q)‖ = ‖eisEmbedding n‖ht:(canonicalNormalizedGauss (fun i => span {q i}) hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))) = FiniteGaussPhase.canonicalProductGauss q ⋯ hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))E:Finset ιhE:E ∈ M.powersetk:GaussGeneratorTransport.O⊢ ↑Y / ↑‖eisEmbedding n‖ * (FiniteGaussPhase.canonicalProductGauss q ⋯ hQ ⋯ j * (↑(moebius (∏ i ∈ E, P i)) * row (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * (star (finiteSexticRow (fun i => span {q i}) ⋯ j k) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))))) = ↑Y / ↑‖eisEmbedding (∏ i, p ↑i)‖ * (↑(moebius (∏ i ∈ E, span {p i})) / ↑(‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2) * (paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2 * ‖eisEmbedding (∏ i, p ↑i)‖ ^ 2)) * activeGaussRowFactor p hp hinj hg T S (primeSubsetGenerator (fun i => span {p i}) E) k)) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhinj:Function.Injective fun i => span {p i}hg:∀ (i : ι), lambda ∉ span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2M:Finset ιS:Finset ιT:Finset ιhd:Disjoint S TW:𝓢(ℝ, ℂ)Y:ℝhY:0 < YP:ι → Ideal O := fun i => span {p i}Q:↥(activeSupport T S) → Ideal ActualEisensteinCubic.O := activePrimes P T Sq:↥(activeSupport T S) → O := fun i => p ↑ihQ:Pairwise (Function.onFun IsCoprime (activePrimes P T S)) := ···j:↥(activeSupport T S) → ℕ := activeExponent T Srow:ActualEisensteinCubic.O → ℂ := finiteSexticRow Q ⋯ jn:O := ∏ i, q ihr:∀ (y : O), star (finiteSquarefreeRow P hg T y) * finiteSquarefreeRow P hg S y = row yh:∑' (u : ActualEisensteinCubic.O), rowCoprimeMask P M u * finiteSexticRow Q ⋯ j u * W (‖eisEmbedding u‖ ^ 2 / Y) = ↑Y * canonicalNormalizedGauss Q hQ ⋯ j / ↑‖eisEmbedding n‖ * ∑ E ∈ M.powerset, ↑(moebius (∏ i ∈ E, P i)) * finiteSexticRow Q ⋯ j (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * ∑' (h : ActualEisensteinCubic.O), star (finiteSexticRow Q ⋯ j h) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))hn:‖eisEmbedding (finitePrimeModulus Q)‖ = ‖eisEmbedding n‖ht:(canonicalNormalizedGauss (fun i => span {q i}) hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))) = FiniteGaussPhase.canonicalProductGauss q ⋯ hQ ⋯ j * ∑ k ∈ M.powerset, (let e := primeSubsetGenerator P k; ↑(moebius (∏ i ∈ k, P i)) * row e / ↑(‖eisEmbedding e‖ ^ 2)) * ∑' (h : GaussGeneratorTransport.O), star (finiteSexticRow (fun i => span {q i}) ⋯ j h) * let e := primeSubsetGenerator P k; paperRadialFourier (⇑W) (Y * ‖eisEmbedding h‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))E:Finset ιhE:E ∈ M.powersetk:GaussGeneratorTransport.O⊢ ↑Y / ↑‖eisEmbedding n‖ * (FiniteGaussPhase.canonicalProductGauss q ⋯ hQ ⋯ j * (↑(moebius (∏ i ∈ E, P i)) * row (primeSubsetGenerator P E) / ↑(‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2) * (star (finiteSexticRow (fun i => span {q i}) ⋯ j k) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator P E)‖ ^ 2 * ‖eisEmbedding n‖ ^ 2))))) = ↑Y / ↑‖eisEmbedding (∏ i, p ↑i)‖ * (↑(moebius (∏ i ∈ E, span {p i})) / ↑(‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2) * (paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding (primeSubsetGenerator (fun i => span {p i}) E)‖ ^ 2 * ‖eisEmbedding (∏ i, p ↑i)‖ ^ 2)) * (FiniteGaussPhase.canonicalProductGauss (fun i => p ↑i) ⋯ ⋯ ⋯ (activeExponent T S) * (finiteSexticRow (activePrimes (fun i => span {p i}) T S) ⋯ (activeExponent T S) (primeSubsetGenerator (fun i => span {p i}) E) * star (finiteSexticRow (activePrimes (fun i => span {p i}) T S) ⋯ (activeExponent T S) k))))) All goals completed! 🐙endopen scoped BigOperators Classical SchwartzMap ContDiffopen MeasureTheoryopen ActualEisensteinCubicopen ConcreteTraceCRT (eisEmbedding)open EisensteinSchwartzPoisson (paperRadialFourier)open FirstCauchyArithmetic (supportMobius activeGaussRowFactor)open FirstPassCubeLabels (columnLog primeProductNorm firstDualScale normalizedColumn)open FourierBridge (logPhase)variable {ι : Type*} [DecidableEq ι] (p : ι → O) (hp : ∀ i, p i ≠ 0) [∀ i, (Ideal.span {p i}).IsMaximal] (hg : ∀ i, lambda ∉ Ideal.span {p i})def secondGaussTerm (hinj : Function.Injective (fun i => Ideal.span {p i})) (Ψ₁ Ψ₂ : O →* ℂ) (m r c d e k : O) (H₁ H₂ : Finset ι → ℂ) (S T : Finset ι) : ℂ := star (supportMobius (fun i => Ideal.span {p i}) S * secondInputCoefficient p hg Ψ₁ (m * (e * r)) c d H₁ S) * (supportMobius (fun i => Ideal.span {p i}) T * secondInputCoefficient p hg Ψ₂ (m * (e * r)) c d H₂ T) * activeGaussRowFactor p hp hinj hg T S e kdef actualSecondKernel (hinj : Function.Injective (fun i => Ideal.span {p i})) (F : Finset ι) (Ψ₁ Ψ₂ : O →* ℂ) (m r c d e k : O) (W : 𝓢(ℝ, ℂ)) (V₁ V₂ : ℝ → ℂ) (X₁ X₂ Y : ℝ) : ℂ := ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, if Disjoint S T then (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T)) * ((Y : ℂ) / (‖eisEmbedding (∏ i ∈ S ∪ T, p i)‖ : ℂ)) * paperRadialFourier W (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun _ => 1) (fun _ => 1) S T else 0def secondLogTest (V : ℝ → ℂ) (X t : ℝ) (negative : Bool) (S : Finset ι) : ℂ := (V (columnLog p X S) * (if negative then star (logPhase t (-(columnLog p X S))) else logPhase t (-(columnLog p X S)))) / (‖eisEmbedding (∏ i ∈ S, p i)‖ : ℂ)theorem secondGaussTerm_log_twist (hinj : Function.Injective (fun i => Ideal.span {p i})) (Ψ₁ Ψ₂ : O →* ℂ) (m r c d e k : O) (V₁ V₂ : ℝ → ℂ) (X₁ X₂ t : ℝ) (S T : Finset ι) : secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T = (logPhase t (-(columnLog p X₁ S)) * logPhase t (-(columnLog p X₂ T))) * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (normalizedColumn p (fun _ => 1)) (normalizedColumn p (fun _ => 1)) S T) := ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}Ψ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OV₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝt:ℝS:Finset ιT:Finset ι⊢ secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T = logPhase t (-columnLog p X₁ S) * logPhase t (-columnLog p X₂ T) * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (normalizedColumn p fun x => 1) (normalizedColumn p fun x => 1) S T) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}Ψ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OV₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝt:ℝS:Finset ιT:Finset ι⊢ logPhase t (-columnLog p X₁ S) * star (V₁ (columnLog p X₁ S)) / star ↑‖eisEmbedding (∏ i ∈ S, p i)‖ * (star (finiteSquarefreeRow (fun i => span {p i}) hg S d) * (star (finiteSquarefreeRow (fun i => span {p i}) hg S c ^ 4) * (star (rowCoprimeMask (fun i => span {p i}) S (m * (e * r))) * star (Ψ₁ (∏ i ∈ S, p i))))) * star (supportMobius (fun i => span {p i}) S) * (supportMobius (fun i => span {p i}) T * (Ψ₂ (∏ i ∈ T, p i) * rowCoprimeMask (fun i => span {p i}) T (m * (e * r)) * finiteSquarefreeRow (fun i => span {p i}) hg T c ^ 4 * finiteSquarefreeRow (fun i => span {p i}) hg T d * (V₂ (columnLog p X₂ T) * logPhase t (-columnLog p X₂ T) / ↑‖eisEmbedding (∏ i ∈ T, p i)‖))) * activeGaussRowFactor p hp hinj hg T S e k = logPhase t (-columnLog p X₁ S) * logPhase t (-columnLog p X₂ T) * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * ((star ↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (star (finiteSquarefreeRow (fun i => span {p i}) hg S d) * (star (finiteSquarefreeRow (fun i => span {p i}) hg S c ^ 4) * (star (rowCoprimeMask (fun i => span {p i}) S (m * (e * r))) * star (Ψ₁ (∏ i ∈ S, p i))))) * star (supportMobius (fun i => span {p i}) S) * (supportMobius (fun i => span {p i}) T * (Ψ₂ (∏ i ∈ T, p i) * rowCoprimeMask (fun i => span {p i}) T (m * (e * r)) * finiteSquarefreeRow (fun i => span {p i}) hg T c ^ 4 * finiteSquarefreeRow (fun i => span {p i}) hg T d * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹)) * activeGaussRowFactor p hp hinj hg T S e k)) All goals completed! 🐙theorem actualSecondKernel_eq_integral (hinj : Function.Injective (fun i => Ideal.span {p i})) (F : Finset ι) (Ψ₁ Ψ₂ : O →* ℂ) (m r c d e k : O) (W : 𝓢(ℝ, ℂ)) (V₁ V₂ : ℝ → ℂ) (X₁ X₂ Y : ℝ) (hX₁ : 0 < X₁) (hX₂ : 0 < X₂) (b : 𝓢(ℝ, ℂ)) (hsep : ∀ x y : ℝ, star (V₁ x) * V₂ y * paperRadialFourier W (firstDualScale p ∅ Y X₁ X₂ e k * Real.exp (-x-y)) = ∫ t : ℝ, (star (V₁ x) * logPhase t (-x)) * (V₂ y * logPhase t (-y)) * b t) : Integrable (fun t : ℝ => b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)) volume ∧ actualSecondKernel p hp hg hinj F Ψ₁ Ψ₂ m r c d e k W V₁ V₂ X₁ X₂ Y = (Y : ℂ) * ∫ t : ℝ, b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) := ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b t⊢ Integrable (fun t => b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)) volume ∧ actualSecondKernel p hp hg hinj F Ψ₁ Ψ₂ m r c d e k W V₁ V₂ X₁ X₂ Y = ↑Y * ∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0⊢ Integrable (fun t => b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)) volume ∧ actualSecondKernel p hp hg hinj F Ψ₁ Ψ₂ m r c d e k W V₁ V₂ X₁ X₂ Y = ↑Y * ∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volume⊢ Integrable (fun t => b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)) volume ∧ actualSecondKernel p hp hg hinj F Ψ₁ Ψ₂ m r c d e k W V₁ V₂ X₁ X₂ Y = ↑Y * ∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)⊢ Integrable (fun t => b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)) volume ∧ actualSecondKernel p hp hg hinj F Ψ₁ Ψ₂ m r c d e k W V₁ V₂ X₁ X₂ Y = ↑Y * ∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volume⊢ Integrable (fun t => b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)) volume ∧ actualSecondKernel p hp hg hinj F Ψ₁ Ψ₂ m r c d e k W V₁ V₂ X₁ X₂ Y = ↑Y * ∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volume⊢ Integrable (fun t => b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)) volumeι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volume⊢ actualSecondKernel p hp hg hinj F Ψ₁ Ψ₂ m r c d e k W V₁ V₂ X₁ X₂ Y = ↑Y * ∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volume⊢ Integrable (fun t => b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)) volume All goals completed! 🐙 ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volume⊢ actualSecondKernel p hp hg hinj F Ψ₁ Ψ₂ m r c d e k W V₁ V₂ X₁ X₂ Y = ↑Y * ∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volume⊢ actualSecondKernel p hp hg hinj F Ψ₁ Ψ₂ m r c d e k W V₁ V₂ X₁ X₂ Y = ↑Y * ∫ (a : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G a S T ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volume⊢ actualSecondKernel p hp hg hinj F Ψ₁ Ψ₂ m r c d e k W V₁ V₂ X₁ X₂ Y = ∑ i ∈ F.powerset, ↑Y * ∫ (a : ℝ), ∑ i_1 ∈ F.powerset, G a i i_1 ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volume⊢ (∑ S ∈ F.powerset, ∑ T ∈ F.powerset, if Disjoint S T then star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * (↑Y / ↑‖eisEmbedding (∏ i ∈ S ∪ T, p i)‖) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T else 0) = ∑ i ∈ F.powerset, ↑Y * ∫ (a : ℝ), ∑ i_1 ∈ F.powerset, G a i i_1 ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volume⊢ ∀ x ∈ F.powerset, (∑ T ∈ F.powerset, if Disjoint x T then star (V₁ (columnLog p X₁ x)) * V₂ (columnLog p X₂ T) * (↑Y / ↑‖eisEmbedding (∏ i ∈ x ∪ T, p i)‖) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (x ∪ T))) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) x T else 0) = ↑Y * ∫ (a : ℝ), ∑ i ∈ F.powerset, G a x i ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powerset⊢ (∑ T ∈ F.powerset, if Disjoint S T then star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * (↑Y / ↑‖eisEmbedding (∏ i ∈ S ∪ T, p i)‖) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T else 0) = ↑Y * ∫ (a : ℝ), ∑ i ∈ F.powerset, G a S i ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powerset⊢ (∑ T ∈ F.powerset, if Disjoint S T then star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * (↑Y / ↑‖eisEmbedding (∏ i ∈ S ∪ T, p i)‖) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T else 0) = ∑ i ∈ F.powerset, ↑Y * ∫ (a : ℝ), G a S i ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powerset⊢ ∀ x ∈ F.powerset, (if Disjoint S x then star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ x) * (↑Y / ↑‖eisEmbedding (∏ i ∈ S ∪ x, p i)‖) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ x))) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S x else 0) = ↑Y * ∫ (a : ℝ), G a S x ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powersetT:Finset ιhT:T ∈ F.powerset⊢ (if Disjoint S T then star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * (↑Y / ↑‖eisEmbedding (∏ i ∈ S ∪ T, p i)‖) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T else 0) = ↑Y * ∫ (a : ℝ), G a S T ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powersetT:Finset ιhT:T ∈ F.powersethd:Disjoint S T⊢ (if Disjoint S T then star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * (↑Y / ↑‖eisEmbedding (∏ i ∈ S ∪ T, p i)‖) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T else 0) = ↑Y * ∫ (a : ℝ), G a S Tι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powersetT:Finset ιhT:T ∈ F.powersethd:¬Disjoint S T⊢ (if Disjoint S T then star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * (↑Y / ↑‖eisEmbedding (∏ i ∈ S ∪ T, p i)‖) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T else 0) = ↑Y * ∫ (a : ℝ), G a S T ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powersetT:Finset ιhT:T ∈ F.powersethd:Disjoint S T⊢ (if Disjoint S T then star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * (↑Y / ↑‖eisEmbedding (∏ i ∈ S ∪ T, p i)‖) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T else 