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97c3405 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 | import OAI.Combinatorics.CompleteCrossing.Polynomials
import Mathlib.LinearAlgebra.BilinearForm.Orthogonal
namespace OAI
/-! The support form, injectivity criterion, and support-space dimension. -/
section
noncomputable section
open scoped BigOperators Classical
open MvPolynomial
namespace Paper170
theorem collision_nonzero_on_support {n : ℕ} (node : Fin n → ℝ)
(hn : Function.Injective node) (J : CrossingTensor n) (a : SupportCoord J) :
evalFour (node a.val.1.1) (node a.val.1.2) (node a.val.2.1) (node a.val.2.2)
collisionFactor ≠ 0 := by
rcases (Finset.mem_filter.mp a.property).2 with ⟨_, _, h02, h03, h12, h13, _⟩
simpa [evalFour, collisionFactor, Matrix.cons_val_two, Matrix.cons_val_three] using
mul_ne_zero (mul_ne_zero (mul_ne_zero (sub_ne_zero.mpr (hn.ne h02))
(sub_ne_zero.mpr (hn.ne h03))) (sub_ne_zero.mpr (hn.ne h12)))
(sub_ne_zero.mpr (hn.ne h13))
lemma support_diagonal_nonzero {n : ℕ} (node : Fin n → ℝ)
(hn : Function.Injective node) (J : CrossingTensor n) (a : SupportCoord J) :
supportDiagonal node J a ≠ 0 := by
rcases (Finset.mem_filter.mp a.property).2 with ⟨h01, h23, _, _, _, _, hJ⟩
have hc := collision_nonzero_on_support node hn J a
have hf : evalFour (node a.val.1.1) (node a.val.1.2) (node a.val.2.1)
(node a.val.2.2) fullDifference ≠ 0 := by
simpa [fullDifference, evalFour, Matrix.cons_val_two, Matrix.cons_val_three] using
mul_ne_zero (mul_ne_zero (sub_ne_zero.mpr (hn.ne (ne_of_lt h01)))
(sub_ne_zero.mpr (hn.ne (ne_of_lt h23)))) hc
exact mul_ne_zero (mul_ne_zero (mul_ne_zero (mul_ne_zero (mul_ne_zero hJ
(weight_nonzero node hn _)) (weight_nonzero node hn _))
(weight_nonzero node hn _)) (weight_nonzero node hn _)) hf
theorem support_bilinear_nondegenerate {n : ℕ} (node : Fin n → ℝ)
(hn : Function.Injective node) (J : CrossingTensor n) :
(supportBilinear node J).Nondegenerate := by
apply LinearMap.BilinForm.nondegenerate_toBilin'_iff_det_ne_zero.mpr
rw [Matrix.det_diagonal]
exact Finset.prod_ne_zero_iff.mpr fun a _ => support_diagonal_nonzero node hn J a
theorem isotropic_dimension {ι : Type*} [Fintype ι]
(B : LinearMap.BilinForm ℝ (ι → ℝ)) (hB : B.Nondegenerate)
(W : Submodule ℝ (ι → ℝ)) (hW : W ≤ B.orthogonal W) :
2 * Module.finrank ℝ W ≤ Fintype.card ι := by
have hle := Submodule.finrank_mono hW
rw [LinearMap.BilinForm.finrank_orthogonal hB] at hle
have hdim := Submodule.finrank_le W
have hv : Module.finrank ℝ (ι → ℝ) = Fintype.card ι := by simp
rw [hv] at hle hdim
omega
end Paper170
end
end
section
noncomputable section
namespace Paper170
@[simp] lemma evalFour_mul (x u y v : ℝ) (P Q : Poly4) :
evalFour x u y v (P * Q) = evalFour x u y v P * evalFour x u y v Q :=
map_mul (MvPolynomial.aeval ![x, u, y, v]) P Q
lemma support_evaluation_injective_of_collision {n s : ℕ} (hs : 2 ≤ s)
(node : Fin n → ℝ) (hn : Function.Injective node) (J : CrossingTensor n)
(hdiv : ∀ P : endpointSymmetricSpace s,
supportEvaluation s node J P = 0 → collisionFactor ∣ P.val) :
Function.Injective (supportEvaluation s node J) := by
have hzero : ∀ d (P : endpointSymmetricSpace s), P.val.degreeOf 0 = d →
supportEvaluation s node J P = 0 → P.val = 0 := by
intro d
induction d using Nat.strong_induction_on with
| h d ih =>
intro P hd hP
by_contra hP0
obtain ⟨Q, hQ0, heq, hQmem, _, hQdeg⟩ := collision_quotient s hs P.val
P.property hP0 (hdiv P hP)
have hQ : supportEvaluation s node J ⟨Q, hQmem⟩ = 0 := by
funext a
have hp := congrFun hP a
change evalFour (node a.val.1.1) (node a.val.1.2)
(node a.val.2.1) (node a.val.2.2) P.val = 0 at hp
rw [heq, evalFour_mul] at hp
exact (mul_eq_zero.mp hp).resolve_left (collision_nonzero_on_support node hn J a)
exact hQ0 (ih (Q.degreeOf 0) (by omega) ⟨Q, hQmem⟩ rfl hQ)
rw [← LinearMap.ker_eq_bot, LinearMap.ker_eq_bot']
intro P hP
exact Subtype.ext (hzero _ P rfl hP)
end Paper170
end
end
section
noncomputable section
namespace Paper170
lemma support_cardinality_of_injective_isotropic {n s : ℕ} (hs : 2 ≤ s)
(node : Fin n → ℝ) (hn : Function.Injective node) (J : CrossingTensor n)
(hi : Function.Injective (supportEvaluation s node J))
(ho : ∀ P Q : endpointSymmetricSpace s,
supportBilinear node J (supportEvaluation s node J P) (supportEvaluation s node J Q) = 0) :
2 * (Nat.choose s 2) ^ 2 ≤ (supportedPairs J).card := by
let := symmetric_space_finite s
let f := supportEvaluation s node J
have hr : f.range ≤ (supportBilinear node J).orthogonal f.range := by
rintro _ ⟨P, rfl⟩ _ ⟨Q, rfl⟩
exact ho Q P
have hd := isotropic_dimension (supportBilinear node J)
(support_bilinear_nondegenerate node hn J) f.range hr
have he : Module.finrank ℝ f.range = Module.finrank ℝ (endpointSymmetricSpace s) :=
(LinearEquiv.ofInjective f hi).finrank_eq.symm
rw [he, symmetric_space_dimension s hs, Fintype.card_coe] at hd
exact hd
end Paper170
end
end
end OAI
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