openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Harmonic /GraphProjectedFunctions.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Harmonic.GraphWalkEnergy | |
| import OAI.Probability.CriticalPercolation.Harmonic.GraphWalkProbability | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function | |
| open scoped Classical BigOperators | |
| namespace CriticalPercolation.HomogeneousAnalysis.Geometry | |
| open HarmonicAnalysis | |
| variable {X : Type*} (W : Geometry X) | |
| def projectedFunction (n : ℕ) (t x : X) : ℝ := | |
| inner ℝ (W.walkFeature n x-W.walkState n) (W.walkFeature n t-W.walkState n) / | |
| (4*(W.degree:ℝ)*W.walkEnergy (W.walkState n)) | |
| lemma exists_large_neighbor_gradient (hD : 0<W.degree) (n : ℕ) : | |
| ∃ t∈W.neighbors W.root, 4*W.walkEnergy (W.walkState n)≤‖W.walkFeature n t-W.walkState n‖^2 := by | |
| by_contra! h | |
| have hh := Finset.sum_lt_sum_of_nonempty (Finset.card_pos.mp hD) (fun t ht => h t ht) | |
| rw [W.walkEnergy_gradient] at hh | |
| simp only [Finset.sum_const,nsmul_eq_mul] at hh | |
| nlinarith | |
| lemma projectedFunction_root (n : ℕ) (t : X) : W.projectedFunction n t W.root=0 := by | |
| simp [projectedFunction,walkState] | |
| lemma projectedFunction_edge_bound [Infinite X] (n : ℕ) (t : X) | |
| (ht : t∈W.neighbors W.root) {x y : X} (hxy : W.graph.Adj x y) : | |
| |W.projectedFunction n t y-W.projectedFunction n t x|≤1 := by | |
| let q := 4*(W.degree:ℝ)*W.walkEnergy (W.walkState n) | |
| have hc : (0:ℝ)<W.degree := by exact_mod_cast W.degree_pos | |
| have hf := W.walkState_energy_pos n | |
| have hq : 0<q := by dsimp [q]; positivity | |
| have he : (W.walkFeature n y-W.walkState n)-(W.walkFeature n x-W.walkState n)= | |
| W.walkFeature n y-W.walkFeature n x := by abel | |
| unfold projectedFunction | |
| rw [←sub_div,←inner_sub_left,he,abs_div,abs_of_pos hq] | |
| apply (div_le_one hq).mpr | |
| calc | |
| _ ≤ ‖W.walkFeature n y-W.walkFeature n x‖*‖W.walkFeature n t-W.walkState n‖ := abs_real_inner_le_norm _ _ | |
| _ ≤ Real.sqrt q*Real.sqrt q := mul_le_mul (W.feature_edge_bound W.degree_pos n hxy) | |
| (W.feature_neighbor_bound W.degree_pos n ht) (norm_nonneg _) (Real.sqrt_nonneg _) | |
| _ = q := Real.mul_self_sqrt hq.le | |
| lemma projectedFunction_length_bound [Infinite X] (n : ℕ) (t : X) | |
| (ht : t∈W.neighbors W.root) (x : X) : |W.projectedFunction n t x|≤(W.length x:ℝ) := by | |
| let q := 4*(W.degree:ℝ)*W.walkEnergy (W.walkState n) | |
| have hc : (0:ℝ)<W.degree := by exact_mod_cast W.degree_pos | |
| have hf := W.walkState_energy_pos n | |
| have hq : 0<q := by dsimp [q]; positivity | |
| unfold projectedFunction | |
| rw [abs_div,abs_of_pos hq] | |
| apply (div_le_iff₀ hq).mpr | |
| calc | |
| _ ≤ ‖W.walkFeature n x-W.walkState n‖*‖W.walkFeature n t-W.walkState n‖ := abs_real_inner_le_norm _ _ | |
| _ ≤ ((W.length x:ℝ)*Real.sqrt q)*Real.sqrt q := | |
| mul_le_mul (W.feature_ball_bound W.degree_pos n _ (W.mem_ball_length x)) | |
| (W.feature_neighbor_bound W.degree_pos n ht) (norm_nonneg _) (by positivity) | |
| _ = (W.length x:ℝ)*q := by rw [mul_assoc,Real.mul_self_sqrt hq.le] | |
| lemma projectedFunction_neighbor_lower [Infinite X] (n : ℕ) (t : X) | |
| (ht : 4*W.walkEnergy (W.walkState n)≤‖W.walkFeature n t-W.walkState n‖^2) : | |
