| import OAI.Combinatorics.CompleteCrossing.Triangle |
| import Mathlib.Algebra.MvPolynomial.Funext |
| import Mathlib.Order.Interval.Finset.Fin |
|
|
| namespace OAI |
|
|
| /-! Triangular evaluation indices, division, and determinant factorization. -/ |
|
|
| section |
|
|
| noncomputable section |
| open scoped Classical |
| namespace Paper170 |
|
|
| def triangleMiddleEmbedding (s : ℕ) (hs : 2≤s) (p : Fin (2*(s-1))) : Fin (2*s) := |
| ⟨if p.val<s-1 then p.val else p.val+2, by split_ifs <;> omega⟩ |
|
|
| def triangleEndEmbedding (s : ℕ) (hs : 2≤s) (p : Fin (2*(s-1))) : Fin (2*s) := |
| ⟨p.val+1, by omega⟩ |
|
|
| theorem triangleMiddleEmbedding_strictMono (s : ℕ) (hs : 2≤s) : |
| StrictMono (triangleMiddleEmbedding s hs) := by |
| intro p q hpq |
| change (if p.val<s-1 then p.val else p.val+2) < |
| (if q.val<s-1 then q.val else q.val+2) |
| change p.val<q.val at hpq |
| split_ifs <;> omega |
|
|
| theorem triangleEndEmbedding_strictMono (s : ℕ) (hs : 2≤s) : |
| StrictMono (triangleEndEmbedding s hs) := by |
| intro p q hpq |
| change p.val+1<q.val+1 |
| exact Nat.add_lt_add_right hpq 1 |
|
|
| theorem triangleMiddleEmbedding_avoids (s : ℕ) (hs : 2≤s) (p : Fin (2*(s-1))) : |
| (triangleMiddleEmbedding s hs p).val ≠ s-1 ∧ |
| (triangleMiddleEmbedding s hs p).val ≠ s := by |
| dsimp [triangleMiddleEmbedding] |
| split_ifs <;> omega |
|
|
| theorem triangleEndEmbedding_avoids (s : ℕ) (hs : 2≤s) (p : Fin (2*(s-1))) : |
| (triangleEndEmbedding s hs p).val ≠ 0 ∧ |
| (triangleEndEmbedding s hs p).val ≠ 2*s-1 := by |
| dsimp [triangleEndEmbedding] |
| omega |
|
|
| theorem evenTriangle_delete_middle (s : ℕ) (hs : 2≤s) (p q : Fin (2*(s-1))) : |
| (triangleMiddleEmbedding s hs p,triangleMiddleEmbedding s hs q)∈evenTriangle s ↔ |
| (p,q)∈evenTriangle (s-1) := by |
| simp only [mem_evenTriangle,triangleMiddleEmbedding,Fin.lt_def] |
| split_ifs <;> omega |
|
|
| theorem evenTriangle_delete_ends (s : ℕ) (hs : 2≤s) (p q : Fin (2*(s-1))) : |
| (triangleEndEmbedding s hs p,triangleEndEmbedding s hs q)∈evenTriangle s ↔ |
| (p,q)∈evenTriangle (s-1) := by |
| simp only [mem_evenTriangle,triangleEndEmbedding,Fin.lt_def] |
| omega |
|
|
| theorem evenTriangle_middle_left_row (s : ℕ) (hs : 2≤s) (q : Fin (2*s)) : |
| (⟨s-1,by omega⟩,q)∈evenTriangle s ↔ q.val<s-1 := by |
| simp only [mem_evenTriangle,Fin.lt_def] |
| omega |
|
|
| theorem evenTriangle_middle_right_row (s : ℕ) (hs : 2≤s) (q : Fin (2*s)) : |
| (⟨s,by omega⟩,q)∈evenTriangle s ↔ s<q.val := by |
| simp only [mem_evenTriangle,Fin.lt_def] |
| omega |
|
|
| theorem evenTriangle_first_column (s : ℕ) (hs : 2≤s) (p : Fin (2*s)) : |
