| import OAI.Combinatorics.CompleteCrossing.Triangle |
| import Mathlib.Data.Finset.Sort |
|
|
| namespace OAI |
|
|
| /-! Order-detecting interpolation and its kernel. -/ |
|
|
| section |
|
|
| /-! The order pairing as a nonzero diagonal pairing on the even two-triangle set. -/ |
| noncomputable section |
| open scoped BigOperators |
| namespace Paper170 |
|
|
| def evenTriangleWeight (s : ℕ) (node : Fin (2*s) → ℝ) (a : EvenTriangleCoord s) : ℝ := |
| (if a.val.1.val < s then -1 else 1) * (node a.val.1-node a.val.2) / |
| (Zarankiewicz.derivativeAtNode node a.val.1 * Zarankiewicz.derivativeAtNode node a.val.2) |
|
|
| theorem evenTriangleWeight_ne_zero (s : ℕ) (node : Fin (2*s) → ℝ) |
| (hn : Function.Injective node) (a : EvenTriangleCoord s) : evenTriangleWeight s node a ≠ 0 := by |
| have ha := (mem_evenTriangle s a.val.1 a.val.2).mp a.property |
| have hne : a.val.1 ≠ a.val.2 := by rcases ha with ha | ha; exact ne_of_gt ha.2; exact ne_of_lt ha.2 |
| unfold evenTriangleWeight |
| apply div_ne_zero |
| · apply mul_ne_zero |
| · split_ifs <;> norm_num |
| · exact sub_ne_zero.mpr (hn.ne hne) |
| · exact mul_ne_zero (Zarankiewicz.derivativeAtNode_ne_zero node hn _) |
| (Zarankiewicz.derivativeAtNode_ne_zero node hn _) |
|
|
| theorem shifted_halfOrder (s : ℕ) (node : Fin (2*s) → ℝ) (p q : Fin (2*s)) : |
| (halfOrder inferInstance p q + (if p.val < s then -(1/2) else 1/2)) * (node p-node q) = |
| if (p,q) ∈ evenTriangle s then |
| (if p.val < s then -1 else 1) * (node p-node q) else 0 := by |
| rcases lt_trichotomy p q with hpq | rfl | hqp |
| · have hqp := not_lt_of_ge (le_of_lt hpq) |
| by_cases hp : p.val < s <;> |
| simp [halfOrder, hpq, hqp, mem_evenTriangle, hp, not_lt.mp] |
| ring |
| · simp [halfOrder] |
| · have hpq := not_lt_of_ge (le_of_lt hqp) |
| by_cases hp : p.val < s <;> |
| simp [halfOrder, hpq, hqp, mem_evenTriangle, hp, not_lt.mp] |
| ring |
|
|
| theorem orderPairing_triangle (s : ℕ) (hs : 2 ≤ s) |
| (node : Fin (2*s) → ℝ) (hn : Function.Injective node) (A B : Poly2) |
| (hA : A ∈ rectangle (s-1) (s-2)) (hB : B ∈ rectangle (s-2) (s-1)) : |
| orderPairing node inferInstance A B = ∑ a : EvenTriangleCoord s, |
| evenTriangleWeight s node a * evalTwo (node a.val.1) (node a.val.2) A * |
| evalTwo (node a.val.1) (node a.val.2) B := by |
| have hshift := pairing_row_column_invariant s hs (by simp) node hn (halfOrder inferInstance) |
| (fun p => if p.val < s then -(1/2) else 1/2) (fun _ => 0) A B hA hB |
| simp only [add_zero] at hshift |
| rw [orderPairing, ← hshift] |
| let F (a : Fin (2*s) × Fin (2*s)) : ℝ := |
| (if a.1.val < s then -1 else 1) * (node a.1-node a.2) / |
| (Zarankiewicz.derivativeAtNode node a.1 * Zarankiewicz.derivativeAtNode node a.2) * |
| evalTwo (node a.1) (node a.2) A * evalTwo (node a.1) (node a.2) B |
| change _ = ∑ a : EvenTriangleCoord s, F a.val |
| calc |
| _ = ∑ a ∈ evenTriangle s, F a := by |
| unfold weightedPairing |
| simp_rw [shifted_halfOrder] |
| rw [evenTriangle, Finset.sum_filter, Fintype.sum_prod_type] |