0) = ↑Y * ∫ (a : ℝ), G a S T ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powersetT:Finset ιhT:T ∈ F.powersethd:Disjoint S T⊢ star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * (↑Y / ↑‖eisEmbedding (∏ i ∈ S ∪ T, p i)‖) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T = ↑Y * ∫ (a : ℝ), b a * (logPhase a (-columnLog p X₁ S) * logPhase a (-columnLog p X₂ T) * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (normalizedColumn p fun x => 1) (normalizedColumn p fun x => 1) S T)) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powersetT:Finset ιhT:T ∈ F.powersethd:Disjoint S Tharg:firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T) = Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T ∪ ∅))⊢ star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * (↑Y / ↑‖eisEmbedding (∏ i ∈ S ∪ T, p i)‖) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T = ↑Y * ∫ (a : ℝ), b a * (logPhase a (-columnLog p X₁ S) * logPhase a (-columnLog p X₂ T) * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (normalizedColumn p fun x => 1) (normalizedColumn p fun x => 1) S T)) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powersetT:Finset ιhT:T ∈ F.powersethd:Disjoint S Tharg:firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T) = Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))⊢ star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * (↑Y / ↑‖eisEmbedding (∏ i ∈ S ∪ T, p i)‖) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T = ↑Y * ∫ (a : ℝ), b a * (logPhase a (-columnLog p X₁ S) * logPhase a (-columnLog p X₂ T) * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (normalizedColumn p fun x => 1) (normalizedColumn p fun x => 1) S T)) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powersetT:Finset ιhT:T ∈ F.powersethd:Disjoint S Tharg:firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T) = Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))⊢ star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * (↑Y / ↑‖eisEmbedding (∏ i ∈ S ∪ T, p i)‖) * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T)) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T = ↑Y * ∫ (a : ℝ), b a * (logPhase a (-columnLog p X₁ S) * logPhase a (-columnLog p X₂ T) * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (normalizedColumn p fun x => 1) (normalizedColumn p fun x => 1) S T)) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powersetT:Finset ιhT:T ∈ F.powersethd:Disjoint S Tharg:firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T) = Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))hn:secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (normalizedColumn p fun x => 1) (normalizedColumn p fun x => 1) S T = (↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T⊢ star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * (↑Y / ↑‖eisEmbedding (∏ i ∈ S ∪ T, p i)‖) * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T)) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T = ↑Y * ∫ (a : ℝ), b a * (logPhase a (-columnLog p X₁ S) * logPhase a (-columnLog p X₂ T) * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (normalizedColumn p fun x => 1) (normalizedColumn p fun x => 1) S T)) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powersetT:Finset ιhT:T ∈ F.powersethd:Disjoint S Tharg:firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T) = Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))hn:secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (normalizedColumn p fun x => 1) (normalizedColumn p fun x => 1) S T = (↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T⊢ star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * (↑Y / ↑‖eisEmbedding (∏ i ∈ S ∪ T, p i)‖) * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T)) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T = ↑Y * ∫ (a : ℝ), b a * (logPhase a (-columnLog p X₁ S) * logPhase a (-columnLog p X₂ T) * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T))) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powersetT:Finset ιhT:T ∈ F.powersethd:Disjoint S Tharg:firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T) = Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))hn:secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (normalizedColumn p fun x => 1) (normalizedColumn p fun x => 1) S T = (↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T⊢ star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * (↑Y / ↑‖eisEmbedding ((∏ x ∈ S, p x) * ∏ x ∈ T, p x)‖) * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T)) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T = ↑Y * ∫ (a : ℝ), b a * (logPhase a (-columnLog p X₁ S) * logPhase a (-columnLog p X₂ T) * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T))) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powersetT:Finset ιhT:T ∈ F.powersethd:Disjoint S Tharg:firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T) = Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))hn:secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (normalizedColumn p fun x => 1) (normalizedColumn p fun x => 1) S T = (↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T⊢ star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * (↑Y * ((↑‖eisEmbedding (∏ x ∈ T, p x)‖)⁻¹ * (↑‖eisEmbedding (∏ x ∈ S, p x)‖)⁻¹)) * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T)) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T = ↑Y * ∫ (a : ℝ), b a * (logPhase a (-columnLog p X₁ S) * logPhase a (-columnLog p X₂ T) * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * ((↑‖eisEmbedding (∏ x ∈ S, p x)‖)⁻¹ * (↑‖eisEmbedding (∏ x ∈ T, p x)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T))) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powersetT:Finset ιhT:T ∈ F.powersethd:Disjoint S Tharg:firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T) = Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))hn:secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (normalizedColumn p fun x => 1) (normalizedColumn p fun x => 1) S T = (↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S Thh:↑Y * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T))) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T = (↑Y * ∫ (t : ℝ), star (V₁ (columnLog p X₁ S)) * logPhase t (-columnLog p X₁ S) * (V₂ (columnLog p X₂ T) * logPhase t (-columnLog p X₂ T)) * b t) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T⊢ star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * (↑Y * ((↑‖eisEmbedding (∏ x ∈ T, p x)‖)⁻¹ * (↑‖eisEmbedding (∏ x ∈ S, p x)‖)⁻¹)) * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T)) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T = ↑Y * ∫ (a : ℝ), b a * (logPhase a (-columnLog p X₁ S) * logPhase a (-columnLog p X₂ T) * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * ((↑‖eisEmbedding (∏ x ∈ S, p x)‖)⁻¹ * (↑‖eisEmbedding (∏ x ∈ T, p x)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T))) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powersetT:Finset ιhT:T ∈ F.powersethd:Disjoint S Tharg:firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T) = Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))hn:secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (normalizedColumn p fun x => 1) (normalizedColumn p fun x => 1) S T = (↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S Thh:↑Y * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T))) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T = ∫ (a : ℝ), ↑Y * (star (V₁ (columnLog p X₁ S)) * logPhase a (-columnLog p X₁ S) * (V₂ (columnLog p X₂ T) * logPhase a (-columnLog p X₂ T)) * b a) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T⊢ star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * (↑Y * ((↑‖eisEmbedding (∏ x ∈ T, p x)‖)⁻¹ * (↑‖eisEmbedding (∏ x ∈ S, p x)‖)⁻¹)) * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T)) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T = ↑Y * ∫ (a : ℝ), b a * (logPhase a (-columnLog p X₁ S) * logPhase a (-columnLog p X₂ T) * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * ((↑‖eisEmbedding (∏ x ∈ S, p x)‖)⁻¹ * (↑‖eisEmbedding (∏ x ∈ T, p x)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T))) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powersetT:Finset ιhT:T ∈ F.powersethd:Disjoint S Tharg:firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T) = Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))hn:secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (normalizedColumn p fun x => 1) (normalizedColumn p fun x => 1) S T = (↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S Thh:↑Y * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T))) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T = ∫ (a : ℝ), ↑Y * (star (V₁ (columnLog p X₁ S)) * logPhase a (-columnLog p X₁ S) * (V₂ (columnLog p X₂ T) * logPhase a (-columnLog p X₂ T)) * b a) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T⊢ star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * (↑Y * ((↑‖eisEmbedding (∏ x ∈ T, p x)‖)⁻¹ * (↑‖eisEmbedding (∏ x ∈ S, p x)‖)⁻¹)) * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T)) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T = ↑Y * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T))) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S Tι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powersetT:Finset ιhT:T ∈ F.powersethd:Disjoint S Tharg:firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T) = Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))hn:secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (normalizedColumn p fun x => 1) (normalizedColumn p fun x => 1) S T = (↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S Thh:↑Y * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T))) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T = ∫ (a : ℝ), ↑Y * (star (V₁ (columnLog p X₁ S)) * logPhase a (-columnLog p X₁ S) * (V₂ (columnLog p X₂ T) * logPhase a (-columnLog p X₂ T)) * b a) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T⊢ ↑Y * ∫ (a : ℝ), b a * (logPhase a (-columnLog p X₁ S) * logPhase a (-columnLog p X₂ T) * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * ((↑‖eisEmbedding (∏ x ∈ S, p x)‖)⁻¹ * (↑‖eisEmbedding (∏ x ∈ T, p x)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T))) = ∫ (a : ℝ), ↑Y * (star (V₁ (columnLog p X₁ S)) * logPhase a (-columnLog p X₁ S) * (V₂ (columnLog p X₂ T) * logPhase a (-columnLog p X₂ T)) * b a) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powersetT:Finset ιhT:T ∈ F.powersethd:Disjoint S Tharg:firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T) = Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))hn:secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (normalizedColumn p fun x => 1) (normalizedColumn p fun x => 1) S T = (↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S Thh:↑Y * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T))) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T = ∫ (a : ℝ), ↑Y * (star (V₁ (columnLog p X₁ S)) * logPhase a (-columnLog p X₁ S) * (V₂ (columnLog p X₂ T) * logPhase a (-columnLog p X₂ T)) * b a) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T⊢ star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * (↑Y * ((↑‖eisEmbedding (∏ x ∈ T, p x)‖)⁻¹ * (↑‖eisEmbedding (∏ x ∈ S, p x)‖)⁻¹)) * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T)) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T = ↑Y * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T))) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T All goals completed! 