4. From Vertex Labellings to Handles🔗

The handles of Theorem 3.4 are produced in two stages: first a combinatorial vertex labelling family on a suitable subdivision, then its affine realization as continuous self-maps.

Definition 4.0🔗

Definition 4.0. Let K be a subdivision of P. A labelling assigns to each vertex q of K a point \ell(q) \in \mathbb R^k. A family \{\ell_\alpha\} of labellings is admissible if:

(L1) every value \ell_\alpha(q) lies in P;

(L2) for every active \alpha and every extreme subset Q of P: every vertex q of K lying in Q has \ell_\alpha(q) \in Q ("face-respecting");

(L3) for every facet D of K there is a set A of at most k indices such that for \alpha \notin A, the labelling \ell_\alpha is constant on the vertex set D.

(As with Definition 3.0, the labelling family's properties appear in Lean as the hypotheses of the assembly theorem handle_family_of_labellings below.)

We take the two stages in the order the Lean development does: first the affine realization of a single labelling, then the convexity lemma and the assembly.

Proposition 4.2🔗

Proposition 4.2 (realization of one labelling). Let K be a subdivision of the convex polytope P and \ell a labelling whose values lie in P. Then there is a continuous map G : P \to P with the carrier-hull property: whenever p \in \operatorname{conv}(\sigma) for a face \sigma of K, one has G(p) \in \operatorname{conv}(\ell(\sigma)).

Proof. For each face \sigma of K, its vertex set is affinely independent, so there is an affine map A_\sigma : \mathbb R^k \to \mathbb R^k with A_\sigma(q) = \ell(q) for every vertex q \in \sigma. (Coordinatewise, one needs an affine functional with prescribed values on an affinely independent finite set: fix a base point p_0 \in \sigma; the difference vectors q - p_0, q \in \sigma \smallsetminus \{p_0\}, are linearly independent, so a linear functional taking prescribed values on them exists — extend to a basis — and one translates.)

Any two of these agree where they must: if p \in \operatorname{conv}(\sigma) \cap \operatorname{conv}(\tau), then by the complex axiom p \in \operatorname{conv}(\sigma \cap \tau), which lies in the affine span of \sigma \cap \tau; the affine maps A_\sigma and A_\tau agree on \sigma \cap \tau (both equal \ell there), hence on its affine span, hence at p.

Since the underlying space of K is P, every p \in P lies in some closed cell; define G(p) := A_\sigma(p) for any face \sigma whose closed cell contains p (well defined by the previous paragraph). The closed cells \operatorname{conv}(\sigma), \sigma ranging over the finitely many faces, form a finite closed cover of P on each member of which G is (affine, hence) continuous, so G is continuous.

Carrier-hull property: if p \in \operatorname{conv}(\sigma), then G(p) = A_\sigma(p) \in A_\sigma(\operatorname{conv} \sigma) = \operatorname{conv}(A_\sigma(\sigma)) = \operatorname{conv}(\ell(\sigma)), as affine maps commute with convex hulls. Finally G maps into P: each \operatorname{conv}(\ell(\sigma)) \subseteq P since \ell(\sigma) \subseteq P and P is convex. \blacksquare

In Lean, the prescribed-values affine maps are built first, then glued:

open Set Geometry
open scoped Topology BigOperators
open _root_.FamiliesOfMapsB01

namespace Submission
namespace FamiliesOfMapsB01

variable {k : ℕ}

/-
A real affine functional taking prescribed values on an affinely independent finite set
`σ` (extend the linearly-independent `vsub` family to a functional, then translate).
-/
lemma exists_affineMap_real_eqOn (σ : Finset (Fin k → ℝ))
    (hσ : AffineIndependent ℝ ((↑) : σ → (Fin k → ℝ)))
    (c : (Fin k → ℝ) → ℝ) :
    ∃ a : (Fin k → ℝ) →ᵃ[ℝ] ℝ, ∀ q ∈ σ, a q = c q := byk:ℕσ:Finset (Fin k → ℝ)hσ:AffineIndependent ℝ Subtype.valc:(Fin k → ℝ) → ℝ⊢ ∃ a, ∀ q ∈ σ, a q = c q
  by_cases hσ' : σ.Nonemptyposk:ℕσ:Finset (Fin k → ℝ)hσ:AffineIndependent ℝ Subtype.valc:(Fin k → ℝ) → ℝhσ':σ.Nonempty⊢ ∃ a, ∀ q ∈ σ, a q = c qnegk:ℕσ:Finset (Fin k → ℝ)hσ:AffineIndependent ℝ Subtype.valc:(Fin k → ℝ) → ℝhσ':¬σ.Nonempty⊢ ∃ a, ∀ q ∈ σ, a q = c q;
  ·posk:ℕσ:Finset (Fin k → ℝ)hσ:AffineIndependent ℝ Subtype.valc:(Fin k → ℝ) → ℝhσ':σ.Nonempty⊢ ∃ a, ∀ q ∈ σ, a q = c q obtain ⟨p₀, hp₀⟩ : ∃ p₀ ∈ σ, True := byk:ℕσ:Finset (Fin k → ℝ)hσ:AffineIndependent ℝ Subtype.valc:(Fin k → ℝ) → ℝhσ':σ.Nonempty⊢ ∃ p₀ ∈ σ, True
      exact ⟨ hσ'.choose, hσ'.choose_spec, trivial ⟩All goals completed! 🐙;
    obtain ⟨ψ, hψ⟩ : ∃ ψ : (Fin k → ℝ) →ₗ[ℝ] ℝ, ∀ q ∈ σ, q ≠ p₀ → ψ (q - p₀) = c q - c p₀ := byk:ℕσ:Finset (Fin k → ℝ)hσ:AffineIndependent ℝ Subtype.valc:(Fin k → ℝ) → ℝhσ':σ.Nonemptyp₀:Fin k → ℝhp₀:p₀ ∈ σ ∧ True⊢ ∃ ψ, ∀ q ∈ σ, q ≠ p₀ → ψ (q - p₀) = c q - c p₀
      have h_lin_ind : LinearIndependent ℝ (fun i : {x : σ // x ≠ ⟨p₀, hp₀.1⟩} => (i : Fin k → ℝ) - p₀) := byk:ℕσ:Finset (Fin k → ℝ)hσ:AffineIndependent ℝ Subtype.valc:(Fin k → ℝ) → ℝhσ':σ.Nonemptyp₀:Fin k → ℝhp₀:p₀ ∈ σ ∧ True⊢ ∃ ψ, ∀ q ∈ σ, q ≠ p₀ → ψ (q - p₀) = c q - c p₀
        have := hσk:ℕσ:Finset (Fin k → ℝ)hσ:AffineIndependent ℝ Subtype.valc:(Fin k → ℝ) → ℝhσ':σ.Nonemptyp₀:Fin k → ℝhp₀:p₀ ∈ σ ∧ Truethis:AffineIndependent ℝ Subtype.val⊢ LinearIndependent ℝ fun i => ↑↑i - p₀;
        rw [ affineIndependent_iff_linearIndependent_vsub ] at thisk:ℕσ:Finset (Fin k → ℝ)hσ:AffineIndependent ℝ Subtype.valc:(Fin k → ℝ) → ℝhσ':σ.Nonemptyp₀:Fin k → ℝhp₀:p₀ ∈ σ ∧ Truethis✝:AffineIndependent ℝ Subtype.valthis:LinearIndependent ℝ fun i => ↑↑i -ᵥ ↑?i1⊢ LinearIndependent ℝ fun i => ↑↑i - p₀i1k:ℕσ:Finset (Fin k → ℝ)hσ:AffineIndependent ℝ Subtype.valc:(Fin k → ℝ) → ℝhσ':σ.Nonemptyp₀:Fin k → ℝhp₀:p₀ ∈ σ ∧ Truethis:AffineIndependent ℝ Subtype.val⊢ ↥σ;
        convert thisAll goals completed! 🐙;
      have := @FamiliesOfMapsB01Regular.exists_linearMap_eq_onk:ℕσ:Finset (Fin k → ℝ)hσ:AffineIndependent ℝ Subtype.valc:(Fin k → ℝ) → ℝhσ':σ.Nonemptyp₀:Fin k → ℝhp₀:p₀ ∈ σ ∧ Trueh_lin_ind:LinearIndependent ℝ fun i => ↑↑i - p₀this:∀ {k : ℕ} {ι : Type ?u.10506} [Fintype ι] (v : ι → Fin k → ℝ),
  LinearIndependent ℝ v → ∀ (c : ι → ℝ), ∃ ψ, ∀ (i : ι), ψ (v i) = c i⊢ ∃ ψ, ∀ q ∈ σ, q ≠ p₀ → ψ (q - p₀) = c q - c p₀;
      convert this ( fun i : { x : σ // x ≠ ⟨ p₀, hp₀.1 ⟩ } => ( i : Fin k → ℝ ) - p₀ ) h_lin_ind ( fun i => c i - c p₀ ) using 1h.e'_2k:ℕσ:Finset (Fin k → ℝ)hσ:AffineIndependent ℝ Subtype.valc:(Fin k → ℝ) → ℝhσ':σ.Nonemptyp₀:Fin k → ℝhp₀:p₀ ∈ σ ∧ Trueh_lin_ind:LinearIndependent ℝ fun i => ↑↑i - p₀this:∀ {k : ℕ} {ι : Type} [Fintype ι] (v : ι → Fin k → ℝ),
  LinearIndependent ℝ v → ∀ (c : ι → ℝ), ∃ ψ, ∀ (i : ι), ψ (v i) = c i⊢ (fun ψ => ∀ q ∈ σ, q ≠ p₀ → ψ (q - p₀) = c q - c p₀) = fun ψ =>
  ∀ (i : { x // x ≠ ⟨p₀, ⋯⟩ }), ψ (↑↑i - p₀) = c ↑↑i - c p₀;
      exth.e'_2.h.ak:ℕσ:Finset (Fin k → ℝ)hσ:AffineIndependent ℝ Subtype.valc:(Fin k → ℝ) → ℝhσ':σ.Nonemptyp₀:Fin k → ℝhp₀:p₀ ∈ σ ∧ Trueh_lin_ind:LinearIndependent ℝ fun i => ↑↑i - p₀this:∀ {k : ℕ} {ι : Type} [Fintype ι] (v : ι → Fin k → ℝ),
  LinearIndependent ℝ v → ∀ (c : ι → ℝ), ∃ ψ, ∀ (i : ι), ψ (v i) = c ix✝:(Fin k → ℝ) →ₗ[ℝ] ℝ⊢ (∀ q ∈ σ, q ≠ p₀ → x✝ (q - p₀) = c q - c p₀) ↔ ∀ (i : { x // x ≠ ⟨p₀, ⋯⟩ }), x✝ (↑↑i - p₀) = c ↑↑i - c p₀; simp +decide [ Subtype.ext_iff ]All goals completed! 🐙 ;
    refine' ⟨ ψ.toAffineMap + AffineMap.const ℝ _ ( c p₀ - ψ p₀ ), fun q hq => _ ⟩posk:ℕσ:Finset (Fin k → ℝ)hσ:AffineIndependent ℝ Subtype.valc:(Fin k → ℝ) → ℝhσ':σ.Nonemptyp₀:Fin k → ℝhp₀:p₀ ∈ σ ∧ Trueψ:(Fin k → ℝ) →ₗ[ℝ] ℝhψ:∀ q ∈ σ, q ≠ p₀ → ψ (q - p₀) = c q - c p₀q:Fin k → ℝhq:q ∈ σ⊢ (ψ.toAffineMap + AffineMap.const ℝ (Fin k → ℝ) (c p₀ - ψ p₀)) q = c q ; by_cases hq₀ : q = p₀posk:ℕσ:Finset (Fin k → ℝ)hσ:AffineIndependent ℝ Subtype.valc:(Fin k → ℝ) → ℝhσ':σ.Nonemptyp₀:Fin k → ℝhp₀:p₀ ∈ σ ∧ Trueψ:(Fin k → ℝ) →ₗ[ℝ] ℝhψ:∀ q ∈ σ, q ≠ p₀ → ψ (q - p₀) = c q - c p₀q:Fin k → ℝhq:q ∈ σhq₀:q = p₀⊢ (ψ.toAffineMap + AffineMap.const ℝ (Fin k → ℝ) (c p₀ - ψ p₀)) q = c qnegk:ℕσ:Finset (Fin k → ℝ)hσ:AffineIndependent ℝ Subtype.valc:(Fin k → ℝ) → ℝhσ':σ.Nonemptyp₀:Fin k → ℝhp₀:p₀ ∈ σ ∧ Trueψ:(Fin k → ℝ) →ₗ[ℝ] ℝhψ:∀ q ∈ σ, q ≠ p₀ → ψ (q - p₀) = c q - c p₀q:Fin k → ℝhq:q ∈ σhq₀:¬q = p₀⊢ (ψ.toAffineMap + AffineMap.const ℝ (Fin k → ℝ) (c p₀ - ψ p₀)) q = c q <;>posk:ℕσ:Finset (Fin k → ℝ)hσ:AffineIndependent ℝ Subtype.valc:(Fin k → ℝ) → ℝhσ':σ.Nonemptyp₀:Fin k → ℝhp₀:p₀ ∈ σ ∧ Trueψ:(Fin k → ℝ) →ₗ[ℝ] ℝhψ:∀ q ∈ σ, q ≠ p₀ → ψ (q - p₀) = c q - c p₀q:Fin k → ℝhq:q ∈ σhq₀:q = p₀⊢ (ψ.toAffineMap + AffineMap.const ℝ (Fin k → ℝ) (c p₀ - ψ p₀)) q = c qnegk:ℕσ:Finset (Fin k → ℝ)hσ:AffineIndependent ℝ Subtype.valc:(Fin k → ℝ) → ℝhσ':σ.Nonemptyp₀:Fin k → ℝhp₀:p₀ ∈ σ ∧ Trueψ:(Fin k → ℝ) →ₗ[ℝ] ℝhψ:∀ q ∈ σ, q ≠ p₀ → ψ (q - p₀) = c q - c p₀q:Fin k → ℝhq:q ∈ σhq₀:¬q = p₀⊢ (ψ.toAffineMap + AffineMap.const ℝ (Fin k → ℝ) (c p₀ - ψ p₀)) q = c q simp_all +decide [ sub_eq_iff_eq_add ]All goals completed! 🐙;
  ·negk:ℕσ:Finset (Fin k → ℝ)hσ:AffineIndependent ℝ Subtype.valc:(Fin k → ℝ) → ℝhσ':¬σ.Nonempty⊢ ∃ a, ∀ q ∈ σ, a q = c q exact ⟨ 0, byk:ℕσ:Finset (Fin k → ℝ)hσ:AffineIndependent ℝ Subtype.valc:(Fin k → ℝ) → ℝhσ':¬σ.Nonempty⊢ ∀ q ∈ σ, 0 q = c q aesopAll goals completed! 🐙 ⟩

