awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
hx : Membership.mem (γ := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) s : Set E) x
k : ℕ := #(minCardFinsetOfMemConvexHull (s := s) hx) - 1
hk : #(minCardFinsetOfMemConvexHull (s := s) hx) = k + 1
by_contra by_contra₁ by_contra₂ :
  ∀
    (h :
      ¬AffineIndependent 𝕜 (P := E) (ι := ↥(minCardFinsetOfMemConvexHull (s := s) hx))
          (Subtype.val (p := fun (x_1 : E) ↦ x_1 ∈ minCardFinsetOfMemConvexHull (s := s) hx))),
    False
by_contra₃ :
  ∀
    (h :
      ¬AffineIndependent 𝕜 (P := E) (ι := ↥(minCardFinsetOfMemConvexHull (s := s) hx))
          (Subtype.val (p := fun (x_1 : E) ↦ x_1 ∈ minCardFinsetOfMemConvexHull (s := s) hx)))
    (p : (↑(↑(minCardFinsetOfMemConvexHull (s := s) hx) : Set E) : Type u))
    (hp :
      Membership.mem (γ := Set E)
        ((convexHull 𝕜 : (a : Set E) → Set E) (↑((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E)) : Set E) :
          Set E)
        x),
    False
⊢ AffineIndependent 𝕜 (P := E) (ι := ↥(minCardFinsetOfMemConvexHull (s := s) hx))
    (Subtype.val (p := fun (x_1 : E) ↦ x_1 ∈ minCardFinsetOfMemConvexHull (s := s) hx))
have by_contra₄ : ∀
  (h :
    ¬AffineIndependent 𝕜 (P := E) (ι := ↥(minCardFinsetOfMemConvexHull (s := s) hx))
        (Subtype.val (p := fun (x_1 : E) ↦ x_1 ∈ minCardFinsetOfMemConvexHull (s := s) hx)))
  (p : (↑(↑(minCardFinsetOfMemConvexHull (s := s) hx) : Set E) : Type u))
  (hp :
    Membership.mem (γ := Set E)
      ((convexHull 𝕜 : (a : Set E) → Set E) (↑((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E)) : Set E) :
        Set E)
      x)
  (contra : #(minCardFinsetOfMemConvexHull (s := s) hx) ≤ #((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E))),
  False
Tactic: classical
  by_contra h
  obtain ⟨p, hp⟩ := mem_convexHull_erase h (mem_minCardFinsetOfMemConvexHull hx)
  have contra := minCardFinsetOfMemConvexHull_card_le_card hx (Set.Subset.trans
    (Finset.erase_subset (p : E) (minCardFinsetOfMemConvexHull hx))
    (minCardFinsetOfMemConvexHull_subseteq hx)) hp
  rw [← not_lt] at contra
  apply contra
  rw [card_erase_of_mem p.2, hk]
  exact lt_add_one _
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
hx : Membership.mem (γ := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) s : Set E) x
k : ℕ := #(minCardFinsetOfMemConvexHull (s := s) hx) - 1
hk : #(minCardFinsetOfMemConvexHull (s := s) hx) = k + 1
by_contra by_contra₁ by_contra₂ :
  ∀
    (h :
      ¬AffineIndependent 𝕜 (P := E) (ι := ↥(minCardFinsetOfMemConvexHull (s := s) hx))
