openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Flow /FlowPathIntegral.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.FlowPointwise | |
| import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | |
| import Mathlib.Topology.Order.ProjIcc | |
| namespace OAI | |
| /-! Integration on the fixed unit interval as a bounded linear operator. | |
| This permits elapsed time to be a parameter of the Picard equation. -/ | |
| noncomputable section | |
| namespace ForcedComputation.Flow | |
| open Set MeasureTheory | |
| open scoped Topology | |
| abbrev UnitTime := Icc (0 : ℝ) 1 | |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] | |
| def extendPath (u : C(UnitTime, E)) : C(ℝ, E) := | |
| u.comp ⟨projIcc 0 1 zero_le_one, continuous_projIcc⟩ | |
| omit [NormedSpace ℝ E] [CompleteSpace E] in | |
| @[simp] theorem extendPath_apply (u : C(UnitTime, E)) (t : ℝ) : | |
| extendPath u t = u (projIcc 0 1 zero_le_one t) := rfl | |
| omit [NormedSpace ℝ E] [CompleteSpace E] in | |
| @[simp] theorem extendPath_coe (u : C(UnitTime, E)) (t : UnitTime) : | |
| extendPath u t = u t := by simp [extendPath] | |
| def primitivePath (u : C(UnitTime, E)) : C(UnitTime, E) := | |
| ⟨fun t => ∫ s in (0 : ℝ)..t.1, extendPath u s, | |
| (continuous_iff_continuousAt.mpr fun t => | |
| ((extendPath u).continuous.integral_hasStrictDerivAt 0 t).hasDerivAt.continuousAt).comp | |
| continuous_subtype_val⟩ | |
| @[simp] theorem primitivePath_apply (u : C(UnitTime, E)) (t : UnitTime) : | |
| primitivePath u t = ∫ s in (0 : ℝ)..t.1, extendPath u s := rfl | |
| theorem primitivePath_norm (u : C(UnitTime, E)) : ‖primitivePath u‖ ≤ ‖u‖ := by | |
| apply (ContinuousMap.norm_le _ (norm_nonneg u)).mpr | |
| intro t | |
| calc | |
| ‖primitivePath u t‖ ≤ ‖u‖ * |t.1 - 0| := | |
| intervalIntegral.norm_integral_le_of_norm_le_const fun s _ => | |
| u.norm_coe_le_norm _ | |
| _ ≤ ‖u‖ := by | |
| rw [sub_zero, abs_of_nonneg t.2.1] | |
| exact mul_le_of_le_one_right (norm_nonneg u) t.2.2 | |
| def primitivePathL : C(UnitTime, E) →L[ℝ] C(UnitTime, E) := | |
| LinearMap.mkContinuous | |
| { toFun := primitivePath | |
| map_add' := by | |
| intro u v | |
| ext t | |
| change (∫ s in (0 : ℝ)..t.1, extendPath u s + extendPath v s) = _ | |
| exact intervalIntegral.integral_add | |
| ((extendPath u).continuous.intervalIntegrable _ _) | |
| ((extendPath v).continuous.intervalIntegrable _ _) | |
| map_smul' := by | |
| intro c u | |
| ext t | |
| change (∫ s in (0 : ℝ)..t.1, c • extendPath u s) = _ | |
| exact intervalIntegral.integral_smul c (extendPath u) } | |
| 1 (by | |
| intro u | |
| change ‖primitivePath u‖ ≤ (1 : ℝ) * ‖u‖ | |
| rw [one_mul] | |
| exact primitivePath_norm u) | |
| @[simp] theorem primitivePathL_apply (u : C(UnitTime, E)) : | |
| primitivePathL u = primitivePath u := rfl | |
| theorem primitivePathL_norm : ‖primitivePathL (E := E)‖ ≤ 1 := | |
| LinearMap.mkContinuous_norm_le _ zero_le_one _ | |
| end ForcedComputation.Flow | |
| end | |
| end OAI | |