/-- An affine self-map of `ℝᵏ` taking prescribed values `lbl` on an affinely independent
finite set `σ` (apply `exists_affineMap_real_eqOn` coordinatewise and bundle with
`AffineMap.pi`). -/
lemma exists_affineMap_eqOn (σ : Finset (Fin k → ℝ))
    (hσ : AffineIndependent ℝ ((↑) : σ → (Fin k → ℝ)))
    (lbl : (Fin k → ℝ) → (Fin k → ℝ)) :
    ∃ A : (Fin k → ℝ) →ᵃ[ℝ] (Fin k → ℝ), ∀ q ∈ σ, A q = lbl q := byk:ℕσ:Finset (Fin k → ℝ)hσ:AffineIndependent ℝ Subtype.vallbl:(Fin k → ℝ) → Fin k → ℝ⊢ ∃ A, ∀ q ∈ σ, A q = lbl q
  choose a ha using fun j : Fin k =>
    exists_affineMap_real_eqOn σ hσ (fun p => lbl p j)k:ℕσ:Finset (Fin k → ℝ)hσ:AffineIndependent ℝ Subtype.vallbl:(Fin k → ℝ) → Fin k → ℝa:Fin k → (Fin k → ℝ) →ᵃ[ℝ] ℝha:∀ (j : Fin k), ∀ q ∈ σ, (a j) q = lbl q j⊢ ∃ A, ∀ q ∈ σ, A q = lbl q
  exact ⟨AffineMap.pi a, fun q hq => funext fun j => ha j q hq⟩All goals completed! 🐙

