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cead56f | 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 | import OAI.Geometry.ProjectionVolume.PolytopeFacets
namespace OAI
universe uι
open Set
open scoped RealInnerProductSpace
noncomputable section
namespace Paper092.HPolytope
variable {d : ℕ} {ι : Type uι} [Fintype ι] (P : HPolytope d ι)
theorem face_vectorSpan_le_normalHyperplane (a : ι) :
vectorSpan ℝ (P.face a) ≤ normalHyperplane (P.normal a) := by
rw [vectorSpan_def]
apply Submodule.span_le.mpr
rintro _ ⟨x, hx, y, hy, rfl⟩
apply Submodule.mem_orthogonal_singleton_iff_inner_right.mpr
change ⟪P.normal a, x - y⟫ = 0
rw [inner_sub_right, hx.2, hy.2, sub_self]
theorem face_index_unique_of_finrank (a b : ι) (hne : (P.face a).Nonempty)
(hdim : Module.finrank ℝ (vectorSpan ℝ (P.face a)) = d - 1)
(heq : P.face a = P.face b) : a = b := by
have han : P.normal a ≠ 0 := by
intro h
have := P.normal_unit a
simp only [h, norm_zero] at this
norm_num at this
have hbn : P.normal b ≠ 0 := by
intro h
have := P.normal_unit b
simp only [h, norm_zero] at this
norm_num at this
have hWa : vectorSpan ℝ (P.face a) = normalHyperplane (P.normal a) :=
Submodule.eq_of_le_of_finrank_eq (P.face_vectorSpan_le_normalHyperplane a)
(hdim.trans (normalHyperplane_finrank (P.normal a) han).symm)
have hWb : vectorSpan ℝ (P.face b) = normalHyperplane (P.normal b) :=
Submodule.eq_of_le_of_finrank_eq (P.face_vectorSpan_le_normalHyperplane b)
((heq ▸ hdim).trans (normalHyperplane_finrank (P.normal b) hbn).symm)
have hH : normalHyperplane (P.normal a) = normalHyperplane (P.normal b) := by
rw [← hWa, ← hWb, heq]
have hspan : Submodule.span ℝ {P.normal a} = Submodule.span ℝ {P.normal b} := by
have h := congrArg Submodule.orthogonal hH
simpa only [normalHyperplane, Submodule.orthogonal_orthogonal] using h
have hbspan : P.normal b ∈ Submodule.span ℝ {P.normal a} := by
rw [hspan]
exact Submodule.subset_span (mem_singleton _)
obtain ⟨c, hc⟩ := Submodule.mem_span_singleton.mp hbspan
have hnorm : |c| = 1 := by
have h := congrArg norm hc
simpa only [norm_smul, Real.norm_eq_abs, P.normal_unit, mul_one] using h
obtain ⟨x, hxa⟩ := hne
have hxb : x ∈ P.face b := heq ▸ hxa
have hoff : P.offset b = c * P.offset a := by
rw [← hxb.2, ← hc, real_inner_smul_left, hxa.2]
obtain ⟨p, hp⟩ := P.strict_feasible
have haPos : 0 < P.offset a - ⟪P.normal a, p⟫ := sub_pos.mpr (hp a)
have hbPos : 0 < c * (P.offset a - ⟪P.normal a, p⟫) := by
have h := sub_pos.mpr (hp b)
rwa [hoff, ← hc, real_inner_smul_left, ← mul_sub] at h
have hcPos : 0 < c := by
by_contra h
exact (not_lt_of_ge (mul_nonpos_of_nonpos_of_nonneg (le_of_not_gt h) haPos.le)) hbPos
have hcOne : c = 1 := by rwa [abs_of_pos hcPos] at hnorm
have hn : P.normal a = P.normal b := by simpa only [hcOne, one_smul] using hc
have ho : P.offset a = P.offset b := by simpa only [hcOne, one_mul] using hoff.symm
exact P.distinct (Prod.ext hn ho)
end Paper092.HPolytope
end
end OAI
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