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37bcf52 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 | 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
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