/-
**Affine realization of a vertex labelling.** For a subdivision `K` of a convex
polytope `P` and a labelling `lbl` whose values on every vertex of `K` lie in `P`, there is
a continuous self-map `G : P → P` such that on every closed simplex `convexHull σ` of `K`,
`G` takes values in `convexHull (lbl '' σ)`.
-/
theorem realization_exists
    {P : Set (Fin k → ℝ)} (hP : Convex ℝ P) (K : Subdivision P)
    (lbl : (Fin k → ℝ) → (Fin k → ℝ))
    (hlbl : ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ P) :
    ∃ G : C(P, P), ∀ σ ∈ K.complex.faces, ∀ p : P,
      (p : Fin k → ℝ) ∈ convexHull ℝ (σ : Set (Fin k → ℝ)) →
      (G p : Fin k → ℝ) ∈ convexHull ℝ (lbl '' (σ : Set (Fin k → ℝ))) := byk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ P⊢ ∃ G, ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑(G p) ∈ (convexHull ℝ) (lbl '' ↑σ)
  obtain ⟨G₀, hG₀⟩ : ∃ G₀ : P → (Fin k → ℝ), (∀ σ ∈ K.complex.faces, ∀ p : P, (p : Fin k → ℝ) ∈ convexHull ℝ (σ : Set (Fin k → ℝ)) → G₀ p ∈ convexHull ℝ (lbl '' (σ : Set (Fin k → ℝ)))) ∧ Continuous G₀ := byk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ P⊢ ∃ G₀, (∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → G₀ p ∈ (convexHull ℝ) (lbl '' ↑σ)) ∧ Continuous G₀
    have h_affine_maps : ∀ σ ∈ K.complex.faces, ∃ A : (Fin k → ℝ) →ᵃ[ℝ] (Fin k → ℝ), ∀ q ∈ σ, A q = lbl q := byk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ P⊢ ∃ G₀, (∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → G₀ p ∈ (convexHull ℝ) (lbl '' ↑σ)) ∧ Continuous G₀
      intro σ hσk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ Pσ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces⊢ ∃ A, ∀ q ∈ σ, A q = lbl q;
      apply exists_affineMap_eqOnhσk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ Pσ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces⊢ AffineIndependent ℝ Subtype.val;
      grind +suggestionsAll goals completed! 🐙;
    choose! A hA using h_affine_mapsk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl q⊢ ∃ G₀, (∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → G₀ p ∈ (convexHull ℝ) (lbl '' ↑σ)) ∧ Continuous G₀;
    -- By the properties of the convex hull and the definition of `A`, we have that for any `σ ∈ K.complex.faces` and `p : P`, if `p ∈ convexHull ℝ σ`, then `A σ p ∈ convexHull ℝ (lbl '' σ)`.
    have h_convex_hull : ∀ σ ∈ K.complex.faces, ∀ p : P, (p : Fin k → ℝ) ∈ convexHull ℝ (σ : Set (Fin k → ℝ)) → (A σ p : Fin k → ℝ) ∈ convexHull ℝ (lbl '' (σ : Set (Fin k → ℝ))) := byk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ P⊢ ∃ G₀, (∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → G₀ p ∈ (convexHull ℝ) (lbl '' ↑σ)) ∧ Continuous G₀
      intros σ hσ p hpk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qσ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.facesp:↑Php:↑p ∈ (convexHull ℝ) ↑σ⊢ (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)
      have h_convex_hull : (A σ p : Fin k → ℝ) ∈ convexHull ℝ (A σ '' (σ : Set (Fin k → ℝ))) := byk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ P⊢ ∃ G₀, (∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → G₀ p ∈ (convexHull ℝ) (lbl '' ↑σ)) ∧ Continuous G₀
        convert AffineMap.image_convexHull ( A σ ) ( σ : Set ( Fin k → ℝ ) ) ▸ Set.mem_image_of_mem _ hp using 1All goals completed! 🐙;
      convert h_convex_hull using 1h.e'_4k:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qσ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.facesp:↑Php:↑p ∈ (convexHull ℝ) ↑σh_convex_hull:(A σ) ↑p ∈ (convexHull ℝ) (⇑(A σ) '' ↑σ)⊢ (convexHull ℝ) (lbl '' ↑σ) = (convexHull ℝ) (⇑(A σ) '' ↑σ);
      exact congr_arg _ ( Set.image_congr fun x hx => hA σ hσ x hx ▸ rfl )All goals completed! 🐙;
    obtain ⟨G₀, hG₀⟩ : ∃ G₀ : P → (Fin k → ℝ), (∀ σ ∈ K.complex.faces, ∀ p : P, (p : Fin k → ℝ) ∈ convexHull ℝ (σ : Set (Fin k → ℝ)) → G₀ p = (A σ p : Fin k → ℝ)) ∧ Continuous G₀ := byk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)⊢ ∃ G₀, (∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → G₀ p = (A σ) ↑p) ∧ Continuous G₀
      have h_agree : ∀ σ τ : Finset (Fin k → ℝ), σ ∈ K.complex.faces → τ ∈ K.complex.faces → ∀ p : P, (p : Fin k → ℝ) ∈ convexHull ℝ (σ : Set (Fin k → ℝ)) → (p : Fin k → ℝ) ∈ convexHull ℝ (τ : Set (Fin k → ℝ)) → (A σ p : Fin k → ℝ) = (A τ p : Fin k → ℝ) := byk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)⊢ ∃ G₀, (∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → G₀ p = (A σ) ↑p) ∧ Continuous G₀
        intros σ τ hσ hτ p hpσ hpτk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)σ:Finset (Fin k → ℝ)τ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faceshτ:τ ∈ K.complex.facesp:↑Phpσ:↑p ∈ (convexHull ℝ) ↑σhpτ:↑p ∈ (convexHull ℝ) ↑τ⊢ (A σ) ↑p = (A τ) ↑p
        have h_eq : Set.EqOn (A σ) (A τ) (σ ∩ τ : Set (Fin k → ℝ)) := byk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)⊢ ∃ G₀, (∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → G₀ p = (A σ) ↑p) ∧ Continuous G₀
          intro q hqk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)σ:Finset (Fin k → ℝ)τ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faceshτ:τ ∈ K.complex.facesp:↑Phpσ:↑p ∈ (convexHull ℝ) ↑σhpτ:↑p ∈ (convexHull ℝ) ↑τq:Fin k → ℝhq:q ∈ ↑σ ∩ ↑τ⊢ (A σ) q = (A τ) q; aesopAll goals completed! 🐙;
        have h_eq_affine : Set.EqOn (A σ) (A τ) (affineSpan ℝ (σ ∩ τ : Set (Fin k → ℝ))) := byk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)⊢ ∃ G₀, (∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → G₀ p = (A σ) ↑p) ∧ Continuous G₀
          apply_rules [ AffineMap.eqOn_affineSpan ]All goals completed! 🐙;
        apply h_eq_affineak:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)σ:Finset (Fin k → ℝ)τ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faceshτ:τ ∈ K.complex.facesp:↑Phpσ:↑p ∈ (convexHull ℝ) ↑σhpτ:↑p ∈ (convexHull ℝ) ↑τh_eq:EqOn (⇑(A σ)) (⇑(A τ)) (↑σ ∩ ↑τ)h_eq_affine:EqOn ⇑(A σ) ⇑(A τ) ↑(affineSpan ℝ (↑σ ∩ ↑τ))⊢ ↑p ∈ ↑(affineSpan ℝ (↑σ ∩ ↑τ));
        have h_eq_affine : (p : Fin k → ℝ) ∈ convexHull ℝ (σ ∩ τ : Set (Fin k → ℝ)) := byk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)⊢ ∃ G₀, (∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → G₀ p = (A σ) ↑p) ∧ Continuous G₀
          apply K.complex.inter_subset_convexHull hσ hτak:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)σ:Finset (Fin k → ℝ)τ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faceshτ:τ ∈ K.complex.facesp:↑Phpσ:↑p ∈ (convexHull ℝ) ↑σhpτ:↑p ∈ (convexHull ℝ) ↑τh_eq:EqOn (⇑(A σ)) (⇑(A τ)) (↑σ ∩ ↑τ)h_eq_affine:EqOn ⇑(A σ) ⇑(A τ) ↑(affineSpan ℝ (↑σ ∩ ↑τ))⊢ ↑p ∈ (convexHull ℝ) ↑σ ∩ (convexHull ℝ) ↑τ;
          exact ⟨ hpσ, hpτ ⟩All goals completed! 🐙;
        exact convexHull_subset_affineSpan _ h_eq_affineAll goals completed! 🐙
      have h_cover : ∀ p : P, ∃ σ ∈ K.complex.faces, (p : Fin k → ℝ) ∈ convexHull ℝ (σ : Set (Fin k → ℝ)) := byk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)⊢ ∃ G₀, (∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → G₀ p = (A σ) ↑p) ∧ Continuous G₀
        intro pk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pp:↑P⊢ ∃ σ ∈ K.complex.faces, ↑p ∈ (convexHull ℝ) ↑σ
        have h_p_in_space : (p : Fin k → ℝ) ∈ K.complex.space := byk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)⊢ ∃ G₀, (∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → G₀ p = (A σ) ↑p) ∧ Continuous G₀
          exact K.space_eq.symm ▸ p.2All goals completed! 🐙;
        exact Set.mem_iUnion₂.mp h_p_in_space |> fun ⟨ σ, hσ₁, hσ₂ ⟩ => ⟨ σ, hσ₁, hσ₂ ⟩All goals completed! 🐙;
      choose! σ hσ₁ hσ₂ using h_coverk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)⊢ ∃ G₀, (∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → G₀ p = (A σ) ↑p) ∧ Continuous G₀;
      refine' ⟨ fun p => ( A ( σ p ) p : Fin k → ℝ ), _, _ ⟩refine'_1k:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)⊢ ∀ σ_1 ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ_1 → (fun p => (A (σ p)) ↑p) p = (A σ_1) ↑prefine'_2k:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)⊢ Continuous fun p => (A (σ p)) ↑p;
      ·refine'_1k:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)⊢ ∀ σ_1 ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ_1 → (fun p => (A (σ p)) ↑p) p = (A σ_1) ↑p exact fun σ' hσ' p hp => h_agree _ _ ( hσ₁ p ) hσ' p ( hσ₂ p ) hpAll goals completed! 🐙;
      ·refine'_2k:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)⊢ Continuous fun p => (A (σ p)) ↑p have h_locally_finite : LocallyFinite (fun σ : K.complex.faces => {p : P | (p : Fin k → ℝ) ∈ convexHull ℝ (σ.val : Set (Fin k → ℝ))}) := byk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)⊢ ∃ G₀, (∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → G₀ p = (A σ) ↑p) ∧ Continuous G₀
          have h_finite : Finite K.complex.faces := byk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)⊢ ∃ G₀, (∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → G₀ p = (A σ) ↑p) ∧ Continuous G₀
            exact Set.Finite.to_subtype K.faces_finiteAll goals completed! 🐙;
          grind +suggestionsAll goals completed! 🐙;
        apply_rules [ h_locally_finite.continuous, continuousOn_iff_continuous_restrict.mpr ]refine'_2.h_covk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)h_locally_finite:LocallyFinite fun σ => {p | ↑p ∈ (convexHull ℝ) ↑↑σ}⊢ ⋃ i, {p | ↑p ∈ (convexHull ℝ) ↑↑i} = univrefine'_2.h_clk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)h_locally_finite:LocallyFinite fun σ => {p | ↑p ∈ (convexHull ℝ) ↑↑σ}⊢ ∀ (i : ↑K.complex.faces), IsClosed {p | ↑p ∈ (convexHull ℝ) ↑↑i}refine'_2.h_contk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)h_locally_finite:LocallyFinite fun σ => {p | ↑p ∈ (convexHull ℝ) ↑↑σ}⊢ ∀ (i : ↑K.complex.faces), ContinuousOn (fun p => (A (σ p)) ↑p) {p | ↑p ∈ (convexHull ℝ) ↑↑i};
        ·refine'_2.h_covk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)h_locally_finite:LocallyFinite fun σ => {p | ↑p ∈ (convexHull ℝ) ↑↑σ}⊢ ⋃ i, {p | ↑p ∈ (convexHull ℝ) ↑↑i} = univ exact Set.eq_univ_of_forall fun p => Set.mem_iUnion.mpr ⟨ ⟨ σ p, hσ₁ p ⟩, hσ₂ p ⟩All goals completed! 🐙;
        ·refine'_2.h_clk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)h_locally_finite:LocallyFinite fun σ => {p | ↑p ∈ (convexHull ℝ) ↑↑σ}⊢ ∀ (i : ↑K.complex.faces), IsClosed {p | ↑p ∈ (convexHull ℝ) ↑↑i} intro σrefine'_2.h_clk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ✝:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)h_locally_finite:LocallyFinite fun σ => {p | ↑p ∈ (convexHull ℝ) ↑↑σ}σ:↑K.complex.faces⊢ IsClosed {p | ↑p ∈ (convexHull ℝ) ↑↑σ};
          exact IsClosed.preimage ( continuous_subtype_val ) ( show IsClosed ( convexHull ℝ ( σ.val : Set ( Fin k → ℝ ) ) ) from byk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ✝:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)h_locally_finite:LocallyFinite fun σ => {p | ↑p ∈ (convexHull ℝ) ↑↑σ}σ:↑K.complex.faces⊢ IsClosed ((convexHull ℝ) ↑↑σ) exact ( IsCompact.isClosed <| byk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ✝:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)h_locally_finite:LocallyFinite fun σ => {p | ↑p ∈ (convexHull ℝ) ↑↑σ}σ:↑K.complex.faces⊢ IsCompact ((convexHull ℝ) ↑↑σ) exact ( (Finset.finite_toSet _).isCompact_convexHull (𝕜 := ℝ) )All goals completed! 🐙 ) );
        ·refine'_2.h_contk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)h_locally_finite:LocallyFinite fun σ => {p | ↑p ∈ (convexHull ℝ) ↑↑σ}⊢ ∀ (i : ↑K.complex.faces), ContinuousOn (fun p => (A (σ p)) ↑p) {p | ↑p ∈ (convexHull ℝ) ↑↑i} intro irefine'_2.h_contk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)h_locally_finite:LocallyFinite fun σ => {p | ↑p ∈ (convexHull ℝ) ↑↑σ}i:↑K.complex.faces⊢ ContinuousOn (fun p => (A (σ p)) ↑p) {p | ↑p ∈ (convexHull ℝ) ↑↑i};
          refine' ContinuousOn.congr _ _refine'_2.h_cont.refine'_1k:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)h_locally_finite:LocallyFinite fun σ => {p | ↑p ∈ (convexHull ℝ) ↑↑σ}i:↑K.complex.faces⊢ ↑P → Fin k → ℝrefine'_2.h_cont.refine'_2k:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)h_locally_finite:LocallyFinite fun σ => {p | ↑p ∈ (convexHull ℝ) ↑↑σ}i:↑K.complex.faces⊢ ContinuousOn ?refine'_2.h_cont.refine'_1 {p | ↑p ∈ (convexHull ℝ) ↑↑i}refine'_2.h_cont.refine'_3k:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)h_locally_finite:LocallyFinite fun σ => {p | ↑p ∈ (convexHull ℝ) ↑↑σ}i:↑K.complex.faces⊢ EqOn (fun p => (A (σ p)) ↑p) ?refine'_2.h_cont.refine'_1 {p | ↑p ∈ (convexHull ℝ) ↑↑i};
          use fun p => ( A i.val p : Fin k → ℝ )refine'_2.h_cont.refine'_2k:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)h_locally_finite:LocallyFinite fun σ => {p | ↑p ∈ (convexHull ℝ) ↑↑σ}i:↑K.complex.faces⊢ ContinuousOn (fun p => (A ↑i) ↑p) {p | ↑p ∈ (convexHull ℝ) ↑↑i}refine'_2.h_cont.refine'_3k:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)h_locally_finite:LocallyFinite fun σ => {p | ↑p ∈ (convexHull ℝ) ↑↑σ}i:↑K.complex.faces⊢ EqOn (fun p => (A (σ p)) ↑p) (fun p => (A ↑i) ↑p) {p | ↑p ∈ (convexHull ℝ) ↑↑i};
          ·refine'_2.h_cont.refine'_2k:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)h_locally_finite:LocallyFinite fun σ => {p | ↑p ∈ (convexHull ℝ) ↑↑σ}i:↑K.complex.faces⊢ ContinuousOn (fun p => (A ↑i) ↑p) {p | ↑p ∈ (convexHull ℝ) ↑↑i} exact Continuous.continuousOn ( byk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)h_locally_finite:LocallyFinite fun σ => {p | ↑p ∈ (convexHull ℝ) ↑↑σ}i:↑K.complex.faces⊢ Continuous fun p => (A ↑i) ↑p exact Continuous.comp ( A i |>.continuous_of_finiteDimensional ) continuous_subtype_valAll goals completed! 🐙 );
          ·refine'_2.h_cont.refine'_3k:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)h_locally_finite:LocallyFinite fun σ => {p | ↑p ∈ (convexHull ℝ) ↑↑σ}i:↑K.complex.faces⊢ EqOn (fun p => (A (σ p)) ↑p) (fun p => (A ↑i) ↑p) {p | ↑p ∈ (convexHull ℝ) ↑↑i} intro p hprefine'_2.h_cont.refine'_3k:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)h_agree:∀ (σ τ : Finset (Fin k → ℝ)),
  σ ∈ K.complex.faces →
    τ ∈ K.complex.faces → ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑p ∈ (convexHull ℝ) ↑τ → (A σ) ↑p = (A τ) ↑pσ:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)h_locally_finite:LocallyFinite fun σ => {p | ↑p ∈ (convexHull ℝ) ↑↑σ}i:↑K.complex.facesp:↑Php:p ∈ {p | ↑p ∈ (convexHull ℝ) ↑↑i}⊢ (fun p => (A (σ p)) ↑p) p = (fun p => (A ↑i) ↑p) p; specialize h_agree ( σ p ) i.val ( hσ₁ p ) i.2 prefine'_2.h_cont.refine'_3k:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PA:Finset (Fin k → ℝ) → (Fin k → ℝ) →ᵃ[ℝ] Fin k → ℝhA:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, (A σ) q = lbl qh_convex_hull:∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → (A σ) ↑p ∈ (convexHull ℝ) (lbl '' ↑σ)σ:↑P → Finset (Fin k → ℝ)hσ₁:∀ (p : ↑P), σ p ∈ K.complex.faceshσ₂:∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑(σ p)h_locally_finite:LocallyFinite fun σ => {p | ↑p ∈ (convexHull ℝ) ↑↑σ}i:↑K.complex.facesp:↑Php:p ∈ {p | ↑p ∈ (convexHull ℝ) ↑↑i}h_agree:↑p ∈ (convexHull ℝ) ↑(σ p) → ↑p ∈ (convexHull ℝ) ↑↑i → (A (σ p)) ↑p = (A ↑i) ↑p⊢ (fun p => (A (σ p)) ↑p) p = (fun p => (A ↑i) ↑p) p; aesopAll goals completed! 🐙;
    exact ⟨ G₀, fun σ hσ p hp => hG₀.1 σ hσ p hp ▸ h_convex_hull σ hσ p hp, hG₀.2 ⟩All goals completed! 🐙;
  have hG₀_P : ∀ p : P, G₀ p ∈ P := byk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ P⊢ ∃ G, ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑(G p) ∈ (convexHull ℝ) (lbl '' ↑σ)
    intro pk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PG₀:↑P → Fin k → ℝhG₀:(∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → G₀ p ∈ (convexHull ℝ) (lbl '' ↑σ)) ∧ Continuous G₀p:↑P⊢ G₀ p ∈ P
    obtain ⟨σ, hσ⟩ : ∃ σ ∈ K.complex.faces, (p : Fin k → ℝ) ∈ convexHull ℝ (σ : Set (Fin k → ℝ)) := byk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PG₀:↑P → Fin k → ℝhG₀:(∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → G₀ p ∈ (convexHull ℝ) (lbl '' ↑σ)) ∧ Continuous G₀p:↑P⊢ ∃ σ ∈ K.complex.faces, ↑p ∈ (convexHull ℝ) ↑σ
      have h_space : p.val ∈ K.complex.space := byk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PG₀:↑P → Fin k → ℝhG₀:(∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → G₀ p ∈ (convexHull ℝ) (lbl '' ↑σ)) ∧ Continuous G₀p:↑P⊢ ∃ σ ∈ K.complex.faces, ↑p ∈ (convexHull ℝ) ↑σ
        exact K.space_eq.symm ▸ p.2All goals completed! 🐙;
      exact Set.mem_iUnion₂.mp h_space |> fun ⟨ σ, hσ₁, hσ₂ ⟩ => ⟨ σ, hσ₁, hσ₂ ⟩All goals completed! 🐙;
    exact hG₀.1 σ hσ.1 p hσ.2 |> fun h => convexHull_min ( Set.image_subset_iff.mpr fun q hq => hlbl σ hσ.1 q hq ) hP hAll goals completed! 🐙;
  exact ⟨ ⟨ fun p => ⟨ G₀ p, hG₀_P p ⟩, byk:ℕP:Set (Fin k → ℝ)hP:Convex ℝ PK:Subdivision Plbl:(Fin k → ℝ) → Fin k → ℝhlbl:∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ PG₀:↑P → Fin k → ℝhG₀:(∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → G₀ p ∈ (convexHull ℝ) (lbl '' ↑σ)) ∧ Continuous G₀hG₀_P:∀ (p : ↑P), G₀ p ∈ P⊢ Continuous fun p => ⟨G₀ p, ⋯⟩ continuityAll goals completed! 🐙 ⟩, hG₀.1 ⟩

end FamiliesOfMapsB01
end Submission

We first record the key convexity fact used to transfer (L2) to the realized handles.

