openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Programs /BalancedCoefficientEvaluation.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.BalancedEvaluation | |
| namespace OAI | |
| /-! Explicit terminating evaluators for the two viscosity-independent force coefficients. -/ | |
| noncomputable section | |
| namespace ForcedComputation.BalancedExpressions | |
| open ShearFlows | |
| def evaluateCoefficient₀ (I : Alternating.MachineInput) (hI : Alternating.ValidInput I) | |
| (α : List (Fin 4)) (b : ℕ → RationalSpaceTime) (ε : ℚ) (hε : 0 < ε) : RationalVector := | |
| evaluate I hI α (fun _ => 0) b ε hε | |
| def evaluateCoefficient₁ (I : Alternating.MachineInput) (hI : Alternating.ValidInput I) | |
| (α : List (Fin 4)) (b : ℕ → RationalSpaceTime) (ε : ℚ) (hε : 0 < ε) : RationalVector := | |
| evaluate I hI α (fun _ => 1) b (ε/2) (by positivity) - | |
| evaluate I hI α (fun _ => 0) b (ε/2) (by positivity) | |
| private theorem constantName (r : ℚ) : IsFastRealName (fun _ => r) (r : ℝ) := by | |
| intro n | |
| simp only [sub_self, abs_zero] | |
| unfold errorTolerance | |
| positivity | |
| theorem evaluateCoefficient₀_spec (I : Alternating.MachineInput) | |
| (hI : Alternating.ValidInput I) (α : List (Fin 4)) | |
| {y : SpaceTime} {b : ℕ → RationalSpaceTime} (hb : IsFastName b y) | |
| (ε : ℚ) (hε : 0 < ε) : | |
| ‖mixedDerivative (BalancedVelocity.force₀ I hI) α y - | |
| rationalVector (evaluateCoefficient₀ I hI α b ε hε)‖ ≤ (ε : ℝ) := by | |
| simpa only [BalancedVelocity.force₀, evaluateCoefficient₀, Rat.cast_zero] using | |
| evaluate_spec I hI α (constantName 0) hb ε hε | |
| theorem evaluateCoefficient₁_spec (I : Alternating.MachineInput) | |
| (hI : Alternating.ValidInput I) (α : List (Fin 4)) | |
| {y : SpaceTime} {b : ℕ → RationalSpaceTime} (hb : IsFastName b y) | |
| (ε : ℚ) (hε : 0 < ε) : | |
| ‖mixedDerivative (BalancedVelocity.force₁ I hI) α y - | |
| rationalVector (evaluateCoefficient₁ I hI α b ε hε)‖ ≤ (ε : ℝ) := by | |
| have hf : BalancedVelocity.force₁ I hI = residual 1 (BalancedVelocity.velocity I hI) - | |
| residual 0 (BalancedVelocity.velocity I hI) := by | |
| have h := BalancedVelocity.force_affine I hI 1 | |
| rw [one_smul] at h | |
| funext z | |
| have hz := congrFun h z | |
| change residual 1 (BalancedVelocity.velocity I hI) z = | |
| residual 0 (BalancedVelocity.velocity I hI) z + BalancedVelocity.force₁ I hI z at hz | |
| exact eq_sub_of_add_eq' hz.symm | |
| have h₁ := evaluate_spec I hI α (constantName 1) hb (ε/2) (by positivity) | |
| have h₀ := evaluate_spec I hI α (constantName 0) hb (ε/2) (by positivity) | |
| simp only [Rat.cast_one] at h₁ | |
| simp only [Rat.cast_zero] at h₀ | |
| rw [hf, mixedDerivative_sub | |
| (residual_smooth (BalancedVelocity.smooth I hI) 1) | |
| (residual_smooth (BalancedVelocity.smooth I hI) 0)] | |
| have hq : rationalVector (evaluateCoefficient₁ I hI α b ε hε) = | |
| rationalVector (evaluate I hI α (fun _ => 1) b (ε/2) (by positivity)) - | |
| rationalVector (evaluate I hI α (fun _ => 0) b (ε/2) (by positivity)) := by | |
| ext j | |
| exact Rat.cast_sub _ _ | |
| rw [hq] | |
| simp only [Pi.sub_apply] | |
| calc | |
| _ = ‖(mixedDerivative (residual 1 (BalancedVelocity.velocity I hI)) α y - | |
| rationalVector (evaluate I hI α (fun _ => 1) b (ε/2) (by positivity))) - | |
| (mixedDerivative (residual 0 (BalancedVelocity.velocity I hI)) α y - | |
| rationalVector (evaluate I hI α (fun _ => 0) b (ε/2) (by positivity)))‖ := by | |
| congr 1 | |
| abel | |
| _ ≤ _ := (norm_sub_le _ _).trans (add_le_add h₁ h₀) |>.trans (by | |
| norm_num only [Rat.cast_div, Rat.cast_ofNat] | |
| linarith) | |
| end ForcedComputation.BalancedExpressions | |
| end | |
| end OAI | |