Step-down estimate assembly from the individual contract producers #
The producer hypotheses are the three part-I contracts and seven relative contracts. Relative producers may use exactly the gate. Independent constants and thresholds are combined before the ingredient instance is introduced.
The independent part-I and relative contract producers.
- initial : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA8Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (hΦ : IsStreamSeq I _Φ) (κ : ℝ) (M : ℕ) (_R : ℝ) (θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (θm _θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => Integration.InitialLayerContract I hΦ m (I.kappaSeq κ M m) θm T C √(l2NormSq θ₀)
- ansatz : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA7Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (hΦ : IsStreamSeq I _Φ) (κ : ℝ) (M : ℕ) (_R : ℝ) (θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (_θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => Integration.SectionFourAnsatzContract I hΦ m (I.kappaSeq κ M m) θ₀ T C
- iterate : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA8Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (_hΦ : IsStreamSeq I _Φ) (_κ : ℝ) (_M : ℕ) (_R : ℝ) (θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (_θm θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => Integration.SectionFourIterateContract I m θ₀ θprev T C
- gradient : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA8Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (_hΦ : IsStreamSeq I _Φ) (κ : ℝ) (M : ℕ) (R : ℝ) (_θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (_θm θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => 6 / 5 ≤ β → epsilon β I.Λ (m - 1) ^ (1 + gamma β / 2 - delta β) ≤ R → Integration.TGradientContract β (I.kappaSeq κ M (m - 1)) T C (√(I.kappaSeq κ M (m - 1)) * √(spaceTimeGradNormSq fun (s : ℝ) (x : Homogenization.Vec 2) => spaceGrad (θprev s) x))
- material : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA8Instances β C₀ C₁ fun (I : Ingredients β) (Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (_hΦ : IsStreamSeq I Φ) (κ : ℝ) (M : ℕ) (R : ℝ) (_θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (_θm θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => 6 / 5 ≤ β → epsilon β I.Λ (m - 1) ^ (1 + gamma β / 2 - delta β) ≤ R → Integration.TMaterialContract I Φ m (I.kappaSeq κ M (m - 1)) T C (√(I.kappaSeq κ M (m - 1)) * √(spaceTimeGradNormSq fun (s : ℝ) (x : Homogenization.Vec 2) => spaceGrad (θprev s) x))
- jets : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA8Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (_hΦ : IsStreamSeq I _Φ) (κ : ℝ) (M : ℕ) (R : ℝ) (_θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (_θm θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => 6 / 5 ≤ β → epsilon β I.Λ (m - 1) ^ (1 + gamma β / 2 - delta β) ≤ R → Integration.TPositiveJetsContract I m T C (√(I.kappaSeq κ M (m - 1)) * √(spaceTimeGradNormSq fun (s : ℝ) (x : Homogenization.Vec 2) => spaceGrad (θprev s) x))
- relative_initial : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA8Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (hΦ : IsStreamSeq I _Φ) (κ : ℝ) (M : ℕ) (R : ℝ) (_θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (θm θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => 6 / 5 ≤ β → epsilon β I.Λ (m - 1) ^ (1 + gamma β / 2 - delta β) ≤ R → Integration.InitialLayerContract I hΦ m (I.kappaSeq κ M m) θm T C (√(I.kappaSeq κ M (m - 1)) * √(spaceTimeGradNormSq fun (s : ℝ) (x : Homogenization.Vec 2) => spaceGrad (θprev s) x))
- terms : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA8Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (hΦ : IsStreamSeq I _Φ) (κ : ℝ) (M : ℕ) (R : ℝ) (_θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (_θm θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => 6 / 5 ≤ β → epsilon β I.Λ (m - 1) ^ (1 + gamma β / 2 - delta β) ≤ R → Integration.RelativeTermsContract I hΦ m (I.kappaSeq κ M m) (I.kappaSeq κ M (m - 1)) T C (√(I.kappaSeq κ M (m - 1)) * √(spaceTimeGradNormSq fun (s : ℝ) (x : Homogenization.Vec 2) => spaceGrad (θprev s) x))
- leading : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA8Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (hΦ : IsStreamSeq I _Φ) (κ : ℝ) (M : ℕ) (R : ℝ) (_θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (_θm θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => 6 / 5 ≤ β → epsilon β I.Λ (m - 1) ^ (1 + gamma β / 2 - delta β) ≤ R → Integration.LeadingErrorContract I hΦ m (I.kappaSeq κ M m) T C (√(I.kappaSeq κ M (m - 1)) * √(spaceTimeGradNormSq fun (s : ℝ) (x : Homogenization.Vec 2) => spaceGrad (θprev s) x))
- temperature : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA8Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (_hΦ : IsStreamSeq I _Φ) (κ : ℝ) (M : ℕ) (R : ℝ) (_θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (_θm θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => 6 / 5 ≤ β → epsilon β I.Λ (m - 1) ^ (1 + gamma β / 2 - delta β) ≤ R → Integration.TemperatureErrorContract I m (I.kappaSeq κ M (m - 1)) θprev T C (√(I.kappaSeq κ M (m - 1)) * √(spaceTimeGradNormSq fun (s : ℝ) (x : Homogenization.Vec 2) => spaceGrad (θprev s) x))
Instances For
theorem
AVenhance.Infra.Section5.Contracts.partI_family_of_contracts
(β C₀ : ℝ)
(h : A8ContractProducers β C₀)
:
Part I from its three independent producers.
theorem
AVenhance.Infra.Section5.Contracts.indy_inputs_of_contracts
(β C₀ : ℝ)
(h7 : A7ContractProducers β C₀)
(h8 : A8ContractProducers β C₀)
:
Assemble all step-down estimate inputs from the individual contracts; no step-down estimate conclusion is assumed.
theorem
AVenhance.Infra.Section5.Contracts.indystepdown_of_contracts
(β C₀ : ℝ)
(h7 : A7ContractProducers β C₀)
(h8 : A8ContractProducers β C₀)
:
The literal step-down estimate statement from only the individual big-bound estimate and step-down estimate contract producers.