openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Programs /BalancedMain.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.BalancedFluid | |
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.BalancedCoefficientEvaluation | |
| namespace OAI | |
| /-! The compact balanced main theorem includes a terminating evaluator | |
| for every mixed derivative of the prescribed force. -/ | |
| noncomputable section | |
| namespace ForcedComputation.Balanced | |
| open ShearFlows Set | |
| open scoped ContDiff | |
| /-- The two finite coefficients prescribe the compact zero-data flow for | |
| every viscosity; named finite queries evaluate every force derivative. -/ | |
| theorem balanced_three_stack_realization : | |
| IsCompact BalancedVelocity.support ∧ | |
| ∀ (I : Alternating.MachineInput) (hI : Alternating.ValidInput I), | |
| (∀ ν : ℝ, force I hI ν = BalancedVelocity.force₀ I hI + | |
| ν • BalancedVelocity.force₁ I hI) ∧ | |
| (∀ (α : List (Fin 4)) (b : ℕ → RationalSpaceTime) (y : SpaceTime), | |
| IsFastName b y → ∀ (ε : ℚ) (hε : 0 < ε), | |
| ‖mixedDerivative (BalancedVelocity.force₀ I hI) α y - | |
| rationalVector (BalancedExpressions.evaluateCoefficient₀ I hI α b ε hε)‖ ≤ (ε : ℝ) ∧ | |
| ‖mixedDerivative (BalancedVelocity.force₁ I hI) α y - | |
| rationalVector (BalancedExpressions.evaluateCoefficient₁ I hI α b ε hε)‖ ≤ (ε : ℝ)) ∧ | |
| (∀ ν : ℝ, 0 < ν → FluidProperties I hI ν) ∧ | |
| (∀ (ν : ℝ) (a : ℕ → ℚ), IsFastRealName a ν → | |
| ∀ (α : List (Fin 4)) (b : ℕ → RationalSpaceTime) (y : SpaceTime), | |
| IsFastName b y → ∀ (ε : ℚ) (hε : 0 < ε), | |
| ‖mixedDerivative (force I hI ν) α y - | |
| rationalVector (BalancedExpressions.evaluate I hI α a b ε hε)‖ ≤ (ε : ℝ)) ∧ | |
| (∀ (ν : ℝ) (a : ℕ → ℚ), IsFastRealName a ν → | |
| ∀ (α : List (Fin 4)) (u v M : ℚ) (y : SpaceTime), | |
| y.1 ∈ Ioo (u : ℝ) v → | |
| (∀ j, |timeSpaceCoord j y| ≤ |(M : ℝ)|) → | |
| ‖mixedDerivative (force I hI ν) α y‖ ≤ | |
| (BalancedExpressions.boxForceBound I hI α u v M a : ℝ)) := by | |
| refine ⟨BalancedVelocity.support_compact,?_⟩ | |
| intro I hI | |
| refine ⟨BalancedVelocity.force_affine I hI,?_, | |
| fun ν hν => fluid_properties I hI hν,?_,?_⟩ | |
| · intro α b y hb ε hε | |
| exact ⟨BalancedExpressions.evaluateCoefficient₀_spec I hI α hb ε hε, | |
| BalancedExpressions.evaluateCoefficient₁_spec I hI α hb ε hε⟩ | |
| · intro ν a ha α b y hb ε hε | |
| exact BalancedExpressions.evaluate_spec I hI α ha hb ε hε | |
| · intro ν a ha α u v M y ht hy | |
| exact BalancedExpressions.boxForceBound_spec I hI α u v M ha y ht hy | |
| end ForcedComputation.Balanced | |
| end | |
| end OAI | |