          (Subtype.val (p := fun (x_1 : E) ↦ x_1 ∈ minCardFinsetOfMemConvexHull (s := s) hx))),
    False
by_contra₃ :
  ∀
    (h :
      ¬AffineIndependent 𝕜 (P := E) (ι := ↥(minCardFinsetOfMemConvexHull (s := s) hx))
          (Subtype.val (p := fun (x_1 : E) ↦ x_1 ∈ minCardFinsetOfMemConvexHull (s := s) hx)))
    (p : (↑(↑(minCardFinsetOfMemConvexHull (s := s) hx) : Set E) : Type u))
    (hp :
      Membership.mem (γ := Set E)
        ((convexHull 𝕜 : (a : Set E) → Set E) (↑((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E)) : Set E) :
          Set E)
        x),
    False
by_contra₄ :
  ∀
    (h :
      ¬AffineIndependent 𝕜 (P := E) (ι := ↥(minCardFinsetOfMemConvexHull (s := s) hx))
          (Subtype.val (p := fun (x_1 : E) ↦ x_1 ∈ minCardFinsetOfMemConvexHull (s := s) hx)))
    (p : (↑(↑(minCardFinsetOfMemConvexHull (s := s) hx) : Set E) : Type u))
    (hp :
      Membership.mem (γ := Set E)
        ((convexHull 𝕜 : (a : Set E) → Set E) (↑((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E)) : Set E) :
          Set E)
        x)
    (contra :
      #(minCardFinsetOfMemConvexHull (s := s) hx) ≤ #((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E))),
    False
⊢ AffineIndependent 𝕜 (P := E) (ι := ↥(minCardFinsetOfMemConvexHull (s := s) hx))
    (Subtype.val (p := fun (x_1 : E) ↦ x_1 ∈ minCardFinsetOfMemConvexHull (s := s) hx))
have by_contra₅ : ∀
  (h :
    ¬AffineIndependent 𝕜 (P := E) (ι := ↥(minCardFinsetOfMemConvexHull (s := s) hx))
        (Subtype.val (p := fun (x_1 : E) ↦ x_1 ∈ minCardFinsetOfMemConvexHull (s := s) hx)))
  (p : (↑(↑(minCardFinsetOfMemConvexHull (s := s) hx) : Set E) : Type u))
  (hp :
    Membership.mem (γ := Set E)
      ((convexHull 𝕜 : (a : Set E) → Set E) (↑((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E)) : Set E) :
        Set E)
      x)
  (contra :
    ¬#((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E)) < #(minCardFinsetOfMemConvexHull (s := s) hx)),
  False
Tactic: classical
  by_contra h
  obtain ⟨p, hp⟩ := mem_convexHull_erase h (mem_minCardFinsetOfMemConvexHull hx)
  have contra := minCardFinsetOfMemConvexHull_card_le_card hx (Set.Subset.trans
    (Finset.erase_subset (p : E) (minCardFinsetOfMemConvexHull hx))
    (minCardFinsetOfMemConvexHull_subseteq hx)) hp
  rw [← not_lt] at contra
  apply contra
  rw [card_erase_of_mem p.2, hk]
  exact lt_add_one _
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
⊢ Eq (α := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) s : Set E)
    (⋃ (t : Finset E),
        ⋃ (_ : (↑t : Set E) ⊆ s),