🐙 ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powersetT:Finset ιhT:T ∈ F.powersethd:Disjoint S Tharg:firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T) = Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))hn:secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (normalizedColumn p fun x => 1) (normalizedColumn p fun x => 1) S T = (↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S Thh:↑Y * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T))) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T = ∫ (a : ℝ), ↑Y * (star (V₁ (columnLog p X₁ S)) * logPhase a (-columnLog p X₁ S) * (V₂ (columnLog p X₂ T) * logPhase a (-columnLog p X₂ T)) * b a) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T⊢ ↑Y * ∫ (a : ℝ), b a * (logPhase a (-columnLog p X₁ S) * logPhase a (-columnLog p X₂ T) * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * ((↑‖eisEmbedding (∏ x ∈ S, p x)‖)⁻¹ * (↑‖eisEmbedding (∏ x ∈ T, p x)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T))) = ∫ (a : ℝ), ↑Y * (star (V₁ (columnLog p X₁ S)) * logPhase a (-columnLog p X₁ S) * (V₂ (columnLog p X₂ T) * logPhase a (-columnLog p X₂ T)) * b a) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powersetT:Finset ιhT:T ∈ F.powersethd:Disjoint S Tharg:firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T) = Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))hn:secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (normalizedColumn p fun x => 1) (normalizedColumn p fun x => 1) S T = (↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S Thh:↑Y * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T))) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T = ∫ (a : ℝ), ↑Y * (star (V₁ (columnLog p X₁ S)) * logPhase a (-columnLog p X₁ S) * (V₂ (columnLog p X₂ T) * logPhase a (-columnLog p X₂ T)) * b a) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T⊢ ∫ (a : ℝ), ↑Y * (b a * (logPhase a (-columnLog p X₁ S) * logPhase a (-columnLog p X₂ T) * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * ((↑‖eisEmbedding (∏ x ∈ S, p x)‖)⁻¹ * (↑‖eisEmbedding (∏ x ∈ T, p x)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T)))) = ∫ (a : ℝ), ↑Y * (star (V₁ (columnLog p X₁ S)) * logPhase a (-columnLog p X₁ S) * (V₂ (columnLog p X₂ T) * logPhase a (-columnLog p X₂ T)) * b a) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powersetT:Finset ιhT:T ∈ F.powersethd:Disjoint S Tharg:firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T) = Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))hn:secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (normalizedColumn p fun x => 1) (normalizedColumn p fun x => 1) S T = (↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S Thh:↑Y * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T))) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T = ∫ (a : ℝ), ↑Y * (star (V₁ (columnLog p X₁ S)) * logPhase a (-columnLog p X₁ S) * (V₂ (columnLog p X₂ T) * logPhase a (-columnLog p X₂ T)) * b a) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T⊢ (fun a => ↑Y * (b a * (logPhase a (-columnLog p X₁ S) * logPhase a (-columnLog p X₂ T) * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * ((↑‖eisEmbedding (∏ x ∈ S, p x)‖)⁻¹ * (↑‖eisEmbedding (∏ x ∈ T, p x)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T))))) =ᵐ[volume] fun a => ↑Y * (star (V₁ (columnLog p X₁ S)) * logPhase a (-columnLog p X₁ S) * (V₂ (columnLog p X₂ T) * logPhase a (-columnLog p X₂ T)) * b a) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powersetT:Finset ιhT:T ∈ F.powersethd:Disjoint S Tharg:firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T) = Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))hn:secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (normalizedColumn p fun x => 1) (normalizedColumn p fun x => 1) S T = (↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S Thh:↑Y * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-columnLog p X₁ S - columnLog p X₂ T))) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T = ∫ (a : ℝ), ↑Y * (star (V₁ (columnLog p X₁ S)) * logPhase a (-columnLog p X₁ S) * (V₂ (columnLog p X₂ T) * logPhase a (-columnLog p X₂ T)) * b a) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S Tt:ℝ⊢ ↑Y * (b t * (logPhase t (-columnLog p X₁ S) * logPhase t (-columnLog p X₂ T) * (star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * ((↑‖eisEmbedding (∏ x ∈ S, p x)‖)⁻¹ * (↑‖eisEmbedding (∏ x ∈ T, p x)‖)⁻¹ * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T)))) = ↑Y * (star (V₁ (columnLog p X₁ S)) * logPhase t (-columnLog p X₁ S) * (V₂ (columnLog p X₂ T) * logPhase t (-columnLog p X₂ T)) * b t) * ((↑‖eisEmbedding (∏ i ∈ S, p i)‖)⁻¹ * (↑‖eisEmbedding (∏ i ∈ T, p i)‖)⁻¹) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T All goals completed! 🐙 ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝY:ℝhX₁:0 < X₁hX₂:0 < X₂b:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * logPhase t (-x) * (V₂ y * logPhase t (-y)) * b tG:ℝ → Finset ι → Finset ι → ℂ := fun t S T => if Disjoint S T then b t * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false) S T else 0hGi:∀ (S T : Finset ι), Integrable (fun t => G t S T) volumehGs:∀ (t : ℝ), ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T = b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)hI:Integrable (fun t => ∑ S ∈ F.powerset, ∑ T ∈ F.powerset, G t S T) volumeS:Finset ιhS:S ∈ F.powersetT:Finset ιhT:T ∈ F.powersethd:¬Disjoint S T⊢ (if Disjoint S T then star (V₁ (columnLog p X₁ S)) * V₂ (columnLog p X₂ T) * (↑Y / ↑‖eisEmbedding (∏ i ∈ S ∪ T, p i)‖) * paperRadialFourier (⇑W) (Y * ‖eisEmbedding k‖ ^ 2 / (‖eisEmbedding e‖ ^ 2 * primeProductNorm p (S ∪ T))) * secondGaussTerm p hp hg hinj Ψ₁ Ψ₂ m r c d e k (fun x => 1) (fun x => 1) S T else 0) = ↑Y * ∫ (a : ℝ), G a S T All goals completed! 🐙endsectionopen scoped BigOperators Classical SchwartzMap ContDiffopen MeasureTheorysectionopen ActualEisensteinCubicvariable {ι : Type*} [DecidableEq ι] (p : ι → O) (hp : ∀ i, p i ≠ 0) [∀ i, (Ideal.span {p i}).IsMaximal] (hcop : Pairwise (Function.onFun IsCoprime (fun i => Ideal.span {p i}))) (hg : ∀ i, lambda ∉ Ideal.span {p i})theorem secondChildSum_continuous (F V : Finset ι) (Ψ : O →* ℂ) (m r c d e k : O) (H : ℝ → Finset ι → ℂ) (hc : ∀ S, Continuous (fun t => H t S)) : Continuous (fun t : ℝ => secondChildSum p hp hcop hg F V Ψ m r c d e k (H t)) := ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιV:Finset ιΨ:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhc:∀ (S : Finset ι), Continuous fun t => H t S⊢ Continuous fun t => secondChildSum p hp hcop hg F V Ψ m r c d e k (H t) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιV:Finset ιΨ:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhc:∀ (S : Finset ι), Continuous fun t => H t S⊢ Continuous fun t => ∑ N ∈ (F \ V).powerset, secondChildColumn p hp hcop hg Ψ (m * r) (c * e * ∏ i ∈ V, p i) (d * e * k) (fun U => H t (V ∪ U)) N ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιV:Finset ιΨ:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhc:∀ (S : Finset ι), Continuous fun t => H t S⊢ ∀ i ∈ (F \ V).powerset, Continuous fun a => secondChildColumn p hp hcop hg Ψ (m * r) (c * e * ∏ i ∈ V, p i) (d * e * k) (fun U => H a (V ∪ U)) i ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιV:Finset ιΨ:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhc:∀ (S : Finset ι), Continuous fun t => H t SN:Finset ιhN:N ∈ (F \ V).powerset⊢ Continuous fun a => secondChildColumn p hp hcop hg Ψ (m * r) (c * e * ∏ i ∈ V, p i) (d * e * k) (fun U => H a (V ∪ U)) N ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιV:Finset ιΨ:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhc:∀ (S : Finset ι), Continuous fun t => H t SN:Finset ιhN:N ∈ (F \ V).powersethh:Continuous fun t => H t (V ∪ N)⊢ Continuous fun a => secondChildColumn p hp hcop hg Ψ (m * r) (c * e * ∏ i ∈ V, p i) (d * e * k) (fun U => H a (V ∪ U)) N ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιV:Finset ιΨ:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhc:∀ (S : Finset ι), Continuous fun t => H t SN:Finset ιhN:N ∈ (F \ V).powersethh:Continuous fun t => H t (V ∪ N)⊢ Continuous fun a => MixedCrossSeparation.columnCoefficient p hp hcop hg N * Ψ (∏ i ∈ N, p i) * rowCoprimeMask (fun i => span {p i}) N (m * r) * finiteSquarefreeRow (fun i => span {p i}) hg N (c * e * ∏ i ∈ V, p i) ^ 4 * finiteSquarefreeRow (fun i => span {p i}) hg N (d * e * k) * (fun U => H a (V ∪ U)) N All goals completed! 🐙theorem secondChildSum_uniform_bound (F V : Finset ι) (Ψ : O →* ℂ) (m r c d e k : O) (H : ℝ → Finset ι → ℂ) (hn : ∀ t S, ‖H t S‖ = ‖H 0 S‖) : ∃ M : ℝ, 0 ≤ M ∧ ∀ t, ‖secondChildSum p hp hcop hg F V Ψ m r c d e k (H t)‖ ≤ M := ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιV:Finset ιΨ:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖⊢ ∃ M, 0 ≤ M ∧ ∀ (t : ℝ), ‖secondChildSum p hp hcop hg F V Ψ m r c d e k (H t)‖ ≤ M refine ⟨∑ N ∈ (F \ V).powerset, ‖secondChildColumn p hp hcop hg Ψ (m * r) (c * e * ∏ i ∈ V, p i) (d * e * k) (fun U => H 0 (V ∪ U)) N‖, ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιV:Finset ιΨ:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖⊢ 0 ≤ ∑ N ∈ (F \ V).powerset, ‖secondChildColumn p hp hcop hg Ψ (m * r) (c * e * ∏ i ∈ V, p i) (d * e * k) (fun U => H 0 (V ∪ U)) N‖ All goals completed! 🐙, ?_⟩ ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιV:Finset ιΨ:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖t:ℝ⊢ ‖secondChildSum p hp hcop hg F V Ψ m r c d e k (H t)‖ ≤ ∑ N ∈ (F \ V).powerset, ‖secondChildColumn p hp hcop hg Ψ (m * r) (c * e * ∏ i ∈ V, p i) (d * e * k) (fun U => H 0 (V ∪ U)) N‖ ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιV:Finset ιΨ:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖t:ℝ⊢ ∑ i ∈ (F \ V).powerset, ‖secondChildColumn p hp hcop hg Ψ (m * r) (c * e * ∏ i ∈ V, p i) (d * e * k) (fun U => H t (V ∪ U)) i‖ ≤ ∑ N ∈ (F \ V).powerset, ‖secondChildColumn p hp hcop hg Ψ (m * r) (c * e * ∏ i ∈ V, p i) (d * e * k) (fun U => H 0 (V ∪ U)) N‖ ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιV:Finset ιΨ:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖t:ℝ⊢ ∀ i ∈ (F \ V).powerset, ‖secondChildColumn p hp hcop hg Ψ (m * r) (c * e * ∏ i ∈ V, p i) (d * e * k) (fun U => H t (V ∪ U)) i‖ ≤ ‖secondChildColumn p hp hcop hg Ψ (m * r) (c * e * ∏ i ∈ V, p i) (d * e * k) (fun U => H 0 (V ∪ U)) i‖ ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιV:Finset ιΨ:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖t:ℝN:Finset ιhN:N ∈ (F \ V).powerset⊢ ‖secondChildColumn p hp hcop hg Ψ (m * r) (c * e * ∏ i ∈ V, p i) (d * e * k) (fun U => H t (V ∪ U)) N‖ ≤ ‖secondChildColumn p hp hcop hg Ψ (m * r) (c * e * ∏ i ∈ V, p i) (d * e * k) (fun U => H 0 (V ∪ U)) N‖ ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιV:Finset ιΨ:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖t:ℝN:Finset ιhN:N ∈ (F \ V).powerset⊢ ‖MixedCrossSeparation.columnCoefficient p hp hcop hg N‖ * ‖Ψ (∏ i ∈ N, p i)‖ * ‖rowCoprimeMask (fun i => span {p i}) N (m * r)‖ * ‖finiteSquarefreeRow (fun i => span {p i}) hg N (c * e * ∏ i ∈ V, p i) ^ 4‖ * ‖finiteSquarefreeRow (fun i => span {p i}) hg N (d * e * k)‖ * ‖H 0 (V ∪ N)‖ ≤ ‖MixedCrossSeparation.columnCoefficient p hp hcop hg N‖ * ‖Ψ (∏ i ∈ N, p i)‖ * ‖rowCoprimeMask (fun i => span {p i}) N (m * r)‖ * ‖finiteSquarefreeRow (fun i => span {p i}) hg N (c * e * ∏ i ∈ V, p i) ^ 4‖ * ‖finiteSquarefreeRow (fun i => span {p i}) hg N (d * e * k)‖ * ‖H 0 (V ∪ N)‖ All goals completed! 🐙theorem secondTotalEnergy_nonneg (F : Finset ι) (Ψ₁ Ψ₂ : O →* ℂ) (m r c d e k : O) (H : Finset ι → ℂ) (negative : Bool) : 0 ≤ secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k H negative := ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:Finset ι → ℂnegative:Bool⊢ 0 ≤ secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k H negative ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:Finset ι → ℂnegative:Bool⊢ 0 ≤ ∑ z, ∑ V ∈ F.powerset, ‖secondTotalWeight p hp hcop hg Ψ₁ Ψ₂ m r c d e k (z, V)‖ * ‖secondChildSum p hp hcop hg F V (if negative = true then secondRayMinus Ψ₁ z else secondRayPlus Ψ₂ z) m r c d e (if negative = true then k else -k) H‖ ^ 2 All goals completed! 🐙theorem secondTotalEnergy_continuous (F : Finset ι) (Ψ₁ Ψ₂ : O →* ℂ) (m r c d e k : O) (H : ℝ → Finset ι → ℂ) (hc : ∀ S, Continuous (fun t => H t S)) (negative : Bool) : Continuous (fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H t) negative) := ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhc:∀ (S : Finset ι), Continuous fun t => H t Snegative:Bool⊢ Continuous fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H t) negative ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhc:∀ (S : Finset ι), Continuous fun t => H t Snegative:Bool⊢ Continuous fun t => ∑ z, ∑ V ∈ F.powerset, ‖secondTotalWeight p hp hcop hg Ψ₁ Ψ₂ m r c d e k (z, V)‖ * ‖secondChildSum p hp hcop hg F V (if negative = true then secondRayMinus Ψ₁ z else secondRayPlus Ψ₂ z) m r c d e (if negative = true then k else -k) (H t)‖ ^ 2 ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhc:∀ (S : Finset ι), Continuous fun t => H t Snegative:Bool⊢ ∀ i ∈ Finset.univ, Continuous fun a => ∑ V ∈ F.powerset, ‖secondTotalWeight p hp hcop hg Ψ₁ Ψ₂ m r c d e k (i, V)‖ * ‖secondChildSum p hp hcop hg F V (if negative = true then secondRayMinus Ψ₁ i else secondRayPlus Ψ₂ i) m r c d e (if negative = true then k else -k) (H a)‖ ^ 2 ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhc:∀ (S : Finset ι), Continuous fun t => H t Snegative:Boolz:SecondRayIndexhz:z ∈ Finset.univ⊢ Continuous fun a => ∑ V ∈ F.powerset, ‖secondTotalWeight p hp hcop hg Ψ₁ Ψ₂ m r c d e k (z, V)‖ * ‖secondChildSum p hp hcop hg F V (if negative = true then secondRayMinus Ψ₁ z else secondRayPlus Ψ₂ z) m r c d e (if negative = true then k else -k) (H a)‖ ^ 2 ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhc:∀ (S : Finset ι), Continuous fun t => H t Snegative:Boolz:SecondRayIndexhz:z ∈ Finset.univ⊢ ∀ i ∈ F.powerset, Continuous fun a => ‖secondTotalWeight p hp hcop hg Ψ₁ Ψ₂ m r c d e k (z, i)‖ * ‖secondChildSum p hp hcop hg F i (if negative = true then secondRayMinus Ψ₁ z else secondRayPlus Ψ₂ z) m r c d e (if negative = true then k else -k) (H a)‖ ^ 2 ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhc:∀ (S : Finset ι), Continuous fun t => H t Snegative:Boolz:SecondRayIndexhz:z ∈ Finset.univV:Finset ιhV:V ∈ F.powerset⊢ Continuous fun a => ‖secondTotalWeight p hp hcop hg Ψ₁ Ψ₂ m r c d e k (z, V)‖ * ‖secondChildSum p hp hcop hg F V (if negative = true then secondRayMinus Ψ₁ z else secondRayPlus Ψ₂ z) m r c d e (if negative = true then k else -k) (H a)‖ ^ 2 All goals completed! 