| import OAI.Combinatorics.CompleteCrossing.Interpolation |
| import Mathlib.Data.Finset.Prod |
| import Mathlib.Data.Set.Finite.Basic |
| import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas |
|
|
| namespace OAI |
|
|
| /-! Even triangular interpolation and unisolvence. -/ |
|
|
| section |
|
|
| /-! The even two-triangle set, as actual ordered interpolation coordinates. -/ |
| noncomputable section |
| namespace Paper170 |
|
|
| theorem mem_evenTriangle (s : ℕ) (p q : Fin (2*s)) : |
| (p,q) ∈ evenTriangle s ↔ |
| (p.val < s ∧ q < p) ∨ (s ≤ p.val ∧ p < q) := by |
| simp [evenTriangle] |
|
|
| def evenTriangleOffDiagEquiv (s : ℕ) : |
| EvenTriangleCoord s ≃ ↥((Finset.univ : Finset (Fin s)).offDiag) where |
| toFun pq := by |
| obtain ⟨⟨p,q⟩, hpq⟩ := pq |
| have hpq := (mem_evenTriangle s p q).mp hpq |
| by_cases h : p.val < s |
| · exact ⟨(⟨p.val, h⟩, ⟨q.val, by rcases hpq with hq | hq <;> omega⟩), by |
| simp only [Finset.mem_offDiag, Finset.mem_univ, true_and, ne_eq, Fin.mk.injEq] |
| rcases hpq with hq | hq <;> omega⟩ |
| · exact ⟨(⟨p.val-s, by omega⟩, ⟨q.val-s, by omega⟩), by |
| simp only [Finset.mem_offDiag, Finset.mem_univ, true_and, ne_eq, Fin.mk.injEq] |
| rcases hpq with hq | hq <;> omega⟩ |
| invFun pq := by |
| obtain ⟨⟨p,q⟩, hpq⟩ := pq |
| have hpq : p ≠ q := (Finset.mem_offDiag.mp hpq).2.2 |
| by_cases h : q < p |
| · exact ⟨(⟨p.val, by omega⟩, ⟨q.val, by omega⟩), |
| (mem_evenTriangle s _ _).mpr (Or.inl ⟨p.isLt,h⟩)⟩ |
| · exact ⟨(⟨p.val+s, by omega⟩, ⟨q.val+s, by omega⟩), |
| (mem_evenTriangle s _ _).mpr (Or.inr ⟨by change s ≤ p.val+s; omega,by |
| have hpq := Fin.val_ne_of_ne hpq |
| change p.val+s < q.val+s |
| omega⟩)⟩ |
| left_inv pq := by |
| obtain ⟨⟨p,q⟩, hpq⟩ := pq |
| have hpq := (mem_evenTriangle s p q).mp hpq |
| apply Subtype.ext |
| dsimp |
| split_ifs with h h' h' |
| · rfl |
| · change ¬q.val < p.val at h' |
| simp only [Fin.lt_def] at hpq |
| omega |
| · change q.val-s < p.val-s at h' |
| simp only [Fin.lt_def] at hpq |
| omega |
| · apply Prod.ext <;> apply Fin.ext <;> dsimp <;> |
| simp only [Fin.lt_def] at hpq <;> omega |
| right_inv pq := by |
| obtain ⟨⟨p,q⟩, hpq⟩ := pq |
| apply Subtype.ext |
| dsimp |
| split_ifs with h h' h' |
| · rfl |
| · change ¬p.val < s at h' |
| omega |
| · change p.val+s < s at h' |
| omega |
| · apply Prod.ext <;> apply Fin.ext <;> dsimp <;> omega |
|
|
| theorem even_triangle_card (s : ℕ) : (evenTriangle s).card = s*(s-1) := by |
| have hc := Fintype.card_congr (evenTriangleOffDiagEquiv s) |
| simp only [EvenTriangleCoord, Fintype.card_coe] at hc |
| rw [Finset.offDiag_card, Finset.card_univ, Fintype.card_fin] at hc |