| (p,⟨0,by omega⟩)∈evenTriangle s ↔ 0<p.val ∧ p.val<s := by |
| simp only [mem_evenTriangle,Fin.lt_def] |
| omega |
|
|
| theorem evenTriangle_last_column (s : ℕ) (hs : 2≤s) (p : Fin (2*s)) : |
| (p,⟨2*s-1,by omega⟩)∈evenTriangle s ↔ s≤p.val ∧ p.val<2*s-1 := by |
| simp only [mem_evenTriangle,Fin.lt_def] |
| omega |
|
|
| theorem triangleMiddleEmbedding_range (s : ℕ) (hs : 2≤s) (i : Fin (2*s)) : |
| (∃p, triangleMiddleEmbedding s hs p=i) ↔ i.val≠s-1 ∧ i.val≠s := by |
| constructor |
| · rintro ⟨p,rfl⟩ |
| exact triangleMiddleEmbedding_avoids s hs p |
| · intro h |
| by_cases hi : i.val<s-1 |
| · refine ⟨⟨i.val,by omega⟩,?_⟩ |
| apply Fin.ext |
| simp [triangleMiddleEmbedding,hi] |
| · refine ⟨⟨i.val-2,by omega⟩,?_⟩ |
| apply Fin.ext |
| simp only [triangleMiddleEmbedding] |
| split_ifs <;> omega |
|
|
| theorem triangleEndEmbedding_range (s : ℕ) (hs : 2≤s) (i : Fin (2*s)) : |
| (∃p, triangleEndEmbedding s hs p=i) ↔ i.val≠0 ∧ i.val≠2*s-1 := by |
| constructor |
| · rintro ⟨p,rfl⟩ |
| exact triangleEndEmbedding_avoids s hs p |
| · intro h |
| refine ⟨⟨i.val-1,by omega⟩,?_⟩ |
| apply Fin.ext |
| dsimp [triangleEndEmbedding] |
| omega |
|
|
| theorem evenTriangle_middle_left_card (s : ℕ) (hs : 2≤s) : |
| (Finset.univ.filter (fun q : Fin (2*s) => (⟨s-1,by omega⟩,q)∈evenTriangle s)).card=s-1 := by |
| have he : Finset.univ.filter (fun q : Fin (2*s) => (⟨s-1,by omega⟩,q)∈evenTriangle s)= |
| Finset.Iio (⟨s-1,by omega⟩ : Fin (2*s)) := by |
| ext q |
| simp only [Finset.mem_filter,Finset.mem_univ,true_and,Finset.mem_Iio, |
| evenTriangle_middle_left_row s hs,Fin.lt_def] |
| rw [he,Fin.card_Iio] |
|
|
| theorem evenTriangle_middle_right_card (s : ℕ) (hs : 2≤s) : |
| (Finset.univ.filter (fun q : Fin (2*s) => (⟨s,by omega⟩,q)∈evenTriangle s)).card=s-1 := by |
| have he : Finset.univ.filter (fun q : Fin (2*s) => (⟨s,by omega⟩,q)∈evenTriangle s)= |
| Finset.Ioi (⟨s,by omega⟩ : Fin (2*s)) := by |
| ext q |
| simp only [Finset.mem_filter,Finset.mem_univ,true_and,Finset.mem_Ioi, |
| evenTriangle_middle_right_row s hs,Fin.lt_def] |
| rw [he,Fin.card_Ioi] |
| dsimp |
| omega |
|
|
| theorem evenTriangle_first_column_card (s : ℕ) (hs : 2≤s) : |
| (Finset.univ.filter (fun p : Fin (2*s) => (p,⟨0,by omega⟩)∈evenTriangle s)).card=s-1 := by |
| have he : Finset.univ.filter (fun p : Fin (2*s) => (p,⟨0,by omega⟩)∈evenTriangle s)= |
| Finset.Ioo (⟨0,by omega⟩ : Fin (2*s)) ⟨s,by omega⟩ := by |
| ext p |
| simp only [Finset.mem_filter,Finset.mem_univ,true_and,Finset.mem_Ioo, |
| evenTriangle_first_column s hs,Fin.lt_def] |
| rw [he,Fin.card_Ioo] |
| simp |
|
|
| theorem evenTriangle_last_column_card (s : ℕ) (hs : 2≤s) : |