          ⋃ (_ : AffineIndependent 𝕜 (P := E) (ι := ↥t) (Subtype.val (p := fun (x : E) ↦ x ∈ t))),
            (convexHull 𝕜 : (a : Set E) → Set E) (↑t : Set E) :
      Set E)
have h₁ : Subset (α := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) s : Set E)
  (⋃ (t : Finset E),
      ⋃ (_ : (↑t : Set E) ⊆ s),
        ⋃ (_ : AffineIndependent 𝕜 (P := E) (ι := ↥t) (Subtype.val (p := fun (x : E) ↦ x ∈ t))),
          (convexHull 𝕜 : (a : Set E) → Set E) (↑t : Set E) :
    Set E)
Tactic: apply Set.Subset.antisymm
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
h₁ :
  Subset (α := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) s : Set E)
    (⋃ (t : Finset E),
        ⋃ (_ : (↑t : Set E) ⊆ s),
          ⋃ (_ : AffineIndependent 𝕜 (P := E) (ι := ↥t) (Subtype.val (p := fun (x : E) ↦ x ∈ t))),
            (convexHull 𝕜 : (a : Set E) → Set E) (↑t : Set E) :
      Set E)
⊢ Eq (α := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) s : Set E)
    (⋃ (t : Finset E),
        ⋃ (_ : (↑t : Set E) ⊆ s),
          ⋃ (_ : AffineIndependent 𝕜 (P := E) (ι := ↥t) (Subtype.val (p := fun (x : E) ↦ x ∈ t))),
            (convexHull 𝕜 : (a : Set E) → Set E) (↑t : Set E) :
      Set E)
have h₂ : Subset.{u} (α := Set E)
  (⋃ (t : Finset E),
    ⋃ (_ : (↑t : Set E) ⊆ s),
      ⋃ (_ : AffineIndependent 𝕜 (P := E) (ι := ↥t) (Subtype.val (p := fun (x : E) ↦ x ∈ t))),
        (convexHull 𝕜 : (a : Set E) → Set E) (↑t : Set E))
  ((convexHull 𝕜 : (a : Set E) → Set E) s)
Tactic: apply Set.Subset.antisymm
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
hx : Membership.mem (γ := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) s : Set E) x
⊢ Membership.mem (γ := Set E)
    (⋃ (t : Finset E),
      ⋃ (_ : (↑t : Set E) ⊆ s),
        ⋃ (_ : AffineIndependent 𝕜 (P := E) (ι := ↥t) (Subtype.val (p := fun (x : E) ↦ x ∈ t))),
          (convexHull 𝕜 : (a : Set E) → Set E) (↑t : Set E))
    x
have h₁ : ∃ (i : Finset E),
  (↑i : Set E) ⊆ s ∧
    AffineIndependent 𝕜 (P := E) (ι := ↥i) (Subtype.val (p := fun (x : E) ↦ x ∈ i)) ∧
      Membership.mem (γ := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) (↑i : Set E) : Set E) x
Tactic: simp only [exists_prop, Set.mem_iUnion]
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
⊢ Subset.{u} (α := Set E)
    (⋃ (t : Finset E),
      ⋃ (_ : (↑t : Set E) ⊆ s),
        ⋃ (_ : AffineIndependent 𝕜 (P := E) (ι := ↥t) (Subtype.val (p := fun (x : E) ↦ x ∈ t))),
          (convexHull 𝕜 : (a : Set E) → Set E) (↑t : Set E))
    ((convexHull 𝕜 : (a : Set E) → Set E) s)
have h₂ : ∀ (i : Finset E),
  Subset.{u} (α := Set E)
    (⋃ (_ : (↑i : Set E) ⊆ s),
