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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 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | import OAI.Geometry.ProjectionVolume.PolytopeDefinitions
namespace OAI
universe uι uκ
open Set MeasureTheory
open scoped RealInnerProductSpace Pointwise
noncomputable section
namespace Paper092.HPolytope
variable {d : ℕ} {ι : Type uι} {κ : Type uκ} [Fintype ι] [Fintype κ]
def reindex (P : HPolytope d ι) (e : κ ≃ ι) : HPolytope d κ where
normal i := P.normal (e i)
offset i := P.offset (e i)
normal_unit i := P.normal_unit (e i)
compact := by
convert P.compact using 1
ext x
constructor
· intro h i
obtain ⟨j, rfl⟩ := e.surjective i
exact h j
· intro h j
exact h (e j)
strict_feasible := by
obtain ⟨x, hx⟩ := P.strict_feasible
exact ⟨x, fun i => hx (e i)⟩
distinct := fun i j h => e.injective (P.distinct h)
theorem reindex_body (P : HPolytope d ι) (e : κ ≃ ι) :
(P.reindex e).body = P.body := by
ext x
constructor
· intro h i
obtain ⟨j, rfl⟩ := e.surjective i
exact h j
· intro h j
exact h (e j)
def translate (P : HPolytope d ι) (v : Euclidean d) : HPolytope d ι where
normal := P.normal
offset i := P.offset i + ⟪P.normal i, v⟫
normal_unit := P.normal_unit
compact := by
have heq : {x : Euclidean d | ∀ i, ⟪P.normal i, x⟫ ≤ P.offset i + ⟪P.normal i, v⟫} =
(fun x => v + x) '' P.body := by
ext x
constructor
· intro hx
refine ⟨x - v, ?_, by abel_nf⟩
intro i
have := hx i
simpa only [inner_sub_right, sub_le_iff_le_add] using this
· rintro ⟨x, hx, rfl⟩ i
simpa only [inner_add_right, add_comm] using add_le_add_right (hx i) ⟪P.normal i, v⟫
rw [heq]
exact P.compact.image (continuous_const.add continuous_id)
strict_feasible := by
obtain ⟨x, hx⟩ := P.strict_feasible
refine ⟨v + x, fun i => ?_⟩
simpa only [inner_add_right, add_comm] using add_lt_add_right (hx i) ⟪P.normal i, v⟫
distinct := by
intro i j h
apply P.distinct
have hn := congrArg Prod.fst h
have ho := congrArg Prod.snd h
dsimp at hn ho
exact Prod.ext hn (by simpa only [hn, add_left_inj] using ho)
theorem translate_body (P : HPolytope d ι) (v : Euclidean d) :
(P.translate v).body = v +ᵥ P.body := by
ext x
constructor
· intro hx
refine ⟨x - v, ?_, by dsimp; abel_nf⟩
intro i
have h := hx i
change ⟪P.normal i, x⟫ ≤ P.offset i + ⟪P.normal i, v⟫ at h
simpa only [inner_sub_right, sub_le_iff_le_add] using h
· rintro ⟨x, hx, rfl⟩ i
change ⟪P.normal i, v + x⟫ ≤ P.offset i + ⟪P.normal i, v⟫
simpa only [inner_add_right, add_comm] using add_le_add_right (hx i) ⟪P.normal i, v⟫
end Paper092.HPolytope
end
end OAI
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