openAI-math-dataset / lean /OAI /Analysis /Conductivity /Variational /CentralUnnormalizedVariational.lean
| import Mathlib | |
| import OAI.Analysis.Conductivity.Sobolev.CentralPhysicalVariational | |
| import OAI.Analysis.Conductivity.Sources.CentralWeightedVariational | |
| namespace OAI | |
| noncomputable section | |
| namespace ScalarConductivity | |
| open Set MeasureTheory | |
| def angularArea : ℝ := (2*Real.pi)^2 | |
| lemma angularArea_ge_one : 1≤angularArea := by | |
| have hp := Real.pi_gt_three | |
| dsimp only [angularArea] | |
| nlinarith [sq_nonneg (Real.pi-3)] | |
| theorem central_unnormalized_variational_exists (s slopes : Fin 3 → ℝ) | |
| (hs : ∑ i,slopes i=0) : | |
| ∃ p : centralEnergySpace s,centralM s p=0 ∧ | |
| (∀ v : centralEnergySpace s, | |
| inner ℝ (centralD s p) (centralD s v)+ | |
| angularArea*(∑ i : Fin 3,inner ℝ (spectralGraphWeight (torusRate s) (centralT s i p)) | |
| (spectralGraphWeight (torusRate s) (centralT s i v)))= | |
| angularArea*(∑ i : Fin 3,slopes i*spectralGraphMean (torusRate s) (centralT s i v))) ∧ | |
| (∀ q : centralEnergySpace s,(∀ v : centralEnergySpace s, | |
| inner ℝ (centralD s q) (centralD s v)+ | |
| angularArea*(∑ i : Fin 3,inner ℝ (spectralGraphWeight (torusRate s) (centralT s i q)) | |
| (spectralGraphWeight (torusRate s) (centralT s i v)))= | |
| angularArea*(∑ i : Fin 3,slopes i*spectralGraphMean (torusRate s) (centralT s i v))) → | |
| q-p=centralM s q • centralOne s) := by | |
| exact @central_variational_exists_graph_weighted (centralEnergySpace s) CentralGradients | |
| inferInstance (centralEnergy_inner s) inferInstance inferInstance inferInstance | |
| TorusModes inferInstance angularArea angularArea_ge_one (torusRate s) (centralD s) (centralT s) | |
| (centralM s) (centralOne s) | |
| (centralAmbientM_one s) (centralAmbientD_one s) (centralAmbientWeight_one s) | |
| (centralOne_mean s) (centralCoercivityConstant s) (centralCoercivityConstant_pos s) | |
| (central_graph_coercivity s) slopes hs | |
| end ScalarConductivity | |
| end | |
| end OAI | |