openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Programs /FixedParticleMain.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.FixedParticleAnalytic | |
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.SlowEffective | |
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.SlowTail | |
| namespace OAI | |
| /-! Full main theorem of A Fixed Particle Test for Computation in a Forced | |
| Viscous Flow, including the terminating evaluator and machine-only tail. -/ | |
| namespace ForcedComputation | |
| open ShearFlows Recorder | |
| def fixedParticleProgram (I : Alternating.MachineInput) (hI : Alternating.ValidInput I) : | |
| Input × Input := | |
| (shiftedLoader fixedStart (initialPointQ I hI) 0, recorderInputAtHeight I.1 hI.1 0) | |
| theorem fixedParticleLoader_valid (I : Alternating.MachineInput) | |
| (hI : Alternating.ValidInput I) : ValidInput (fixedParticleProgram I hI).1 := | |
| shiftedLoader_valid _ _ _ (by intro j; fin_cases j <;> norm_num [fixedStart]) | |
| (initialPointQ_bounds I hI) | |
| theorem fixedParticleBody_valid (I : Alternating.MachineInput) | |
| (hI : Alternating.ValidInput I) : ValidInput (fixedParticleProgram I hI).2 := | |
| recorderInputAtHeight_valid I.1 (Alternating.ValidInput.machine_wellFormed hI) 0 | |
| noncomputable def fixedParticleForce (ν : ℝ) (I : Alternating.MachineInput) | |
| (hI : Alternating.ValidInput I) : Velocity := force ν (fixedParticleVelocity I hI) | |
| theorem fixed_particle_effective_computation (I : Alternating.MachineInput) | |
| (hI : Alternating.ValidInput I) (ν : ℝ) (hν : 0 < ν) : | |
| SlowFluidProperties 1 ν (fixedParticleVelocity I hI) ∧ | |
| (∀ t x, Real.exp 1 - 1 ≤ t → | |
| fixedParticleVelocity I hI (t, x) = | |
| slowVelocity (recorderInputAtHeight I.1 hI.1 0).realizingVelocity (t, x) ∧ | |
| fixedParticleForce ν I hI (t, x) = | |
| force ν (slowVelocity (recorderInputAtHeight I.1 hI.1 0).realizingVelocity) (t, x)) ∧ | |
| (∀ (α : List (Fin 4)) (a : ℕ → ℚ) (b : ℕ → RationalSpaceTime) (y : SpaceTime) | |
| (ha : IsFastRealName a ν) (hb : IsFastName b y) (ε : ℚ) (hε : 0 < ε), | |
| ‖mixedDerivative (fixedParticleForce ν I hI) α y - | |
| rationalVector (evaluateSlowForce (fixedParticleProgram I hI).1 | |
| (fixedParticleProgram I hI).2 (fixedParticleLoader_valid I hI) | |
| (fixedParticleBody_valid I hI) α a ha b hb ε hε)‖ ≤ (ε : ℝ)) ∧ | |
| ∃ 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 | |
| obtain ⟨hp, he⟩ := fixed_particle_analytic_computation I hI ν hν | |
| refine ⟨hp, ?_, ?_, he⟩ | |
| · intro t x ht | |
| exact slow_initialized_tail _ _ ν ht x | |
| · intro α a b y ha hb ε hε | |
| exact evaluateSlowForce_spec (fixedParticleLoader_valid I hI) | |
| (fixedParticleBody_valid I hI) α a ha b hb ε hε | |
| end ForcedComputation | |
| end OAI | |