| apply Finset.sum_congr rfl |
| intro p _ |
| apply Finset.sum_congr rfl |
| intro q _ |
| by_cases ha : (p,q) ∈ evenTriangle s |
| · have hh := (mem_evenTriangle s p q).mp ha |
| simp only [Finset.mem_filter, Finset.mem_univ, true_and, ite_eq_left hh, F] |
| · have hh := (mem_evenTriangle s p q).not.mp ha |
| simp only [Finset.mem_filter, Finset.mem_univ, true_and, ite_eq_right hh, zero_div, zero_mul] |
| _ = ∑ a : EvenTriangleCoord s, F a.val := (Finset.sum_coe_sort (evenTriangle s) F).symm |
|
|
| theorem derivativeAtNode_equiv {ι κ : Type*} [Fintype ι] [Fintype κ] |
| (node : ι → ℝ) (e : κ ≃ ι) (p : κ) : |
| Zarankiewicz.derivativeAtNode (node ∘ e) p = Zarankiewicz.derivativeAtNode node (e p) := by |
| classical |
| have h : Zarankiewicz.nodePolynomial (node ∘ e) = Zarankiewicz.nodePolynomial node := by |
| unfold Zarankiewicz.nodePolynomial |
| exact e.prod_comp (fun i => (Polynomial.X-Polynomial.C (node i) : Polynomial ℝ)) |
| simp only [Zarankiewicz.derivativeAtNode, h, Function.comp_apply] |
|
|
| theorem weightedPairing_equiv {ι κ : Type*} [Fintype ι] [Fintype κ] |
| (node : ι → ℝ) (e : κ ≃ ι) (K : ι → ι → ℝ) (A B : Poly2) : |
| weightedPairing (node ∘ e) (fun p q => K (e p) (e q)) A B = weightedPairing node K A B := by |
| unfold weightedPairing |
| simp_rw [derivativeAtNode_equiv, Function.comp_apply] |
| let F (p q : ι) : ℝ := K p q * (node p-node q) / |
| (Zarankiewicz.derivativeAtNode node p*Zarankiewicz.derivativeAtNode node q) * |
| evalTwo (node p) (node q) A * evalTwo (node p) (node q) B |
| change (∑p : κ, ∑q : κ, F (e p) (e q)) = ∑p : ι, ∑q : ι, F p q |
| calc |
| _ = ∑p : κ, ∑q : ι, F (e p) q := by |
| apply Finset.sum_congr rfl |
| intro p _ |
| exact e.sum_comp (F (e p)) |
| _ = _ := e.sum_comp (fun p => ∑q, F p q) |
|
|
| theorem orderPairing_orderIso {ι κ : Type*} [Fintype ι] [Fintype κ] |
| [LinearOrder ι] [LinearOrder κ] (node : ι → ℝ) (e : κ ≃o ι) (A B : Poly2) : |
| orderPairing (node ∘ e) inferInstance A B = orderPairing node inferInstance A B := by |
| have h : (fun p q => halfOrder inferInstance (e p) (e q)) = halfOrder (inferInstance : LinearOrder κ) := by |
| funext p q |
| simp only [halfOrder, e.lt_iff_lt] |
| rw [orderPairing, ← h] |
| exact weightedPairing_equiv node e.toEquiv _ A B |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Nonsingularity of the weighted half-order pairing for every node order. -/ |
| noncomputable section |
| open scoped BigOperators |
| namespace Paper170 |
|
|
| theorem order_pairing_nonsingular_fin (s : ℕ) (hs : 2 ≤ s) |
| (node : Fin (2*s) → ℝ) (hn : Function.Injective node) : |
| (∀ A ∈ rectangle (s-1) (s-2), |
| (∀ B ∈ rectangle (s-2) (s-1), orderPairing node inferInstance A B = 0) → A=0) ∧ |
| (∀ B ∈ rectangle (s-2) (s-1), |
| (∀ A ∈ rectangle (s-1) (s-2), orderPairing node inferInstance A B = 0) → B=0) := by |
| classical |
| obtain ⟨hfirst,hsecond⟩ := two_triangle_interpolation s hs node hn |
| constructor |
| · intro A hA hz |