🐙theorem secondTotalEnergy_uniform_bound (F : Finset ι) (Ψ₁ Ψ₂ : O →* ℂ) (m r c d e k : O) (H : ℝ → Finset ι → ℂ) (hn : ∀ t S, ‖H t S‖ = ‖H 0 S‖) (negative : Bool) : ∃ M : ℝ, 0 ≤ M ∧ ∀ t, secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H t) negative ≤ M := ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖negative:Bool⊢ ∃ M, 0 ≤ M ∧ ∀ (t : ℝ), secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H t) negative ≤ M ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖negative:BoolM:SecondRayIndex → Finset ι → ℝhM:∀ (z : SecondRayIndex) (V : Finset ι), 0 ≤ M z Vhbound:∀ (z : SecondRayIndex) (V : Finset ι) (t : ℝ), ‖secondChildSum p hp hcop hg F V (if negative = true then secondRayMinus Ψ₁ z else secondRayPlus Ψ₂ z) m r c d e (if negative = true then k else -k) (H t)‖ ≤ M z V⊢ ∃ M, 0 ≤ M ∧ ∀ (t : ℝ), secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H t) negative ≤ M refine ⟨∑ z : SecondRayIndex, ∑ V ∈ F.powerset, ‖secondTotalWeight p hp hcop hg Ψ₁ Ψ₂ m r c d e k (z,V)‖ * (M z V)^2, ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖negative:BoolM:SecondRayIndex → Finset ι → ℝhM:∀ (z : SecondRayIndex) (V : Finset ι), 0 ≤ M z Vhbound:∀ (z : SecondRayIndex) (V : Finset ι) (t : ℝ), ‖secondChildSum p hp hcop hg F V (if negative = true then secondRayMinus Ψ₁ z else secondRayPlus Ψ₂ z) m r c d e (if negative = true then k else -k) (H t)‖ ≤ M z V⊢ 0 ≤ ∑ z, ∑ V ∈ F.powerset, ‖secondTotalWeight p hp hcop hg Ψ₁ Ψ₂ m r c d e k (z, V)‖ * M z V ^ 2 All goals completed! 🐙, ?_⟩ ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖negative:BoolM:SecondRayIndex → Finset ι → ℝhM:∀ (z : SecondRayIndex) (V : Finset ι), 0 ≤ M z Vhbound:∀ (z : SecondRayIndex) (V : Finset ι) (t : ℝ), ‖secondChildSum p hp hcop hg F V (if negative = true then secondRayMinus Ψ₁ z else secondRayPlus Ψ₂ z) m r c d e (if negative = true then k else -k) (H t)‖ ≤ M z Vt:ℝ⊢ secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H t) negative ≤ ∑ z, ∑ V ∈ F.powerset, ‖secondTotalWeight p hp hcop hg Ψ₁ Ψ₂ m r c d e k (z, V)‖ * M z V ^ 2 ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖negative:BoolM:SecondRayIndex → Finset ι → ℝhM:∀ (z : SecondRayIndex) (V : Finset ι), 0 ≤ M z Vhbound:∀ (z : SecondRayIndex) (V : Finset ι) (t : ℝ), ‖secondChildSum p hp hcop hg F V (if negative = true then secondRayMinus Ψ₁ z else secondRayPlus Ψ₂ z) m r c d e (if negative = true then k else -k) (H t)‖ ≤ M z Vt:ℝ⊢ ∀ i ∈ Finset.univ, ∑ V ∈ F.powerset, ‖secondTotalWeight p hp hcop hg Ψ₁ Ψ₂ m r c d e k (i, V)‖ * ‖secondChildSum p hp hcop hg F V (if negative = true then secondRayMinus Ψ₁ i else secondRayPlus Ψ₂ i) m r c d e (if negative = true then k else -k) (H t)‖ ^ 2 ≤ ∑ V ∈ F.powerset, ‖secondTotalWeight p hp hcop hg Ψ₁ Ψ₂ m r c d e k (i, V)‖ * M i V ^ 2 ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖negative:BoolM:SecondRayIndex → Finset ι → ℝhM:∀ (z : SecondRayIndex) (V : Finset ι), 0 ≤ M z Vhbound:∀ (z : SecondRayIndex) (V : Finset ι) (t : ℝ), ‖secondChildSum p hp hcop hg F V (if negative = true then secondRayMinus Ψ₁ z else secondRayPlus Ψ₂ z) m r c d e (if negative = true then k else -k) (H t)‖ ≤ M z Vt:ℝz:SecondRayIndexhz:z ∈ Finset.univ⊢ ∑ V ∈ F.powerset, ‖secondTotalWeight p hp hcop hg Ψ₁ Ψ₂ m r c d e k (z, V)‖ * ‖secondChildSum p hp hcop hg F V (if negative = true then secondRayMinus Ψ₁ z else secondRayPlus Ψ₂ z) m r c d e (if negative = true then k else -k) (H t)‖ ^ 2 ≤ ∑ V ∈ F.powerset, ‖secondTotalWeight p hp hcop hg Ψ₁ Ψ₂ m r c d e k (z, V)‖ * M z V ^ 2 ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖negative:BoolM:SecondRayIndex → Finset ι → ℝhM:∀ (z : SecondRayIndex) (V : Finset ι), 0 ≤ M z Vhbound:∀ (z : SecondRayIndex) (V : Finset ι) (t : ℝ), ‖secondChildSum p hp hcop hg F V (if negative = true then secondRayMinus Ψ₁ z else secondRayPlus Ψ₂ z) m r c d e (if negative = true then k else -k) (H t)‖ ≤ M z Vt:ℝz:SecondRayIndexhz:z ∈ Finset.univ⊢ ∀ i ∈ F.powerset, ‖secondTotalWeight p hp hcop hg Ψ₁ Ψ₂ m r c d e k (z, i)‖ * ‖secondChildSum p hp hcop hg F i (if negative = true then secondRayMinus Ψ₁ z else secondRayPlus Ψ₂ z) m r c d e (if negative = true then k else -k) (H t)‖ ^ 2 ≤ ‖secondTotalWeight p hp hcop hg Ψ₁ Ψ₂ m r c d e k (z, i)‖ * M z i ^ 2 ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖negative:BoolM:SecondRayIndex → Finset ι → ℝhM:∀ (z : SecondRayIndex) (V : Finset ι), 0 ≤ M z Vhbound:∀ (z : SecondRayIndex) (V : Finset ι) (t : ℝ), ‖secondChildSum p hp hcop hg F V (if negative = true then secondRayMinus Ψ₁ z else secondRayPlus Ψ₂ z) m r c d e (if negative = true then k else -k) (H t)‖ ≤ M z Vt:ℝz:SecondRayIndexhz:z ∈ Finset.univV:Finset ιhV:V ∈ F.powerset⊢ ‖secondTotalWeight p hp hcop hg Ψ₁ Ψ₂ m r c d e k (z, V)‖ * ‖secondChildSum p hp hcop hg F V (if negative = true then secondRayMinus Ψ₁ z else secondRayPlus Ψ₂ z) m r c d e (if negative = true then k else -k) (H t)‖ ^ 2 ≤ ‖secondTotalWeight p hp hcop hg Ψ₁ Ψ₂ m r c d e k (z, V)‖ * M z V ^ 2 All goals completed! 🐙theorem secondTotalEnergy_integrable (F : Finset ι) (Ψ₁ Ψ₂ : O →* ℂ) (m r c d e k : O) (H : ℝ → Finset ι → ℂ) (hc : ∀ S, Continuous (fun t => H t S)) (hn : ∀ t S, ‖H t S‖ = ‖H 0 S‖) (negative : Bool) (b : 𝓢(ℝ, ℂ)) : Integrable (fun t => ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H t) negative) volume := ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhc:∀ (S : Finset ι), Continuous fun t => H t Shn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖negative:Boolb:𝓢(ℝ, ℂ)⊢ Integrable (fun t => ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H t) negative) volume ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhc:∀ (S : Finset ι), Continuous fun t => H t Shn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖negative:Boolb:𝓢(ℝ, ℂ)M:ℝhM:0 ≤ Mhb:∀ (t : ℝ), secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H t) negative ≤ M⊢ Integrable (fun t => ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H t) negative) volume ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhc:∀ (S : Finset ι), Continuous fun t => H t Shn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖negative:Boolb:𝓢(ℝ, ℂ)M:ℝhM:0 ≤ Mhb:∀ (t : ℝ), secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H t) negative ≤ M⊢ AEStronglyMeasurable (fun x => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H x) negative) volumeι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhc:∀ (S : Finset ι), Continuous fun t => H t Shn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖negative:Boolb:𝓢(ℝ, ℂ)M:ℝhM:0 ≤ Mhb:∀ (t : ℝ), secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H t) negative ≤ M⊢ ∀ᵐ (x : ℝ), ‖secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H x) negative‖ ≤ M ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhc:∀ (S : Finset ι), Continuous fun t => H t Shn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖negative:Boolb:𝓢(ℝ, ℂ)M:ℝhM:0 ≤ Mhb:∀ (t : ℝ), secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H t) negative ≤ M⊢ AEStronglyMeasurable (fun x => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H x) negative) volume All goals completed! 🐙 ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhc:∀ (S : Finset ι), Continuous fun t => H t Shn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖negative:Boolb:𝓢(ℝ, ℂ)M:ℝhM:0 ≤ Mhb:∀ (t : ℝ), secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H t) negative ≤ M⊢ ∀ᵐ (x : ℝ), ‖secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H x) negative‖ ≤ M exact Filter.Eventually.of_forall (fun t => ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhc:∀ (S : Finset ι), Continuous fun t => H t Shn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖negative:Boolb:𝓢(ℝ, ℂ)M:ℝhM:0 ≤ Mhb:∀ (t : ℝ), secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H t) negative ≤ Mt:ℝ⊢ ‖secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H t) negative‖ ≤ M ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH:ℝ → Finset ι → ℂhc:∀ (S : Finset ι), Continuous fun t => H t Shn:∀ (t : ℝ) (S : Finset ι), ‖H t S‖ = ‖H 0 S‖negative:Boolb:𝓢(ℝ, ℂ)M:ℝhM:0 ≤ Mhb:∀ (t : ℝ), secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H t) negative ≤ Mt:ℝ⊢ secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H t) negative ≤ M All goals completed! 🐙)theorem second_integral_norm_le_energy (hinj : Function.Injective (fun i => Ideal.span {p i})) (hc : ∀ i, ringChar (O ⧸ Ideal.span {p i}) ≠ 2) (hpr : ∀ i, lambda ^ 2 ∣ p i - 1) (F : Finset ι) (Ψ₁ Ψ₂ : O →* ℂ) (m r c d e k : O) (H₁ H₂ : ℝ → Finset ι → ℂ) (hc₁ : ∀ S, Continuous (fun t => H₁ t S)) (hc₂ : ∀ S, Continuous (fun t => H₂ t S)) (hn₁ : ∀ t S, ‖H₁ t S‖ = ‖H₁ 0 S‖) (hn₂ : ∀ t S, ‖H₂ t S‖ = ‖H₂ 0 S‖) (b : 𝓢(ℝ, ℂ)) : ‖∫ t : ℝ, b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ Real.sqrt (∫ t : ℝ, ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true) * Real.sqrt (∫ t : ℝ, ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false) := ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH₁:ℝ → Finset ι → ℂH₂:ℝ → Finset ι → ℂhc₁:∀ (S : Finset ι), Continuous fun t => H₁ t Shc₂:∀ (S : Finset ι), Continuous fun t => H₂ t Shn₁:∀ (t : ℝ) (S : Finset ι), ‖H₁ t S‖ = ‖H₁ 0 S‖hn₂:∀ (t : ℝ) (S : Finset ι), ‖H₂ t S‖ = ‖H₂ 0 S‖b:𝓢(ℝ, ℂ)⊢ ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH₁:ℝ → Finset ι → ℂH₂:ℝ → Finset ι → ℂhc₁:∀ (S : Finset ι), Continuous fun t => H₁ t Shc₂:∀ (S : Finset ι), Continuous fun t => H₂ t Shn₁:∀ (t : ℝ) (S : Finset ι), ‖H₁ t S‖ = ‖H₁ 0 S‖hn₂:∀ (t : ℝ) (S : Finset ι), ‖H₂ t S‖ = ‖H₂ 0 S‖b:𝓢(ℝ, ℂ)E₁:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true⊢ ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH₁:ℝ → Finset ι → ℂH₂:ℝ → Finset ι → ℂhc₁:∀ (S : Finset ι), Continuous fun t => H₁ t Shc₂:∀ (S : Finset ι), Continuous fun t => H₂ t Shn₁:∀ (t : ℝ) (S : Finset ι), ‖H₁ t S‖ = ‖H₁ 0 S‖hn₂:∀ (t : ℝ) (S : Finset ι), ‖H₂ t S‖ = ‖H₂ 0 S‖b:𝓢(ℝ, ℂ)E₁:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) trueE₂:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false⊢ ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH₁:ℝ → Finset ι → ℂH₂:ℝ → Finset ι → ℂhc₁:∀ (S : Finset ι), Continuous fun t => H₁ t Shc₂:∀ (S : Finset ι), Continuous fun t => H₂ t Shn₁:∀ (t : ℝ) (S : Finset ι), ‖H₁ t S‖ = ‖H₁ 0 S‖hn₂:∀ (t : ℝ) (S : Finset ι), ‖H₂ t S‖ = ‖H₂ 0 S‖b:𝓢(ℝ, ℂ)E₁:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) trueE₂:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) falsew:ℝ → ℝ := fun t => ‖b t‖⊢ ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH₁:ℝ → Finset ι → ℂH₂:ℝ → Finset ι → ℂhc₁:∀ (S : Finset ι), Continuous fun t => H₁ t Shc₂:∀ (S : Finset ι), Continuous fun t => H₂ t Shn₁:∀ (t : ℝ) (S : Finset ι), ‖H₁ t S‖ = ‖H₁ 0 S‖hn₂:∀ (t : ℝ) (S : Finset ι), ‖H₂ t S‖ = ‖H₂ 0 S‖b:𝓢(ℝ, ℂ)E₁:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) trueE₂:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) falsew:ℝ → ℝ := fun t => ‖b t‖hE₁:∀ (t : ℝ), 0 ≤ E₁ t⊢ ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH₁:ℝ → Finset ι → ℂH₂:ℝ → Finset ι → ℂhc₁:∀ (S : Finset ι), Continuous fun t => H₁ t Shc₂:∀ (S : Finset ι), Continuous fun t => H₂ t Shn₁:∀ (t : ℝ) (S : Finset ι), ‖H₁ t S‖ = ‖H₁ 0 S‖hn₂:∀ (t : ℝ) (S : Finset ι), ‖H₂ t S‖ = ‖H₂ 0 S‖b:𝓢(ℝ, ℂ)E₁:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) trueE₂:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) falsew:ℝ → ℝ := fun t => ‖b t‖hE₁:∀ (t : ℝ), 0 ≤ E₁ thE₂:∀ (t : ℝ), 0 ≤ E₂ t⊢ ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH₁:ℝ → Finset ι → ℂH₂:ℝ → Finset ι → ℂhc₁:∀ (S : Finset ι), Continuous fun t => H₁ t Shc₂:∀ (S : Finset ι), Continuous fun t => H₂ t Shn₁:∀ (t : ℝ) (S : Finset ι), ‖H₁ t S‖ = ‖H₁ 0 S‖hn₂:∀ (t : ℝ) (S : Finset ι), ‖H₂ t S‖ = ‖H₂ 0 S‖b:𝓢(ℝ, ℂ)E₁:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) trueE₂:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) falsew:ℝ → ℝ := fun t => ‖b t‖hE₁:∀ (t : ℝ), 0 ≤ E₁ thE₂:∀ (t : ℝ), 0 ≤ E₂ thw:∀ (t : ℝ), 0 ≤ w t⊢ ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH₁:ℝ → Finset ι → ℂH₂:ℝ → Finset ι → ℂhc₁:∀ (S : Finset ι), Continuous fun t => H₁ t Shc₂:∀ (S : Finset ι), Continuous fun t => H₂ t Shn₁:∀ (t : ℝ) (S : Finset ι), ‖H₁ t S‖ = ‖H₁ 0 S‖hn₂:∀ (t : ℝ) (S : Finset ι), ‖H₂ t S‖ = ‖H₂ 0 S‖b:𝓢(ℝ, ℂ)E₁:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) trueE₂:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) falsew:ℝ → ℝ := fun t => ‖b t‖hE₁:∀ (t : ℝ), 0 ≤ E₁ thE₂:∀ (t : ℝ), 0 ≤ E₂ thw:∀ (t : ℝ), 0 ≤ w ti₁:Integrable (fun t => w t * E₁ t) volume⊢ ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH₁:ℝ → Finset ι → ℂH₂:ℝ → Finset ι → ℂhc₁:∀ (S : Finset ι), Continuous fun t => H₁ t Shc₂:∀ (S : Finset ι), Continuous fun t => H₂ t Shn₁:∀ (t : ℝ) (S : Finset ι), ‖H₁ t S‖ = ‖H₁ 0 S‖hn₂:∀ (t : ℝ) (S : Finset ι), ‖H₂ t S‖ = ‖H₂ 0 S‖b:𝓢(ℝ, ℂ)E₁:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) trueE₂:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) falsew:ℝ → ℝ := fun t => ‖b t‖hE₁:∀ (t : ℝ), 0 ≤ E₁ thE₂:∀ (t : ℝ), 0 ≤ E₂ thw:∀ (t : ℝ), 0 ≤ w ti₁:Integrable (fun t => w t * E₁ t) volumei₂:Integrable (fun t => w t * E₂ t) volume⊢ ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH₁:ℝ → Finset ι → ℂH₂:ℝ → Finset ι → ℂhc₁:∀ (S : Finset ι), Continuous fun t => H₁ t Shc₂:∀ (S : Finset ι), Continuous fun t => H₂ t Shn₁:∀ (t : ℝ) (S : Finset ι), ‖H₁ t S‖ = ‖H₁ 0 S‖hn₂:∀ (t : ℝ) (S : Finset ι), ‖H₂ t S‖ = ‖H₂ 0 S‖b:𝓢(ℝ, ℂ)E₁:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) trueE₂:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) falsew:ℝ → ℝ := fun t => ‖b t‖hE₁:∀ (t : ℝ), 0 ≤ E₁ thE₂:∀ (t : ℝ), 0 ≤ E₂ thw:∀ (t : ℝ), 0 ≤ w ti₁:Integrable (fun t => w t * E₁ t) volumei₂:Integrable (fun t => w t * E₂ t) volumef:ℝ → ℝ := fun t => √(w t * E₁ t)⊢ ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH₁:ℝ → Finset ι → ℂH₂:ℝ → Finset ι → ℂhc₁:∀ (S : Finset ι), Continuous fun t => H₁ t Shc₂:∀ (S : Finset ι), Continuous fun t => H₂ t Shn₁:∀ (t : ℝ) (S : Finset ι), ‖H₁ t S‖ = ‖H₁ 0 S‖hn₂:∀ (t : ℝ) (S : Finset ι), ‖H₂ t S‖ = ‖H₂ 0 S‖b:𝓢(ℝ, ℂ)E₁:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) trueE₂:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) falsew:ℝ → ℝ := fun t => ‖b t‖hE₁:∀ (t : ℝ), 0 ≤ E₁ thE₂:∀ (t : ℝ), 0 ≤ E₂ thw:∀ (t : ℝ), 0 ≤ w ti₁:Integrable (fun t => w t * E₁ t) volumei₂:Integrable (fun t => w t * E₂ t) volumef:ℝ → ℝ := fun t => √(w t * E₁ t)g:ℝ → ℝ := fun t => √(w t * E₂ t)⊢ ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH₁:ℝ → Finset ι → ℂH₂:ℝ → Finset ι → ℂhc₁:∀ (S : Finset ι), Continuous fun t => H₁ t Shc₂:∀ (S : Finset ι), Continuous fun t => H₂ t Shn₁:∀ (t : ℝ) (S : Finset ι), ‖H₁ t S‖ = ‖H₁ 0 S‖hn₂:∀ (t : ℝ) (S : Finset ι), ‖H₂ t S‖ = ‖H₂ 0 S‖b:𝓢(ℝ, ℂ)E₁:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) trueE₂:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) falsew:ℝ → ℝ := fun t => ‖b t‖hE₁:∀ (t : ℝ), 0 ≤ E₁ thE₂:∀ (t : ℝ), 0 ≤ E₂ thw:∀ (t : ℝ), 0 ≤ w ti₁:Integrable (fun t => w t * E₁ t) volumei₂:Integrable (fun t => w t * E₂ t) volumef:ℝ → ℝ := fun t => √(w t * E₁ t)g:ℝ → ℝ := fun t => √(w t * E₂ t)hf:Continuous f⊢ ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH₁:ℝ → Finset ι → ℂH₂:ℝ → Finset ι → ℂhc₁:∀ (S : Finset ι), Continuous fun t => H₁ t Shc₂:∀ (S : Finset ι), Continuous fun t => H₂ t Shn₁:∀ (t : ℝ) (S : Finset ι), ‖H₁ t S‖ = ‖H₁ 0 S‖hn₂:∀ (t : ℝ) (S : Finset ι), ‖H₂ t S‖ = ‖H₂ 0 S‖b:𝓢(ℝ, ℂ)E₁:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) trueE₂:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) falsew:ℝ → ℝ := fun t => ‖b t‖hE₁:∀ (t : ℝ), 0 ≤ E₁ thE₂:∀ (t : ℝ), 0 ≤ E₂ thw:∀ (t : ℝ), 0 ≤ w ti₁:Integrable (fun t => w t * E₁ t) volumei₂:Integrable (fun t => w t * E₂ t) volumef:ℝ → ℝ := fun t => √(w t * E₁ t)g:ℝ → ℝ := fun t => √(w t * E₂ t)hf:Continuous fhg':Continuous g⊢ ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH₁:ℝ → Finset ι → ℂH₂:ℝ → Finset ι → ℂhc₁:∀ (S : Finset ι), Continuous fun t => H₁ t Shc₂:∀ (S : Finset ι), Continuous fun t => H₂ t Shn₁:∀ (t : ℝ) (S : Finset ι), ‖H₁ t S‖ = ‖H₁ 0 S‖hn₂:∀ (t : ℝ) (S : Finset ι), ‖H₂ t S‖ = ‖H₂ 0 S‖b:𝓢(ℝ, ℂ)E₁:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) trueE₂:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) falsew:ℝ → ℝ := fun t => ‖b t‖hE₁:∀ (t : ℝ), 0 ≤ E₁ thE₂:∀ (t : ℝ), 0 ≤ E₂ thw:∀ (t : ℝ), 0 ≤ w ti₁:Integrable (fun t => w t * E₁ t) volumei₂:Integrable (fun t => w t * E₂ t) volumef:ℝ → ℝ := fun t => √(w t * E₁ t)g:ℝ → ℝ := fun t => √(w t * E₂ t)hf:Continuous fhg':Continuous ghfsq:∀ (t : ℝ), f t ^ 2 = w t * E₁ t⊢ ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH₁:ℝ → Finset ι → ℂH₂:ℝ → Finset ι → ℂhc₁:∀ (S : Finset ι), Continuous fun t => H₁ t Shc₂:∀ (S : Finset ι), Continuous fun t => H₂ t Shn₁:∀ (t : ℝ) (S : Finset ι), ‖H₁ t S‖ = ‖H₁ 0 S‖hn₂:∀ (t : ℝ) (S : Finset ι), ‖H₂ t S‖ = ‖H₂ 0 S‖b:𝓢(ℝ, ℂ)E₁:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) trueE₂:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) falsew:ℝ → ℝ := fun t => ‖b t‖hE₁:∀ (t : ℝ), 0 ≤ E₁ thE₂:∀ (t : ℝ), 0 ≤ E₂ thw:∀ (t : ℝ), 0 ≤ w ti₁:Integrable (fun t => w t * E₁ t) volumei₂:Integrable (fun t => w t * E₂ t) volumef:ℝ → ℝ := fun t => √(w t * E₁ t)g:ℝ → ℝ := fun t => √(w t * E₂ t)hf:Continuous fhg':Continuous ghfsq:∀ (t : ℝ), f t ^ 2 = w t * E₁ thgsq:∀ (t : ℝ), g t ^ 2 = w t * E₂ t⊢ ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH₁:ℝ → Finset ι → ℂH₂:ℝ → Finset ι → ℂhc₁:∀ (S : Finset ι), Continuous fun t => H₁ t Shc₂:∀ (S : Finset ι), Continuous fun t => H₂ t Shn₁:∀ (t : ℝ) (S : Finset ι), ‖H₁ t S‖ = ‖H₁ 0 S‖hn₂:∀ (t : ℝ) (S : Finset ι), ‖H₂ t S‖ = ‖H₂ 0 S‖b:𝓢(ℝ, ℂ)E₁:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) trueE₂:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) falsew:ℝ → ℝ := fun t => ‖b t‖hE₁:∀ (t : ℝ), 0 ≤ E₁ thE₂:∀ (t : ℝ), 0 ≤ E₂ thw:∀ (t : ℝ), 0 ≤ w ti₁:Integrable (fun t => w t * E₁ t) volumei₂:Integrable (fun t => w t * E₂ t) volumef:ℝ → ℝ := fun t => √(w t * E₁ t)g:ℝ → ℝ := fun t => √(w t * E₂ t)hf:Continuous fhg':Continuous ghfsq:∀ (t : ℝ), f t ^ 2 = w t * E₁ thgsq:∀ (t : ℝ), g t ^ 2 = w t * E₂ tmf:MemLp f 2 volume⊢ ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH₁:ℝ → Finset ι → ℂH₂:ℝ → Finset ι → ℂhc₁:∀ (S : Finset ι), Continuous fun t => H₁ t Shc₂:∀ (S : Finset ι), Continuous fun t => H₂ t Shn₁:∀ (t : ℝ) (S : Finset ι), ‖H₁ t S‖ = ‖H₁ 0 S‖hn₂:∀ (t : ℝ) (S : Finset ι), ‖H₂ t S‖ = ‖H₂ 0 S‖b:𝓢(ℝ, ℂ)E₁:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) trueE₂:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) falsew:ℝ → ℝ := fun t => ‖b t‖hE₁:∀ (t : ℝ), 0 ≤ E₁ thE₂:∀ (t : ℝ), 0 ≤ E₂ thw:∀ (t : ℝ), 0 ≤ w ti₁:Integrable (fun t => w t * E₁ t) volumei₂:Integrable (fun t => w t * E₂ t) volumef:ℝ → ℝ := fun t => √(w t * E₁ t)g:ℝ → ℝ := fun t => √(w t * E₂ t)hf:Continuous fhg':Continuous ghfsq:∀ (t : ℝ), f t ^ 2 = w t * E₁ thgsq:∀ (t : ℝ), g t ^ 2 = w t * E₂ tmf:MemLp f 2 volumemg:MemLp g 2 volume⊢ ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH₁:ℝ → Finset ι → ℂH₂:ℝ → Finset ι → ℂhc₁:∀ (S : Finset ι), Continuous fun t => H₁ t Shc₂:∀ (S : Finset ι), Continuous fun t => H₂ t Shn₁:∀ (t : ℝ) (S : Finset ι), ‖H₁ t S‖ = ‖H₁ 0 S‖hn₂:∀ (t : ℝ) (S : Finset ι), ‖H₂ t S‖ = ‖H₂ 0 S‖b:𝓢(ℝ, ℂ)E₁:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) trueE₂:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) falsew:ℝ → ℝ := fun t => ‖b t‖hE₁:∀ (t : ℝ), 0 ≤ E₁ thE₂:∀ (t : ℝ), 0 ≤ E₂ thw:∀ (t : ℝ), 0 ≤ w ti₁:Integrable (fun t => w t * E₁ t) volumei₂:Integrable (fun t => w t * E₂ t) volumef:ℝ → ℝ := fun t => √(w t * E₁ t)g:ℝ → ℝ := fun t => √(w t * E₂ t)hf:Continuous fhg':Continuous ghfsq:∀ (t : ℝ), f t ^ 2 = w t * E₁ thgsq:∀ (t : ℝ), g t ^ 2 = w t * E₂ tmf:MemLp f 2 volumemg:MemLp g 2 volumehfg:∀ (t : ℝ), f t * g t = w t * (√(E₁ t) * √(E₂ t))⊢ ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH₁:ℝ → Finset ι → ℂH₂:ℝ → Finset ι → ℂhc₁:∀ (S : Finset ι), Continuous fun t => H₁ t Shc₂:∀ (S : Finset ι), Continuous fun t => H₂ t Shn₁:∀ (t : ℝ) (S : Finset ι), ‖H₁ t S‖ = ‖H₁ 0 S‖hn₂:∀ (t : ℝ) (S : Finset ι), ‖H₂ t S‖ = ‖H₂ 0 S‖b:𝓢(ℝ, ℂ)E₁:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) trueE₂:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) falsew:ℝ → ℝ := fun t => ‖b t‖hE₁:∀ (t : ℝ), 0 ≤ E₁ thE₂:∀ (t : ℝ), 0 ≤ E₂ thw:∀ (t : ℝ), 0 ≤ w ti₁:Integrable (fun t => w t * E₁ t) volumei₂:Integrable (fun t => w t * E₂ t) volumef:ℝ → ℝ := fun t => √(w t * E₁ t)g:ℝ → ℝ := fun t => √(w t * E₂ t)hf:Continuous fhg':Continuous ghfsq:∀ (t : ℝ), f t ^ 2 = w t * E₁ thgsq:∀ (t : ℝ), g t ^ 2 = w t * E₂ tmf:MemLp f 2 volumemg:MemLp g 2 volumehfg:∀ (t : ℝ), f t * g t = w t * (√(E₁ t) * √(E₂ t))hnorm:‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ ∫ (t : ℝ), f t * g t⊢ ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH₁:ℝ → Finset ι → ℂH₂:ℝ → Finset ι → ℂhc₁:∀ (S : Finset ι), Continuous fun t => H₁ t Shc₂:∀ (S : Finset ι), Continuous fun t => H₂ t Shn₁:∀ (t : ℝ) (S : Finset ι), ‖H₁ t S‖ = ‖H₁ 0 S‖hn₂:∀ (t : ℝ) (S : Finset ι), ‖H₂ t S‖ = ‖H₂ 0 S‖b:𝓢(ℝ, ℂ)E₁:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) trueE₂:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) falsew:ℝ → ℝ := fun t => ‖b t‖hE₁:∀ (t : ℝ), 0 ≤ E₁ thE₂:∀ (t : ℝ), 0 ≤ E₂ thw:∀ (t : ℝ), 0 ≤ w ti₁:Integrable (fun t => w t * E₁ t) volumei₂:Integrable (fun t => w t * E₂ t) volumef:ℝ → ℝ := fun t => √(w t * E₁ t)g:ℝ → ℝ := fun t => √(w t * E₂ t)hf:Continuous fhg':Continuous ghfsq:∀ (t : ℝ), f t ^ 2 = w t * E₁ thgsq:∀ (t : ℝ), g t ^ 2 = w t * E₂ tmf:MemLp f 2 volumemg:MemLp g 2 volumehfg:∀ (t : ℝ), f t * g t = w t * (√(E₁ t) * √(E₂ t))hnorm:‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ ∫ (t : ℝ), f t * g thh:∫ (a : ℝ), √(w a * E₁ a) * √(w a * E₂ a) ≤ (∫ (a : ℝ), √(w a * E₁ a) ^ 2) ^ (1 / 2) * (∫ (a : ℝ), √(w a * E₂ a) ^ 2) ^ (1 / 2)⊢ ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Oe:Ok:OH₁:ℝ → Finset ι → ℂH₂:ℝ → Finset ι → ℂhc₁:∀ (S : Finset ι), Continuous fun t => H₁ t Shc₂:∀ (S : Finset ι), Continuous fun t => H₂ t Shn₁:∀ (t : ℝ) (S : Finset ι), ‖H₁ t S‖ = ‖H₁ 0 S‖hn₂:∀ (t : ℝ) (S : Finset ι), ‖H₂ t S‖ = ‖H₂ 0 S‖b:𝓢(ℝ, ℂ)E₁:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) trueE₂:ℝ → ℝ := fun t => secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) falsew:ℝ → ℝ := fun t => ‖b t‖hE₁:∀ (t : ℝ), 0 ≤ E₁ thE₂:∀ (t : ℝ), 0 ≤ E₂ thw:∀ (t : ℝ), 0 ≤ w ti₁:Integrable (fun t => w t * E₁ t) volumei₂:Integrable (fun t => w t * E₂ t) volumef:ℝ → ℝ := fun t => √(w t * E₁ t)g:ℝ → ℝ := fun t => √(w t * E₂ t)hf:Continuous fhg':Continuous ghfsq:∀ (t : ℝ), f t ^ 2 = w t * E₁ thgsq:∀ (t : ℝ), g t ^ 2 = w t * E₂ tmf:MemLp f 2 volumemg:MemLp g 2 volumehfg:∀ (t : ℝ), f t * g t = w t * (√(E₁ t) * √(E₂ t))hnorm:‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ ∫ (t : ℝ), f t * g thh:∫ (a : ℝ), √(w a * E₁ a) * √(w a * E₂ a) ≤ (∫ (a : ℝ), √(w a * E₁ a) ^ 2) ^ (1 / 2) * (∫ (a : ℝ), √(w a * E₂ a) ^ 2) ^ (1 / 2)hfinal:∫ (t : ℝ), f t * g t ≤ √(∫ (t : ℝ), w t * E₁ t) * √(∫ (t : ℝ), w t * E₂ t)⊢ ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (H₁ t) (H₂ t)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₁ t) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (H₂ t) false) All goals completed! 🐙endopen ActualEisensteinCubicopen ConcreteTraceCRT (eisEmbedding eisEmbedding_ne_zero)open FirstPassCubeLabels (firstDualScale firstLogDensity primeProductNorm_pos)variable {ι : Type*} [DecidableEq ι] (p : ι → O) (hp : ∀ i, p i ≠ 0) [∀ i, (Ideal.span {p i}).IsMaximal] (hcop : Pairwise (Function.onFun IsCoprime (fun i => Ideal.span {p i}))) (hg : ∀ i, lambda ∉ Ideal.span {p i})omit [DecidableEq ι] [∀ (i : ι), (span {p i}).IsMaximal] in theorem secondLogTest_continuous (V : ℝ → ℂ) (X : ℝ) (negative : Bool) (S : Finset ι) : Continuous (fun t : ℝ => secondLogTest p V X t negative S) := ι:Type u_1p:ι → OV:ℝ → ℂX:ℝnegative:BoolS:Finset ι⊢ Continuous fun t => secondLogTest p V X t negative S ι:Type u_1p:ι → OV:ℝ → ℂX:ℝnegative:BoolS:Finset ιh:Continuous fun t => FourierBridge.logPhase t (-FirstPassCubeLabels.columnLog p X S)⊢ Continuous fun t => secondLogTest p V X t negative S ι:Type u_1p:ι → OV:ℝ → ℂX:ℝS:Finset ιh:Continuous fun t => FourierBridge.logPhase t (-FirstPassCubeLabels.columnLog p X S)⊢ Continuous fun t => secondLogTest p V X t false Sι:Type u_1p:ι → OV:ℝ → ℂX:ℝS:Finset ιh:Continuous fun t => FourierBridge.logPhase t (-FirstPassCubeLabels.columnLog p X S)⊢ Continuous fun t => secondLogTest p V X t true S ι:Type u_1p:ι → OV:ℝ → ℂX:ℝS:Finset ιh:Continuous fun t => FourierBridge.logPhase t (-FirstPassCubeLabels.columnLog p X S)⊢ Continuous fun t => secondLogTest p V X t false Sι:Type u_1p:ι → OV:ℝ → ℂX:ℝS:Finset ιh:Continuous fun t => FourierBridge.logPhase t (-FirstPassCubeLabels.columnLog p X S)⊢ Continuous fun t => secondLogTest p V X t true S ι:Type u_1p:ι → OV:ℝ → ℂX:ℝS:Finset ιh:Continuous fun t => FourierBridge.logPhase t (-FirstPassCubeLabels.columnLog p X S)⊢ Continuous fun t => V (FirstPassCubeLabels.columnLog p X S) * star (FourierBridge.logPhase t (-FirstPassCubeLabels.columnLog p X S)) / ↑‖eisEmbedding (∏ i ∈ S, p i)‖ ι:Type u_1p:ι → OV:ℝ → ℂX:ℝS:Finset ιh:Continuous fun t => FourierBridge.logPhase t (-FirstPassCubeLabels.columnLog p X S)⊢ Continuous fun t => V (FirstPassCubeLabels.columnLog p X S) * FourierBridge.logPhase t (-FirstPassCubeLabels.columnLog p X S) / ↑‖eisEmbedding (∏ i ∈ S, p i)‖ι:Type u_1p:ι → OV:ℝ → ℂX:ℝS:Finset ιh:Continuous fun t => FourierBridge.logPhase t (-FirstPassCubeLabels.columnLog p X S)⊢ Continuous fun t => V (FirstPassCubeLabels.columnLog p X S) * star (FourierBridge.logPhase t (-FirstPassCubeLabels.columnLog p X S)) / ↑‖eisEmbedding (∏ i ∈ S, p i)‖ All goals completed! 🐙omit [DecidableEq ι] [∀ (i : ι), (span {p i}).IsMaximal] in theorem secondLogTest_norm (V : ℝ → ℂ) (X t : ℝ) (negative : Bool) (S : Finset ι) : ‖secondLogTest p V X t negative S‖ = ‖secondLogTest p V X 0 negative S‖ := ι:Type u_1p:ι → OV:ℝ → ℂX:ℝt:ℝnegative:BoolS:Finset ι⊢ ‖secondLogTest p V X t negative S‖ = ‖secondLogTest p V X 0 negative S‖ ι:Type u_1p:ι → OV:ℝ → ℂX:ℝt:ℝS:Finset ι⊢ ‖secondLogTest p V X t false S‖ = ‖secondLogTest p V X 0 false S‖ι:Type u_1p:ι → OV:ℝ → ℂX:ℝt:ℝS:Finset ι⊢ ‖secondLogTest p V X t true S‖ = ‖secondLogTest p V X 0 true S‖ ι:Type u_1p:ι → OV:ℝ → ℂX:ℝt:ℝS:Finset ι⊢ ‖secondLogTest p V X t false S‖ = ‖secondLogTest p V X 0 false S‖ι:Type u_1p:ι → OV:ℝ → ℂX:ℝt:ℝS:Finset ι⊢ ‖secondLogTest p V X t true S‖ = ‖secondLogTest p V X 0 true S‖ All goals completed! 