| (W.degree:ℝ)⁻¹≤W.projectedFunction n t t := by | |
| have hc : (0:ℝ)<W.degree := by exact_mod_cast W.degree_pos | |
| have hf := W.walkState_energy_pos n | |
| have hq : 0<4*(W.degree:ℝ)*W.walkEnergy (W.walkState n) := by positivity | |
| unfold projectedFunction | |
| rw [real_inner_self_eq_norm_sq] | |
| apply (le_div_iff₀ hq).mpr | |
| have he : (W.degree:ℝ)⁻¹*(4*(W.degree:ℝ)*W.walkEnergy (W.walkState n))=4*W.walkEnergy (W.walkState n) := by field_simp | |
| rw [he]; exact ht | |
| lemma sum_feature_displacement (n : ℕ) (x : X) : | |
| (∑ y∈W.neighbors x, (W.walkFeature n x-W.walkFeature n y))= | |
| (2*(W.degree:ℝ)) • (W.walkFeature n x-W.lazyWalk (W.walkFeature n x)) := by | |
| rw [Finset.sum_sub_distrib,Finset.sum_const,←Nat.cast_smul_eq_nsmul ℝ, | |
| W.neighbors_card,W.walkFeature_average,W.lazyWalk_apply] | |
| module | |
| lemma projectedFunction_residual (n : ℕ) (t x : X) : | |
| (∑ y∈W.neighbors x, (W.projectedFunction n t x-W.projectedFunction n t y))= | |
| (2*(W.degree:ℝ))*inner ℝ (W.walkFeature n x-W.lazyWalk (W.walkFeature n x)) | |
| (W.walkFeature n t-W.walkState n)/(4*(W.degree:ℝ)*W.walkEnergy (W.walkState n)) := by | |
| simp only [projectedFunction,←sub_div,←Finset.sum_div,←inner_sub_left,←sum_inner] | |
| have he : (∑ y∈W.neighbors x, (W.walkFeature n x-W.walkState n-(W.walkFeature n y-W.walkState n)))= | |
| ∑ y∈W.neighbors x, (W.walkFeature n x-W.walkFeature n y) := by | |
| apply Finset.sum_congr rfl; intros; abel | |
| rw [he,W.sum_feature_displacement,real_inner_smul_left (F:=FH X)] | |
| lemma feature_residual_norm (n : ℕ) (x : X) : | |
| ‖W.walkFeature n x-W.lazyWalk (W.walkFeature n x)‖=‖W.walkState n-W.lazyWalk (W.walkState n)‖ := by | |
| obtain ⟨g,rfl⟩ := W.transitive W.root x | |
| rw [←W.transport_walkFeature,←W.transport_lazyWalk,←map_sub,LinearIsometryEquiv.norm_map] | |
| lemma projectedFunction_residual_sq [Infinite X] (n : ℕ) (t : X) | |
| (ht : t∈W.neighbors W.root) (x : X) : | |
| (∑ y∈W.neighbors x, (W.projectedFunction n t x-W.projectedFunction n t y))^2≤ | |
| (W.degree:ℝ)*‖W.walkState n-W.lazyWalk (W.walkState n)‖^2/W.walkEnergy (W.walkState n) := by | |
| have hc : (0:ℝ)<W.degree := by exact_mod_cast W.degree_pos | |
| have hf := W.walkState_energy_pos n | |
| have hgrad := (Finset.single_le_sum (f:=fun y => ‖W.walkFeature n y-W.walkState n‖^2) | |
| (fun y hy => sq_nonneg _) ht).trans_eq (W.walkEnergy_gradient n) | |
| have hi := abs_real_inner_le_norm (W.walkFeature n x-W.lazyWalk (W.walkFeature n x)) | |
| (W.walkFeature n t-W.walkState n) | |
| rw [W.feature_residual_norm] at hi | |
| have hisq := pow_le_pow_left₀ (abs_nonneg _) hi 2 | |
| rw [sq_abs,mul_pow] at hisq | |
| have hb := mul_le_mul_of_nonneg_left hgrad (sq_nonneg ‖W.walkState n-W.lazyWalk (W.walkState n)‖) | |
| rw [W.projectedFunction_residual,div_pow] | |
| apply (div_le_div_iff₀ (sq_pos_of_pos (show 0<4*(W.degree:ℝ)*W.walkEnergy (W.walkState n) by positivity)) hf).mpr | |
| have hm := mul_le_mul_of_nonneg_left (hisq.trans hb) | |
| (show 0≤4*(W.degree:ℝ)^2*W.walkEnergy (W.walkState n) by positivity) | |
| nlinarith only [hm] | |
| end CriticalPercolation.HomogeneousAnalysis.Geometry | |
| end | |
| end | |
| end OAI | |