      ⋃ (_ : AffineIndependent 𝕜 (P := E) (ι := ↥i) (Subtype.val (p := fun (x : E) ↦ x ∈ i))),
        (convexHull 𝕜 : (a : Set E) → Set E) (↑i : Set E))
    ((convexHull 𝕜 : (a : Set E) → Set E) s)
Tactic: convert! Set.iUnion_subset _
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
i : Finset E
⊢ Subset.{u} (α := Set E)
    (⋃ (_ : (↑i : Set E) ⊆ s),
      ⋃ (_ : AffineIndependent 𝕜 (P := E) (ι := ↥i) (Subtype.val (p := fun (x : E) ↦ x ∈ i))),
        (convexHull 𝕜 : (a : Set E) → Set E) (↑i : Set E))
    ((convexHull 𝕜 : (a : Set E) → Set E) s)
have h₂ : ∀ (i_1 : (↑i : Set E) ⊆ s),
  Subset.{u} (α := Set E)
    (⋃ (_ : AffineIndependent 𝕜 (P := E) (ι := ↥i) (Subtype.val (p := fun (x : E) ↦ x ∈ i))),
      (convexHull 𝕜 : (a : Set E) → Set E) (↑i : Set E))
    ((convexHull 𝕜 : (a : Set E) → Set E) s)
Tactic: convert! Set.iUnion_subset _
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
i : Finset E
i_1 : (↑i : Set E) ⊆ s
⊢ Subset.{u} (α := Set E)
    (⋃ (_ : AffineIndependent 𝕜 (P := E) (ι := ↥i) (Subtype.val (p := fun (x : E) ↦ x ∈ i))),
      (convexHull 𝕜 : (a : Set E) → Set E) (↑i : Set E))
    ((convexHull 𝕜 : (a : Set E) → Set E) s)
have h₂ : ∀ (i_2 : AffineIndependent 𝕜 (P := E) (ι := ↥i) (Subtype.val (p := fun (x : E) ↦ x ∈ i))),
  Subset (α := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) (↑i : Set E) : Set E)
    ((convexHull 𝕜 : (a : Set E) → Set E) s : Set E)
Tactic: convert! Set.iUnion_subset _
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
⊢ Subset.{u} (α := Set E)
    (⋃ (t : Finset E),
      ⋃ (_ : (↑t : Set E) ⊆ s),
        ⋃ (_ : AffineIndependent 𝕜 (P := E) (ι := ↥t) (Subtype.val (p := fun (x : E) ↦ x ∈ t))),
          (convexHull 𝕜 : (a : Set E) → Set E) (↑t : Set E))
    ((convexHull 𝕜 : (a : Set E) → Set E) s)
have h₂ : ∀ (i : Finset E),
  (↑i : Set E) ⊆ s →
    AffineIndependent 𝕜 (P := E) (ι := ↥i) (Subtype.val (p := fun (x : E) ↦ x ∈ i)) →
      Subset (α := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) (↑i : Set E) : Set E)
        ((convexHull 𝕜 : (a : Set E) → Set E) s : Set E)
Tactic: iterate 3 convert! Set.iUnion_subset _; intro
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
hx : Membership.mem (γ := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) s : Set E) x
⊢ ∃ (ι : Type u) (x_1 : Fintype ι) (z : ι → E) (w : ι → 𝕜),
    Set.range z ⊆ s ∧
      AffineIndependent 𝕜 z ∧ (∀ (i : ι), 0 < w i) ∧ Eq.{u_1 + 1} (α := 𝕜) (∑ i : ι, w i) 1 ∧ ∑ i : ι, w i • z i = x
have hx₁ : Membership.mem (γ := Set E)
  (⋃ (t : Finset E),
    ⋃ (_ : (↑t : Set E) ⊆ s),