🐙def secondKernelEnergy (F : Finset ι) (Ψ₁ Ψ₂ : O →* ℂ) (m r c d e k : O) (V : ℝ → ℂ) (X : ℝ) (negative : Bool) (b : 𝓢(ℝ, ℂ)) : ℝ := ∫ t : ℝ, ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V X t negative) negativetheorem actual_second_kernel_uniform_energy_envelope (hinj : Function.Injective (fun i => Ideal.span {p i})) (hc : ∀ i, ringChar (O ⧸ Ideal.span {p i}) ≠ 2) (hpr : ∀ i, lambda ^ 2 ∣ p i - 1) (F : Finset ι) (W : 𝓢(ℝ, ℂ)) (V₁ V₂ : ℝ → ℂ) (X₁ X₂ : ℝ) (hX₁ : 0 < X₁) (hX₂ : 0 < X₂) (M₁ M₂ : ℝ) (hM₁ : 0 ≤ M₁) (hM₂ : 0 ≤ M₂) (hV₁ : ∀ x, V₁ x ≠ 0 → |x| ≤ M₁) (hV₂ : ∀ x, V₂ x ≠ 0 → |x| ≤ M₂) (A J : ℕ) : ∃ K : ℝ, 0 ≤ K ∧ ∀ Y : ℝ, 0 < Y → ∀ e k : O, e ≠ 0 → k ≠ 0 → ∃ b : 𝓢(ℝ, ℂ), (∀ t : ℝ, (1 + firstDualScale p ∅ Y X₁ X₂ e k) ^ A * ‖b t‖ ≤ K * firstLogDensity J t) ∧ ∀ (Ψ₁ Ψ₂ : O →* ℂ) (m r c d : O), ‖actualSecondKernel p hp hg hinj F Ψ₁ Ψ₂ m r c d e k W V₁ V₂ X₁ X₂ Y‖ ≤ Y * Real.sqrt (secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₁ X₁ true b) * Real.sqrt (secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₂ X₂ false b) := ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝhX₁:0 < X₁hX₂:0 < X₂M₁:ℝM₂:ℝhM₁:0 ≤ M₁hM₂:0 ≤ M₂hV₁:∀ (x : ℝ), V₁ x ≠ 0 → |x| ≤ M₁hV₂:∀ (x : ℝ), V₂ x ≠ 0 → |x| ≤ M₂A:ℕJ:ℕ⊢ ∃ K, 0 ≤ K ∧ ∀ (Y : ℝ), 0 < Y → ∀ (e k : O), e ≠ 0 → k ≠ 0 → ∃ b, (∀ (t : ℝ), (1 + firstDualScale p ∅ Y X₁ X₂ e k) ^ A * ‖b t‖ ≤ K * firstLogDensity J t) ∧ ∀ (Ψ₁ Ψ₂ : O →* ℂ) (m r c d : O), ‖actualSecondKernel p hp hg hinj F Ψ₁ Ψ₂ m r c d e k W V₁ V₂ X₁ X₂ Y‖ ≤ Y * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₁ X₁ true b) * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₂ X₂ false b) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝhX₁:0 < X₁hX₂:0 < X₂M₁:ℝM₂:ℝhM₁:0 ≤ M₁hM₂:0 ≤ M₂hV₁:∀ (x : ℝ), V₁ x ≠ 0 → |x| ≤ M₁hV₂:∀ (x : ℝ), V₂ x ≠ 0 → |x| ≤ M₂A:ℕJ:ℕhV₁':∀ (x : ℝ), star (V₁ x) ≠ 0 → |x| ≤ M₁⊢ ∃ K, 0 ≤ K ∧ ∀ (Y : ℝ), 0 < Y → ∀ (e k : O), e ≠ 0 → k ≠ 0 → ∃ b, (∀ (t : ℝ), (1 + firstDualScale p ∅ Y X₁ X₂ e k) ^ A * ‖b t‖ ≤ K * firstLogDensity J t) ∧ ∀ (Ψ₁ Ψ₂ : O →* ℂ) (m r c d : O), ‖actualSecondKernel p hp hg hinj F Ψ₁ Ψ₂ m r c d e k W V₁ V₂ X₁ X₂ Y‖ ≤ Y * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₁ X₁ true b) * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₂ X₂ false b) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝhX₁:0 < X₁hX₂:0 < X₂M₁:ℝM₂:ℝhM₁:0 ≤ M₁hM₂:0 ≤ M₂hV₁:∀ (x : ℝ), V₁ x ≠ 0 → |x| ≤ M₁hV₂:∀ (x : ℝ), V₂ x ≠ 0 → |x| ≤ M₂A:ℕJ:ℕhV₁':∀ (x : ℝ), star (V₁ x) ≠ 0 → |x| ≤ M₁K:ℝhK:0 ≤ Khs:∀ (R : ℝ), 0 < R → ∃ b, (∀ (x y : ℝ), star (V₁ x) * V₂ y * EisensteinSchwartzPoisson.paperRadialFourier (⇑W) (R * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * FourierBridge.logPhase t (-x) * (V₂ y * FourierBridge.logPhase t (-y)) * b t) ∧ ∀ (t : ℝ), (1 + R) ^ A * ‖b t‖ ≤ K * firstLogDensity J t⊢ ∃ K, 0 ≤ K ∧ ∀ (Y : ℝ), 0 < Y → ∀ (e k : O), e ≠ 0 → k ≠ 0 → ∃ b, (∀ (t : ℝ), (1 + firstDualScale p ∅ Y X₁ X₂ e k) ^ A * ‖b t‖ ≤ K * firstLogDensity J t) ∧ ∀ (Ψ₁ Ψ₂ : O →* ℂ) (m r c d : O), ‖actualSecondKernel p hp hg hinj F Ψ₁ Ψ₂ m r c d e k W V₁ V₂ X₁ X₂ Y‖ ≤ Y * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₁ X₁ true b) * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₂ X₂ false b) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝhX₁:0 < X₁hX₂:0 < X₂M₁:ℝM₂:ℝhM₁:0 ≤ M₁hM₂:0 ≤ M₂hV₁:∀ (x : ℝ), V₁ x ≠ 0 → |x| ≤ M₁hV₂:∀ (x : ℝ), V₂ x ≠ 0 → |x| ≤ M₂A:ℕJ:ℕhV₁':∀ (x : ℝ), star (V₁ x) ≠ 0 → |x| ≤ M₁K:ℝhK:0 ≤ Khs:∀ (R : ℝ), 0 < R → ∃ b, (∀ (x y : ℝ), star (V₁ x) * V₂ y * EisensteinSchwartzPoisson.paperRadialFourier (⇑W) (R * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * FourierBridge.logPhase t (-x) * (V₂ y * FourierBridge.logPhase t (-y)) * b t) ∧ ∀ (t : ℝ), (1 + R) ^ A * ‖b t‖ ≤ K * firstLogDensity J t⊢ ∀ (Y : ℝ), 0 < Y → ∀ (e k : O), e ≠ 0 → k ≠ 0 → ∃ b, (∀ (t : ℝ), (1 + firstDualScale p ∅ Y X₁ X₂ e k) ^ A * ‖b t‖ ≤ K * firstLogDensity J t) ∧ ∀ (Ψ₁ Ψ₂ : O →* ℂ) (m r c d : O), ‖actualSecondKernel p hp hg hinj F Ψ₁ Ψ₂ m r c d e k W V₁ V₂ X₁ X₂ Y‖ ≤ Y * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₁ X₁ true b) * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₂ X₂ false b) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝhX₁:0 < X₁hX₂:0 < X₂M₁:ℝM₂:ℝhM₁:0 ≤ M₁hM₂:0 ≤ M₂hV₁:∀ (x : ℝ), V₁ x ≠ 0 → |x| ≤ M₁hV₂:∀ (x : ℝ), V₂ x ≠ 0 → |x| ≤ M₂A:ℕJ:ℕhV₁':∀ (x : ℝ), star (V₁ x) ≠ 0 → |x| ≤ M₁K:ℝhK:0 ≤ Khs:∀ (R : ℝ), 0 < R → ∃ b, (∀ (x y : ℝ), star (V₁ x) * V₂ y * EisensteinSchwartzPoisson.paperRadialFourier (⇑W) (R * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * FourierBridge.logPhase t (-x) * (V₂ y * FourierBridge.logPhase t (-y)) * b t) ∧ ∀ (t : ℝ), (1 + R) ^ A * ‖b t‖ ≤ K * firstLogDensity J tY:ℝhY:0 < Ye:Ok:Ohe:e ≠ 0hk:k ≠ 0⊢ ∃ b, (∀ (t : ℝ), (1 + firstDualScale p ∅ Y X₁ X₂ e k) ^ A * ‖b t‖ ≤ K * firstLogDensity J t) ∧ ∀ (Ψ₁ Ψ₂ : O →* ℂ) (m r c d : O), ‖actualSecondKernel p hp hg hinj F Ψ₁ Ψ₂ m r c d e k W V₁ V₂ X₁ X₂ Y‖ ≤ Y * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₁ X₁ true b) * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₂ X₂ false b) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝhX₁:0 < X₁hX₂:0 < X₂M₁:ℝM₂:ℝhM₁:0 ≤ M₁hM₂:0 ≤ M₂hV₁:∀ (x : ℝ), V₁ x ≠ 0 → |x| ≤ M₁hV₂:∀ (x : ℝ), V₂ x ≠ 0 → |x| ≤ M₂A:ℕJ:ℕhV₁':∀ (x : ℝ), star (V₁ x) ≠ 0 → |x| ≤ M₁K:ℝhK:0 ≤ Khs:∀ (R : ℝ), 0 < R → ∃ b, (∀ (x y : ℝ), star (V₁ x) * V₂ y * EisensteinSchwartzPoisson.paperRadialFourier (⇑W) (R * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * FourierBridge.logPhase t (-x) * (V₂ y * FourierBridge.logPhase t (-y)) * b t) ∧ ∀ (t : ℝ), (1 + R) ^ A * ‖b t‖ ≤ K * firstLogDensity J tY:ℝhY:0 < Ye:Ok:Ohe:e ≠ 0hk:k ≠ 0hR:0 < firstDualScale p ∅ Y X₁ X₂ e k⊢ ∃ b, (∀ (t : ℝ), (1 + firstDualScale p ∅ Y X₁ X₂ e k) ^ A * ‖b t‖ ≤ K * firstLogDensity J t) ∧ ∀ (Ψ₁ Ψ₂ : O →* ℂ) (m r c d : O), ‖actualSecondKernel p hp hg hinj F Ψ₁ Ψ₂ m r c d e k W V₁ V₂ X₁ X₂ Y‖ ≤ Y * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₁ X₁ true b) * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₂ X₂ false b) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝhX₁:0 < X₁hX₂:0 < X₂M₁:ℝM₂:ℝhM₁:0 ≤ M₁hM₂:0 ≤ M₂hV₁:∀ (x : ℝ), V₁ x ≠ 0 → |x| ≤ M₁hV₂:∀ (x : ℝ), V₂ x ≠ 0 → |x| ≤ M₂A:ℕJ:ℕhV₁':∀ (x : ℝ), star (V₁ x) ≠ 0 → |x| ≤ M₁K:ℝhK:0 ≤ Khs:∀ (R : ℝ), 0 < R → ∃ b, (∀ (x y : ℝ), star (V₁ x) * V₂ y * EisensteinSchwartzPoisson.paperRadialFourier (⇑W) (R * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * FourierBridge.logPhase t (-x) * (V₂ y * FourierBridge.logPhase t (-y)) * b t) ∧ ∀ (t : ℝ), (1 + R) ^ A * ‖b t‖ ≤ K * firstLogDensity J tY:ℝhY:0 < Ye:Ok:Ohe:e ≠ 0hk:k ≠ 0hR:0 < firstDualScale p ∅ Y X₁ X₂ e kb:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * EisensteinSchwartzPoisson.paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * FourierBridge.logPhase t (-x) * (V₂ y * FourierBridge.logPhase t (-y)) * b thb:∀ (t : ℝ), (1 + firstDualScale p ∅ Y X₁ X₂ e k) ^ A * ‖b t‖ ≤ K * firstLogDensity J t⊢ ∃ b, (∀ (t : ℝ), (1 + firstDualScale p ∅ Y X₁ X₂ e k) ^ A * ‖b t‖ ≤ K * firstLogDensity J t) ∧ ∀ (Ψ₁ Ψ₂ : O →* ℂ) (m r c d : O), ‖actualSecondKernel p hp hg hinj F Ψ₁ Ψ₂ m r c d e k W V₁ V₂ X₁ X₂ Y‖ ≤ Y * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₁ X₁ true b) * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₂ X₂ false b) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝhX₁:0 < X₁hX₂:0 < X₂M₁:ℝM₂:ℝhM₁:0 ≤ M₁hM₂:0 ≤ M₂hV₁:∀ (x : ℝ), V₁ x ≠ 0 → |x| ≤ M₁hV₂:∀ (x : ℝ), V₂ x ≠ 0 → |x| ≤ M₂A:ℕJ:ℕhV₁':∀ (x : ℝ), star (V₁ x) ≠ 0 → |x| ≤ M₁K:ℝhK:0 ≤ Khs:∀ (R : ℝ), 0 < R → ∃ b, (∀ (x y : ℝ), star (V₁ x) * V₂ y * EisensteinSchwartzPoisson.paperRadialFourier (⇑W) (R * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * FourierBridge.logPhase t (-x) * (V₂ y * FourierBridge.logPhase t (-y)) * b t) ∧ ∀ (t : ℝ), (1 + R) ^ A * ‖b t‖ ≤ K * firstLogDensity J tY:ℝhY:0 < Ye:Ok:Ohe:e ≠ 0hk:k ≠ 0hR:0 < firstDualScale p ∅ Y X₁ X₂ e kb:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * EisensteinSchwartzPoisson.paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * FourierBridge.logPhase t (-x) * (V₂ y * FourierBridge.logPhase t (-y)) * b thb:∀ (t : ℝ), (1 + firstDualScale p ∅ Y X₁ X₂ e k) ^ A * ‖b t‖ ≤ K * firstLogDensity J t⊢ ∀ (Ψ₁ Ψ₂ : O →* ℂ) (m r c d : O), ‖actualSecondKernel p hp hg hinj F Ψ₁ Ψ₂ m r c d e k W V₁ V₂ X₁ X₂ Y‖ ≤ Y * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₁ X₁ true b) * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₂ X₂ false b) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝhX₁:0 < X₁hX₂:0 < X₂M₁:ℝM₂:ℝhM₁:0 ≤ M₁hM₂:0 ≤ M₂hV₁:∀ (x : ℝ), V₁ x ≠ 0 → |x| ≤ M₁hV₂:∀ (x : ℝ), V₂ x ≠ 0 → |x| ≤ M₂A:ℕJ:ℕhV₁':∀ (x : ℝ), star (V₁ x) ≠ 0 → |x| ≤ M₁K:ℝhK:0 ≤ Khs:∀ (R : ℝ), 0 < R → ∃ b, (∀ (x y : ℝ), star (V₁ x) * V₂ y * EisensteinSchwartzPoisson.paperRadialFourier (⇑W) (R * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * FourierBridge.logPhase t (-x) * (V₂ y * FourierBridge.logPhase t (-y)) * b t) ∧ ∀ (t : ℝ), (1 + R) ^ A * ‖b t‖ ≤ K * firstLogDensity J tY:ℝhY:0 < Ye:Ok:Ohe:e ≠ 0hk:k ≠ 0hR:0 < firstDualScale p ∅ Y X₁ X₂ e kb:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * EisensteinSchwartzPoisson.paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * FourierBridge.logPhase t (-x) * (V₂ y * FourierBridge.logPhase t (-y)) * b thb:∀ (t : ℝ), (1 + firstDualScale p ∅ Y X₁ X₂ e k) ^ A * ‖b t‖ ≤ K * firstLogDensity J tΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:O⊢ ‖actualSecondKernel p hp hg hinj F Ψ₁ Ψ₂ m r c d e k W V₁ V₂ X₁ X₂ Y‖ ≤ Y * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₁ X₁ true b) * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₂ X₂ false b) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝhX₁:0 < X₁hX₂:0 < X₂M₁:ℝM₂:ℝhM₁:0 ≤ M₁hM₂:0 ≤ M₂hV₁:∀ (x : ℝ), V₁ x ≠ 0 → |x| ≤ M₁hV₂:∀ (x : ℝ), V₂ x ≠ 0 → |x| ≤ M₂A:ℕJ:ℕhV₁':∀ (x : ℝ), star (V₁ x) ≠ 0 → |x| ≤ M₁K:ℝhK:0 ≤ Khs:∀ (R : ℝ), 0 < R → ∃ b, (∀ (x y : ℝ), star (V₁ x) * V₂ y * EisensteinSchwartzPoisson.paperRadialFourier (⇑W) (R * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * FourierBridge.logPhase t (-x) * (V₂ y * FourierBridge.logPhase t (-y)) * b t) ∧ ∀ (t : ℝ), (1 + R) ^ A * ‖b t‖ ≤ K * firstLogDensity J tY:ℝhY:0 < Ye:Ok:Ohe:e ≠ 0hk:k ≠ 0hR:0 < firstDualScale p ∅ Y X₁ X₂ e kb:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * EisensteinSchwartzPoisson.paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * FourierBridge.logPhase t (-x) * (V₂ y * FourierBridge.logPhase t (-y)) * b thb:∀ (t : ℝ), (1 + firstDualScale p ∅ Y X₁ X₂ e k) ^ A * ‖b t‖ ≤ K * firstLogDensity J tΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:O⊢ Y * ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)‖ ≤ Y * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₁ X₁ true b) * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₂ X₂ false b) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝhX₁:0 < X₁hX₂:0 < X₂M₁:ℝM₂:ℝhM₁:0 ≤ M₁hM₂:0 ≤ M₂hV₁:∀ (x : ℝ), V₁ x ≠ 0 → |x| ≤ M₁hV₂:∀ (x : ℝ), V₂ x ≠ 0 → |x| ≤ M₂A:ℕJ:ℕhV₁':∀ (x : ℝ), star (V₁ x) ≠ 0 → |x| ≤ M₁K:ℝhK:0 ≤ Khs:∀ (R : ℝ), 0 < R → ∃ b, (∀ (x y : ℝ), star (V₁ x) * V₂ y * EisensteinSchwartzPoisson.paperRadialFourier (⇑W) (R * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * FourierBridge.logPhase t (-x) * (V₂ y * FourierBridge.logPhase t (-y)) * b t) ∧ ∀ (t : ℝ), (1 + R) ^ A * ‖b t‖ ≤ K * firstLogDensity J tY:ℝhY:0 < Ye:Ok:Ohe:e ≠ 0hk:k ≠ 0hR:0 < firstDualScale p ∅ Y X₁ X₂ e kb:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * EisensteinSchwartzPoisson.paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * FourierBridge.logPhase t (-x) * (V₂ y * FourierBridge.logPhase t (-y)) * b thb:∀ (t : ℝ), (1 + firstDualScale p ∅ Y X₁ X₂ e k) ^ A * ‖b t‖ ≤ K * firstLogDensity J tΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Ohi:‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₂ X₂ t false) false)⊢ Y * ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)‖ ≤ Y * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₁ X₁ true b) * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₂ X₂ false b) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝhX₁:0 < X₁hX₂:0 < X₂M₁:ℝM₂:ℝhM₁:0 ≤ M₁hM₂:0 ≤ M₂hV₁:∀ (x : ℝ), V₁ x ≠ 0 → |x| ≤ M₁hV₂:∀ (x : ℝ), V₂ x ≠ 0 → |x| ≤ M₂A:ℕJ:ℕhV₁':∀ (x : ℝ), star (V₁ x) ≠ 0 → |x| ≤ M₁K:ℝhK:0 ≤ Khs:∀ (R : ℝ), 0 < R → ∃ b, (∀ (x y : ℝ), star (V₁ x) * V₂ y * EisensteinSchwartzPoisson.paperRadialFourier (⇑W) (R * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * FourierBridge.logPhase t (-x) * (V₂ y * FourierBridge.logPhase t (-y)) * b t) ∧ ∀ (t : ℝ), (1 + R) ^ A * ‖b t‖ ≤ K * firstLogDensity J tY:ℝhY:0 < Ye:Ok:Ohe:e ≠ 0hk:k ≠ 0hR:0 < firstDualScale p ∅ Y X₁ X₂ e kb:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * EisensteinSchwartzPoisson.paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * FourierBridge.logPhase t (-x) * (V₂ y * FourierBridge.logPhase t (-y)) * b thb:∀ (t : ℝ), (1 + firstDualScale p ∅ Y X₁ X₂ e k) ^ A * ‖b t‖ ≤ K * firstLogDensity J tΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Ohi:‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₂ X₂ t false) false)⊢ Y * ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)‖ ≤ Y * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₁ X₁ true b) * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₂ X₂ false b) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}hinj:Function.Injective fun i => span {p i}hc:∀ (i : ι), ringChar (O ⧸ span {p i}) ≠ 2hpr:∀ (i : ι), lambda ^ 2 ∣ p i - 1F:Finset ιW:𝓢(ℝ, ℂ)V₁:ℝ → ℂV₂:ℝ → ℂX₁:ℝX₂:ℝhX₁:0 < X₁hX₂:0 < X₂M₁:ℝM₂:ℝhM₁:0 ≤ M₁hM₂:0 ≤ M₂hV₁:∀ (x : ℝ), V₁ x ≠ 0 → |x| ≤ M₁hV₂:∀ (x : ℝ), V₂ x ≠ 0 → |x| ≤ M₂A:ℕJ:ℕhV₁':∀ (x : ℝ), star (V₁ x) ≠ 0 → |x| ≤ M₁K:ℝhK:0 ≤ Khs:∀ (R : ℝ), 0 < R → ∃ b, (∀ (x y : ℝ), star (V₁ x) * V₂ y * EisensteinSchwartzPoisson.paperRadialFourier (⇑W) (R * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * FourierBridge.logPhase t (-x) * (V₂ y * FourierBridge.logPhase t (-y)) * b t) ∧ ∀ (t : ℝ), (1 + R) ^ A * ‖b t‖ ≤ K * firstLogDensity J tY:ℝhY:0 < Ye:Ok:Ohe:e ≠ 0hk:k ≠ 0hR:0 < firstDualScale p ∅ Y X₁ X₂ e kb:𝓢(ℝ, ℂ)hsep:∀ (x y : ℝ), star (V₁ x) * V₂ y * EisensteinSchwartzPoisson.paperRadialFourier (⇑W) (firstDualScale p ∅ Y X₁ X₂ e k * rexp (-x - y)) = ∫ (t : ℝ), star (V₁ x) * FourierBridge.logPhase t (-x) * (V₂ y * FourierBridge.logPhase t (-y)) * b thb:∀ (t : ℝ), (1 + firstDualScale p ∅ Y X₁ X₂ e k) ^ A * ‖b t‖ ≤ K * firstLogDensity J tΨ₁:O →* ℂΨ₂:O →* ℂm:Or:Oc:Od:Ohi:‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)‖ ≤ √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) true) * √(∫ (t : ℝ), ‖b t‖ * secondTotalEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₂ X₂ t false) false)⊢ Y * ‖∫ (t : ℝ), b t * secondActualPair p hp hg hinj F Ψ₁ Ψ₂ m r c d e k (secondLogTest p V₁ X₁ t true) (secondLogTest p V₂ X₂ t false)‖ ≤ Y * (√(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₁ X₁ true b) * √(secondKernelEnergy p hp hcop hg F Ψ₁ Ψ₂ m r c d e k V₂ X₂ false b)) All goals completed! 