| (Finset.univ.filter (fun p : Fin (2*s) => (p,⟨2*s-1,by omega⟩)∈evenTriangle s)).card=s-1 := by |
| have he : Finset.univ.filter (fun p : Fin (2*s) => (p,⟨2*s-1,by omega⟩)∈evenTriangle s)= |
| Finset.Ico (⟨s,by omega⟩ : Fin (2*s)) ⟨2*s-1,by omega⟩ := by |
| ext p |
| simp only [Finset.mem_filter,Finset.mem_univ,true_and,Finset.mem_Ico, |
| evenTriangle_last_column s hs,Fin.le_def,Fin.lt_def] |
| rw [he,Fin.card_Ico] |
| dsimp |
| omega |
|
|
| theorem evenTriangle_two : evenTriangle 2 = {((1 : Fin 4),0),(2,3)} := by |
| ext ⟨p,q⟩ |
| fin_cases p <;> fin_cases q <;> decide |
|
|
| theorem evenTriangle_two_distinct_coordinates (node : Fin 4 → ℝ) |
| (hn : Function.Injective node) : node 1≠node 2 ∧ node 0≠node 3 := by |
| constructor <;> exact hn.ne (by decide) |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! The literal row and column elimination used in the even-node proof. -/ |
| noncomputable section |
| open PolynomialModule Zarankiewicz |
| namespace Paper170 |
|
|
| theorem mem_rectangle_of_toBiPolynomial_bounded (a b : ℕ) (P : Poly2) |
| (hP : BiBounded (a+1) (b+1) (toBiPolynomial P)) : P∈rectangle a b := by |
| rw [rectangle_membership] |
| constructor |
| · rw [MvPolynomial.degreeOf_le_iff] |
| intro m hm |
| by_contra h |
| have hz := hP.1 (m 0) (by omega) |
| have hc : biCoeff (toBiPolynomial P) (m 0) (m 1)=0 := by |
| simp [biCoeff,hz] |
| rw [toBiPolynomial_coeff] at hc |
| have he : exponentPair.symm (m 0,m 1)=m := exponentPair.symm_apply_apply m |
| rw [he] at hc |
| exact (MvPolynomial.mem_support_iff.mp hm) hc |
| · rw [MvPolynomial.degreeOf_le_iff] |
| intro m hm |
| by_contra h |
| have hc := hP.2 (m 0) (m 1) (by omega) |
| change biCoeff (toBiPolynomial P) (m 0) (m 1)=0 at hc |
| rw [toBiPolynomial_coeff] at hc |
| have he : exponentPair.symm (m 0,m 1)=m := exponentPair.symm_apply_apply m |
| rw [he] at hc |
| exact (MvPolynomial.mem_support_iff.mp hm) hc |
|
|
| theorem bi_factor_two_rows (r s : ℕ) (P : BiPoly) |
| (hP : BiBounded (r+2) s P) (a b : ℝ) (hab : a≠b) |
| (ha : eval a P=0) (hb : eval b P=0) : |
| ∃R : BiPoly, BiBounded r s R ∧ mulRoot a (mulRoot b R)=P := by |
| obtain ⟨Q,hQ,hQP⟩ := bi_factor_row (r+1) s P hP a ha |
| have hbQ : eval b Q=0 := by |
| rw [←hQP,eval_mulRoot] at hb |
| exact (smul_eq_zero.mp hb).resolve_left (sub_ne_zero.mpr hab.symm) |
| obtain ⟨R,hR,hRQ⟩ := bi_factor_row r s Q hQ b hbQ |
| exact ⟨R,hR,by rw [hRQ,hQP]⟩ |
|
|
| theorem evenTriangle_middle_rows_zero (s : ℕ) (hs : 2≤s) |
| (node : Fin (2*s)→ℝ) (hn : Function.Injective node) |
| (P : BiPoly) (hP : BiBounded s (s-1) P) |
| (hv : ∀p q, (p,q)∈evenTriangle s → biEval (node p) (node q) P=0) : |