      ⋃ (_ : AffineIndependent 𝕜 (P := E) (ι := ↥t) (Subtype.val (p := fun (x : E) ↦ x ∈ t))),
        (convexHull 𝕜 : (a : Set E) → Set E) (↑t : Set E))
  x
Tactic: rw [convexHull_eq_union] at hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
hx :
  Membership.mem (γ := Set E)
    (⋃ (t : Finset E),
      ⋃ (_ : (↑t : Set E) ⊆ s),
        ⋃ (_ : AffineIndependent 𝕜 (P := E) (ι := ↥t) (Subtype.val (p := fun (x : E) ↦ x ∈ t))),
          (convexHull 𝕜 : (a : Set E) → Set E) (↑t : Set E))
    x
⊢ ∃ (ι : Type u) (x_1 : Fintype ι) (z : ι → E) (w : ι → 𝕜),
    Set.range z ⊆ s ∧
      AffineIndependent 𝕜 z ∧ (∀ (i : ι), 0 < w i) ∧ Eq.{u_1 + 1} (α := 𝕜) (∑ i : ι, w i) 1 ∧ ∑ i : ι, w i • z i = x
have hx₁ : ∃ (i : Finset E),
  (↑i : Set E) ⊆ s ∧
    AffineIndependent 𝕜 (P := E) (ι := ↥i) (Subtype.val (p := fun (x : E) ↦ x ∈ i)) ∧
      Membership.mem (γ := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) (↑i : Set E) : Set E) x
Tactic: simp only [exists_prop, Set.mem_iUnion] at hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
hx :
  ∃ (i : Finset E),
    (↑i : Set E) ⊆ s ∧
      AffineIndependent 𝕜 (P := E) (ι := ↥i) (Subtype.val (p := fun (x : E) ↦ x ∈ i)) ∧
        Membership.mem (γ := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) (↑i : Set E) : Set E) x
⊢ ∃ (ι : Type u) (x_1 : Fintype ι) (z : ι → E) (w : ι → 𝕜),
    Set.range z ⊆ s ∧
      AffineIndependent 𝕜 z ∧ (∀ (i : ι), 0 < w i) ∧ Eq.{u_1 + 1} (α := 𝕜) (∑ i : ι, w i) 1 ∧ ∑ i : ι, w i • z i = x
have obtain : ∀ (t : Finset E) (ht₁ : (↑t : Set E) ⊆ s)
  (ht₂ : AffineIndependent 𝕜 (P := E) (ι := ↥t) (Subtype.val (p := fun (x : E) ↦ x ∈ t)))
  (ht₃ : Membership.mem (γ := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) (↑t : Set E) : Set E) x),
  ∃ (ι : Type u) (x_1 : Fintype ι) (z : ι → E) (w : ι → 𝕜),
    Set.range z ⊆ s ∧
      AffineIndependent 𝕜 z ∧ (∀ (i : ι), 0 < w i) ∧ Eq.{u_1 + 1} (α := 𝕜) (∑ i : ι, w i) 1 ∧ ∑ i : ι, w i • z i = x
Tactic: obtain ⟨t, ht₁, ht₂, ht₃⟩ := hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
t : Finset E
ht₁ : (↑t : Set E) ⊆ s
ht₂ : AffineIndependent 𝕜 (P := E) (ι := ↥t) (Subtype.val (p := fun (x : E) ↦ x ∈ t))
ht₃ : Membership.mem (γ := Set E) ((convexHull 𝕜 : (a : Set E) → Set E) (↑t : Set E) : Set E) x
⊢ ∃ (ι : Type u) (x_1 : Fintype ι) (z : ι → E) (w : ι → 𝕜),
    Set.range z ⊆ s ∧
      AffineIndependent 𝕜 z ∧ (∀ (i : ι), 0 < w i) ∧ Eq.{u_1 + 1} (α := 𝕜) (∑ i : ι, w i) 1 ∧ ∑ i : ι, w i • z i = x
have ht₄ : ∃ (w : E → 𝕜), (∀ y ∈ t, 0 ≤ w y) ∧ Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, w y) 1 ∧ t.centerMass w id = x