🐙endsectionopen scoped BigOperators Classicalopen ActualEisensteinCubicopen SecondPassFiber (OldTuple newLabel newRow Valid)theorem second_energy_selected_family {κ : Type*} [Fintype κ] (ε : ℝ) (hε : 0 < ε) : ∃ C : ℝ, 0 < C ∧ ∀ (s : Finset OldTuple) (t : Finset (Ideal O × O)) (b0 : Ideal O) (w : OldTuple → ℂ) (B lengthScale : ℝ) (P : κ → Ideal O × O → ℂ) (sel : OldTuple → κ), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ lengthScale → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, (Ideal.absNorm z.1 : ℝ) ≤ lengthScale) → (∀ x ∈ s, ‖w x‖ ≤ B) → (∑ x ∈ s, ‖w x‖ * ‖P (sel x) (newLabel x, newRow x)‖ ^ 2) ≤ B * C * (lengthScale * Ideal.absNorm b0) ^ ε * ∑ u : κ, ∑ z ∈ t, ‖P u z‖ ^ 2 := κ:Type u_1inst✝:Fintype κε:ℝhε:0 < ε⊢ ∃ C, 0 < C ∧ ∀ (s : Finset OldTuple) (t : Finset (Ideal O × O)) (b0 : Ideal O) (w : OldTuple → ℂ) (B lengthScale : ℝ) (P : κ → Ideal O × O → ℂ) (sel : OldTuple → κ), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ lengthScale → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScale) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖P (sel x) (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ u, ∑ z ∈ t, ‖P u z‖ ^ 2 κ:Type u_1inst✝:Fintype κε:ℝhε:0 < εC:ℝhC:0 < Chb:∀ (s : Finset OldTuple) (t : Finset (Ideal SecondPassFiber.O × SecondPassFiber.O)) (b0 : Ideal SecondPassFiber.O) (w : OldTuple → ℂ) (P : Ideal SecondPassFiber.O × SecondPassFiber.O → ℂ) (B H : ℝ), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ H → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ H) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖P (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (H * ↑(absNorm b0)) ^ ε * ∑ z ∈ t, ‖P z‖ ^ 2⊢ ∃ C, 0 < C ∧ ∀ (s : Finset OldTuple) (t : Finset (Ideal O × O)) (b0 : Ideal O) (w : OldTuple → ℂ) (B lengthScale : ℝ) (P : κ → Ideal O × O → ℂ) (sel : OldTuple → κ), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ lengthScale → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScale) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖P (sel x) (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ u, ∑ z ∈ t, ‖P u z‖ ^ 2 κ:Type u_1inst✝:Fintype κε:ℝhε:0 < εC:ℝhC:0 < Chb:∀ (s : Finset OldTuple) (t : Finset (Ideal SecondPassFiber.O × SecondPassFiber.O)) (b0 : Ideal SecondPassFiber.O) (w : OldTuple → ℂ) (P : Ideal SecondPassFiber.O × SecondPassFiber.O → ℂ) (B H : ℝ), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ H → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ H) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖P (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (H * ↑(absNorm b0)) ^ ε * ∑ z ∈ t, ‖P z‖ ^ 2⊢ ∀ (s : Finset OldTuple) (t : Finset (Ideal O × O)) (b0 : Ideal O) (w : OldTuple → ℂ) (B lengthScale : ℝ) (P : κ → Ideal O × O → ℂ) (sel : OldTuple → κ), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ lengthScale → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScale) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖P (sel x) (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ u, ∑ z ∈ t, ‖P u z‖ ^ 2 κ:Type u_1inst✝:Fintype κε:ℝhε:0 < εC:ℝhC:0 < Chb:∀ (s : Finset OldTuple) (t : Finset (Ideal SecondPassFiber.O × SecondPassFiber.O)) (b0 : Ideal SecondPassFiber.O) (w : OldTuple → ℂ) (P : Ideal SecondPassFiber.O × SecondPassFiber.O → ℂ) (B H : ℝ), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ H → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ H) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖P (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (H * ↑(absNorm b0)) ^ ε * ∑ z ∈ t, ‖P z‖ ^ 2s:Finset OldTuplet:Finset (Ideal O × O)b0:Ideal Ow:OldTuple → ℂB:ℝlengthScale:ℝP:κ → Ideal O × O → ℂsel:OldTuple → κhb0:b0 ≠ ⊥hB:0 ≤ BhL:0 ≤ lengthScalehvalid:∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)hmap:∀ x ∈ s, (newLabel x, newRow x) ∈ tht:∀ z ∈ t, z.1 ≠ ⊥hnorm:∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScalehw:∀ x ∈ s, ‖w x‖ ≤ B⊢ ∑ x ∈ s, ‖w x‖ * ‖P (sel x) (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ u, ∑ z ∈ t, ‖P u z‖ ^ 2 calc _ ≤ ∑ x ∈ s, ‖w x‖ * ∑ u : κ, ‖P u (newLabel x, newRow x)‖ ^ 2 := κ:Type u_1inst✝:Fintype κε:ℝhε:0 < εC:ℝhC:0 < Chb:∀ (s : Finset OldTuple) (t : Finset (Ideal SecondPassFiber.O × SecondPassFiber.O)) (b0 : Ideal SecondPassFiber.O) (w : OldTuple → ℂ) (P : Ideal SecondPassFiber.O × SecondPassFiber.O → ℂ) (B H : ℝ), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ H → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ H) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖P (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (H * ↑(absNorm b0)) ^ ε * ∑ z ∈ t, ‖P z‖ ^ 2s:Finset OldTuplet:Finset (Ideal O × O)b0:Ideal Ow:OldTuple → ℂB:ℝlengthScale:ℝP:κ → Ideal O × O → ℂsel:OldTuple → κhb0:b0 ≠ ⊥hB:0 ≤ BhL:0 ≤ lengthScalehvalid:∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)hmap:∀ x ∈ s, (newLabel x, newRow x) ∈ tht:∀ z ∈ t, z.1 ≠ ⊥hnorm:∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScalehw:∀ x ∈ s, ‖w x‖ ≤ B⊢ ∑ x ∈ s, ‖w x‖ * ‖P (sel x) (newLabel x, newRow x)‖ ^ 2 ≤ ∑ x ∈ s, ‖w x‖ * ∑ u, ‖P u (newLabel x, newRow x)‖ ^ 2 κ:Type u_1inst✝:Fintype κε:ℝhε:0 < εC:ℝhC:0 < Chb:∀ (s : Finset OldTuple) (t : Finset (Ideal SecondPassFiber.O × SecondPassFiber.O)) (b0 : Ideal SecondPassFiber.O) (w : OldTuple → ℂ) (P : Ideal SecondPassFiber.O × SecondPassFiber.O → ℂ) (B H : ℝ), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ H → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ H) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖P (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (H * ↑(absNorm b0)) ^ ε * ∑ z ∈ t, ‖P z‖ ^ 2s:Finset OldTuplet:Finset (Ideal O × O)b0:Ideal Ow:OldTuple → ℂB:ℝlengthScale:ℝP:κ → Ideal O × O → ℂsel:OldTuple → κhb0:b0 ≠ ⊥hB:0 ≤ BhL:0 ≤ lengthScalehvalid:∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)hmap:∀ x ∈ s, (newLabel x, newRow x) ∈ tht:∀ z ∈ t, z.1 ≠ ⊥hnorm:∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScalehw:∀ x ∈ s, ‖w x‖ ≤ B⊢ ∀ i ∈ s, ‖w i‖ * ‖P (sel i) (newLabel i, newRow i)‖ ^ 2 ≤ ‖w i‖ * ∑ u, ‖P u (newLabel i, newRow i)‖ ^ 2 κ:Type u_1inst✝:Fintype κε:ℝhε:0 < εC:ℝhC:0 < Chb:∀ (s : Finset OldTuple) (t : Finset (Ideal SecondPassFiber.O × SecondPassFiber.O)) (b0 : Ideal SecondPassFiber.O) (w : OldTuple → ℂ) (P : Ideal SecondPassFiber.O × SecondPassFiber.O → ℂ) (B H : ℝ), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ H → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ H) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖P (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (H * ↑(absNorm b0)) ^ ε * ∑ z ∈ t, ‖P z‖ ^ 2s:Finset OldTuplet:Finset (Ideal O × O)b0:Ideal Ow:OldTuple → ℂB:ℝlengthScale:ℝP:κ → Ideal O × O → ℂsel:OldTuple → κhb0:b0 ≠ ⊥hB:0 ≤ BhL:0 ≤ lengthScalehvalid:∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)hmap:∀ x ∈ s, (newLabel x, newRow x) ∈ tht:∀ z ∈ t, z.1 ≠ ⊥hnorm:∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScalehw:∀ x ∈ s, ‖w x‖ ≤ Bx:OldTuplehx:x ∈ s⊢ ‖w x‖ * ‖P (sel x) (newLabel x, newRow x)‖ ^ 2 ≤ ‖w x‖ * ∑ u, ‖P u (newLabel x, newRow x)‖ ^ 2 κ:Type u_1inst✝:Fintype κε:ℝhε:0 < εC:ℝhC:0 < Chb:∀ (s : Finset OldTuple) (t : Finset (Ideal SecondPassFiber.O × SecondPassFiber.O)) (b0 : Ideal SecondPassFiber.O) (w : OldTuple → ℂ) (P : Ideal SecondPassFiber.O × SecondPassFiber.O → ℂ) (B H : ℝ), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ H → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ H) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖P (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (H * ↑(absNorm b0)) ^ ε * ∑ z ∈ t, ‖P z‖ ^ 2s:Finset OldTuplet:Finset (Ideal O × O)b0:Ideal Ow:OldTuple → ℂB:ℝlengthScale:ℝP:κ → Ideal O × O → ℂsel:OldTuple → κhb0:b0 ≠ ⊥hB:0 ≤ BhL:0 ≤ lengthScalehvalid:∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)hmap:∀ x ∈ s, (newLabel x, newRow x) ∈ tht:∀ z ∈ t, z.1 ≠ ⊥hnorm:∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScalehw:∀ x ∈ s, ‖w x‖ ≤ Bx:OldTuplehx:x ∈ s⊢ ‖P (sel x) (newLabel x, newRow x)‖ ^ 2 ≤ ∑ u, ‖P u (newLabel x, newRow x)‖ ^ 2 All goals completed! 🐙 _ = ∑ u : κ, ∑ x ∈ s, ‖w x‖ * ‖P u (newLabel x, newRow x)‖ ^ 2 := κ:Type u_1inst✝:Fintype κε:ℝhε:0 < εC:ℝhC:0 < Chb:∀ (s : Finset OldTuple) (t : Finset (Ideal SecondPassFiber.O × SecondPassFiber.O)) (b0 : Ideal SecondPassFiber.O) (w : OldTuple → ℂ) (P : Ideal SecondPassFiber.O × SecondPassFiber.O → ℂ) (B H : ℝ), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ H → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ H) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖P (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (H * ↑(absNorm b0)) ^ ε * ∑ z ∈ t, ‖P z‖ ^ 2s:Finset OldTuplet:Finset (Ideal O × O)b0:Ideal Ow:OldTuple → ℂB:ℝlengthScale:ℝP:κ → Ideal O × O → ℂsel:OldTuple → κhb0:b0 ≠ ⊥hB:0 ≤ BhL:0 ≤ lengthScalehvalid:∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)hmap:∀ x ∈ s, (newLabel x, newRow x) ∈ tht:∀ z ∈ t, z.1 ≠ ⊥hnorm:∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScalehw:∀ x ∈ s, ‖w x‖ ≤ B⊢ ∑ x ∈ s, ‖w x‖ * ∑ u, ‖P u (newLabel x, newRow x)‖ ^ 2 = ∑ u, ∑ x ∈ s, ‖w x‖ * ‖P u (newLabel x, newRow x)‖ ^ 2 κ:Type u_1inst✝:Fintype κε:ℝhε:0 < εC:ℝhC:0 < Chb:∀ (s : Finset OldTuple) (t : Finset (Ideal SecondPassFiber.O × SecondPassFiber.O)) (b0 : Ideal SecondPassFiber.O) (w : OldTuple → ℂ) (P : Ideal SecondPassFiber.O × SecondPassFiber.O → ℂ) (B H : ℝ), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ H → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ H) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖P (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (H * ↑(absNorm b0)) ^ ε * ∑ z ∈ t, ‖P z‖ ^ 2s:Finset OldTuplet:Finset (Ideal O × O)b0:Ideal Ow:OldTuple → ℂB:ℝlengthScale:ℝP:κ → Ideal O × O → ℂsel:OldTuple → κhb0:b0 ≠ ⊥hB:0 ≤ BhL:0 ≤ lengthScalehvalid:∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)hmap:∀ x ∈ s, (newLabel x, newRow x) ∈ tht:∀ z ∈ t, z.1 ≠ ⊥hnorm:∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScalehw:∀ x ∈ s, ‖w x‖ ≤ B⊢ ∑ x ∈ s, ∑ i, ‖w x‖ * ‖P i (newLabel x, newRow x)‖ ^ 2 = ∑ u, ∑ x ∈ s, ‖w x‖ * ‖P u (newLabel x, newRow x)‖ ^ 2 All goals completed! 🐙 _ ≤ ∑ u : κ, B * C * (lengthScale * Ideal.absNorm b0) ^ ε * ∑ z ∈ t, ‖P u z‖ ^ 2 := κ:Type u_1inst✝:Fintype κε:ℝhε:0 < εC:ℝhC:0 < Chb:∀ (s : Finset OldTuple) (t : Finset (Ideal SecondPassFiber.O × SecondPassFiber.O)) (b0 : Ideal SecondPassFiber.O) (w : OldTuple → ℂ) (P : Ideal SecondPassFiber.O × SecondPassFiber.O → ℂ) (B H : ℝ), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ H → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ H) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖P (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (H * ↑(absNorm b0)) ^ ε * ∑ z ∈ t, ‖P z‖ ^ 2s:Finset OldTuplet:Finset (Ideal O × O)b0:Ideal Ow:OldTuple → ℂB:ℝlengthScale:ℝP:κ → Ideal O × O → ℂsel:OldTuple → κhb0:b0 ≠ ⊥hB:0 ≤ BhL:0 ≤ lengthScalehvalid:∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)hmap:∀ x ∈ s, (newLabel x, newRow x) ∈ tht:∀ z ∈ t, z.1 ≠ ⊥hnorm:∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScalehw:∀ x ∈ s, ‖w x‖ ≤ B⊢ ∑ u, ∑ x ∈ s, ‖w x‖ * ‖P u (newLabel x, newRow x)‖ ^ 2 ≤ ∑ u, B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ z ∈ t, ‖P u z‖ ^ 2 All goals completed! 🐙 _ = _ := κ:Type u_1inst✝:Fintype κε:ℝhε:0 < εC:ℝhC:0 < Chb:∀ (s : Finset OldTuple) (t : Finset (Ideal SecondPassFiber.O × SecondPassFiber.O)) (b0 : Ideal SecondPassFiber.O) (w : OldTuple → ℂ) (P : Ideal SecondPassFiber.O × SecondPassFiber.O → ℂ) (B H : ℝ), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ H → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ H) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖P (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (H * ↑(absNorm b0)) ^ ε * ∑ z ∈ t, ‖P z‖ ^ 2s:Finset OldTuplet:Finset (Ideal O × O)b0:Ideal Ow:OldTuple → ℂB:ℝlengthScale:ℝP:κ → Ideal O × O → ℂsel:OldTuple → κhb0:b0 ≠ ⊥hB:0 ≤ BhL:0 ≤ lengthScalehvalid:∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)hmap:∀ x ∈ s, (newLabel x, newRow x) ∈ tht:∀ z ∈ t, z.1 ≠ ⊥hnorm:∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScalehw:∀ x ∈ s, ‖w x‖ ≤ B⊢ ∑ u, B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ z ∈ t, ‖P u z‖ ^ 2 = B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ u, ∑ z ∈ t, ‖P u z‖ ^ 2 All goals completed! 🐙endopen scoped BigOperators Classicalopen ActualEisensteinCubicopen ConcretePrimeRowBridge (idealGenerator span_idealGenerator)open SecondPassFiber (OldTuple newLabel newRow Valid)local instance : Fintype Oˣ := @Fintype.ofFinite _ PrimaryIdealUnitReindex.finite_unitsdef tupleLabelElement (x : OldTuple) : O := idealGenerator (x.core 0) * idealGenerator (x.core 1) * idealGenerator (x.core 2) * idealGenerator x.vtheorem tupleLabelElement_span (x : OldTuple) : Ideal.span {tupleLabelElement x} = newLabel x := x:OldTuple⊢ span {tupleLabelElement x} = newLabel x All goals completed! 🐙theorem exists_label_unit (f : O) (I : Ideal O) (hf : Ideal.span {f} = I) : ∃ u : Oˣ, f = (u : O) * idealGenerator I := f:OI:Ideal Ohf:span {f} = I⊢ ∃ u, f = ↑u * idealGenerator I f:OI:Ideal Ohf:span {f} = Ihs:span {idealGenerator I} = span {f}⊢ ∃ u, f = ↑u * idealGenerator I f:OI:Ideal Ohf:span {f} = Ihs:span {idealGenerator I} = span {f}u:ConcretePrimeRowBridge.Oˣhu:idealGenerator I * ↑u = f⊢ ∃ u, f = ↑u * idealGenerator I exact ⟨u, f:OI:Ideal Ohf:span {f} = Ihs:span {idealGenerator I} = span {f}u:ConcretePrimeRowBridge.Oˣhu:idealGenerator I * ↑u = f⊢ f = ↑u * idealGenerator I All goals completed! 