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37bcf52 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 | 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
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