Tactic: simp only [t.convexHull_eq, Set.mem_setOf_eq] at ht₃
Cleared hypotheses: ht₃
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
t : Finset E
ht₁ : (↑t : Set E) ⊆ s
ht₂ : AffineIndependent 𝕜 (P := E) (ι := ↥t) (Subtype.val (p := fun (x : E) ↦ x ∈ t))
ht₃ : ∃ (w : E → 𝕜), (∀ y ∈ t, 0 ≤ w y) ∧ Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, w y) 1 ∧ t.centerMass w id = x
⊢ ∃ (ι : Type u) (x_1 : Fintype ι) (z : ι → E) (w : ι → 𝕜),
    Set.range z ⊆ s ∧
      AffineIndependent 𝕜 z ∧ (∀ (i : ι), 0 < w i) ∧ Eq.{u_1 + 1} (α := 𝕜) (∑ i : ι, w i) 1 ∧ ∑ i : ι, w i • z i = x
have obtain : ∀ (w : E → 𝕜) (hw₁ : ∀ y ∈ t, 0 ≤ w y) (hw₂ : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, w y) 1) (hw₃ : t.centerMass w id = x),
  ∃ (ι : Type u) (x_1 : Fintype ι) (z : ι → E) (w : ι → 𝕜),
    Set.range z ⊆ s ∧
      AffineIndependent 𝕜 z ∧ (∀ (i : ι), 0 < w i) ∧ Eq.{u_1 + 1} (α := 𝕜) (∑ i : ι, w i) 1 ∧ ∑ i : ι, w i • z i = x
Tactic: obtain ⟨w, hw₁, hw₂, hw₃⟩ := ht₃
Cleared hypotheses: ht₃
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
t : Finset E
ht₁ : (↑t : Set E) ⊆ s
ht₂ : AffineIndependent 𝕜 (P := E) (ι := ↥t) (Subtype.val (p := fun (x : E) ↦ x ∈ t))
w : E → 𝕜
hw₁ : ∀ y ∈ t, 0 ≤ w y
hw₂ : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, w y) 1
hw₃ : t.centerMass w id = x
t' : Finset E := {i ∈ t | w i ≠ 0}
⊢ ∃ (ι : Type u) (x_1 : Fintype ι) (z : ι → E) (w : ι → 𝕜),
    Set.range z ⊆ s ∧
      AffineIndependent 𝕜 z ∧ (∀ (i : ι), 0 < w i) ∧ Eq.{u_1 + 1} (α := 𝕜) (∑ i : ι, w i) 1 ∧ ∑ i : ι, w i • z i = x
have refine_1 : Set.range (ι := ↥t') (Subtype.val (p := fun (x : E) ↦ x ∈ t')) ⊆ s
Tactic: refine ⟨t', t'.fintypeCoeSort, ((↑) : t' → E), w ∘ ((↑) : t' → E), ?_, ?_, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
t : Finset E
ht₁ : (↑t : Set E) ⊆ s
ht₂ : AffineIndependent 𝕜 (P := E) (ι := ↥t) (Subtype.val (p := fun (x : E) ↦ x ∈ t))
w : E → 𝕜
hw₁ : ∀ y ∈ t, 0 ≤ w y
hw₂ : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, w y) 1
hw₃ : t.centerMass w id = x
t' : Finset E := {i ∈ t | w i ≠ 0}
refine_1 : Set.range (ι := ↥t') (Subtype.val (p := fun (x : E) ↦ x ∈ t')) ⊆ s
⊢ ∃ (ι : Type u) (x_1 : Fintype ι) (z : ι → E) (w : ι → 𝕜),
    Set.range z ⊆ s ∧
      AffineIndependent 𝕜 z ∧ (∀ (i : ι), 0 < w i) ∧ Eq.{u_1 + 1} (α := 𝕜) (∑ i : ι, w i) 1 ∧ ∑ i : ι, w i • z i = x
have refine_2 : AffineIndependent 𝕜 (P := E) (ι := ↥t') (Subtype.val (p := fun (x : E) ↦ x ∈ t'))
Tactic: refine ⟨t', t'.fintypeCoeSort, ((↑) : t' → E), w ∘ ((↑) : t' → E), ?_, ?_, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
