openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Modular /ActionMassTransport.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Modular.ActionOrbitTransport | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set | |
| open scoped Classical BigOperators | |
| namespace ActionOrbitTransport | |
| variable (Λ X : Type*) [Group Λ] [MulAction Λ X] | |
| namespace Representatives | |
| variable (R : Representatives Λ X) | |
| def weightedRow (w : X → ℝ) (F : X → X → ℝ) (p : R.RootedPair) : ℝ := | |
| w p.1 * F p.1 p.2 | |
| lemma weightedRow_finite (w : X → ℝ) (F : X → X → ℝ) | |
| (hF : ∀ x, (Function.support (F x)).Finite) : | |
| (Function.support (weightedRow Λ X R w F)).Finite := by | |
| apply (Set.finite_iUnion (fun x : {x // x∈R.roots} => | |
| (hF x).image (Prod.mk x))).subset | |
| intro p hp | |
| refine Set.mem_iUnion.mpr ⟨p.1,p.2,?_,rfl⟩ | |
| intro hz | |
| exact hp (by simp only [weightedRow,hz,mul_zero]) | |
| lemma fiber_sum_root (hfin : ∀ x y, (stabilizerOrbit Λ X x y).Finite) | |
| (w : X → ℝ) (F : X → X → ℝ) | |
| (hF : ∀ (φ : Λ) x y, F (φ • x) (φ • y) = F x y) | |
| (x : {x // x∈R.roots}) (y : X) : | |
| (∑' p : (pairMap Λ X R) ⁻¹' {pairOrbit Λ X x y}, weightedRow Λ X R w F p) = | |
| w x * (suborbitCard Λ X x y : ℝ) * F x y := by | |
| let := (hfin x y).fintype | |
| rw [← (fiberEquiv Λ X R x y).symm.tsum_eq] | |
| have hv (z : stabilizerOrbit Λ X x y) : | |
| weightedRow Λ X R w F ((fiberEquiv Λ X R x y).symm z) = w x * F x y := by | |
| change w x * F x z = w x * F x y | |
| obtain ⟨φ,hx,hy⟩ := (mem_stabilizerOrbit Λ X).mp z.property | |
| have hh := hF φ x y | |
| rw [hx,hy] at hh | |
| rw [hh] | |
| simp only [hv,tsum_fintype,Finset.sum_const,Finset.card_univ,nsmul_eq_mul] | |
| have hcard : Fintype.card (stabilizerOrbit Λ X x y) = suborbitCard Λ X x y := by | |
| rw [← Nat.card_eq_fintype_card,Nat.card_coe_set_eq] | |
| rfl | |
| rw [hcard] | |
| ring | |
| lemma fiber_sum (hfin : ∀ x y, (stabilizerOrbit Λ X x y).Finite) | |
| (w : X → ℝ) (hw : ∀ (φ : Λ) x, w (φ • x) = w x) | |
| (F : X → X → ℝ) (hF : ∀ (φ : Λ) x y, F (φ • x) (φ • y) = F x y) | |
| (x y : X) : | |
| (∑' p : (pairMap Λ X R) ⁻¹' {pairOrbit Λ X x y}, weightedRow Λ X R w F p) = | |
| w x * (suborbitCard Λ X x y : ℝ) * F x y := by | |
| obtain ⟨u,hu,φ,rfl⟩ := R.cover x | |
| let v := φ⁻¹ • y | |
| have hy : φ • v = y := smul_inv_smul φ y | |
| have hq : pairOrbit Λ X (φ • u) y = pairOrbit Λ X u v := by | |
| rw [←hy,smul_pairOrbit] | |
| rw [hq,fiber_sum_root Λ X R hfin w F hF ⟨u,hu⟩ v] | |
| change w u * (suborbitCard Λ X u v : ℝ) * F u v = _ | |
| rw [←hy,hw,smul_suborbitCard,hF] | |
| lemma tsum_weightedRow (w : X → ℝ) (F : X → X → ℝ) | |
| (hF : ∀ x, (Function.support (F x)).Finite) : | |
| (∑' p, weightedRow Λ X R w F p) = ∑ x ∈ R.roots, w x * ∑' y, F x y := by | |
| have hs : Summable (weightedRow Λ X R w F) := summable_of_hasFiniteSupport (weightedRow_finite Λ X R w F hF) | |
| rw [hs.tsum_prod] | |
| simp only [weightedRow,tsum_mul_left,tsum_fintype] | |
| exact Finset.sum_coe_sort R.roots (fun x => w x * ∑' y, F x y) | |
| lemma massTransport (hfin : ∀ x y, (stabilizerOrbit Λ X x y).Finite) | |
| (w : X → ℝ) (hw : ∀ (φ : Λ) x, w (φ • x) = w x) | |
| (hbal : ∀ x y, w x * (suborbitCard Λ X x y : ℝ) = w y * (suborbitCard Λ X y x : ℝ)) | |
| (F : X → X → ℝ) (hF : ∀ (φ : Λ) x y, F (φ • x) (φ • y) = F x y) | |
| (hrow : ∀ x, (Function.support (F x)).Finite) | |
| (hcol : ∀ y, (Function.support (fun x => F x y)).Finite) : | |
| (∑ x ∈ R.roots, w x * ∑' y, F x y) = ∑ y ∈ R.roots, w y * ∑' x, F x y := by | |
| let F' := fun x y => F y x | |
| have hF' (φ : Λ) (x y : X) : F' (φ • x) (φ • y) = F' x y := hF φ y x | |
| have hs : Summable (weightedRow Λ X R w F) := summable_of_hasFiniteSupport (weightedRow_finite Λ X R w F hrow) | |
| have ht : Summable (weightedRow Λ X R w F') := summable_of_hasFiniteSupport (weightedRow_finite Λ X R w F' hcol) | |
| rw [←tsum_weightedRow Λ X R w F hrow,←tsum_weightedRow Λ X R w F' hcol] | |
| calc | |
| _ = ∑' q : PairOrbit Λ X, ∑' p : (pairMap Λ X R) ⁻¹' {q}, weightedRow Λ X R w F p := | |
| (hs.hasSum.tsum_fiberwise (pairMap Λ X R)).tsum_eq.symm | |
| _ = ∑' q : PairOrbit Λ X, ∑' p : (pairMap Λ X R) ⁻¹' {swapPairOrbit Λ X q}, | |
| weightedRow Λ X R w F' p := by | |
| apply tsum_congr | |
| intro q | |
| induction q using Quotient.inductionOn' with | |
| | h p => | |
| change (∑' z : (pairMap Λ X R) ⁻¹' {pairOrbit Λ X p.1 p.2}, weightedRow Λ X R w F z) = | |
| ∑' z : (pairMap Λ X R) ⁻¹' {pairOrbit Λ X p.2 p.1}, weightedRow Λ X R w F' z | |
| rw [fiber_sum Λ X R hfin w hw F hF,fiber_sum Λ X R hfin w hw F' hF',hbal] | |
| _ = ∑' p, weightedRow Λ X R w F' p := by | |
| exact ((swapPairOrbit Λ X).tsum_eq (fun q => | |
| ∑' p : (pairMap Λ X R) ⁻¹' {q}, weightedRow Λ X R w F' p)).trans | |
| (ht.hasSum.tsum_fiberwise (pairMap Λ X R)).tsum_eq | |
| end Representatives | |
| end ActionOrbitTransport | |
| end | |
| end | |
| end OAI | |