| simpa only [Nat.mul_sub_left_distrib, mul_one] using hc |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Even-node interpolation by adjoining one fresh node to the checked odd-node theorem. -/ |
| noncomputable section |
| open PolynomialModule |
| open Zarankiewicz |
| namespace Paper170 |
|
|
| theorem biBounded_mulRoot {r s : ℕ} {P : BiPoly} (hP : BiBounded r s P) (t : ℝ) : |
| BiBounded (r+1) s (mulRoot t P) := by |
| refine ⟨bounded_mulRoot t hP.1, ?_⟩ |
| intro a |
| cases a with |
| | zero => |
| rw [mulRoot_coeff_zero] |
| exact (boundedSpace s).smul_mem _ (hP.2 0) |
| | succ a => |
| rw [mulRoot_coeff_succ] |
| exact bounded_sub (hP.2 a) ((boundedSpace s).smul_mem _ (hP.2 (a+1))) |
|
|
| theorem insertNode_injective {n : ℕ} (r : Fin (n+1)) (node : Fin n → ℝ) |
| (hn : Function.Injective node) (t : ℝ) (ht : ∀i, node i ≠ t) : |
| Function.Injective (Fin.insertNth r t node) := by |
| intro a b |
| refine Fin.succAboveCases r ?_ (fun a => ?_) a |
| · refine Fin.succAboveCases r ?_ (fun b => ?_) b |
| · intro |
| rfl |
| · simp only [Fin.insertNth_apply_same, Fin.insertNth_apply_succAbove] |
| intro h |
| exact (ht b h.symm).elim |
| · refine Fin.succAboveCases r ?_ (fun b => ?_) b |
| · simp only [Fin.insertNth_apply_same, Fin.insertNth_apply_succAbove] |
| intro h |
| exact (ht a h).elim |
| · simp only [Fin.insertNth_apply_succAbove] |
| intro h |
| exact congrArg r.succAbove (hn h) |
|
|
| theorem oddTriangle_unisolvent_fin (s : ℕ) (node : Fin (2*s+1) → ℝ) |
| (hn : Function.Injective node) (P : BiPoly) (hP : BiBounded s s P) |
| (hv : ∀a b : Fin (2*s+1), |
| (b < a ∧ a.val ≤ s) ∨ (s < a.val ∧ a < b) → biEval (node a) (node b) P = 0) : |
| P = 0 := by |
| let x : ℕ → ℝ := fun k => if h : 1 ≤ k ∧ k ≤ 2*s+1 then node ⟨k-1,by omega⟩ else 0 |
| have hx : Set.InjOn x (Set.Icc 1 (2*s+1)) := by |
| intro a ha b hb he |
| simp only [Set.mem_Icc] at ha hb |
| simp only [x, dite_eq_left ha, dite_eq_left hb] at he |
| have := congrArg Fin.val (hn he) |
| dsimp at this |
| omega |
| apply twoTriangle_unisolvent s x hx P hP |
| intro a b hab |
| rw [mem_twoTriangle] at hab |
| have ha : 1 ≤ a ∧ a ≤ 2*s+1 := by omega |
| have hb : 1 ≤ b ∧ b ≤ 2*s+1 := by omega |
| simp only [x, dite_eq_left ha, dite_eq_left hb] |
| apply hv |
| change (b-1 < a-1 ∧ a-1 ≤ s) ∨ (s < a-1 ∧ a-1 < b-1) |
| omega |
|
|
| theorem evenTriangle_unisolvent_first (s : ℕ) (hs : 2 ≤ s) |
| (node : Fin (2*s) → ℝ) (hn : Function.Injective node) |
| (P : BiPoly) (hP : BiBounded s (s-1) P) |
| (hv : ∀a b, (a,b) ∈ evenTriangle s → biEval (node a) (node b) P = 0) : P = 0 := by |
| obtain ⟨t,ht⟩ := (Set.finite_range node).exists_notMem |
| have hnt : ∀i, node i ≠ t := fun i h => ht ⟨i,h⟩ |
| let x : Fin (2*s+1) → ℝ := Fin.insertNth (0 : Fin (2*s+1)) t node |
