| import Mathlib |
| import OAI.Probability.LatticeSurface.Dynamics.ProjectionMixture |
|
|
| namespace OAI |
|
|
| namespace ProjectionComparison |
|
|
| open Set |
|
|
| variable {V E : Type*} [Fintype V] |
|
|
| lemma mem_faceComponent (S : System V E) (F : Finset E) (x y : V) : |
| y∈faceComponent S F x ↔ (faceGraph S F).Reachable x y := by |
| classical exact Set.mem_toFinset |
|
|
| lemma faceComponent_eq (S : System V E) (F : Finset E) (x y : V) |
| (hxy : (faceGraph S F).Reachable x y) : faceComponent S F x=faceComponent S F y := by |
| ext z |
| simp only [mem_faceComponent] |
| exact ⟨fun hx => hxy.symm.trans hx,fun hy => hxy.trans hy⟩ |
|
|
| omit [Fintype V] in |
| lemma faceFree_iff (S : System V E) (F : Finset E) (x y : V) |
| (hxy : (faceGraph S F).Reachable x y) : |
| (∀p∈S.pins,¬(faceGraph S F).Reachable x p) ↔ (∀p∈S.pins,¬(faceGraph S F).Reachable y p) := by |
| constructor <;> intro h p hp hr |
| · exact h p hp (hxy.trans hr) |
| · exact h p hp (hxy.symm.trans hr) |
|
|
| noncomputable def faceKernel (S : System V E) (F : Finset E) (x y : V) : ℝ := by |
| classical exact if (∀p∈S.pins,¬(faceGraph S F).Reachable x p) ∧ (faceGraph S F).Reachable x y |
| then ((faceComponent S F x).card:ℝ)⁻¹ else 0 |
|
|
| lemma faceKernel_nonneg (S : System V E) (F : Finset E) (x y : V) : 0≤faceKernel S F x y := by |
| classical |
| unfold faceKernel |
| split_ifs <;> positivity |
|
|
| lemma faceKernel_symm (S : System V E) (F : Finset E) (x y : V) : faceKernel S F x y=faceKernel S F y x := by |
| classical |
| unfold faceKernel |
| by_cases hr : (faceGraph S F).Reachable x y |
| · simp only [faceComponent_eq S F x y hr,faceFree_iff S F x y hr,hr,hr.symm,and_true] |
| · have hn : ¬(faceGraph S F).Reachable y x := fun hy => hr hy.symm |
| simp only [hr,hn,and_false,ite_false] |
|
|
| lemma faceAverage_eq_sum (S : System V E) (F : Finset E) (d : V → ℝ) (x : V) : |
| faceAverage S F d x=∑y,faceKernel S F x y*d y := by |
| classical |
| unfold faceAverage faceKernel |
| by_cases hf : ∀p∈S.pins,¬(faceGraph S F).Reachable x p |
| · simp only [ite_and,ite_eq_left hf,div_eq_mul_inv] |
| rw [Finset.sum_mul] |
| calc |
| (∑ y∈faceComponent S F x,d y*((faceComponent S F x).card:ℝ)⁻¹)= |
| ∑ y∈faceComponent S F x,(if (faceGraph S F).Reachable x y then ((faceComponent S F x).card:ℝ)⁻¹ else 0)*d y := by |
| apply Finset.sum_congr rfl |
| intro y hy |
| rw [ite_eq_left ((mem_faceComponent S F x y).mp hy),mul_comm] |
| _ = _ := by |
| apply Finset.sum_subset (Finset.subset_univ _) |
| intro y hy hn |
| have hn' : ¬(faceGraph S F).Reachable x y := by simpa only [mem_faceComponent] using hn |
| rw [ite_eq_right hn',zero_mul] |
| · simp only [ite_and,ite_eq_right hf,zero_mul,Finset.sum_const_zero] |
|
|
| lemma faceKernel_row (S : System V E) (F : Finset E) (x : V) : (∑y,faceKernel S F x y)≤1 := by |
| classical |
| have h := faceAverage_eq_sum S F (fun _ => 1) x |
| simp only [mul_one] at h |
| rw [← h] |
| unfold faceAverage |
| split_ifs with hf |
| · have hx : x∈faceComponent S F x := (mem_faceComponent S F x x).mpr (.refl x) |
| have hp : (0:ℝ)<(faceComponent S F x).card := Nat.cast_pos.mpr (Finset.card_pos.mpr ⟨x,hx⟩) |
| simp only [Finset.sum_const,nsmul_eq_mul,mul_one,div_self hp.ne'] |
| rfl |
| · norm_num |
|
|
| lemma faceKernel_column (S : System V E) (F : Finset E) (y : V) : (∑x,faceKernel S F x y)≤1 := by |
| simpa only [faceKernel_symm S F] using faceKernel_row S F y |
|
|
| lemma faceAverage_constant (S : System V E) (F : Finset E) (d : V → ℝ) (x y : V) |
| (hxy : (faceGraph S F).Reachable x y) : faceAverage S F d x=faceAverage S F d y := by |
| classical |
| unfold faceAverage |
| simp only [faceFree_iff S F x y hxy,faceComponent_eq S F x y hxy] |
|
|
| lemma faceAverage_idempotent (S : System V E) (F : Finset E) (d : V → ℝ) (x : V) : |
| faceAverage S F (faceAverage S F d) x=faceAverage S F d x := by |
| classical |
| unfold faceAverage at ⊢ |
| split_ifs with hf |
| · have hsum : (∑ y∈faceComponent S F x,faceAverage S F d y)=(faceComponent S F x).card*faceAverage S F d x := by |
| trans ∑ _y∈faceComponent S F x,faceAverage S F d x |
| · apply Finset.sum_congr rfl |
| intro y hy |
| exact (faceAverage_constant S F d x y ((mem_faceComponent S F x y).mp hy)).symm |
| · simp only [Finset.sum_const,nsmul_eq_mul] |
| have hx : x∈faceComponent S F x := (mem_faceComponent S F x x).mpr (.refl x) |
| have hp : (0:ℝ)<(faceComponent S F x).card := Nat.cast_pos.mpr (Finset.card_pos.mpr ⟨x,hx⟩) |
| change (∑ y∈faceComponent S F x,faceAverage S F d y)/(faceComponent S F x).card=_ |
| rw [hsum,mul_div_cancel_left₀ _ hp.ne'] |
| simp only [faceAverage,ite_eq_left hf] |
| · rfl |
|
|
| end ProjectionComparison |
|
|
| end OAI |
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|