| eval (node ⟨s-1,by omega⟩) P=0 ∧ eval (node ⟨s,by omega⟩) P=0 := by |
| constructor |
| · let f : Fin (s-1)→Fin (2*s) := fun i => ⟨i.val,by omega⟩ |
| apply eq_zero_of_node_roots (node ∘ f) (hn.comp (by |
| intro i j h |
| exact Fin.ext (show i.val=j.val from congrArg (fun x : Fin (2*s) => x.val) h))) Finset.univ |
| · simpa only [Finset.card_univ,Fintype.card_fin] using |
| bounded_eval_outer hP (node ⟨s-1,by omega⟩) |
| · intro i _ |
| apply hv |
| rw [evenTriangle_middle_left_row s hs] |
| exact i.isLt |
| · let f : Fin (s-1)→Fin (2*s) := fun i => ⟨s+1+i.val,by omega⟩ |
| apply eq_zero_of_node_roots (node ∘ f) (hn.comp (by |
| intro i j h |
| apply Fin.ext |
| have := congrArg Fin.val h |
| dsimp [f] at this |
| omega)) Finset.univ |
| · simpa only [Finset.card_univ,Fintype.card_fin] using |
| bounded_eval_outer hP (node ⟨s,by omega⟩) |
| · intro i _ |
| apply hv |
| rw [evenTriangle_middle_right_row s hs] |
| dsimp [f] |
| omega |
|
|
| theorem evenTriangle_end_columns_zero (s : ℕ) (hs : 2≤s) |
| (node : Fin (2*s)→ℝ) (hn : Function.Injective node) |
| (P : BiPoly) (hP : BiBounded (s-1) s P) |
| (hv : ∀p q, (p,q)∈evenTriangle s → biEval (node p) (node q) P=0) : |
| eval (node ⟨0,by omega⟩) (swapBi P)=0 ∧ |
| eval (node ⟨2*s-1,by omega⟩) (swapBi P)=0 := by |
| constructor |
| · let f : Fin (s-1)→Fin (2*s) := fun i => ⟨i.val+1,by omega⟩ |
| apply eq_zero_of_node_roots (node ∘ f) (hn.comp (by |
| intro i j h |
| apply Fin.ext |
| have := congrArg Fin.val h |
| dsimp [f] at this |
| omega)) Finset.univ |
| · simpa only [Finset.card_univ,Fintype.card_fin] using |
| bounded_eval_outer (biBounded_swap hP) (node ⟨0,by omega⟩) |
| · intro i _ |
| change biEval _ _ (swapBi P)=0 |
| rw [biEval_swap] |
| apply hv |
| rw [evenTriangle_first_column s hs] |
| dsimp [f] |
| omega |
| · let f : Fin (s-1)→Fin (2*s) := fun i => ⟨s+i.val,by omega⟩ |
| apply eq_zero_of_node_roots (node ∘ f) (hn.comp (by |
| intro i j h |
| apply Fin.ext |
| have := congrArg Fin.val h |
| dsimp [f] at this |
| omega)) Finset.univ |
| · simpa only [Finset.card_univ,Fintype.card_fin] using |
| bounded_eval_outer (biBounded_swap hP) (node ⟨2*s-1,by omega⟩) |
| · intro i _ |
| change biEval _ _ (swapBi P)=0 |
| rw [biEval_swap] |
| apply hv |
| rw [evenTriangle_last_column s hs] |
| dsimp [f] |
| omega |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open PolynomialModule Zarankiewicz MvPolynomial |
| namespace Paper170 |
|
|
| theorem evalTwo_ext {P Q : Poly2} (h : ∀u v, evalTwo u v P=evalTwo u v Q) : P=Q := by |
| apply MvPolynomial.funext |
| intro t |
| have ht : t = ![t 0,t 1] := by ext i; fin_cases i <;> rfl |
| rw [ht] |
| exact h (t 0) (t 1) |