Lemma 4.1🔗

Lemma 4.1 (support of a convex combination inside a face). Let Q be an extreme subset of the convex set P, and suppose a point of Q is written as a convex combination, with all weights strictly positive, of finitely many points of P. Then all those points lie in Q.

Proof. Induction on the number of points. One point is trivial. Given y_0 = a\,q + b\,z \in Q with a, b > 0, a + b = 1, q \in P, and z the renormalized convex combination of the remaining points (which lies in P by convexity): extremality applied to this expression gives q \in Q and z \in Q (if q = z this is immediate, since then y_0 = q = z); apply the inductive hypothesis to z with the renormalized (still strictly positive) weights. \blacksquare

open Set Geometry
open scoped Topology BigOperators
open _root_.FamiliesOfMapsB01

namespace Submission
namespace FamiliesOfMapsB01

variable {k : ℕ} {ι X : Type*} [TopologicalSpace X]

/-
**Support of a convex combination inside an extreme set.** If `F` is an extreme subset
of a convex set `P`, and a point of `F` is written as a convex combination (all weights
strictly positive) of points of `P`, then every point used lies in `F`.  Proof by induction
on the (finite) support, splitting off one point via `openSegment`.
-/
lemma extreme_convexHull_support
    {P F : Set (Fin k → ℝ)} (hF : IsExtreme ℝ P F) (hPc : Convex ℝ P)
    (s : Finset (Fin k → ℝ)) (hsP : (s : Set (Fin k → ℝ)) ⊆ P)
    (w : (Fin k → ℝ) → ℝ) (hw : ∀ q ∈ s, 0 < w q) (hsum : ∑ q ∈ s, w q = 1)
    (hx : (∑ q ∈ s, w q • q) ∈ F) :
    (s : Set (Fin k → ℝ)) ⊆ F := byk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Ps:Finset (Fin k → ℝ)hsP:↑s ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q ∈ s, 0 < w qhsum:∑ q ∈ s, w q = 1hx:∑ q ∈ s, w q • q ∈ F⊢ ↑s ⊆ F
  induction' s using Finset.induction with q s hs ih generalizing wemptyk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ PhsP:↑∅ ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q ∈ ∅, 0 < w qhsum:∑ q ∈ ∅, w q = 1hx:∑ q ∈ ∅, w q • q ∈ F⊢ ↑∅ ⊆ Finsertk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ F⊢ ↑(insert q s) ⊆ F;
  ·emptyk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ PhsP:↑∅ ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q ∈ ∅, 0 < w qhsum:∑ q ∈ ∅, w q = 1hx:∑ q ∈ ∅, w q • q ∈ F⊢ ↑∅ ⊆ F norm_num at hsumAll goals completed! 🐙;
  ·insertk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ F⊢ ↑(insert q s) ⊆ F by_cases hs_empty : s.Nonemptyposk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonempty⊢ ↑(insert q s) ⊆ Fnegk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:¬s.Nonempty⊢ ↑(insert q s) ⊆ F;
    ·posk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonempty⊢ ↑(insert q s) ⊆ F -- Let $a = w q \in (0,1)$ (since $0 < w q$ and $\sum_{s} w = 1$ with other positive terms, $w q < 1$). Let $b = 1 - a > 0$.
      set a := w qposk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w q⊢ ↑(insert q s) ⊆ F
      set b := 1 - aposk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - a⊢ ↑(insert q s) ⊆ F
      have ha_pos : 0 < a := byk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Ps:Finset (Fin k → ℝ)hsP:↑s ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q ∈ s, 0 < w qhsum:∑ q ∈ s, w q = 1hx:∑ q ∈ s, w q • q ∈ F⊢ ↑s ⊆ F
        exact hw q ( Finset.mem_insert_self _ _ )All goals completed! 🐙
      have hb_pos : 0 < b := byk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Ps:Finset (Fin k → ℝ)hsP:↑s ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q ∈ s, 0 < w qhsum:∑ q ∈ s, w q = 1hx:∑ q ∈ s, w q • q ∈ F⊢ ↑s ⊆ F
        simp +zetaDelta at *k:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ Fw:(Fin k → ℝ) → ℝhsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < w qhsP:insert q ↑s ⊆ Phw:0 < w q ∧ ∀ a ∈ s, 0 < w a⊢ w q < 1;
        rw [ ← hsum, Finset.sum_insert hs ]k:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ Fw:(Fin k → ℝ) → ℝhsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < w qhsP:insert q ↑s ⊆ Phw:0 < w q ∧ ∀ a ∈ s, 0 < w a⊢ w q < w q + ∑ x ∈ s, w x ; linarith [ Finset.sum_pos ( fun x hx => hw.2 x hx ) hs_empty ]All goals completed! 🐙
      have ha_lt_one : a < 1 := byk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Ps:Finset (Fin k → ℝ)hsP:↑s ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q ∈ s, 0 < w qhsum:∑ q ∈ s, w q = 1hx:∑ q ∈ s, w q • q ∈ F⊢ ↑s ⊆ F
        exact lt_of_sub_pos hb_posAll goals completed! 🐙;
      -- Define $y = \sum_{q \in s} (w q / b) • q$; the weights $w q / b$ are positive and sum to $1$ (since $\sum_{s} w = b$), so $y \in P$ ($Convex.sum_mem hPc$) and $y \in convexHull$ of $s$.
      set y := ∑ q ∈ s, (w q / b) • qposk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • q⊢ ↑(insert q s) ⊆ F
      have hyP : y ∈ P := byk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Ps:Finset (Fin k → ℝ)hsP:↑s ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q ∈ s, 0 < w qhsum:∑ q ∈ s, w q = 1hx:∑ q ∈ s, w q • q ∈ F⊢ ↑s ⊆ F
        convert hPc.sum_mem _ _ _convert_5k:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • q⊢ ∀ i ∈ s, 0 ≤ w i / bconvert_6k:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • q⊢ ∑ i ∈ s, w i / b = 1convert_7k:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • q⊢ ∀ i ∈ s, i ∈ P;
        ·convert_5k:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • q⊢ ∀ i ∈ s, 0 ≤ w i / b exact fun x hx => div_nonneg ( le_of_lt ( hw x ( Finset.mem_insert_of_mem hx ) ) ) hb_pos.leAll goals completed! 🐙;
        ·convert_6k:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • q⊢ ∑ i ∈ s, w i / b = 1 rw [ ← Finset.sum_div, div_eq_iff ]convert_6k:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • q⊢ ∑ i ∈ s, w i = 1 * bconvert_6k:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • q⊢ b ≠ 0 <;>convert_6k:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • q⊢ ∑ i ∈ s, w i = 1 * bconvert_6k:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • q⊢ b ≠ 0 first | positivityAll goals completed! 🐙 | rw [ Finset.sum_insert hs ] at hsumconvert_6k:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:w q + ∑ x ∈ s, w x = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • q⊢ ∑ i ∈ s, w i = 1 * b ; linarithAll goals completed! 🐙;
        ·convert_7k:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • q⊢ ∀ i ∈ s, i ∈ P exact fun x hx => hsP <| Finset.mem_insert_of_mem hxAll goals completed! 🐙
      have hyF : y ∈ F := byk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Ps:Finset (Fin k → ℝ)hsP:↑s ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q ∈ s, 0 < w qhsum:∑ q ∈ s, w q = 1hx:∑ q ∈ s, w q • q ∈ F⊢ ↑s ⊆ F
        have hyF : (a • q + b • y) ∈ F := byk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Ps:Finset (Fin k → ℝ)hsP:↑s ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q ∈ s, 0 < w qhsum:∑ q ∈ s, w q = 1hx:∑ q ∈ s, w q • q ∈ F⊢ ↑s ⊆ F
          convert hx using 1h.e'_5k:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • qhyP:y ∈ P⊢ a • q + b • y = ∑ q ∈ insert q s, w q • q;
          rw [ Finset.sum_insert hs, Finset.smul_sum ]h.e'_5k:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • qhyP:y ∈ P⊢ a • q + ∑ x ∈ s, b • (w x / b) • x = w q • q + ∑ x ∈ s, w x • x;
          exact congrArg _ ( Finset.sum_congr rfl fun x hx => byk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx✝:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • qhyP:y ∈ Px:Fin k → ℝhx:x ∈ s⊢ b • (w x / b) • x = w x • x rw [ smul_smul, mul_div_cancel₀ _ ( by linarith ) ]All goals completed! 🐙 );
        have := hF.2 hyP ( hsP ( Finset.mem_insert_self _ _ ) ) ( show ( 1 - a ) • y + a • q ∈ F from byk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • qhyP:y ∈ PhyF:a • q + b • y ∈ F⊢ (1 - a) • y + a • q ∈ F simpa only [ add_comm ] using hyFAll goals completed! 🐙 );
        exact this ⟨ 1 - a, a, hb_pos, ha_pos, byk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • qhyP:y ∈ PhyF:a • q + b • y ∈ Fthis:(1 - a) • y + a • q ∈ openSegment ℝ y q → y ∈ F⊢ 1 - a + a = 1 linarithAll goals completed! 🐙, byk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • qhyP:y ∈ PhyF:a • q + b • y ∈ Fthis:(1 - a) • y + a • q ∈ openSegment ℝ y q → y ∈ F⊢ (1 - a) • y + a • q = (1 - a) • y + a • q simp +decide [ add_comm ]All goals completed! 🐙 ⟩;
      have hqF : q ∈ F := byk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Ps:Finset (Fin k → ℝ)hsP:↑s ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q ∈ s, 0 < w qhsum:∑ q ∈ s, w q = 1hx:∑ q ∈ s, w q • q ∈ F⊢ ↑s ⊆ F
        have hqF : q ∈ F := byk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Ps:Finset (Fin k → ℝ)hsP:↑s ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q ∈ s, 0 < w qhsum:∑ q ∈ s, w q = 1hx:∑ q ∈ s, w q • q ∈ F⊢ ↑s ⊆ F
          have hqF : q ∈ P := hsP (byk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • qhyP:y ∈ PhyF:y ∈ F⊢ q ∈ ↑(insert q s) simpAll goals completed! 🐙)
          have hqF : y ∈ P := hyPk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • qhyP:y ∈ PhyF:y ∈ FhqF✝:q ∈ PhqF:y ∈ P⊢ q ∈ F
          have hqF : ∑ q ∈ insert q s, w q • q = a • q + b • y := byk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Ps:Finset (Fin k → ℝ)hsP:↑s ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q ∈ s, 0 < w qhsum:∑ q ∈ s, w q = 1hx:∑ q ∈ s, w q • q ∈ F⊢ ↑s ⊆ F
            simp +zetaDelta at *k:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ Fw:(Fin k → ℝ) → ℝhsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < w qha_lt_one:w q < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • qhqF✝:q ∈ PhsP:insert q ↑s ⊆ Phw:0 < w q ∧ ∀ a ∈ s, 0 < w ahb_pos:w q < 1hyP:∑ x ∈ s, (w x / (1 - w q)) • x ∈ PhyF:∑ x ∈ s, (w x / (1 - w q)) • x ∈ FhqF:∑ x ∈ s, (w x / (1 - w q)) • x ∈ P⊢ ∑ q ∈ insert q s, w q • q = w q • q + (1 - w q) • ∑ x ∈ s, (w x / (1 - w q)) • x;
            simp +decide [ Finset.sum_insert hs, Finset.smul_sum, smul_smul, div_eq_inv_mul, ne_of_gt ( sub_pos.mpr hb_pos ) ]All goals completed! 🐙
          have := hF.2 ‹q ∈ P› ‹y ∈ P›k:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • qhyP:y ∈ PhyF:y ∈ FhqF✝¹:q ∈ PhqF✝:y ∈ PhqF:∑ q ∈ insert q s, w q • q = a • q + b • ythis:∀ ⦃z : Fin k → ℝ⦄, z ∈ F → z ∈ openSegment ℝ q y → q ∈ F⊢ q ∈ F;
          exact this hx ⟨ a, b, ha_pos, hb_pos, byk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • qhyP:y ∈ PhyF:y ∈ FhqF✝¹:q ∈ PhqF✝:y ∈ PhqF:∑ q ∈ insert q s, w q • q = a • q + b • ythis:∀ ⦃z : Fin k → ℝ⦄, z ∈ F → z ∈ openSegment ℝ q y → q ∈ F⊢ a + b = 1 linarithAll goals completed! 🐙, byk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:s.Nonemptya:ℝ := w qb:ℝ := 1 - aha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1y:Fin k → ℝ := ∑ q ∈ s, (w q / b) • qhyP:y ∈ PhyF:y ∈ FhqF✝¹:q ∈ PhqF✝:y ∈ PhqF:∑ q ∈ insert q s, w q • q = a • q + b • ythis:∀ ⦃z : Fin k → ℝ⦄, z ∈ F → z ∈ openSegment ℝ q y → q ∈ F⊢ a • q + b • y = ∑ q ∈ insert q s, w q • q aesopAll goals completed! 🐙 ⟩;
        exact hqFAll goals completed! 🐙;
      simp_all +decide [ Finset.sum_div _ _ _ ]posk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ Fw:(Fin k → ℝ) → ℝa:ℝ := w qb:ℝ := 1 - ay:Fin k → ℝ := ∑ q ∈ s, (w q / b) • qhsP:insert q ↑s ⊆ Phw:0 < w q ∧ ∀ a ∈ s, 0 < w ahsum:w q + ∑ x ∈ s, w x = 1hx:w q • q + ∑ q ∈ s, w q • q ∈ Fhs_empty:s.Nonemptyha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1hyP:y ∈ PhyF:y ∈ FhqF:q ∈ F⊢ insert q ↑s ⊆ F;
      exact Set.insert_subset_iff.mpr ⟨ hqF, ih ( fun x hx => hsP ( Set.mem_insert_of_mem _ hx ) ) ( fun x => w x / b ) ( fun x hx => div_pos ( hw.2 x hx ) hb_pos ) ( byk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ Fw:(Fin k → ℝ) → ℝa:ℝ := w qb:ℝ := 1 - ay:Fin k → ℝ := ∑ q ∈ s, (w q / b) • qhsP:insert q ↑s ⊆ Phw:0 < w q ∧ ∀ a ∈ s, 0 < w ahsum:w q + ∑ x ∈ s, w x = 1hx:w q • q + ∑ q ∈ s, w q • q ∈ Fhs_empty:s.Nonemptyha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1hyP:y ∈ PhyF:y ∈ FhqF:q ∈ F⊢ ∑ q ∈ s, (fun x => w x / b) q = 1 rw [ ← Finset.sum_div _ _ _, div_eq_iff hb_pos.ne' ]k:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ Fw:(Fin k → ℝ) → ℝa:ℝ := w qb:ℝ := 1 - ay:Fin k → ℝ := ∑ q ∈ s, (w q / b) • qhsP:insert q ↑s ⊆ Phw:0 < w q ∧ ∀ a ∈ s, 0 < w ahsum:w q + ∑ x ∈ s, w x = 1hx:w q • q + ∑ q ∈ s, w q • q ∈ Fhs_empty:s.Nonemptyha_pos:0 < ahb_pos:0 < bha_lt_one:a < 1hyP:y ∈ PhyF:y ∈ FhqF:q ∈ F⊢ ∑ i ∈ s, w i = 1 * b ; linarithAll goals completed! 🐙 ) hyF ⟩;
    ·negk:ℕP:Set (Fin k → ℝ)F:Set (Fin k → ℝ)hF:IsExtreme ℝ P FhPc:Convex ℝ Pq:Fin k → ℝs:Finset (Fin k → ℝ)hs:q ∉ sih:↑s ⊆ P → ∀ (w : (Fin k → ℝ) → ℝ), (∀ q ∈ s, 0 < w q) → ∑ q ∈ s, w q = 1 → ∑ q ∈ s, w q • q ∈ F → ↑s ⊆ FhsP:↑(insert q s) ⊆ Pw:(Fin k → ℝ) → ℝhw:∀ q_1 ∈ insert q s, 0 < w q_1hsum:∑ q ∈ insert q s, w q = 1hx:∑ q ∈ insert q s, w q • q ∈ Fhs_empty:¬s.Nonempty⊢ ↑(insert q s) ⊆ F simp_all +decide [ Finset.not_nonempty_iff_eq_empty.mp hs_empty ]All goals completed! 🐙