t : Finset E
ht₁ : (↑t : Set E) ⊆ s
ht₂ : AffineIndependent 𝕜 (P := E) (ι := ↥t) (Subtype.val (p := fun (x : E) ↦ x ∈ t))
w : E → 𝕜
hw₁ : ∀ y ∈ t, 0 ≤ w y
hw₂ : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, w y) 1
hw₃ : t.centerMass w id = x
t' : Finset E := {i ∈ t | w i ≠ 0}
refine_1 : Set.range (ι := ↥t') (Subtype.val (p := fun (x : E) ↦ x ∈ t')) ⊆ s
refine_2 : AffineIndependent 𝕜 (P := E) (ι := ↥t') (Subtype.val (p := fun (x : E) ↦ x ∈ t'))
⊢ ∃ (ι : Type u) (x_1 : Fintype ι) (z : ι → E) (w : ι → 𝕜),
    Set.range z ⊆ s ∧
      AffineIndependent 𝕜 z ∧ (∀ (i : ι), 0 < w i) ∧ Eq.{u_1 + 1} (α := 𝕜) (∑ i : ι, w i) 1 ∧ ∑ i : ι, w i • z i = x
have refine_3 : ∀ (i : ↥t'), 0 < (w ∘ Subtype.val (p := fun (x : E) ↦ x ∈ t')) i
Tactic: refine ⟨t', t'.fintypeCoeSort, ((↑) : t' → E), w ∘ ((↑) : t' → E), ?_, ?_, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
t : Finset E
ht₁ : (↑t : Set E) ⊆ s
ht₂ : AffineIndependent 𝕜 (P := E) (ι := ↥t) (Subtype.val (p := fun (x : E) ↦ x ∈ t))
w : E → 𝕜
hw₁ : ∀ y ∈ t, 0 ≤ w y
hw₂ : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, w y) 1
hw₃ : t.centerMass w id = x
t' : Finset E := {i ∈ t | w i ≠ 0}
refine_1 : Set.range (ι := ↥t') (Subtype.val (p := fun (x : E) ↦ x ∈ t')) ⊆ s
refine_2 : AffineIndependent 𝕜 (P := E) (ι := ↥t') (Subtype.val (p := fun (x : E) ↦ x ∈ t'))
refine_3 : ∀ (i : ↥t'), 0 < (w ∘ Subtype.val (p := fun (x : E) ↦ x ∈ t')) i
⊢ ∃ (ι : Type u) (x_1 : Fintype ι) (z : ι → E) (w : ι → 𝕜),
    Set.range z ⊆ s ∧
      AffineIndependent 𝕜 z ∧ (∀ (i : ι), 0 < w i) ∧ Eq.{u_1 + 1} (α := 𝕜) (∑ i : ι, w i) 1 ∧ ∑ i : ι, w i • z i = x
have refine_4 : Eq.{u_1 + 1} (α := 𝕜) (∑ i : ↥t', (w ∘ Subtype.val (p := fun (x : E) ↦ x ∈ t')) i) 1
Tactic: refine ⟨t', t'.fintypeCoeSort, ((↑) : t' → E), w ∘ ((↑) : t' → E), ?_, ?_, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
t : Finset E
ht₁ : (↑t : Set E) ⊆ s
ht₂ : AffineIndependent 𝕜 (P := E) (ι := ↥t) (Subtype.val (p := fun (x : E) ↦ x ∈ t))
w : E → 𝕜
hw₁ : ∀ y ∈ t, 0 ≤ w y
hw₂ : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, w y) 1
hw₃ : t.centerMass w id = x
t' : Finset E := {i ∈ t | w i ≠ 0}
refine_1 : Set.range (ι := ↥t') (Subtype.val (p := fun (x : E) ↦ x ∈ t')) ⊆ s
refine_2 : AffineIndependent 𝕜 (P := E) (ι := ↥t') (Subtype.val (p := fun (x : E) ↦ x ∈ t'))
refine_3 : ∀ (i : ↥t'), 0 < (w ∘ Subtype.val (p := fun (x : E) ↦ x ∈ t')) i
refine_4 : Eq.{u_1 + 1} (α := 𝕜) (∑ i : ↥t', (w ∘ Subtype.val (p := fun (x : E) ↦ x ∈ t')) i) 1
⊢ ∃ (ι : Type u) (x_1 : Fintype ι) (z : ι → E) (w : ι → 𝕜),
    Set.range z ⊆ s ∧