| apply evenTriangle_vanish_first s hs node hn A hA |
| intro i j hij |
| let a : EvenTriangleCoord s := ⟨(i,j),hij⟩ |
| obtain ⟨B,hB⟩ := hsecond.surjective (Pi.single a (1:ℝ)) |
| have he (b : EvenTriangleCoord s) : evalTwo (node b.val.1) (node b.val.2) B.val = |
| (Pi.single a (1:ℝ) : EvenTriangleCoord s → ℝ) b := congrFun hB b |
| have hzero := hz B.val B.property |
| rw [orderPairing_triangle s hs node hn A B.val hA B.property] at hzero |
| simp_rw [he] at hzero |
| have hv : evenTriangleWeight s node a * evalTwo (node i) (node j) A = 0 := by |
| simpa [Pi.single_apply, mul_ite, a] using hzero |
| exact (mul_eq_zero.mp hv).resolve_left (evenTriangleWeight_ne_zero s node hn a) |
| · intro B hB hz |
| apply evenTriangle_vanish_second s hs node hn B hB |
| intro i j hij |
| let a : EvenTriangleCoord s := ⟨(i,j),hij⟩ |
| obtain ⟨A,hA⟩ := hfirst.surjective (Pi.single a (1:ℝ)) |
| have he (b : EvenTriangleCoord s) : evalTwo (node b.val.1) (node b.val.2) A.val = |
| (Pi.single a (1:ℝ) : EvenTriangleCoord s → ℝ) b := congrFun hA b |
| have hzero := hz A.val A.property |
| rw [orderPairing_triangle s hs node hn A.val B A.property hB] at hzero |
| simp_rw [he] at hzero |
| have hv : evenTriangleWeight s node a * evalTwo (node i) (node j) B = 0 := by |
| simpa [Pi.single_apply, mul_ite, ite_mul, a] using hzero |
| exact (mul_eq_zero.mp hv).resolve_left (evenTriangleWeight_ne_zero s node hn a) |
|
|
| theorem order_pairing_nonsingular {ι : Type*} [Fintype ι] |
| (s : ℕ) (hs : 2 ≤ s) (hc : Fintype.card ι = 2*s) |
| (node : ι → ℝ) (hn : Function.Injective node) (order : LinearOrder ι) : |
| (∀ A ∈ rectangle (s-1) (s-2), |
| (∀ B ∈ rectangle (s-2) (s-1), orderPairing node order A B = 0) → A=0) ∧ |
| (∀ B ∈ rectangle (s-2) (s-1), |
| (∀ A ∈ rectangle (s-1) (s-2), orderPairing node order A B = 0) → B=0) := by |
| let := order |
| let e : Fin (2*s) ≃o ι := Fintype.orderIsoFinOfCardEq ι hc |
| obtain ⟨hfirst,hsecond⟩ := order_pairing_nonsingular_fin s hs (node ∘ e) (hn.comp e.injective) |
| constructor |
| · intro A hA hz |
| apply hfirst A hA |
| intro B hB |
| rw [orderPairing_orderIso node e] |
| exact hz B hB |
| · intro B hB hz |
| apply hsecond B hB |
| intro A hA |
| rw [orderPairing_orderIso node e] |
| exact hz A hA |
|
|
| theorem order_detector {ι : Type*} [Fintype ι] |
| (s : ℕ) (hs : 2 ≤ s) (hc : Fintype.card ι = 2*s) |
| (node : ι → ℝ) (hn : Function.Injective node) (order : LinearOrder ι) |
| (A : Poly2) (hA : A ∈ rectangle (s-2) (s-2)) |
| (hzero : ∀ B ∈ rectangle (s-1) (s-1), orderPairing node order A B = 0) : A=0 := by |
| apply (order_pairing_nonsingular s hs hc node hn order).1 A |
| · rw [rectangle_membership] at hA ⊢ |
| exact ⟨hA.1.trans (by omega), hA.2⟩ |
| · intro B hB |
| apply hzero B |
| rw [rectangle_membership] at hB ⊢ |
| exact ⟨hB.1.trans (by omega), hB.2⟩ |
|
|
| end Paper170 |
|
|
| end |
| end |
|
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| end OAI |
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|