| let Q := swapBi (mulRoot t (swapBi P)) |
| have hQ : BiBounded s s Q := by |
| have hb := biBounded_swap (biBounded_mulRoot (biBounded_swap hP) t) |
| simpa only [Nat.sub_add_cancel (by omega : 1 ≤ s)] using hb |
| have hev (u v : ℝ) : biEval u v Q = (v-t)*biEval u v P := by |
| simp only [Q, biEval_swap, biEval_mulRoot] |
| have hzero : Q = 0 := by |
| apply oddTriangle_unisolvent_fin s x (insertNode_injective _ node hn t hnt) Q hQ |
| intro a b |
| refine Fin.succAboveCases (0 : Fin (2*s+1)) ?_ (fun a => ?_) a |
| · intro h |
| have : False := by simp only [Fin.lt_def, Fin.val_zero] at h; omega |
| exact this.elim |
| · refine Fin.succAboveCases (0 : Fin (2*s+1)) ?_ (fun b => ?_) b |
| · intro _ |
| simp only [hev, x, Fin.insertNth_apply_same, sub_self, zero_mul] |
| · intro hab |
| simp only [x, Fin.insertNth_apply_succAbove, hev] |
| rw [hv a b] |
| · exact mul_zero _ |
| · rw [mem_evenTriangle] |
| simp only [Fin.succAbove_zero, Fin.val_succ, Fin.lt_def] at hab |
| simp only [Fin.lt_def] |
| omega |
| have hroot : mulRoot t (swapBi P) = 0 := by |
| apply swapBi.injective |
| simpa only [map_zero] using hzero |
| have hpzero := eq_zero_of_mulRoot_eq_zero t (s-1) (swapBi P) (biBounded_swap hP).1 hroot |
| apply swapBi.injective |
| simpa only [map_zero] using hpzero |
|
|
| theorem evenTriangle_unisolvent_second (s : ℕ) (hs : 2 ≤ s) |
| (node : Fin (2*s) → ℝ) (hn : Function.Injective node) |
| (P : BiPoly) (hP : BiBounded (s-1) s P) |
| (hv : ∀a b, (a,b) ∈ evenTriangle s → biEval (node a) (node b) P = 0) : P = 0 := by |
| obtain ⟨t,ht⟩ := (Set.finite_range node).exists_notMem |
| have hnt : ∀i, node i ≠ t := fun i h => ht ⟨i,h⟩ |
| let r : Fin (2*s+1) := ⟨s,by omega⟩ |
| let x : Fin (2*s+1) → ℝ := Fin.insertNth r t node |
| have hQ : BiBounded s s (mulRoot t P) := by |
| simpa only [Nat.sub_add_cancel (by omega : 1 ≤ s)] using biBounded_mulRoot hP t |
| apply eq_zero_of_mulRoot_eq_zero t (s-1) P hP.1 |
| apply oddTriangle_unisolvent_fin s x (insertNode_injective r node hn t hnt) _ hQ |
| intro a b |
| refine Fin.succAboveCases r ?_ (fun a => ?_) a |
| · intro _ |
| simp only [biEval_mulRoot, x, Fin.insertNth_apply_same, sub_self, zero_mul] |
| · refine Fin.succAboveCases r ?_ (fun b => ?_) b |
| · intro hab |
| have : False := by |
| simp only [Fin.lt_def, r] at hab |
| omega |
| exact this.elim |
| · intro hab |
| simp only [x, Fin.insertNth_apply_succAbove, biEval_mulRoot] |
| rw [hv a b] |
| · exact mul_zero _ |
| · rw [mem_evenTriangle] |
| simp only [Fin.lt_def, r, Fin.succAbove, Fin.val_castSucc] at hab |
| simp only [Fin.lt_def] |