Proposition 4.3🔗

Proposition 4.3 (handles from labellings). Let K be a subdivision of P and \{\ell_\alpha\} an admissible labelling family. Let g_\alpha : P \to P be the realization of \ell_\alpha from Proposition 4.2. Then (K, \{g_\alpha\}) is a handle family.

Proof. (b), constancy: Fix a facet D and the set A from (L3). For \alpha \notin A, \ell_\alpha is constant on D, say with value z; the carrier-hull property gives, for every p \in \operatorname{conv}(D), g_\alpha(p) \in \operatorname{conv}(\{z\}) = \{z\}; so g_\alpha is constant on the closed cell.

(a), face preservation: Let \alpha be active (otherwise there is nothing to prove), let Q be a boundary subpolyhedron (a polytope, and an extreme subset of P), and let p \in Q. Choose a face \sigma of K with p \in \operatorname{conv}(\sigma) and write p as a convex combination of the vertices of \sigma; discard the vertices of weight zero, leaving a subset \tau \subseteq \sigma (again a face of K, being a nonempty subset of a face) with p \in \operatorname{conv}(\tau) and all weights strictly positive. By Lemma 4.1 (applied to the extreme subset Q and this positive combination), every vertex of \tau lies in Q. By (L2), \ell_\alpha maps every vertex of \tau into Q. By the carrier-hull property applied to \tau, g_\alpha(p) \in \operatorname{conv}(\ell_\alpha(\tau)) \subseteq \operatorname{conv}(Q) = Q. \blacksquare

/-
**Assembly of the handle family from labellings.** Given a subdivision `K` of `P` and a
family of labellings `lbl` such that

* each labelling sends every face-vertex of `K` into `P` (`hlblP`);
* each *active* labelling is *face-respecting*: it keeps every vertex of `K` lying in an
  extreme subset `Q` of `P` inside `Q` (`hface`);
* on every facet `D` of `K`, all but `≤ k` labellings are constant on the vertices of `D`
  (`hcount`),