      AffineIndependent 𝕜 z ∧ (∀ (i : ι), 0 < w i) ∧ Eq.{u_1 + 1} (α := 𝕜) (∑ i : ι, w i) 1 ∧ ∑ i : ι, w i • z i = x
have refine_5 : ∑ i : ↥t', (w ∘ Subtype.val (p := fun (x : E) ↦ x ∈ t')) i • (↑i : E) = x
Tactic: refine ⟨t', t'.fintypeCoeSort, ((↑) : t' → E), w ∘ ((↑) : t' → E), ?_, ?_, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
t : Finset E
ht₁ : (↑t : Set E) ⊆ s
ht₂ : AffineIndependent 𝕜 (P := E) (ι := ↥t) (Subtype.val (p := fun (x : E) ↦ x ∈ t))
w : E → 𝕜
hw₁ : ∀ y ∈ t, 0 ≤ w y
hw₂ : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, w y) 1
hw₃ : t.centerMass w id = x
t' : Finset E := {i ∈ t | w i ≠ 0}
⊢ Set.range (ι := ↥t') (Subtype.val (p := fun (x : E) ↦ x ∈ t')) ⊆ s
have refine_1 : {x : E | x ∈ t'} ⊆ s
Tactic: rw [Subtype.range_coe_subtype]
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
t : Finset E
ht₁ : (↑t : Set E) ⊆ s
ht₂ : AffineIndependent 𝕜 (P := E) (ι := ↥t) (Subtype.val (p := fun (x : E) ↦ x ∈ t))
w : E → 𝕜
hw₁ : ∀ y ∈ t, 0 ≤ w y
hw₂ : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, w y) 1
hw₃ : t.centerMass w id = x
t' : Finset E := {i ∈ t | w i ≠ 0}
⊢ Eq.{u_1 + 1} (α := 𝕜) (∑ i : ↥t', (w ∘ Subtype.val (p := fun (x : E) ↦ x ∈ t')) i) 1
have refine_4 : Eq.{u_1 + 1} (α := 𝕜) (∑ x ∈ t'.attach, w (↑x : E)) 1
Tactic: simp only [univ_eq_attach, Function.comp_apply]
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
t : Finset E
ht₁ : (↑t : Set E) ⊆ s
ht₂ : AffineIndependent 𝕜 (P := E) (ι := ↥t) (Subtype.val (p := fun (x : E) ↦ x ∈ t))
w : E → 𝕜
hw₁ : ∀ y ∈ t, 0 ≤ w y
hw₂ : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, w y) 1
hw₃ : t.centerMass w id = x
t' : Finset E := {i ∈ t | w i ≠ 0}
⊢ Eq.{u_1 + 1} (α := 𝕜) (∑ x ∈ t'.attach, w (↑x : E)) 1
have refine_4 : Eq.{u_1 + 1} (α := 𝕜) (∑ x ∈ t', w x) 1
Tactic: rw [Finset.sum_attach, Finset.sum_filter_ne_zero, hw₂]
-- Lean 4 proof step from Mathlib.Analysis.Convex.Caratheodory (goal)
Goal:
𝕜 : Type u_1
E : Type u
inst : Field 𝕜
inst_1 : LinearOrder 𝕜
inst_2 : IsStrictOrderedRing 𝕜
inst_3 : AddCommGroup E
inst_4 : Module 𝕜 E
s : Set E
x : E
t : Finset E
ht₁ : (↑t : Set E) ⊆ s
ht₂ : AffineIndependent 𝕜 (P := E) (ι := ↥t) (Subtype.val (p := fun (x : E) ↦ x ∈ t))
w : E → 𝕜
hw₁ : ∀ y ∈ t, 0 ≤ w y
hw₂ : Eq.{u_1 + 1} (α := 𝕜) (∑ y ∈ t, w y) 1
hw₃ : t.centerMass w id = x
t' : Finset E := {i ∈ t | w i ≠ 0}
refine_4 : Eq.{u_1 + 1} (α := 𝕜) (∑ x ∈ t', w x) 1
refine_4₁ : Eq.{u_1 + 1} (α := 𝕜) (∑ x ∈ t, w x) 1
⊢ Eq.{u_1 + 1} (α := 𝕜) (∑ x ∈ t'.attach, w (↑x : E)) 1
have refine_4₂ : Eq.{u_1 + 1} (α := 𝕜) 1 1
Tactic: rw [Finset.sum_attach, Finset.sum_filter_ne_zero, hw₂]