openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Programs /BalancedEvaluation.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.BalancedFiniteCode | |
| namespace OAI | |
| /-! The zeroth time approximation selects a finite translate window with | |
| one unit of spare width. Evaluation of every residual jet is then the | |
| existing terminating finite-expression evaluator, including a viscosity | |
| name when viscosity is not rational. -/ | |
| noncomputable section | |
| namespace ForcedComputation.BalancedExpressions | |
| open ShearFlows Set Filter | |
| open scoped Topology | |
| def windowLower (b : ℕ → RationalSpaceTime) : ℚ := (b 0).1 - 2 | |
| def windowUpper (b : ℕ → RationalSpaceTime) : ℚ := (b 0).1 + 2 | |
| theorem named_window {b : ℕ → RationalSpaceTime} {y : SpaceTime} | |
| (hb : IsFastName b y) : y.1 ∈ Ioo (windowLower b : ℝ) (windowUpper b) := by | |
| have h := (norm_fst_le (y-rationalPoint (b 0))).trans (hb 0) | |
| change |y.1 - ((b 0).1 : ℝ)| ≤ errorTolerance 0 at h | |
| norm_num [errorTolerance] at h | |
| obtain ⟨hl,hu⟩ := abs_le.mp h | |
| simp only [windowLower, windowUpper, Rat.cast_sub, Rat.cast_add, Rat.cast_ofNat] | |
| constructor <;> linarith | |
| def evaluate (I : Alternating.MachineInput) (hI : Alternating.ValidInput I) | |
| (α : List (Fin 4)) (a : ℕ → ℚ) (b : ℕ → RationalSpaceTime) | |
| (ε : ℚ) (hε : 0 < ε) : RationalVector := | |
| NonperiodicResidual.evaluate (code I hI (windowLower b) (windowUpper b)) α a b ε hε | |
| theorem evaluate_spec (I : Alternating.MachineInput) (hI : Alternating.ValidInput I) | |
| (α : List (Fin 4)) {ν : ℝ} {a : ℕ → ℚ} (ha : IsFastRealName a ν) | |
| {y : SpaceTime} {b : ℕ → RationalSpaceTime} (hb : IsFastName b y) | |
| (ε : ℚ) (hε : 0 < ε) : | |
| ‖mixedDerivative (residual ν (BalancedVelocity.velocity I hI)) α y - | |
| rationalVector (evaluate I hI α a b ε hε)‖ ≤ (ε : ℝ) := by | |
| have hg := code_germ I hI (windowLower b) (windowUpper b) (named_window hb) | |
| have hj := (mixedDerivative_eventuallyEq (residual_eventuallyEq hg ν) α).self_of_nhds | |
| rw [← hj] | |
| exact NonperiodicResidual.evaluate_spec _ α ha hb ε hε | |
| def boxForceBound (I : Alternating.MachineInput) (hI : Alternating.ValidInput I) | |
| (α : List (Fin 4)) (u v M : ℚ) (a : ℕ → ℚ) : ℕ := | |
| let c := code I hI u v | |
| ⌈(NonperiodicResidual.code c 0).bound α M + | |
| (|a 0|+1) * NonperiodicResidual.laplaceBound c α M⌉₊ | |
| theorem boxForceBound_spec (I : Alternating.MachineInput) (hI : Alternating.ValidInput I) | |
| (α : List (Fin 4)) (u v M : ℚ) {ν : ℝ} {a : ℕ → ℚ} (ha : IsFastRealName a ν) | |
| (y : SpaceTime) (ht : y.1 ∈ Ioo (u : ℝ) v) | |
| (hy : ∀ j, |timeSpaceCoord j y| ≤ |(M : ℝ)|) : | |
| ‖mixedDerivative (residual ν (BalancedVelocity.velocity I hI)) α y‖ ≤ | |
| (boxForceBound I hI α u v M a : ℝ) := by | |
| let c := code I hI u v | |
| have hg := code_germ I hI u v ht | |
| have hj := (mixedDerivative_eventuallyEq (residual_eventuallyEq hg ν) α).self_of_nhds | |
| have hν : |ν| ≤ |((a 0) : ℝ)|+1 := by | |
| have ha0 := ha 0 | |
| norm_num [errorTolerance] at ha0 | |
| linarith [abs_sub_abs_le_abs_sub ν (a 0)] | |
| rw [← hj] | |
| calc | |
| _ ≤ ‖mixedDerivative (residual 0 c.val) α y‖ + | |
| ‖mixedDerivative (residual ν c.val) α y - mixedDerivative (residual 0 c.val) α y‖ := by | |
| simpa only [norm_sub_rev] using norm_le_norm_add_norm_sub | |
| (mixedDerivative (residual 0 c.val) α y) (mixedDerivative (residual ν c.val) α y) | |
| _ ≤ ((NonperiodicResidual.code c 0).bound α M : ℝ) + | |
| (|((a 0) : ℝ)|+1) * (NonperiodicResidual.laplaceBound c α M : ℝ) := by | |
| apply add_le_add | |
| · simpa only [NonperiodicResidual.val, Rat.cast_zero] using | |
| (NonperiodicResidual.code c 0).val_bound α M y hy | |
| · exact (NonperiodicResidual.coefficient_error c α M y hy ν 0).trans | |
| (by simpa only [sub_zero] using (mul_le_mul_of_nonneg_right hν | |
| (Rat.cast_nonneg.mpr (NonperiodicResidual.laplaceBound_nonneg c α M)))) | |
| _ ≤ _ := by | |
| exact_mod_cast Nat.le_ceil ((NonperiodicResidual.code c 0).bound α M + | |
| (|a 0|+1) * NonperiodicResidual.laplaceBound c α M) | |
| end ForcedComputation.BalancedExpressions | |
| end | |
| end OAI | |