the affine realizations of the labellings form a handle family with the face-preservation
and per-cell `≤ k` properties required by `handle_family_exists`.
-/
set_option maxHeartbeats 1600000 in
theorem handle_family_of_labellings
    {P : Set (Fin k → ℝ)} (hP : IsPolyhedron P)
    (ρ : PartitionOfUnity ι X univ)
    (K : Subdivision P)
    (lbl : ι → (Fin k → ℝ) → (Fin k → ℝ))
    (hlblP : ∀ α, ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ P)
    (hface : ∀ α, ρ α ≠ 0 → ∀ Q : Set (Fin k → ℝ), IsExtreme ℝ P Q →
        ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Q)
    (hcount : ∀ D ∈ K.complex.facets, ∃ A : Finset ι, A.card ≤ k ∧
        ∀ α, α ∉ A → ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q') :
    ∃ (g : ι → C(P, P)),
      (∀ α, ρ α = 0 ∨
        (∀ Q : Set P, IsBoundarySubpolyhedron Q → ∀ p : P, p ∈ Q → g α p ∈ Q)) ∧
      (∀ D ∈ K.complex.facets, ∃ A : Finset ι, A.card ≤ k ∧
        ∀ α, α ∉ A → ∀ p p' : P,
          ((p : Fin k → ℝ) ∈ convexHull ℝ (D : Set (Fin k → ℝ))) →
          ((p' : Fin k → ℝ) ∈ convexHull ℝ (D : Set (Fin k → ℝ))) →
          g α p = g α p') ∧
      (∀ α, ∀ σ ∈ K.complex.faces, ∀ p : P,
          (p : Fin k → ℝ) ∈ convexHull ℝ (σ : Set (Fin k → ℝ)) →
          (g α p : Fin k → ℝ) ∈ convexHull ℝ (lbl α '' (σ : Set (Fin k → ℝ)))) := byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'⊢ ∃ g,
  (∀ (α : ι), ρ α = 0 ∨ ∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ p ∈ Q, (g α) p ∈ Q) ∧
    (∀ D ∈ K.complex.facets,
        ∃ A,
          A.card ≤ k ∧ ∀ α ∉ A, ∀ (p p' : ↑P), ↑p ∈ (convexHull ℝ) ↑D → ↑p' ∈ (convexHull ℝ) ↑D → (g α) p = (g α) p') ∧
      ∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)
  obtain ⟨g, hg⟩ : ∃ g : ι → C(P, P), ∀ α, ∀ σ ∈ K.complex.faces, ∀ p : P, (p : Fin k → ℝ) ∈ convexHull ℝ (σ : Set (Fin k → ℝ)) → (g α p : Fin k → ℝ) ∈ convexHull ℝ (lbl α '' (σ : Set (Fin k → ℝ))) := byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'⊢ ∃ g, ∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)
    have := @Submission.FamiliesOfMapsB01.realization_existsk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'this:∀ {k : ℕ} {P : Set (Fin k → ℝ)},
  Convex ℝ P →
    ∀ (K : Subdivision P) (lbl : (Fin k → ℝ) → Fin k → ℝ),
      (∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ P) →
        ∃ G, ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑(G p) ∈ (convexHull ℝ) (lbl '' ↑σ)⊢ ∃ g, ∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ);
    exact ⟨ fun α => Classical.choose ( this ( show Convex ℝ P from byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'this:∀ {k : ℕ} {P : Set (Fin k → ℝ)},
  Convex ℝ P →
    ∀ (K : Subdivision P) (lbl : (Fin k → ℝ) → Fin k → ℝ),
      (∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ P) →
        ∃ G, ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑(G p) ∈ (convexHull ℝ) (lbl '' ↑σ)α:ι⊢ Convex ℝ P rcases hP with ⟨ S, rfl ⟩k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace Xρ:PartitionOfUnity ι Xlbl:ι → (Fin k → ℝ) → Fin k → ℝthis:∀ {k : ℕ} {P : Set (Fin k → ℝ)},
  Convex ℝ P →
    ∀ (K : Subdivision P) (lbl : (Fin k → ℝ) → Fin k → ℝ),
      (∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ P) →
        ∃ G, ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑(G p) ∈ (convexHull ℝ) (lbl '' ↑σ)α:ιS:Finset (Fin k → ℝ)K:Subdivision ((convexHull ℝ) ↑S)hlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ (convexHull ℝ) ↑Shface:∀ (α : ι),
  ρ α ≠ 0 →
    ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ ((convexHull ℝ) ↑S) Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'⊢ Convex ℝ ((convexHull ℝ) ↑S) ; exact convex_convexHull ℝ _All goals completed! 🐙 ) K ( lbl α ) ( hlblP α ) ), fun α => Classical.choose_spec ( this ( show Convex ℝ P from byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'this:∀ {k : ℕ} {P : Set (Fin k → ℝ)},
  Convex ℝ P →
    ∀ (K : Subdivision P) (lbl : (Fin k → ℝ) → Fin k → ℝ),
      (∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ P) →
        ∃ G, ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑(G p) ∈ (convexHull ℝ) (lbl '' ↑σ)α:ι⊢ Convex ℝ P rcases hP with ⟨ S, rfl ⟩k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace Xρ:PartitionOfUnity ι Xlbl:ι → (Fin k → ℝ) → Fin k → ℝthis:∀ {k : ℕ} {P : Set (Fin k → ℝ)},
  Convex ℝ P →
    ∀ (K : Subdivision P) (lbl : (Fin k → ℝ) → Fin k → ℝ),
      (∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl q ∈ P) →
        ∃ G, ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑(G p) ∈ (convexHull ℝ) (lbl '' ↑σ)α:ιS:Finset (Fin k → ℝ)K:Subdivision ((convexHull ℝ) ↑S)hlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ (convexHull ℝ) ↑Shface:∀ (α : ι),
  ρ α ≠ 0 →
    ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ ((convexHull ℝ) ↑S) Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'⊢ Convex ℝ ((convexHull ℝ) ↑S) ; exact convex_convexHull ℝ _All goals completed! 🐙 ) K ( lbl α ) ( hlblP α ) ) ⟩;
  refine' ⟨ g, _, _, hg ⟩refine'_1k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)⊢ ∀ (α : ι), ρ α = 0 ∨ ∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ p ∈ Q, (g α) p ∈ Qrefine'_2k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)⊢ ∀ D ∈ K.complex.facets,
  ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ (p p' : ↑P), ↑p ∈ (convexHull ℝ) ↑D → ↑p' ∈ (convexHull ℝ) ↑D → (g α) p = (g α) p';
  ·refine'_1k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)⊢ ∀ (α : ι), ρ α = 0 ∨ ∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ p ∈ Q, (g α) p ∈ Q intro αrefine'_1k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ι⊢ ρ α = 0 ∨ ∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ p ∈ Q, (g α) p ∈ Q;
    refine' Classical.or_iff_not_imp_left.2 fun h => _refine'_1k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0⊢ ∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ p ∈ Q, (g α) p ∈ Q;
    intro Q hQrefine'_1k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑PhQ:IsBoundarySubpolyhedron Q⊢ ∀ p ∈ Q, (g α) p ∈ Q prefine'_1k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑PhQ:IsBoundarySubpolyhedron Qp:↑P⊢ p ∈ Q → (g α) p ∈ Q hprefine'_1k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑PhQ:IsBoundarySubpolyhedron Qp:↑Php:p ∈ Q⊢ (g α) p ∈ Q
    obtain ⟨F, hF⟩ := hQrefine'_1k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)⊢ (g α) p ∈ Q;
    obtain ⟨σ, hσ⟩ : ∃ σ ∈ K.complex.faces, (p : Fin k → ℝ) ∈ convexHull ℝ (σ : Set (Fin k → ℝ)) := byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)⊢ ∃ σ ∈ K.complex.faces, ↑p ∈ (convexHull ℝ) ↑σ
      have h_space : p.val ∈ K.complex.space := byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)⊢ ∃ σ ∈ K.complex.faces, ↑p ∈ (convexHull ℝ) ↑σ
        exact K.space_eq.symm ▸ p.2All goals completed! 🐙;
      exact Set.mem_iUnion₂.mp h_space |> fun ⟨ σ, hσ₁, hσ₂ ⟩ => ⟨ σ, hσ₁, hσ₂ ⟩All goals completed! 🐙;
    -- Write `(p:_) = ∑_{q∈σ} w q • q` with `w ≥ 0`, `∑ w = 1` (from `Finset.convexHull_eq`).
    obtain ⟨w, hw_nonneg, hw_sum, hw_eq⟩ : ∃ w : (Fin k → ℝ) → ℝ, (∀ q ∈ σ, 0 ≤ w q) ∧ (∑ q ∈ σ, w q = 1) ∧ (p : Fin k → ℝ) = ∑ q ∈ σ, w q • q := byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σ⊢ ∃ w, (∀ q ∈ σ, 0 ≤ w q) ∧ ∑ q ∈ σ, w q = 1 ∧ ↑p = ∑ q ∈ σ, w q • q
      rw [ mem_convexHull_iff_exists_fintype ] at hσk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ∃ ι x w z, (∀ (i : ι), 0 ≤ w i) ∧ ∑ i, w i = 1 ∧ (∀ (i : ι), z i ∈ ↑σ) ∧ ∑ i, w i • z i = ↑p⊢ ∃ w, (∀ q ∈ σ, 0 ≤ w q) ∧ ∑ q ∈ σ, w q = 1 ∧ ↑p = ∑ q ∈ σ, w q • q;
      obtain ⟨ ι, x, w, z, hw, hw', hz, hp ⟩ := hσ.2k:ℕι✝:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php✝:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ∃ ι x w z, (∀ (i : ι), 0 ≤ w i) ∧ ∑ i, w i = 1 ∧ (∀ (i : ι), z i ∈ ↑σ) ∧ ∑ i, w i • z i = ↑pι:Typex:Fintype ιw:ι → ℝz:ι → Fin k → ℝhw:∀ (i : ι), 0 ≤ w ihw':∑ i, w i = 1hz:∀ (i : ι), z i ∈ ↑σhp:∑ i, w i • z i = ↑p⊢ ∃ w, (∀ q ∈ σ, 0 ≤ w q) ∧ ∑ q ∈ σ, w q = 1 ∧ ↑p = ∑ q ∈ σ, w q • q;
      refine' ⟨ fun q => ∑ i ∈ Finset.univ.filter ( fun i => z i = q ), w i, _, _, _ ⟩refine'_1k:ℕι✝:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php✝:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ∃ ι x w z, (∀ (i : ι), 0 ≤ w i) ∧ ∑ i, w i = 1 ∧ (∀ (i : ι), z i ∈ ↑σ) ∧ ∑ i, w i • z i = ↑pι:Typex:Fintype ιw:ι → ℝz:ι → Fin k → ℝhw:∀ (i : ι), 0 ≤ w ihw':∑ i, w i = 1hz:∀ (i : ι), z i ∈ ↑σhp:∑ i, w i • z i = ↑p⊢ ∀ q ∈ σ, 0 ≤ (fun q => ∑ i with z i = q, w i) qrefine'_2k:ℕι✝:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php✝:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ∃ ι x w z, (∀ (i : ι), 0 ≤ w i) ∧ ∑ i, w i = 1 ∧ (∀ (i : ι), z i ∈ ↑σ) ∧ ∑ i, w i • z i = ↑pι:Typex:Fintype ιw:ι → ℝz:ι → Fin k → ℝhw:∀ (i : ι), 0 ≤ w ihw':∑ i, w i = 1hz:∀ (i : ι), z i ∈ ↑σhp:∑ i, w i • z i = ↑p⊢ ∑ q ∈ σ, (fun q => ∑ i with z i = q, w i) q = 1refine'_3k:ℕι✝:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php✝:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ∃ ι x w z, (∀ (i : ι), 0 ≤ w i) ∧ ∑ i, w i = 1 ∧ (∀ (i : ι), z i ∈ ↑σ) ∧ ∑ i, w i • z i = ↑pι:Typex:Fintype ιw:ι → ℝz:ι → Fin k → ℝhw:∀ (i : ι), 0 ≤ w ihw':∑ i, w i = 1hz:∀ (i : ι), z i ∈ ↑σhp:∑ i, w i • z i = ↑p⊢ ↑p = ∑ q ∈ σ, (fun q => ∑ i with z i = q, w i) q • q;
      ·refine'_1k:ℕι✝:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php✝:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ∃ ι x w z, (∀ (i : ι), 0 ≤ w i) ∧ ∑ i, w i = 1 ∧ (∀ (i : ι), z i ∈ ↑σ) ∧ ∑ i, w i • z i = ↑pι:Typex:Fintype ιw:ι → ℝz:ι → Fin k → ℝhw:∀ (i : ι), 0 ≤ w ihw':∑ i, w i = 1hz:∀ (i : ι), z i ∈ ↑σhp:∑ i, w i • z i = ↑p⊢ ∀ q ∈ σ, 0 ≤ (fun q => ∑ i with z i = q, w i) q exact fun q hq => Finset.sum_nonneg fun i hi => hw iAll goals completed! 🐙;
      ·refine'_2k:ℕι✝:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php✝:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ∃ ι x w z, (∀ (i : ι), 0 ≤ w i) ∧ ∑ i, w i = 1 ∧ (∀ (i : ι), z i ∈ ↑σ) ∧ ∑ i, w i • z i = ↑pι:Typex:Fintype ιw:ι → ℝz:ι → Fin k → ℝhw:∀ (i : ι), 0 ≤ w ihw':∑ i, w i = 1hz:∀ (i : ι), z i ∈ ↑σhp:∑ i, w i • z i = ↑p⊢ ∑ q ∈ σ, (fun q => ∑ i with z i = q, w i) q = 1 rw [ ← hw', Finset.sum_fiberwise_of_maps_to ]refine'_2.hk:ℕι✝:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php✝:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ∃ ι x w z, (∀ (i : ι), 0 ≤ w i) ∧ ∑ i, w i = 1 ∧ (∀ (i : ι), z i ∈ ↑σ) ∧ ∑ i, w i • z i = ↑pι:Typex:Fintype ιw:ι → ℝz:ι → Fin k → ℝhw:∀ (i : ι), 0 ≤ w ihw':∑ i, w i = 1hz:∀ (i : ι), z i ∈ ↑σhp:∑ i, w i • z i = ↑p⊢ ∀ i ∈ Finset.univ, z i ∈ σ;
