openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Programs /FixedParticleAnalytic.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.LoadedMachine | |
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.SlowBundle | |
| namespace OAI | |
| /-! Analytic properties and the fixed-particle observation for the | |
| square-integrable forcing construction. -/ | |
| noncomputable section | |
| namespace ForcedComputation | |
| open ShearFlows Recorder Set | |
| open scoped ContDiff NNReal | |
| def fixedParticleVelocity (I : Alternating.MachineInput) (hI : Alternating.ValidInput I) : Velocity := | |
| slowFromRest (loadedMachineVelocity fixedStart 0 I hI) | |
| theorem fixed_particle_analytic_computation (I : Alternating.MachineInput) | |
| (hI : Alternating.ValidInput I) (ν : ℝ) (hν : 0 < ν) : | |
| SlowFluidProperties 1 ν (fixedParticleVelocity I hI) ∧ | |
| ∃ Ψ : ℝ → Space → Space, IsMaterialFlow 1 (fixedParticleVelocity I hI) Ψ ∧ | |
| (Alternating.Halts I ↔ ∃ t : ℝ, 0 ≤ t ∧ | |
| 1 / 2 < Ψ t ![1 / 8, 3 / 8, 0] 0 ∧ Ψ t ![1 / 8, 3 / 8, 0] 0 < 1) := by | |
| let loader := shiftedLoader fixedStart (initialPointQ I hI) 0 | |
| let body := recorderInputAtHeight I.1 hI.1 0 | |
| let V := loadedMachineVelocity fixedStart 0 I hI | |
| have hl : ValidInput loader := shiftedLoader_valid _ _ _ | |
| (by intro j; fin_cases j <;> norm_num [fixedStart]) (initialPointQ_bounds I hI) | |
| have hb : ValidInput body := recorderInputAtHeight_valid _ _ _ | |
| have hp : loader.period = body.period := rfl | |
| have hbodyPeriod : (body.period : ℝ) = 1 := recorderInputAtHeight_period _ _ _ | |
| have hprops := slow_initialized_bundle hl hb hp ν hν.le | |
| rw [hbodyPeriod] at hprops | |
| refine ⟨hprops, ?_⟩ | |
| obtain ⟨_, Φ, hΦ, hobs⟩ := loadedMachine_computation fixedStart 0 | |
| (by intro j; fin_cases j <;> norm_num [fixedStart]) (by norm_num [fixedStart]) I hI ν hν | |
| simp only [Rat.cast_zero] at hobs | |
| obtain ⟨K, hK⟩ := initializedProgram_spatial_lipschitz hl hb | |
| have hs : ContDiff ℝ ∞ V := initializedProgram_smooth hl hb | |
| have hper : SpatiallyPeriodic 1 V := by | |
| change SpatiallyPeriodic 1 (initializedProgram loader body) | |
| simpa only [hbodyPeriod] using initializedProgram_periodic hp | |
| have hz : ∀ t, t < (1 / 32 : ℝ) → ∀ x, V (t, x) = 0 := | |
| fun _ ht x => initializedProgram_zero_extend loader body ht x | |
| obtain ⟨B, hB, hbound⟩ := initializedProgram_bounded hl hb hp [] | |
| obtain ⟨Ψ, hΨ⟩ := slowFromRest_materialFlow hs hper hz hK hB hbound | |
| refine ⟨Ψ, hΨ, ?_⟩ | |
| have he := slowFromRest_reaches_iff (fun x => hz 0 (by norm_num) x) hΦ hΨ hK | |
| (atHeight (fun j => (fixedStart j : ℝ)) 0) {x : Space | 1 / 2 < x 0 ∧ x 0 < 1} | |
| simp only [Reaches, mem_ofPred_eq] at he | |
| have hevent := hobs.trans he.symm | |
| simpa only [Reaches, mem_ofPred_eq, fixedStart, atHeight, | |
| Matrix.cons_val_zero, Matrix.cons_val_one, Rat.cast_div, Rat.cast_one, | |
| Rat.cast_ofNat, Rat.cast_zero] using hevent | |
| end ForcedComputation | |
| end | |
| end OAI | |