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e5a97c6 | 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 | import Mathlib
namespace OAI
/-!
Finite polyhedra, local perturbations, and quotient dimension bounds.
-/
noncomputable section
open scoped BigOperators Topology
open Filter
namespace MatchingEntropy
section PolyhedralPrelude
variable {I J : Type*} [Fintype I] [Fintype J]
def finitePolyhedron (L : I → (J → ℝ) →ₗ[ℝ] ℝ) (b : I → ℝ) : Set (J → ℝ) :=
{x | ∀ i, L i x ≤ b i}
/-- Finite slack, with every active equation preserved, permits both signs. -/
theorem finitePolyhedron_eventually_line (L : I → (J → ℝ) →ₗ[ℝ] ℝ) (b : I → ℝ)
{x : J → ℝ} (hx : x ∈ finitePolyhedron L b) (u : J → ℝ)
(hu : ∀ i, L i x = b i → L i u = 0) :
∀ᶠ t : ℝ in 𝓝 0, x+t • u ∈ finitePolyhedron L b := by
apply Filter.eventually_all.mpr
intro i
by_cases hi : L i x=b i
· apply Filter.Eventually.of_forall
intro t
rw [map_add, map_smul, hu i hi, smul_zero, add_zero, hi]
· have hlt : L i x < b i := lt_of_le_of_ne (hx i) hi
have hc : Continuous (fun t : ℝ => L i (x+t • u)) :=
(L i).continuous_of_finiteDimensional.comp (by fun_prop)
have hh : ∀ᶠ t : ℝ in 𝓝 0, L i (x+t • u) < b i :=
(isOpen_lt hc continuous_const).mem_nhds (by simpa using hlt : L i (x+(0:ℝ) • u) < b i)
exact hh.mono fun t ht => le_of_lt ht
/-- Radial expansion about x preserves every active inequality for arbitrary
nearby feasible points. Inactive inequalities are handled by finite slack. -/
theorem finitePolyhedron_eventually_expand (L : I → (J → ℝ) →ₗ[ℝ] ℝ) (b : I → ℝ)
{x : J → ℝ} (hx : x ∈ finitePolyhedron L b) {t : ℝ} (ht : 0 < t) :
∀ᶠ y in 𝓝[finitePolyhedron L b] x, x+t⁻¹ • (y-x) ∈ finitePolyhedron L b := by
apply Filter.eventually_all.mpr
intro i
by_cases hi : L i x=b i
· filter_upwards [eventually_mem_nhdsWithin] with y hy
rw [map_add, map_smul, map_sub, hi]
change b i+t⁻¹ * (L i y-b i) ≤ b i
have hh := mul_nonpos_of_nonneg_of_nonpos (inv_nonneg.mpr ht.le) (sub_nonpos.mpr (hy i))
linarith
· have hlt : L i x < b i := lt_of_le_of_ne (hx i) hi
have hc : Continuous (fun y : J → ℝ => L i (x+t⁻¹ • (y-x))) :=
(L i).continuous_of_finiteDimensional.comp (by fun_prop)
have hh : ∀ᶠ y : J → ℝ in 𝓝 x, L i (x+t⁻¹ • (y-x)) < b i :=
(isOpen_lt hc continuous_const).mem_nhds (by simpa using hlt : L i (x+t⁻¹ • (x-x)) < b i)
exact (hh.filter_mono nhdsWithin_le_nhds).mono fun y hy => le_of_lt hy
omit [Fintype J] in
theorem extreme_point_direction_zero (P : Set (J → ℝ)) {x : J → ℝ}
(hx : x ∈ Set.extremePoints ℝ P) (u : J → ℝ)
(hu : ∀ᶠ t : ℝ in 𝓝 0, x+t • u ∈ P) : u=0 := by
have hu' : ∀ᶠ t : ℝ in 𝓝 0, x+(-t) • u ∈ P := by
have hc : Tendsto (fun t : ℝ => -t) (𝓝 0) (𝓝 0) := by simpa using (continuous_neg.tendsto (0:ℝ))
exact hc.eventually hu
-- choose a strictly positive point using the right-hand punctured neighborhood
have hh : ∀ᶠ t : ℝ in 𝓝[>] 0, (x+t • u ∈ P ∧ x+(-t) • u ∈ P) ∧ 0<t :=
((hu.and hu').filter_mono nhdsWithin_le_nhds).and eventually_mem_nhdsWithin
obtain ⟨a,⟨ha,hna⟩,ha0⟩ := hh.exists
have hseg : x ∈ openSegment ℝ (x+a • u) (x+(-a) • u) := by
refine ⟨1/2,1/2,by norm_num,by norm_num,by norm_num,?_⟩
ext j
simp only [Pi.add_apply, Pi.smul_apply, smul_eq_mul]
ring
have he := (mem_extremePoints.mp hx).2 _ ha _ hna hseg |>.1
ext j
have hj := congrFun he j
change x j+a*u j=x j at hj
change u j=0
exact (mul_eq_zero.mp (by linarith : a*u j=0)).resolve_left (ne_of_gt ha0)
section LinearHelper
open Module
variable {A B C : Type*} [AddCommGroup A] [AddCommGroup B] [AddCommGroup C]
[Module ℝ A] [Module ℝ B] [Module ℝ C]
[FiniteDimensional ℝ A] [FiniteDimensional ℝ B] [FiniteDimensional ℝ C]
theorem quotient_finrank_le_of_ker_le (U : Submodule ℝ A) (f : A →ₗ[ℝ] B)
(h : f.ker ≤ U) : finrank ℝ (A ⧸ U) ≤ finrank ℝ B := by
have h1 := U.finrank_quotient_add_finrank
have h2 := f.finrank_range_add_finrank_ker
have h3 := Submodule.finrank_mono h
have h4 : finrank ℝ f.range ≤ finrank ℝ B := Submodule.finrank_le _
omega
theorem quotient_finrank_le_pair (U : Submodule ℝ A) (W : Submodule ℝ B) (Z : Submodule ℝ C)
(f : A →ₗ[ℝ] B) (g : A →ₗ[ℝ] C)
(h : ∀ u, f u ∈ W → g u ∈ Z → u ∈ U) :
finrank ℝ (A ⧸ U) ≤ finrank ℝ (B ⧸ W)+finrank ℝ (C ⧸ Z) := by
let F := (W.mkQ.comp f).prod (Z.mkQ.comp g)
have hk : F.ker ≤ U := by
intro u hu
have hh : (W.mkQ (f u), Z.mkQ (g u))=0 := hu
have hW : f u ∈ W := by
have hq := congrArg Prod.fst hh
simpa using hq
have hZ : g u ∈ Z := by
have hq := congrArg Prod.snd hh
simpa using hq
exact h u hW hZ
simpa using quotient_finrank_le_of_ker_le U F hk
end LinearHelper
end PolyhedralPrelude
end MatchingEntropy
end
end OAI
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