|
|
| theorem evenTriangle_row_factorization (s : ℕ) (hs : 3≤s) |
| (node : Fin (2*s)→ℝ) (hn : Function.Injective node) |
| (P : Poly2) (hP : P∈rectangle (s-1) (s-2)) |
| (hv : ∀p q, (p,q)∈evenTriangle s → evalTwo (node p) (node q) P=0) : |
| ∃R : Poly2, R∈rectangle (s-3) (s-2) ∧ |
| P=(X 0-C (node ⟨s-1,by omega⟩))*(X 0-C (node ⟨s,by omega⟩))*R ∧ |
| ∀p q, (p,q)∈evenTriangle (s-1) → |
| evalTwo (node (triangleMiddleEmbedding s (by omega) p)) |
| (node (triangleMiddleEmbedding s (by omega) q)) R=0 := by |
| have hB : BiBounded s (s-1) (toBiPolynomial P) := by |
| convert toBiPolynomial_bounded (s-1) (s-2) P hP using 1 <;> omega |
| have hV : ∀p q, (p,q)∈evenTriangle s → |
| biEval (node p) (node q) (toBiPolynomial P)=0 := by |
| simpa only [toBiPolynomial_eval] using hv |
| obtain ⟨hleft,hright⟩ := evenTriangle_middle_rows_zero s (by omega) node hn _ hB hV |
| obtain ⟨Q,hQ,hQP⟩ := bi_factor_two_rows (s-2) (s-1) (toBiPolynomial P) |
| (by convert hB using 1; omega) (node ⟨s-1,by omega⟩) (node ⟨s,by omega⟩) |
| (hn.ne (by intro h; have := congrArg Fin.val h; dsimp at this; omega)) hleft hright |
| let R := toBiPolynomial.symm Q |
| have hR : R∈rectangle (s-3) (s-2) := by |
| apply mem_rectangle_of_toBiPolynomial_bounded |
| dsimp [R] |
| rw [LinearEquiv.apply_symm_apply] |
| convert hQ using 1 <;> omega |
| have hf : P=(X 0-C (node ⟨s-1,by omega⟩))*(X 0-C (node ⟨s,by omega⟩))*R := by |
| apply evalTwo_ext |
| intro u v |
| rw [←toBiPolynomial_eval,hQP.symm,biEval_mulRoot,biEval_mulRoot] |
| have hQR : biEval u v Q=evalTwo u v R := by |
| rw [←toBiPolynomial_eval] |
| simp only [R,LinearEquiv.apply_symm_apply] |
| rw [hQR] |
| simp [evalTwo,mul_assoc] |
| refine ⟨R,hR,hf,?_⟩ |
| intro p q hpq |
| have hzero := hv (triangleMiddleEmbedding s (by omega) p) |
| (triangleMiddleEmbedding s (by omega) q) |
| ((evenTriangle_delete_middle s (by omega) p q).mpr hpq) |
| rw [hf] at hzero |
| have ha := triangleMiddleEmbedding_avoids s (by omega) p |
| have h₁ : node (triangleMiddleEmbedding s (by omega) p)-node ⟨s-1,by omega⟩≠0 := |
| sub_ne_zero.mpr (hn.ne (by intro h; exact ha.1 (congrArg Fin.val h))) |
| have h₂ : node (triangleMiddleEmbedding s (by omega) p)-node ⟨s,by omega⟩≠0 := |
| sub_ne_zero.mpr (hn.ne (by intro h; exact ha.2 (congrArg Fin.val h))) |
| change (MvPolynomial.aeval _ _) = 0 at hzero |
| simp only [map_mul,map_sub,MvPolynomial.aeval_X,MvPolynomial.aeval_C, |
| Matrix.cons_val_zero,Algebra.algebraMap_self,RingHom.id_apply] at hzero |
| exact (mul_eq_zero.mp hzero).resolve_left (mul_ne_zero h₁ h₂) |
|
|
| theorem evenTriangle_column_factorization (s : ℕ) (hs : 3≤s) |
