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dd14c26 | 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 | import Mathlib.Basic.Real.Basic
import Mathlib.Combinatorics.SetFamily.Shatter
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Tactic.Linarith
namespace OAI
namespace MatroidProphet
namespace Pivots
open scoped BigOperators
variable {ι X : Type*} [Fintype ι] [DecidableEq ι]
variable [AddCommGroup X] [Module ℝ X]
def RealizesSigns (v : ι → X →ₗ[ℝ] ℝ) (c : ι → ℝ)
(t : Finset ι) (x : X) : Prop :=
∀ i, if i ∈ t then 0 < v i x + c i else v i x + c i < 0
noncomputable def affinePatterns (v : ι → X →ₗ[ℝ] ℝ) (c : ι → ℝ) :
Finset (Finset ι) := by
classical
exact Finset.univ.filter (fun t => ∃ x, RealizesSigns v c t x)
lemma mem_affinePatterns {v : ι → X →ₗ[ℝ] ℝ} {c : ι → ℝ} {t : Finset ι} :
t ∈ affinePatterns v c ↔ ∃ x, RealizesSigns v c t x := by
classical
simp [affinePatterns]
private lemma exists_positive_combination
(v : ι → X →ₗ[ℝ] ℝ) (c : ι → ℝ)
(hreal : ∀ t : Finset ι, ∃ x, RealizesSigns v c t x)
(a : ι → ℝ) (ha : ∃ i, a i ≠ 0) :
∃ x, 0 < ∑ i, a i * (v i x + c i) := by
classical
obtain ⟨x, hx⟩ := hreal (Finset.univ.filter (fun i => 0 < a i))
have hpos (i : ι) (hai : 0 < a i) : 0 < v i x + c i := by
simpa [RealizesSigns, hai] using hx i
have hneg (i : ι) (hai : ¬ 0 < a i) : v i x + c i < 0 := by
simpa [RealizesSigns, hai] using hx i
refine ⟨x, Finset.sum_pos' (fun i _ => ?_) ?_⟩
· by_cases hai : 0 < a i
· exact mul_nonneg (le_of_lt hai) (le_of_lt (hpos i hai))
· exact mul_nonneg_of_nonpos_of_nonpos (le_of_not_gt hai) (le_of_lt (hneg i hai))
· obtain ⟨i, hi⟩ := ha
refine ⟨i, Finset.mem_univ i, ?_⟩
by_cases hai : 0 < a i
· exact mul_pos hai (hpos i hai)
· exact mul_pos_of_neg_of_neg (lt_of_le_of_ne (le_of_not_gt hai) hi) (hneg i hai)
omit [DecidableEq ι] in
private lemma sum_affine_eq_of_linear_combination_zero
{v : ι → X →ₗ[ℝ] ℝ} (c : ι → ℝ) {a : ι → ℝ}
(ha : ∑ i, a i • v i = 0) (x : X) :
(∑ i, a i * (v i x + c i)) = ∑ i, a i * c i := by
have hz := congrArg (fun f : X →ₗ[ℝ] ℝ => f x) ha
simp only [LinearMap.sum_apply, LinearMap.smul_apply, smul_eq_mul,
LinearMap.zero_apply] at hz
simp_rw [mul_add, Finset.sum_add_distrib]
rw [hz, zero_add]
lemma linearIndependent_of_all_signs
(v : ι → X →ₗ[ℝ] ℝ) (c : ι → ℝ)
(hreal : ∀ t : Finset ι, ∃ x, RealizesSigns v c t x) :
LinearIndependent ℝ v := by
classical
rw [Fintype.linearIndependent_iff]
intro a ha i
by_contra hi
obtain ⟨x, hx⟩ := exists_positive_combination v c hreal a ⟨i, hi⟩
obtain ⟨y, hy⟩ := exists_positive_combination v c hreal (fun j => -a j)
⟨i, neg_ne_zero.mpr hi⟩
simp only [neg_mul, Finset.sum_neg_distrib] at hy
rw [sum_affine_eq_of_linear_combination_zero c ha x] at hx
rw [sum_affine_eq_of_linear_combination_zero c ha y] at hy
linarith
lemma all_signs_of_shatters {v : ι → X →ₗ[ℝ] ℝ} {c : ι → ℝ}
{s : Finset ι} (hs : (affinePatterns v c).Shatters s) :
∀ t : Finset s, ∃ x,
RealizesSigns (fun i : s => v i) (fun i : s => c i) t x := by
classical
intro t
obtain ⟨u, hu, hsu⟩ := hs (t := t.image Subtype.val) (by
intro i hi
obtain ⟨j, _, rfl⟩ := Finset.mem_image.mp hi
exact j.property)
obtain ⟨x, hx⟩ := mem_affinePatterns.mp hu
refine ⟨x, fun i => ?_⟩
have hi : (i : ι) ∈ u ↔ i ∈ t := by
have heq := congrArg (fun z : Finset ι => (i : ι) ∈ z) hsu
simpa [Finset.mem_image, i.property] using heq
simpa only [hi] using hx i
lemma shatters_card_le_finrank [FiniteDimensional ℝ X]
{v : ι → X →ₗ[ℝ] ℝ} {c : ι → ℝ} {s : Finset ι}
(hs : (affinePatterns v c).Shatters s) : s.card ≤ Module.finrank ℝ X := by
have hli := linearIndependent_of_all_signs
(fun i : s => v i) (fun i : s => c i) (all_signs_of_shatters hs)
have hc := hli.fintype_card_le_finrank
simpa using hc
lemma card_affinePatterns_le [FiniteDimensional ℝ X]
(v : ι → X →ₗ[ℝ] ℝ) (c : ι → ℝ) :
(affinePatterns v c).card ≤
∑ k ∈ Finset.Iic (Module.finrank ℝ X), (Fintype.card ι).choose k := by
classical
have hdim : (affinePatterns v c).vcDim ≤ Module.finrank ℝ X := by
unfold Finset.vcDim
exact Finset.sup_le (fun s hs => shatters_card_le_finrank (Finset.mem_shatterer.mp hs))
exact (Finset.card_le_card_shatterer _).trans
(Finset.card_shatterer_le_sum_vcDim.trans
(Finset.sum_le_sum_of_subset_of_nonneg (Finset.Iic_subset_Iic.mpr hdim)
(fun _ _ _ => Nat.zero_le _)))
end Pivots
end MatroidProphet
end OAI
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