openAI-math-dataset / lean /OAI /Analysis /BiholderTransport /Regularity /FiniteSimplexRepresentation.lean
| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open Set | |
| namespace WeakMTWTransport | |
| lemma finite_simplex_representation {E : Type*} [AddCommGroup E] [Module ℝ E] | |
| [FiniteDimensional ℝ E] {S : Set E} {x : E} (hx : x ∈ convexHull ℝ S) : | |
| ∃ m : ℕ, 0 < m ∧ m ≤ Module.finrank ℝ E + 1 ∧ | |
| ∃ z : Fin m → E, ∃ w : Fin m → ℝ, | |
| (∀ i, z i ∈ S) ∧ (∀ i, 0 ≤ w i) ∧ | |
| (∑ i, w i = 1) ∧ (∑ i, w i • z i = x) := by | |
| classical | |
| obtain ⟨ι,inst,z,w,hz,hi,hw,hsum,heq⟩ := eq_pos_convex_span_of_mem_convexHull hx | |
| let : Fintype ι := inst | |
| have : Nonempty ι := by | |
| by_contra h | |
| have : IsEmpty ι := not_nonempty_iff.mp h | |
| simp at hsum | |
| let e := Fintype.equivFin ι | |
| refine ⟨Fintype.card ι,Fintype.card_pos, | |
| hi.card_le_finrank_succ.trans (Nat.add_le_add_right (Submodule.finrank_le _) 1), | |
| fun i => z (e.symm i),fun i => w (e.symm i),?_,?_,?_,?_⟩ | |
| · intro i | |
| exact hz (mem_range_self _) | |
| · intro i | |
| exact (hw _).le | |
| · simpa only [e.symm.sum_comp] using hsum | |
| · exact (e.symm.sum_comp (fun i => w i • z i)).trans heq | |
| end WeakMTWTransport | |
| end | |
| end OAI | |