openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Programs /ClockedVector.lean
| 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 | |