| split_ifs at hab <;> simp only [Fin.val_castSucc, Fin.val_succ] at hab <;> omega |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Bijective interpolation on both even-node degree rectangles. -/ |
| noncomputable section |
| namespace Paper170 |
|
|
| theorem evenTriangle_vanish_first (s : ℕ) (hs : 2 ≤ s) |
| (node : Fin (2*s) → ℝ) (hn : Function.Injective node) |
| (P : Poly2) (hP : P ∈ rectangle (s-1) (s-2)) |
| (hv : ∀a b, (a,b) ∈ evenTriangle s → evalTwo (node a) (node b) P = 0) : P = 0 := by |
| apply toBiPolynomial.injective |
| rw [map_zero] |
| apply evenTriangle_unisolvent_first s hs node hn (toBiPolynomial P) |
| · convert toBiPolynomial_bounded (s-1) (s-2) P hP using 1 <;> omega |
| · intro a b hab |
| rw [toBiPolynomial_eval] |
| exact hv a b hab |
|
|
| theorem evenTriangle_vanish_second (s : ℕ) (hs : 2 ≤ s) |
| (node : Fin (2*s) → ℝ) (hn : Function.Injective node) |
| (P : Poly2) (hP : P ∈ rectangle (s-2) (s-1)) |
| (hv : ∀a b, (a,b) ∈ evenTriangle s → evalTwo (node a) (node b) P = 0) : P = 0 := by |
| apply toBiPolynomial.injective |
| rw [map_zero] |
| apply evenTriangle_unisolvent_second s hs node hn (toBiPolynomial P) |
| · convert toBiPolynomial_bounded (s-2) (s-1) P hP using 1 <;> omega |
| · intro a b hab |
| rw [toBiPolynomial_eval] |
| exact hv a b hab |
|
|
| theorem triangleEvaluation_bijective_of_unisolvent (s a b : ℕ) (node : Fin (2*s) → ℝ) |
| (hdim : (a+1)*(b+1) = s*(s-1)) |
| (hv : ∀P ∈ rectangle a b, |
| (∀i j, (i,j) ∈ evenTriangle s → evalTwo (node i) (node j) P = 0) → P=0) : |
| Function.Bijective (triangleEvaluation s a b node) := by |
| let := rectangle_finite a b |
| have hi : Function.Injective (triangleEvaluation s a b node) := by |
| rw [← LinearMap.ker_eq_bot, LinearMap.ker_eq_bot'] |
| intro P hP |
| apply Subtype.ext |
| apply hv P.val P.property |
| intro i j hij |
| exact congrFun hP ⟨(i,j),hij⟩ |
| refine ⟨hi, (LinearMap.injective_iff_surjective_of_finrank_eq_finrank ?_).mp hi⟩ |
| rw [rectangle_dimension, Module.finrank_fintype_fun_eq_card] |
| have hc : Fintype.card (EvenTriangleCoord s) = s*(s-1) := by |
| rw [Fintype.card_coe, even_triangle_card] |
| rw [hc,hdim] |
|
|
| theorem two_triangle_interpolation (s : ℕ) (hs : 2 ≤ s) |
| (node : Fin (2*s) → ℝ) (hn : Function.Injective node) : |
| Function.Bijective (triangleEvaluation s (s-1) (s-2) node) ∧ |
| Function.Bijective (triangleEvaluation s (s-2) (s-1) node) := by |
| constructor |
| · apply triangleEvaluation_bijective_of_unisolvent |
| · congr 1 <;> omega |
| · exact evenTriangle_vanish_first s hs node hn |
| · apply triangleEvaluation_bijective_of_unisolvent |
| · rw [Nat.mul_comm] |
| congr 1 <;> omega |
| · exact evenTriangle_vanish_second s hs node hn |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| end OAI |
|
|