| (node : Fin (2*s)→ℝ) (hn : Function.Injective node) |
| (P : Poly2) (hP : P∈rectangle (s-2) (s-1)) |
| (hv : ∀p q, (p,q)∈evenTriangle s → evalTwo (node p) (node q) P=0) : |
| ∃R : Poly2, R∈rectangle (s-2) (s-3) ∧ |
| P=(X 1-C (node ⟨0,by omega⟩))*(X 1-C (node ⟨2*s-1,by omega⟩))*R ∧ |
| ∀p q, (p,q)∈evenTriangle (s-1) → |
| evalTwo (node (triangleEndEmbedding s (by omega) p)) |
| (node (triangleEndEmbedding s (by omega) q)) R=0 := by |
| have hB : BiBounded (s-1) s (toBiPolynomial P) := by |
| convert toBiPolynomial_bounded (s-2) (s-1) P hP using 1 <;> omega |
| have hV : ∀p q, (p,q)∈evenTriangle s → |
| biEval (node p) (node q) (toBiPolynomial P)=0 := by |
| simpa only [toBiPolynomial_eval] using hv |
| obtain ⟨hleft,hright⟩ := evenTriangle_end_columns_zero s (by omega) node hn _ hB hV |
| obtain ⟨Q,hQ,hQP⟩ := bi_factor_two_rows (s-2) (s-1) (swapBi (toBiPolynomial P)) |
| (by convert biBounded_swap hB using 1; omega) |
| (node ⟨0,by omega⟩) (node ⟨2*s-1,by omega⟩) |
| (hn.ne (by intro h; have := congrArg Fin.val h; dsimp at this; omega)) hleft hright |
| let R := toBiPolynomial.symm (swapBi Q) |
| have hR : R∈rectangle (s-2) (s-3) := by |
| apply mem_rectangle_of_toBiPolynomial_bounded |
| dsimp [R] |
| rw [LinearEquiv.apply_symm_apply] |
| convert biBounded_swap hQ using 1 <;> omega |
| have hf : P=(X 1-C (node ⟨0,by omega⟩))*(X 1-C (node ⟨2*s-1,by omega⟩))*R := by |
| apply evalTwo_ext |
| intro u v |
| rw [←toBiPolynomial_eval,←biEval_swap (toBiPolynomial P),hQP.symm, |
| biEval_mulRoot,biEval_mulRoot] |
| have hQR : biEval v u Q=evalTwo u v R := by |
| rw [←toBiPolynomial_eval] |
| simp only [R,LinearEquiv.apply_symm_apply,biEval_swap] |
| rw [hQR] |
| simp [evalTwo,mul_assoc] |
| refine ⟨R,hR,hf,?_⟩ |
| intro p q hpq |
| have hzero := hv (triangleEndEmbedding s (by omega) p) |
| (triangleEndEmbedding s (by omega) q) |
| ((evenTriangle_delete_ends s (by omega) p q).mpr hpq) |
| rw [hf] at hzero |
| have ha := triangleEndEmbedding_avoids s (by omega) q |
| have h₁ : node (triangleEndEmbedding s (by omega) q)-node ⟨0,by omega⟩≠0 := |
| sub_ne_zero.mpr (hn.ne (by intro h; exact ha.1 (congrArg Fin.val h))) |
| have h₂ : node (triangleEndEmbedding s (by omega) q)-node ⟨2*s-1,by omega⟩≠0 := |
| sub_ne_zero.mpr (hn.ne (by intro h; exact ha.2 (congrArg Fin.val h))) |
| change (MvPolynomial.aeval _ _) = 0 at hzero |
| simp only [map_mul,map_sub,MvPolynomial.aeval_X,MvPolynomial.aeval_C, |
| Matrix.cons_val_one,Matrix.cons_val_zero,Algebra.algebraMap_self,RingHom.id_apply] at hzero |
| exact (mul_eq_zero.mp hzero).resolve_left (mul_ne_zero h₁ h₂) |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| end OAI |
|
|
