File size: 3,120 Bytes
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 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.ClockedEvaluation
namespace OAI
/-! Vector-valued finite clocked programs, with certified evaluation and
bounds for every ordered mixed derivative on finite time slabs. -/
noncomputable section
namespace ForcedComputation
open ShearFlows
open scoped ContDiff BigOperators
abbrev ClockedVector := Fin 3 → ClockedExpr
namespace ClockedVector
def Valid (c : ClockedVector) : Prop := ∀ j, (c j).Valid
def val (c : ClockedVector) : Velocity := fun y j => (c j).val y
def diffWord (c : ClockedVector) (α : List (Fin 4)) : ClockedVector :=
fun j => (c j).diffWord α
theorem valid_diffWord {c : ClockedVector} (hc : c.Valid) (α : List (Fin 4)) :
(c.diffWord α).Valid := fun j => (c j).valid_diffWord (hc j) α
theorem smooth {c : ClockedVector} (hc : c.Valid) : ContDiff ℝ ∞ c.val :=
contDiff_pi.mpr (fun j => (c j).smooth (hc j))
theorem val_diffWord {c : ClockedVector} (hc : c.Valid) (α : List (Fin 4)) :
(c.diffWord α).val = mixedDerivative c.val α := by
induction α with
| nil => rfl
| cons i α ih =>
funext y j
rw [mixedDerivative, ← ih]
have h := fderiv_pi (fun k =>
(((c k).diffWord α).smooth ((c k).valid_diffWord (hc k) α)).differentiable (by simp) y)
change ((c j).diffWord (i :: α)).val y =
(fderiv ℝ (c.diffWord α).val y (spaceTimeDirection i)) j
change fderiv ℝ (c.diffWord α).val y = _ at h
rw [h]
exact ((c j).diffWord α).val_diff ((c j).valid_diffWord (hc j) α) i y
def bound (c : ClockedVector) (α : List (Fin 4)) (M : ℚ) : ℚ :=
∑ j, ((c j).diffWord α).bound M
theorem bound_nonneg (c : ClockedVector) (α : List (Fin 4)) (M : ℚ) :
0 ≤ c.bound α M :=
Finset.sum_nonneg (fun j _ => ((c j).diffWord α).bound_nonneg M)
theorem val_bound {c : ClockedVector} (hc : c.Valid) (α : List (Fin 4))
(M : ℚ) (y : SpaceTime) (hy : |y.1| ≤ |(M : ℝ)|) :
‖mixedDerivative c.val α y‖ ≤ (c.bound α M : ℝ) := by
rw [← val_diffWord hc]
apply (pi_norm_le_iff_of_nonneg (Rat.cast_nonneg.mpr (c.bound_nonneg α M))).mpr
intro j
have hj := ((c j).diffWord α).val_bound ((c j).valid_diffWord (hc j) α) M y hy
apply hj.trans
exact_mod_cast Finset.single_le_sum
(fun k _ => ((c k).diffWord α).bound_nonneg M) (Finset.mem_univ j)
def evaluate (c : ClockedVector) (hc : c.Valid) (α : List (Fin 4))
(a : ℕ → RationalSpaceTime) (ε : ℚ) (hε : 0 < ε) : RationalVector :=
fun j => ((c j).diffWord α).evaluate ((c j).valid_diffWord (hc j) α) a ε hε
theorem evaluate_spec {c : ClockedVector} (hc : c.Valid) (α : List (Fin 4))
(a : ℕ → RationalSpaceTime) {y : SpaceTime} (ha : IsFastName a y)
(ε : ℚ) (hε : 0 < ε) :
‖mixedDerivative c.val α y - rationalVector (c.evaluate hc α a ε hε)‖ ≤ (ε : ℝ) := by
rw [← val_diffWord hc]
apply (pi_norm_le_iff_of_nonneg (Rat.cast_nonneg.mpr hε.le)).mpr
intro j
exact ((c j).diffWord α).evaluate_spec ((c j).valid_diffWord (hc j) α) a ha ε hε
end ClockedVector
end ForcedComputation
end
end OAI
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