🐙⟩def tupleLabelUnit (x : OldTuple) : Oˣ := Classical.choose (exists_label_unit (tupleLabelElement x) (newLabel x) (tupleLabelElement_span x))theorem tupleLabelUnit_spec (x : OldTuple) : tupleLabelElement x = (tupleLabelUnit x : O) * idealGenerator (newLabel x) := Classical.choose_spec (exists_label_unit (tupleLabelElement x) (newLabel x) (tupleLabelElement_span x))variable {ι : Type*} [DecidableEq ι] (p : ι → O) (hp : ∀ i, p i ≠ 0) [∀ i, (Ideal.span {p i}).IsMaximal] (hcop : Pairwise (Function.onFun IsCoprime (fun i => Ideal.span {p i}))) (hg : ∀ i, lambda ∉ Ideal.span {p i})def fixedChildRow (F : Finset ι) (Ψ : O →* ℂ) (m : O) (H : Finset ι → ℂ) (f y : O) : ℂ := ∑ N ∈ F.powerset, secondChildColumn p hp hcop hg Ψ m f y H Ndef idealChildRow (F : Finset ι) (Ψ : O →* ℂ) (m : O) (H : Finset ι → ℂ) (negative : Bool) (u : Oˣ) (z : Ideal O × O) : ℂ := fixedChildRow p hp hcop hg F Ψ m H ((u : O) * idealGenerator z.1) (if negative then -z.2 else z.2)theorem tuple_child_eq_unit_sector (F : Finset ι) (Ψ : O →* ℂ) (m : O) (H : Finset ι → ℂ) (negative : Bool) (x : OldTuple) : fixedChildRow p hp hcop hg F Ψ m H (tupleLabelElement x) (if negative then -newRow x else newRow x) = idealChildRow p hp hcop hg F Ψ m H negative (tupleLabelUnit x) (newLabel x, newRow x) := ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ:O →* ℂm:OH:Finset ι → ℂnegative:Boolx:OldTuple⊢ fixedChildRow p hp hcop hg F Ψ m H (tupleLabelElement x) (if negative = true then -newRow x else newRow x) = idealChildRow p hp hcop hg F Ψ m H negative (tupleLabelUnit x) (newLabel x, newRow x) ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ:O →* ℂm:OH:Finset ι → ℂnegative:Boolx:OldTuple⊢ fixedChildRow p hp hcop hg F Ψ m H (↑(tupleLabelUnit x) * idealGenerator (newLabel x)) (if negative = true then -newRow x else newRow x) = idealChildRow p hp hcop hg F Ψ m H negative (tupleLabelUnit x) (newLabel x, newRow x) All goals completed! 🐙theorem fixed_child_second_energy_pushforward (ε : ℝ) (hε : 0 < ε) : ∃ C : ℝ, 0 < C ∧ ∀ (s : Finset OldTuple) (t : Finset (Ideal O × O)) (b0 : Ideal O) (w : OldTuple → ℂ) (B lengthScale : ℝ) (F : Finset ι) (Ψ : O →* ℂ) (m : O) (H : Finset ι → ℂ) (negative : Bool), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ lengthScale → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, (Ideal.absNorm z.1 : ℝ) ≤ lengthScale) → (∀ x ∈ s, ‖w x‖ ≤ B) → (∑ x ∈ s, ‖w x‖ * ‖fixedChildRow p hp hcop hg F Ψ m H (tupleLabelElement x) (if negative then -newRow x else newRow x)‖ ^ 2) ≤ B * C * (lengthScale * Ideal.absNorm b0) ^ ε * ∑ u : Oˣ, ∑ z ∈ t, ‖idealChildRow p hp hcop hg F Ψ m H negative u z‖ ^ 2 := ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}ε:ℝhε:0 < ε⊢ ∃ C, 0 < C ∧ ∀ (s : Finset OldTuple) (t : Finset (Ideal O × O)) (b0 : Ideal O) (w : OldTuple → ℂ) (B lengthScale : ℝ) (F : Finset ι) (Ψ : O →* ℂ) (m : O) (H : Finset ι → ℂ) (negative : Bool), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ lengthScale → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScale) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖fixedChildRow p hp hcop hg F Ψ m H (tupleLabelElement x) (if negative = true then -newRow x else newRow x)‖ ^ 2 ≤ B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ u, ∑ z ∈ t, ‖idealChildRow p hp hcop hg F Ψ m H negative u z‖ ^ 2 ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}ε:ℝhε:0 < εC:ℝhC:0 < Chb:∀ (s : Finset OldTuple) (t : Finset (Ideal O × O)) (b0 : Ideal O) (w : OldTuple → ℂ) (B lengthScale : ℝ) (P : Oˣ → Ideal O × O → ℂ) (sel : OldTuple → Oˣ), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ lengthScale → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScale) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖P (sel x) (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ u, ∑ z ∈ t, ‖P u z‖ ^ 2⊢ ∃ C, 0 < C ∧ ∀ (s : Finset OldTuple) (t : Finset (Ideal O × O)) (b0 : Ideal O) (w : OldTuple → ℂ) (B lengthScale : ℝ) (F : Finset ι) (Ψ : O →* ℂ) (m : O) (H : Finset ι → ℂ) (negative : Bool), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ lengthScale → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScale) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖fixedChildRow p hp hcop hg F Ψ m H (tupleLabelElement x) (if negative = true then -newRow x else newRow x)‖ ^ 2 ≤ B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ u, ∑ z ∈ t, ‖idealChildRow p hp hcop hg F Ψ m H negative u z‖ ^ 2 ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}ε:ℝhε:0 < εC:ℝhC:0 < Chb:∀ (s : Finset OldTuple) (t : Finset (Ideal O × O)) (b0 : Ideal O) (w : OldTuple → ℂ) (B lengthScale : ℝ) (P : Oˣ → Ideal O × O → ℂ) (sel : OldTuple → Oˣ), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ lengthScale → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScale) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖P (sel x) (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ u, ∑ z ∈ t, ‖P u z‖ ^ 2⊢ ∀ (s : Finset OldTuple) (t : Finset (Ideal O × O)) (b0 : Ideal O) (w : OldTuple → ℂ) (B lengthScale : ℝ) (F : Finset ι) (Ψ : O →* ℂ) (m : O) (H : Finset ι → ℂ) (negative : Bool), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ lengthScale → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScale) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖fixedChildRow p hp hcop hg F Ψ m H (tupleLabelElement x) (if negative = true then -newRow x else newRow x)‖ ^ 2 ≤ B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ u, ∑ z ∈ t, ‖idealChildRow p hp hcop hg F Ψ m H negative u z‖ ^ 2 ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}ε:ℝhε:0 < εC:ℝhC:0 < Chb:∀ (s : Finset OldTuple) (t : Finset (Ideal O × O)) (b0 : Ideal O) (w : OldTuple → ℂ) (B lengthScale : ℝ) (P : Oˣ → Ideal O × O → ℂ) (sel : OldTuple → Oˣ), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ lengthScale → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScale) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖P (sel x) (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ u, ∑ z ∈ t, ‖P u z‖ ^ 2s:Finset OldTuplet:Finset (Ideal O × O)b0:Ideal Ow:OldTuple → ℂB:ℝlengthScale:ℝF:Finset ιΨ:O →* ℂm:OH:Finset ι → ℂnegative:Boolhb0:b0 ≠ ⊥hB:0 ≤ BhL:0 ≤ lengthScalehvalid:∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)hmap:∀ x ∈ s, (newLabel x, newRow x) ∈ tht:∀ z ∈ t, z.1 ≠ ⊥hnorm:∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScalehw:∀ x ∈ s, ‖w x‖ ≤ B⊢ ∑ x ∈ s, ‖w x‖ * ‖fixedChildRow p hp hcop hg F Ψ m H (tupleLabelElement x) (if negative = true then -newRow x else newRow x)‖ ^ 2 ≤ B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ u, ∑ z ∈ t, ‖idealChildRow p hp hcop hg F Ψ m H negative u z‖ ^ 2 ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}ε:ℝhε:0 < εC:ℝhC:0 < Chb:∀ (s : Finset OldTuple) (t : Finset (Ideal O × O)) (b0 : Ideal O) (w : OldTuple → ℂ) (B lengthScale : ℝ) (P : Oˣ → Ideal O × O → ℂ) (sel : OldTuple → Oˣ), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ lengthScale → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScale) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖P (sel x) (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ u, ∑ z ∈ t, ‖P u z‖ ^ 2s:Finset OldTuplet:Finset (Ideal O × O)b0:Ideal Ow:OldTuple → ℂB:ℝlengthScale:ℝF:Finset ιΨ:O →* ℂm:OH:Finset ι → ℂnegative:Boolhb0:b0 ≠ ⊥hB:0 ≤ BhL:0 ≤ lengthScalehvalid:∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)hmap:∀ x ∈ s, (newLabel x, newRow x) ∈ tht:∀ z ∈ t, z.1 ≠ ⊥hnorm:∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScalehw:∀ x ∈ s, ‖w x‖ ≤ Bhh:∑ x ∈ s, ‖w x‖ * ‖idealChildRow p hp hcop hg F Ψ m H negative (tupleLabelUnit x) (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ u, ∑ z ∈ t, ‖idealChildRow p hp hcop hg F Ψ m H negative u z‖ ^ 2⊢ ∑ x ∈ s, ‖w x‖ * ‖fixedChildRow p hp hcop hg F Ψ m H (tupleLabelElement x) (if negative = true then -newRow x else newRow x)‖ ^ 2 ≤ B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ u, ∑ z ∈ t, ‖idealChildRow p hp hcop hg F Ψ m H negative u z‖ ^ 2 ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}ε:ℝhε:0 < εC:ℝhC:0 < Chb:∀ (s : Finset OldTuple) (t : Finset (Ideal O × O)) (b0 : Ideal O) (w : OldTuple → ℂ) (B lengthScale : ℝ) (P : Oˣ → Ideal O × O → ℂ) (sel : OldTuple → Oˣ), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ lengthScale → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScale) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖P (sel x) (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ u, ∑ z ∈ t, ‖P u z‖ ^ 2s:Finset OldTuplet:Finset (Ideal O × O)b0:Ideal Ow:OldTuple → ℂB:ℝlengthScale:ℝF:Finset ιΨ:O →* ℂm:OH:Finset ι → ℂnegative:Boolhb0:b0 ≠ ⊥hB:0 ≤ BhL:0 ≤ lengthScalehvalid:∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)hmap:∀ x ∈ s, (newLabel x, newRow x) ∈ tht:∀ z ∈ t, z.1 ≠ ⊥hnorm:∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScalehw:∀ x ∈ s, ‖w x‖ ≤ Bhh:∑ x ∈ s, ‖w x‖ * ‖idealChildRow p hp hcop hg F Ψ m H negative (tupleLabelUnit x) (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ u, ∑ z ∈ t, ‖idealChildRow p hp hcop hg F Ψ m H negative u z‖ ^ 2⊢ ∑ x ∈ s, ‖w x‖ * ‖fixedChildRow p hp hcop hg F Ψ m H (tupleLabelElement x) (if negative = true then -newRow x else newRow x)‖ ^ 2 = ∑ x ∈ s, ‖w x‖ * ‖idealChildRow p hp hcop hg F Ψ m H negative (tupleLabelUnit x) (newLabel x, newRow x)‖ ^ 2 ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}ε:ℝhε:0 < εC:ℝhC:0 < Chb:∀ (s : Finset OldTuple) (t : Finset (Ideal O × O)) (b0 : Ideal O) (w : OldTuple → ℂ) (B lengthScale : ℝ) (P : Oˣ → Ideal O × O → ℂ) (sel : OldTuple → Oˣ), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ lengthScale → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScale) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖P (sel x) (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ u, ∑ z ∈ t, ‖P u z‖ ^ 2s:Finset OldTuplet:Finset (Ideal O × O)b0:Ideal Ow:OldTuple → ℂB:ℝlengthScale:ℝF:Finset ιΨ:O →* ℂm:OH:Finset ι → ℂnegative:Boolhb0:b0 ≠ ⊥hB:0 ≤ BhL:0 ≤ lengthScalehvalid:∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)hmap:∀ x ∈ s, (newLabel x, newRow x) ∈ tht:∀ z ∈ t, z.1 ≠ ⊥hnorm:∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScalehw:∀ x ∈ s, ‖w x‖ ≤ Bhh:∑ x ∈ s, ‖w x‖ * ‖idealChildRow p hp hcop hg F Ψ m H negative (tupleLabelUnit x) (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ u, ∑ z ∈ t, ‖idealChildRow p hp hcop hg F Ψ m H negative u z‖ ^ 2⊢ ∀ x ∈ s, ‖w x‖ * ‖fixedChildRow p hp hcop hg F Ψ m H (tupleLabelElement x) (if negative = true then -newRow x else newRow x)‖ ^ 2 = ‖w x‖ * ‖idealChildRow p hp hcop hg F Ψ m H negative (tupleLabelUnit x) (newLabel x, newRow x)‖ ^ 2 ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}ε:ℝhε:0 < εC:ℝhC:0 < Chb:∀ (s : Finset OldTuple) (t : Finset (Ideal O × O)) (b0 : Ideal O) (w : OldTuple → ℂ) (B lengthScale : ℝ) (P : Oˣ → Ideal O × O → ℂ) (sel : OldTuple → Oˣ), b0 ≠ ⊥ → 0 ≤ B → 0 ≤ lengthScale → (∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)) → (∀ x ∈ s, (newLabel x, newRow x) ∈ t) → (∀ z ∈ t, z.1 ≠ ⊥) → (∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScale) → (∀ x ∈ s, ‖w x‖ ≤ B) → ∑ x ∈ s, ‖w x‖ * ‖P (sel x) (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ u, ∑ z ∈ t, ‖P u z‖ ^ 2s:Finset OldTuplet:Finset (Ideal O × O)b0:Ideal Ow:OldTuple → ℂB:ℝlengthScale:ℝF:Finset ιΨ:O →* ℂm:OH:Finset ι → ℂnegative:Boolhb0:b0 ≠ ⊥hB:0 ≤ BhL:0 ≤ lengthScalehvalid:∀ x ∈ s, Valid x (newLabel x) b0 (newRow x)hmap:∀ x ∈ s, (newLabel x, newRow x) ∈ tht:∀ z ∈ t, z.1 ≠ ⊥hnorm:∀ z ∈ t, ↑(absNorm z.1) ≤ lengthScalehw:∀ x ∈ s, ‖w x‖ ≤ Bhh:∑ x ∈ s, ‖w x‖ * ‖idealChildRow p hp hcop hg F Ψ m H negative (tupleLabelUnit x) (newLabel x, newRow x)‖ ^ 2 ≤ B * C * (lengthScale * ↑(absNorm b0)) ^ ε * ∑ u, ∑ z ∈ t, ‖idealChildRow p hp hcop hg F Ψ m H negative u z‖ ^ 2x:OldTuplehx:x ∈ s⊢ ‖w x‖ * ‖fixedChildRow p hp hcop hg F Ψ m H (tupleLabelElement x) (if negative = true then -newRow x else newRow x)‖ ^ 2 = ‖w x‖ * ‖idealChildRow p hp hcop hg F Ψ m H negative (tupleLabelUnit x) (newLabel x, newRow x)‖ ^ 2 All goals completed! 🐙theorem idealChildRow_unit_energy_le (F : Finset ι) (Ψ : O →* ℂ) (m : O) (H : Finset ι → ℂ) (negative : Bool) (t : Finset (Ideal O × O)) (E : ℝ) (hE : ∀ u : Oˣ, (∑ z ∈ t, ‖idealChildRow p hp hcop hg F Ψ m H negative u z‖ ^ 2) ≤ E) : (∑ u : Oˣ, ∑ z ∈ t, ‖idealChildRow p hp hcop hg F Ψ m H negative u z‖ ^ 2) ≤ 6 * E := ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ:O →* ℂm:OH:Finset ι → ℂnegative:Boolt:Finset (Ideal O × O)E:ℝhE:∀ (u : Oˣ), ∑ z ∈ t, ‖idealChildRow p hp hcop hg F Ψ m H negative u z‖ ^ 2 ≤ E⊢ ∑ u, ∑ z ∈ t, ‖idealChildRow p hp hcop hg F Ψ m H negative u z‖ ^ 2 ≤ 6 * E ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ:O →* ℂm:OH:Finset ι → ℂnegative:Boolt:Finset (Ideal O × O)E:ℝhE:∀ (u : Oˣ), ∑ z ∈ t, ‖idealChildRow p hp hcop hg F Ψ m H negative u z‖ ^ 2 ≤ Ehc:Fintype.card Oˣ = 6⊢ ∑ u, ∑ z ∈ t, ‖idealChildRow p hp hcop hg F Ψ m H negative u z‖ ^ 2 ≤ 6 * E calc _ ≤ ∑ u : Oˣ, E := Finset.sum_le_sum (fun u _ => hE u) _ = _ := ι:Type u_1inst✝¹:DecidableEq ιp:ι → Ohp:∀ (i : ι), p i ≠ 0inst✝:∀ (i : ι), (span {p i}).IsMaximalhcop:Pairwise (Function.onFun IsCoprime fun i => span {p i})hg:∀ (i : ι), lambda ∉ span {p i}F:Finset ιΨ:O →* ℂm:OH:Finset ι → ℂnegative:Boolt:Finset (Ideal O × O)E:ℝhE:∀ (u : Oˣ), ∑ z ∈ t, ‖idealChildRow p hp hcop hg F Ψ m H negative u z‖ ^ 2 ≤ Ehc:Fintype.card Oˣ = 6⊢ ∑ u, E = 6 * E All goals completed! 🐙end SecondPassArithmeticendend OAI