        exact fun i _ => Finset.mem_coe.mp ( hz i )All goals completed! 🐙;
      ·refine'_3k:ℕι✝:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php✝:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ∃ ι x w z, (∀ (i : ι), 0 ≤ w i) ∧ ∑ i, w i = 1 ∧ (∀ (i : ι), z i ∈ ↑σ) ∧ ∑ i, w i • z i = ↑pι:Typex:Fintype ιw:ι → ℝz:ι → Fin k → ℝhw:∀ (i : ι), 0 ≤ w ihw':∑ i, w i = 1hz:∀ (i : ι), z i ∈ ↑σhp:∑ i, w i • z i = ↑p⊢ ↑p = ∑ q ∈ σ, (fun q => ∑ i with z i = q, w i) q • q simp +decide [ ← hp, Finset.sum_filter ]refine'_3k:ℕι✝:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php✝:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ∃ ι x w z, (∀ (i : ι), 0 ≤ w i) ∧ ∑ i, w i = 1 ∧ (∀ (i : ι), z i ∈ ↑σ) ∧ ∑ i, w i • z i = ↑pι:Typex:Fintype ιw:ι → ℝz:ι → Fin k → ℝhw:∀ (i : ι), 0 ≤ w ihw':∑ i, w i = 1hz:∀ (i : ι), z i ∈ ↑σhp:∑ i, w i • z i = ↑p⊢ ∑ i, w i • z i = ∑ x_1 ∈ σ, (∑ a, if z a = x_1 then w a else 0) • x_1;
        simp +decide [ Finset.sum_comm, Finset.sum_smul ]refine'_3k:ℕι✝:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php✝:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ∃ ι x w z, (∀ (i : ι), 0 ≤ w i) ∧ ∑ i, w i = 1 ∧ (∀ (i : ι), z i ∈ ↑σ) ∧ ∑ i, w i • z i = ↑pι:Typex:Fintype ιw:ι → ℝz:ι → Fin k → ℝhw:∀ (i : ι), 0 ≤ w ihw':∑ i, w i = 1hz:∀ (i : ι), z i ∈ ↑σhp:∑ i, w i • z i = ↑p⊢ ∑ i, w i • z i = ∑ x, if z x ∈ σ then w x • z x else 0;
        grind +localsAll goals completed! 🐙;
    -- Let `τ := σ.filter (fun q => 0 < w q)`; then `(p:_) = ∑_{q∈τ} w q • q` (zero-weight terms drop), the weights on `τ` are strictly positive and sum to 1, `(p:_) ∈ convexHull ℝ ↑τ`, and `τ ∈ faces` (sub-Finset of a face, nonempty, by `isRelLowerSet_faces`).
    set τ := σ.filter (fun q => 0 < w q) with hτ_defrefine'_1k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}⊢ (g α) p ∈ Q
    have hτ_pos : ∀ q ∈ τ, 0 < w q := byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'⊢ ∃ g,
  (∀ (α : ι), ρ α = 0 ∨ ∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ p ∈ Q, (g α) p ∈ Q) ∧
    (∀ D ∈ K.complex.facets,
        ∃ A,
          A.card ≤ k ∧ ∀ α ∉ A, ∀ (p p' : ↑P), ↑p ∈ (convexHull ℝ) ↑D → ↑p' ∈ (convexHull ℝ) ↑D → (g α) p = (g α) p') ∧
      ∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)
      exact fun q hq => Finset.mem_filter.mp hq |>.2All goals completed! 🐙
    have hτ_sum : ∑ q ∈ τ, w q = 1 := byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'⊢ ∃ g,
  (∀ (α : ι), ρ α = 0 ∨ ∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ p ∈ Q, (g α) p ∈ Q) ∧
    (∀ D ∈ K.complex.facets,
        ∃ A,
          A.card ≤ k ∧ ∀ α ∉ A, ∀ (p p' : ↑P), ↑p ∈ (convexHull ℝ) ↑D → ↑p' ∈ (convexHull ℝ) ↑D → (g α) p = (g α) p') ∧
      ∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)
      rw [ ← hw_sum, Finset.sum_filter_of_ne ]k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w q⊢ ∀ x ∈ σ, w x ≠ 0 → 0 < w x;
      exact fun q hq hq' => lt_of_le_of_ne ( hw_nonneg q hq ) ( Ne.symm hq' )All goals completed! 🐙
    have hτ_eq : (p : Fin k → ℝ) = ∑ q ∈ τ, w q • q := byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'⊢ ∃ g,
  (∀ (α : ι), ρ α = 0 ∨ ∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ p ∈ Q, (g α) p ∈ Q) ∧
    (∀ D ∈ K.complex.facets,
        ∃ A,
          A.card ≤ k ∧ ∀ α ∉ A, ∀ (p p' : ↑P), ↑p ∈ (convexHull ℝ) ↑D → ↑p' ∈ (convexHull ℝ) ↑D → (g α) p = (g α) p') ∧
      ∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)
      rw [ hw_eq, Finset.sum_filter_of_ne ]k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1⊢ ∀ x ∈ σ, w x • x ≠ 0 → 0 < w x;
      exact fun q hq hq' => lt_of_le_of_ne ( hw_nonneg q hq ) ( Ne.symm <| byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1q:Fin k → ℝhq:q ∈ σhq':w q • q ≠ 0⊢ w q ≠ 0 aesopAll goals completed! 🐙 )
    have hτ_face : τ ∈ K.complex.faces := byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'⊢ ∃ g,
  (∀ (α : ι), ρ α = 0 ∨ ∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ p ∈ Q, (g α) p ∈ Q) ∧
    (∀ D ∈ K.complex.facets,
        ∃ A,
          A.card ≤ k ∧ ∀ α ∉ A, ∀ (p p' : ↑P), ↑p ∈ (convexHull ℝ) ↑D → ↑p' ∈ (convexHull ℝ) ↑D → (g α) p = (g α) p') ∧
      ∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)
      have hτ_nonempty : τ.Nonempty := byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'⊢ ∃ g,
  (∀ (α : ι), ρ α = 0 ∨ ∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ p ∈ Q, (g α) p ∈ Q) ∧
    (∀ D ∈ K.complex.facets,
        ∃ A,
          A.card ≤ k ∧ ∀ α ∉ A, ∀ (p p' : ↑P), ↑p ∈ (convexHull ℝ) ↑D → ↑p' ∈ (convexHull ℝ) ↑D → (g α) p = (g α) p') ∧
      ∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)
        exact Finset.nonempty_of_ne_empty ( byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • q⊢ τ ≠ ∅ rintro hk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh✝:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qh:τ = ∅⊢ False; simp +decide [ h ] at hτ_sumAll goals completed! 🐙 );
      grind +suggestionsAll goals completed! 🐙;
    -- By `extreme_convexHull_support hFext hPc τ (τ⊆σ⊆P) w (positivity) (sum=1) ((p:_)∈F)` we get `↑τ ⊆ F`, i.e. every `q ∈ τ` lies in `F`.
    have hτ_subset_F : (τ : Set (Fin k → ℝ)) ⊆ Subtype.val '' Q := byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'⊢ ∃ g,
  (∀ (α : ι), ρ α = 0 ∨ ∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ p ∈ Q, (g α) p ∈ Q) ∧
    (∀ D ∈ K.complex.facets,
        ∃ A,
          A.card ≤ k ∧ ∀ α ∉ A, ∀ (p p' : ↑P), ↑p ∈ (convexHull ℝ) ↑D → ↑p' ∈ (convexHull ℝ) ↑D → (g α) p = (g α) p') ∧
      ∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)
      apply extreme_convexHull_support hF.right (byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faces⊢ Convex ℝ P
      obtain ⟨ S, rfl ⟩ := hPk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace Xρ:PartitionOfUnity ι Xlbl:ι → (Fin k → ℝ) → Fin k → ℝα:ιh:¬ρ α = 0σ:Finset (Fin k → ℝ)w:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1τ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1S:Finset (Fin k → ℝ)K:Subdivision ((convexHull ℝ) ↑S)hlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ (convexHull ℝ) ↑Shface:∀ (α : ι),
  ρ α ≠ 0 →
    ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ ((convexHull ℝ) ↑S) Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑((convexHull ℝ) ↑S), ↑((convexHull ℝ) ↑S))hg:∀ (α : ι),
  ∀ σ ∈ K.complex.faces,
    ∀ (p : ↑((convexHull ℝ) ↑S)), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)Q:Set ↑((convexHull ℝ) ↑S)p:↑((convexHull ℝ) ↑S)hp:p ∈ QF:∃ S_1, Subtype.val '' Q = (convexHull ℝ) ↑S_1hF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ ((convexHull ℝ) ↑S) ∧ IsExtreme ℝ ((convexHull ℝ) ↑S) (Subtype.val '' Q)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σhw_eq:↑p = ∑ q ∈ σ, w q • qhτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faces⊢ Convex ℝ ((convexHull ℝ) ↑S); exact convex_convexHull ℝ _All goals completed! 🐙;) τ (byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faces⊢ ↑τ ⊆ P
      have := K.space_eqk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.facesthis:K.complex.space = P⊢ ↑τ ⊆ P;
      exact this ▸ K.complex.subset_space hτ_faceAll goals completed! 🐙) w (byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faces⊢ ∀ q ∈ τ, 0 < w q
      exact hτ_posAll goals completed! 🐙) hτ_sum (byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faces⊢ ∑ q ∈ τ, w q • q ∈ Subtype.val '' Q
      grind +qliaAll goals completed! 🐙);
    -- Hence `lbl α '' ↑τ ⊆ F`, so `convexHull ℝ (lbl α '' ↑τ) ⊆ convexHull ℝ F = F` (`convexHull_min`, `F` convex as a convexHull).
    have hlbl_subset_F : lbl α '' (τ : Set (Fin k → ℝ)) ⊆ Subtype.val '' Q := byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'⊢ ∃ g,
  (∀ (α : ι), ρ α = 0 ∨ ∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ p ∈ Q, (g α) p ∈ Q) ∧
    (∀ D ∈ K.complex.facets,
        ∃ A,
          A.card ≤ k ∧ ∀ α ∉ A, ∀ (p p' : ↑P), ↑p ∈ (convexHull ℝ) ↑D → ↑p' ∈ (convexHull ℝ) ↑D → (g α) p = (g α) p') ∧
      ∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)
      grind +splitImpAll goals completed! 🐙;
    have hconvexHull_subset_F : convexHull ℝ (lbl α '' (τ : Set (Fin k → ℝ))) ⊆ Subtype.val '' Q := byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'⊢ ∃ g,
  (∀ (α : ι), ρ α = 0 ∨ ∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ p ∈ Q, (g α) p ∈ Q) ∧
    (∀ D ∈ K.complex.facets,
        ∃ A,
          A.card ≤ k ∧ ∀ α ∉ A, ∀ (p p' : ↑P), ↑p ∈ (convexHull ℝ) ↑D → ↑p' ∈ (convexHull ℝ) ↑D → (g α) p = (g α) p') ∧
      ∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)
      exact convexHull_min hlbl_subset_F ( byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faceshτ_subset_F:↑τ ⊆ Subtype.val '' Qhlbl_subset_F:lbl α '' ↑τ ⊆ Subtype.val '' Q⊢ Convex ℝ (Subtype.val '' Q) rcases F with ⟨ S, hS ⟩k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QhF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faceshτ_subset_F:↑τ ⊆ Subtype.val '' Qhlbl_subset_F:lbl α '' ↑τ ⊆ Subtype.val '' QS:Finset (Fin k → ℝ)hS:Subtype.val '' Q = (convexHull ℝ) ↑S⊢ Convex ℝ (Subtype.val '' Q) ; exact hS.symm ▸ convex_convexHull ℝ _All goals completed! 🐙 );
    have := hg α τ hτ_face p ?_refine'_1.refine_2k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faceshτ_subset_F:↑τ ⊆ Subtype.val '' Qhlbl_subset_F:lbl α '' ↑τ ⊆ Subtype.val '' QhconvexHull_subset_F:(convexHull ℝ) (lbl α '' ↑τ) ⊆ Subtype.val '' Qthis:↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑τ)⊢ (g α) p ∈ Qrefine'_1.refine_1k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faceshτ_subset_F:↑τ ⊆ Subtype.val '' Qhlbl_subset_F:lbl α '' ↑τ ⊆ Subtype.val '' QhconvexHull_subset_F:(convexHull ℝ) (lbl α '' ↑τ) ⊆ Subtype.val '' Q⊢ ↑p ∈ (convexHull ℝ) ↑τ;
    ·refine'_1.refine_2k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faceshτ_subset_F:↑τ ⊆ Subtype.val '' Qhlbl_subset_F:lbl α '' ↑τ ⊆ Subtype.val '' QhconvexHull_subset_F:(convexHull ℝ) (lbl α '' ↑τ) ⊆ Subtype.val '' Qthis:↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑τ)⊢ (g α) p ∈ Q grind +revertAll goals completed! 🐙;
