openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Flow /PlanarCompactBridge.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.CompactPlanarProcessor | |
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.NormalizedPlanarProgram | |
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.PlanarPerturbations | |
| namespace OAI | |
| /-! Agreement of the compact planar field with the finite periodic scalar | |
| expression on the open unit-square chart, at every phase. -/ | |
| noncomputable section | |
| namespace ForcedComputation | |
| open ShearFlows Set | |
| theorem periodic_eq_of_eqOn_unit {E : Type*} {f g : ℝ → E} | |
| (hf : Function.Periodic f 1) (hg : Function.Periodic g 1) | |
| (he : EqOn f g (Icc (0 : ℝ) 1)) : f = g := by | |
| funext t | |
| have hf' : f (Int.fract t) = f t := by | |
| simpa only [Int.fract, mul_one] using hf.sub_int_mul_eq (x := t) ⌊t⌋ | |
| have hg' : g (Int.fract t) = g t := by | |
| simpa only [Int.fract, mul_one] using hg.sub_int_mul_eq (x := t) ⌊t⌋ | |
| exact hf'.symm.trans ((he ⟨Int.fract_nonneg t, (Int.fract_lt_one t).le⟩).trans hg') | |
| theorem planarSlice_periodic {H : FieldExpr} | |
| (hP : CubePeriodic 1 (SpatialExpression.spatialValue H)) (x : Plane) : | |
| Function.Periodic (fun s => planarSlice H s x) 1 := by | |
| intro s | |
| have he : (fun Y => SpatialExpression.spatialValue H (atHeight Y (s + 1))) = | |
| (fun Y => SpatialExpression.spatialValue H (atHeight Y s)) := by | |
| funext Y | |
| have hv : atHeight Y (s + 1) = atHeight Y s + latticeVector 1 ![0, 0, 1] := by | |
| ext j | |
| fin_cases j <;> simp [atHeight, latticeVector] | |
| rw [hv] | |
| exact hP _ _ | |
| change PlanarHamiltonian.field _ x = PlanarHamiltonian.field _ x | |
| rw [he] | |
| namespace Recorder.Planar | |
| open PlanarHamiltonian | |
| theorem normalized_planar_field_on_chart (I : Alternating.MachineInput) | |
| (hI : Alternating.ValidInput I) (s : ℝ) (x : Plane) | |
| (hx₀ : x 0 ∈ Ioo (0 : ℝ) 1) (hx₁ : x 1 ∈ Ioo (0 : ℝ) 1) : | |
| PlanarHamiltonian.periodicVelocity (normalizedPulse I hI) s x = | |
| planarSlice (normalizedHamiltonian I hI) s x := by | |
| have he : (fun t => PlanarHamiltonian.periodicVelocity (normalizedPulse I hI) t x) = | |
| (fun t => planarSlice (normalizedHamiltonian I hI) t x) := by | |
| apply periodic_eq_of_eqOn_unit (PlanarHamiltonian.periodicVelocity_periodic _ x) | |
| (planarSlice_periodic (normalizedHamiltonian_periodic I hI) x) | |
| intro t ht | |
| exact (PlanarHamiltonian.periodicVelocity_on_unit (normalizedPulse_valid I hI) | |
| (normalizedPulse_inUnit I hI) ht x).trans | |
| (processorExpression_planar_field (normalizedPulse_valid I hI) | |
| (normalizedPulse_inUnit I hI) ht hx₀ hx₁).symm | |
| exact congrFun he s | |
| end Recorder.Planar | |
| end ForcedComputation | |
| end | |
| end OAI | |