openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Programs /ClockedBounds.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.ClockedExpressions | |
| import OAI.MathematicalPhysics.NavierStokes.ShearFlows.EffectiveInterface | |
| namespace OAI | |
| /-! Local effective bounds for clocked expressions. Periodic field factors | |
| have global bounds; only the explicit clock profiles use a time radius. -/ | |
| namespace ForcedComputation.ClockedExpr | |
| open ShearFlows | |
| open scoped ContDiff BigOperators | |
| def bound (M : ℚ) : ClockedExpr → ℚ | |
| | .const r => |r| | |
| | .profile q => q.bound M | |
| | .field _ e => e.bound | |
| | .add e f => e.bound M + f.bound M | |
| | .mul e f => e.bound M * f.bound M | |
| theorem bound_nonneg (e : ClockedExpr) (M : ℚ) : 0 ≤ e.bound M := by | |
| induction e with | |
| | const => exact abs_nonneg _ | |
| | profile q => exact q.bound_nonneg M | |
| | field q e => exact e.bound_nonneg | |
| | add e f he hf => exact add_nonneg he hf | |
| | mul e f he hf => exact mul_nonneg he hf | |
| theorem val_bound {e : ClockedExpr} (he : e.Valid) (M : ℚ) (y : SpaceTime) | |
| (hy : |y.1| ≤ |(M : ℝ)|) : |e.val y| ≤ (e.bound M : ℝ) := by | |
| induction e with | |
| | const r => simp [val, bound] | |
| | profile q => exact q.val_bound hy | |
| | field q e => exact e.val_bound he _ | |
| | add e f ih₁ ih₂ => | |
| simpa only [val, bound, Rat.cast_add] using | |
| (abs_add_le _ _).trans (add_le_add (ih₁ he.1) (ih₂ he.2)) | |
| | mul e f ih₁ ih₂ => | |
| simpa only [val, bound, Rat.cast_mul, abs_mul] using | |
| mul_le_mul (ih₁ he.1) (ih₂ he.2) (abs_nonneg _) | |
| (Rat.cast_nonneg.mpr (e.bound_nonneg M)) | |
| def diffWord (e : ClockedExpr) : List (Fin 4) → ClockedExpr | |
| | [] => e | |
| | j :: α => (e.diffWord α).diff j | |
| theorem valid_diffWord {e : ClockedExpr} (he : e.Valid) (α : List (Fin 4)) : | |
| (e.diffWord α).Valid := by | |
| induction α with | |
| | nil => exact he | |
| | cons j α ih => exact valid_diff ih j | |
| noncomputable def scalarMixed (f : SpaceTime → ℝ) : List (Fin 4) → SpaceTime → ℝ := | |
| List.rec f (fun j _ g y => fderiv ℝ g y (spaceTimeDirection j)) | |
| theorem val_diffWord {e : ClockedExpr} (he : e.Valid) (α : List (Fin 4)) : | |
| (e.diffWord α).val = scalarMixed e.val α := by | |
| induction α with | |
| | nil => rfl | |
| | cons j α ih => | |
| funext y | |
| change ((e.diffWord α).diff j).val y = _ | |
| rw [val_diff (valid_diffWord he α), ih] | |
| rfl | |
| theorem mixed_bound {e : ClockedExpr} (he : e.Valid) (α : List (Fin 4)) | |
| (M : ℚ) (y : SpaceTime) (hy : |y.1| ≤ |(M : ℝ)|) : | |
| |scalarMixed e.val α y| ≤ ((e.diffWord α).bound M : ℝ) := by | |
| rw [← val_diffWord he α] | |
| exact val_bound (valid_diffWord he α) M y hy | |
| end ForcedComputation.ClockedExpr | |
| end OAI | |