    ·refine'_1.refine_1k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faceshτ_subset_F:↑τ ⊆ Subtype.val '' Qhlbl_subset_F:lbl α '' ↑τ ⊆ Subtype.val '' QhconvexHull_subset_F:(convexHull ℝ) (lbl α '' ↑τ) ⊆ Subtype.val '' Q⊢ ↑p ∈ (convexHull ℝ) ↑τ rw [hτ_eq]refine'_1.refine_1k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faceshτ_subset_F:↑τ ⊆ Subtype.val '' Qhlbl_subset_F:lbl α '' ↑τ ⊆ Subtype.val '' QhconvexHull_subset_F:(convexHull ℝ) (lbl α '' ↑τ) ⊆ Subtype.val '' Q⊢ ∑ q ∈ τ, w q • q ∈ (convexHull ℝ) ↑τ;
      rw [ convexHull_eq ]refine'_1.refine_1k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faceshτ_subset_F:↑τ ⊆ Subtype.val '' Qhlbl_subset_F:lbl α '' ↑τ ⊆ Subtype.val '' QhconvexHull_subset_F:(convexHull ℝ) (lbl α '' ↑τ) ⊆ Subtype.val '' Q⊢ ∑ q ∈ τ, w q • q ∈ {x | ∃ ι t w z, (∀ i ∈ t, 0 ≤ w i) ∧ ∑ i ∈ t, w i = 1 ∧ (∀ i ∈ t, z i ∈ ↑τ) ∧ t.centerMass w z = x};
      refine' ⟨ _, τ, w, fun q => q, _, _, _, _ ⟩refine'_1.refine_1.refine'_1k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faceshτ_subset_F:↑τ ⊆ Subtype.val '' Qhlbl_subset_F:lbl α '' ↑τ ⊆ Subtype.val '' QhconvexHull_subset_F:(convexHull ℝ) (lbl α '' ↑τ) ⊆ Subtype.val '' Q⊢ ∀ i ∈ τ, 0 ≤ w irefine'_1.refine_1.refine'_2k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faceshτ_subset_F:↑τ ⊆ Subtype.val '' Qhlbl_subset_F:lbl α '' ↑τ ⊆ Subtype.val '' QhconvexHull_subset_F:(convexHull ℝ) (lbl α '' ↑τ) ⊆ Subtype.val '' Q⊢ ∑ i ∈ τ, w i = 1refine'_1.refine_1.refine'_3k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faceshτ_subset_F:↑τ ⊆ Subtype.val '' Qhlbl_subset_F:lbl α '' ↑τ ⊆ Subtype.val '' QhconvexHull_subset_F:(convexHull ℝ) (lbl α '' ↑τ) ⊆ Subtype.val '' Q⊢ ∀ i ∈ τ, (fun q => q) i ∈ ↑τrefine'_1.refine_1.refine'_4k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faceshτ_subset_F:↑τ ⊆ Subtype.val '' Qhlbl_subset_F:lbl α '' ↑τ ⊆ Subtype.val '' QhconvexHull_subset_F:(convexHull ℝ) (lbl α '' ↑τ) ⊆ Subtype.val '' Q⊢ (τ.centerMass w fun q => q) = ∑ q ∈ τ, w q • q <;>refine'_1.refine_1.refine'_1k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faceshτ_subset_F:↑τ ⊆ Subtype.val '' Qhlbl_subset_F:lbl α '' ↑τ ⊆ Subtype.val '' QhconvexHull_subset_F:(convexHull ℝ) (lbl α '' ↑τ) ⊆ Subtype.val '' Q⊢ ∀ i ∈ τ, 0 ≤ w irefine'_1.refine_1.refine'_2k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faceshτ_subset_F:↑τ ⊆ Subtype.val '' Qhlbl_subset_F:lbl α '' ↑τ ⊆ Subtype.val '' QhconvexHull_subset_F:(convexHull ℝ) (lbl α '' ↑τ) ⊆ Subtype.val '' Q⊢ ∑ i ∈ τ, w i = 1refine'_1.refine_1.refine'_3k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faceshτ_subset_F:↑τ ⊆ Subtype.val '' Qhlbl_subset_F:lbl α '' ↑τ ⊆ Subtype.val '' QhconvexHull_subset_F:(convexHull ℝ) (lbl α '' ↑τ) ⊆ Subtype.val '' Q⊢ ∀ i ∈ τ, (fun q => q) i ∈ ↑τrefine'_1.refine_1.refine'_4k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faceshτ_subset_F:↑τ ⊆ Subtype.val '' Qhlbl_subset_F:lbl α '' ↑τ ⊆ Subtype.val '' QhconvexHull_subset_F:(convexHull ℝ) (lbl α '' ↑τ) ⊆ Subtype.val '' Q⊢ (τ.centerMass w fun q => q) = ∑ q ∈ τ, w q • q simp +decide [ *, Finset.centerMass ]refine'_1.refine_1.refine'_4k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faceshτ_subset_F:↑τ ⊆ Subtype.val '' Qhlbl_subset_F:lbl α '' ↑τ ⊆ Subtype.val '' QhconvexHull_subset_F:(convexHull ℝ) (lbl α '' ↑τ) ⊆ Subtype.val '' Q⊢ (∑ x ∈ σ with 0 < w x, w x)⁻¹ • ∑ x ∈ σ with 0 < w x, w x • x = ∑ x ∈ σ with 0 < w x, w x • x;
      ·refine'_1.refine_1.refine'_1k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faceshτ_subset_F:↑τ ⊆ Subtype.val '' Qhlbl_subset_F:lbl α '' ↑τ ⊆ Subtype.val '' QhconvexHull_subset_F:(convexHull ℝ) (lbl α '' ↑τ) ⊆ Subtype.val '' Q⊢ ∀ i ∈ σ, 0 < w i → 0 ≤ w i exact fun q hq hq' => le_of_lt hq'All goals completed! 🐙;
      ·refine'_1.refine_1.refine'_2k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faceshτ_subset_F:↑τ ⊆ Subtype.val '' Qhlbl_subset_F:lbl α '' ↑τ ⊆ Subtype.val '' QhconvexHull_subset_F:(convexHull ℝ) (lbl α '' ↑τ) ⊆ Subtype.val '' Q⊢ ∑ x ∈ σ with 0 < w x, w x = 1 exact hτ_sumAll goals completed! 🐙;
      ·refine'_1.refine_1.refine'_4k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)α:ιh:¬ρ α = 0Q:Set ↑Pp:↑Php:p ∈ QF:∃ S, Subtype.val '' Q = (convexHull ℝ) ↑ShF:Subtype.val '' Q ⊆ intrinsicFrontier ℝ P ∧ IsExtreme ℝ P (Subtype.val '' Q)σ:Finset (Fin k → ℝ)hσ:σ ∈ K.complex.faces ∧ ↑p ∈ (convexHull ℝ) ↑σw:(Fin k → ℝ) → ℝhw_nonneg:∀ q ∈ σ, 0 ≤ w qhw_sum:∑ q ∈ σ, w q = 1hw_eq:↑p = ∑ q ∈ σ, w q • qτ:Finset (Fin k → ℝ) := {q ∈ σ | 0 < w q}hτ_def:τ = {q ∈ σ | 0 < w q}hτ_pos:∀ q ∈ τ, 0 < w qhτ_sum:∑ q ∈ τ, w q = 1hτ_eq:↑p = ∑ q ∈ τ, w q • qhτ_face:τ ∈ K.complex.faceshτ_subset_F:↑τ ⊆ Subtype.val '' Qhlbl_subset_F:lbl α '' ↑τ ⊆ Subtype.val '' QhconvexHull_subset_F:(convexHull ℝ) (lbl α '' ↑τ) ⊆ Subtype.val '' Q⊢ (∑ x ∈ σ with 0 < w x, w x)⁻¹ • ∑ x ∈ σ with 0 < w x, w x • x = ∑ x ∈ σ with 0 < w x, w x • x rw [ hτ_sum, inv_one, one_smul ]All goals completed! 🐙;
  ·refine'_2k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)⊢ ∀ D ∈ K.complex.facets,
  ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ (p p' : ↑P), ↑p ∈ (convexHull ℝ) ↑D → ↑p' ∈ (convexHull ℝ) ↑D → (g α) p = (g α) p' intro D hDrefine'_2k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)D:Finset (Fin k → ℝ)hD:D ∈ K.complex.facets⊢ ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ (p p' : ↑P), ↑p ∈ (convexHull ℝ) ↑D → ↑p' ∈ (convexHull ℝ) ↑D → (g α) p = (g α) p';
    obtain ⟨ A, hA₁, hA₂ ⟩ := hcount D hDrefine'_2k:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)D:Finset (Fin k → ℝ)hD:D ∈ K.complex.facetsA:Finset ιhA₁:A.card ≤ khA₂:∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'⊢ ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ (p p' : ↑P), ↑p ∈ (convexHull ℝ) ↑D → ↑p' ∈ (convexHull ℝ) ↑D → (g α) p = (g α) p'
    use A, hA₁rightk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)D:Finset (Fin k → ℝ)hD:D ∈ K.complex.facetsA:Finset ιhA₁:A.card ≤ khA₂:∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'⊢ ∀ α ∉ A, ∀ (p p' : ↑P), ↑p ∈ (convexHull ℝ) ↑D → ↑p' ∈ (convexHull ℝ) ↑D → (g α) p = (g α) p'
    intro α hαrightk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)D:Finset (Fin k → ℝ)hD:D ∈ K.complex.facetsA:Finset ιhA₁:A.card ≤ khA₂:∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'α:ιhα:α ∉ A⊢ ∀ (p p' : ↑P), ↑p ∈ (convexHull ℝ) ↑D → ↑p' ∈ (convexHull ℝ) ↑D → (g α) p = (g α) p' prightk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)D:Finset (Fin k → ℝ)hD:D ∈ K.complex.facetsA:Finset ιhA₁:A.card ≤ khA₂:∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'α:ιhα:α ∉ Ap:↑P⊢ ∀ (p' : ↑P), ↑p ∈ (convexHull ℝ) ↑D → ↑p' ∈ (convexHull ℝ) ↑D → (g α) p = (g α) p' p'rightk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)D:Finset (Fin k → ℝ)hD:D ∈ K.complex.facetsA:Finset ιhA₁:A.card ≤ khA₂:∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'α:ιhα:α ∉ Ap:↑Pp':↑P⊢ ↑p ∈ (convexHull ℝ) ↑D → ↑p' ∈ (convexHull ℝ) ↑D → (g α) p = (g α) p' hprightk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)D:Finset (Fin k → ℝ)hD:D ∈ K.complex.facetsA:Finset ιhA₁:A.card ≤ khA₂:∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'α:ιhα:α ∉ Ap:↑Pp':↑Php:↑p ∈ (convexHull ℝ) ↑D⊢ ↑p' ∈ (convexHull ℝ) ↑D → (g α) p = (g α) p' hp'rightk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)D:Finset (Fin k → ℝ)hD:D ∈ K.complex.facetsA:Finset ιhA₁:A.card ≤ khA₂:∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'α:ιhα:α ∉ Ap:↑Pp':↑Php:↑p ∈ (convexHull ℝ) ↑Dhp':↑p' ∈ (convexHull ℝ) ↑D⊢ (g α) p = (g α) p'
    have h_const : lbl α '' (D : Set (Fin k → ℝ)) = {lbl α (Classical.choose (Finset.card_pos.mp (byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)D:Finset (Fin k → ℝ)hD:D ∈ K.complex.facetsA:Finset ιhA₁:A.card ≤ khA₂:∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'α:ιhα:α ∉ Ap:↑Pp':↑Php:↑p ∈ (convexHull ℝ) ↑Dhp':↑p' ∈ (convexHull ℝ) ↑D⊢ 0 < D.card
    contrapose! hpk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)D:Finset (Fin k → ℝ)hD:D ∈ K.complex.facetsA:Finset ιhA₁:A.card ≤ khA₂:∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'α:ιhα:α ∉ Ap:↑Pp':↑Php':↑p' ∈ (convexHull ℝ) ↑Dhp:D.card ≤ 0⊢ ↑p ∉ (convexHull ℝ) ↑D; aesopAll goals completed! 🐙; : 0 < D.card)))} := byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'⊢ ∃ g,
  (∀ (α : ι), ρ α = 0 ∨ ∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ p ∈ Q, (g α) p ∈ Q) ∧
    (∀ D ∈ K.complex.facets,
        ∃ A,
          A.card ≤ k ∧ ∀ α ∉ A, ∀ (p p' : ↑P), ↑p ∈ (convexHull ℝ) ↑D → ↑p' ∈ (convexHull ℝ) ↑D → (g α) p = (g α) p') ∧
      ∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)
      grindAll goals completed! 🐙
    generalize_proofs at *rightk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'g:ι → C(↑P, ↑P)hg:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)D:Finset (Fin k → ℝ)hD:D ∈ K.complex.facetsA:Finset ιhA₁:A.card ≤ khA₂:∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'α:ιhα:α ∉ Ap:↑Pp':↑Php:↑p ∈ (convexHull ℝ) ↑Dhp':↑p' ∈ (convexHull ℝ) ↑Dpf✝:∃ x, x ∈ Dh_const:lbl α '' ↑D = {lbl α (Classical.choose pf✝)}⊢ (g α) p = (g α) p';
    have h_const : convexHull ℝ (lbl α '' (D : Set (Fin k → ℝ))) = {lbl α (Classical.choose ‹∃ x, x ∈ D›)} := byk:ℕι:Type u_1X:Type u_2inst✝:TopologicalSpace XP:Set (Fin k → ℝ)hP:IsPolyhedron Pρ:PartitionOfUnity ι XK:Subdivision Plbl:ι → (Fin k → ℝ) → Fin k → ℝhlblP:∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, lbl α q ∈ Phface:∀ (α : ι), ρ α ≠ 0 → ∀ (Q : Set (Fin k → ℝ)), IsExtreme ℝ P Q → ∀ σ ∈ K.complex.faces, ∀ q ∈ σ, q ∈ Q → lbl α q ∈ Qhcount:∀ D ∈ K.complex.facets, ∃ A, A.card ≤ k ∧ ∀ α ∉ A, ∀ q ∈ D, ∀ q' ∈ D, lbl α q = lbl α q'⊢ ∃ g,
  (∀ (α : ι), ρ α = 0 ∨ ∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ p ∈ Q, (g α) p ∈ Q) ∧
    (∀ D ∈ K.complex.facets,
        ∃ A,
          A.card ≤ k ∧ ∀ α ∉ A, ∀ (p p' : ↑P), ↑p ∈ (convexHull ℝ) ↑D → ↑p' ∈ (convexHull ℝ) ↑D → (g α) p = (g α) p') ∧
      ∀ (α : ι), ∀ σ ∈ K.complex.faces, ∀ (p : ↑P), ↑p ∈ (convexHull ℝ) ↑σ → ↑((g α) p) ∈ (convexHull ℝ) (lbl α '' ↑σ)
      rw [ h_const, convexHull_singleton ]All goals completed! 🐙;
    grind +suggestionsAll goals completed! 🐙

end FamiliesOfMapsB01
end Submission

So Theorem 3.4 reduces to: if N > k, an admissible labelling family exists on some subdivision of P. This is Step 5.