awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Analysis.Convex.Caratheodory
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
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)
apply : #((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E)) < #(minCardFinsetOfMemConvexHull (s := s) hx)
⊢ False

Tactic:
apply contra
-- Module: Mathlib.Analysis.Convex.Caratheodory
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
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)
rw : k + 1 - 1 < k + 1
⊢ #((minCardFinsetOfMemConvexHull (s := s) hx).erase (↑p : E)) < #(minCardFinsetOfMemConvexHull (s := s) hx)

Tactic:
rw [card_erase_of_mem p.2, hk]

Premises used: Finset.card_erase_of_mem, Subtype.property
-- Module: Mathlib.Analysis.Convex.Caratheodory
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
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)
⊢ k + 1 - 1 < k + 1

Tactic:
lt_add_one _

Premises used: lt_add_one, Finset.card, Multiset.card, Nat.pred
-- Module: Mathlib.Analysis.Convex.Caratheodory
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
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)
⊢ k + 1 - 1 < k + 1

Tactic:
exact lt_add_one _

Premises used: lt_add_one
-- Module: Mathlib.Analysis.Convex.Caratheodory
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)
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)
⊢ 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)

Tactic:
apply Set.Subset.antisymm

Premises used: Set.Subset.antisymm
-- Module: Mathlib.Analysis.Convex.Caratheodory
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
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
⊢ 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:
simp only [exists_prop, Set.mem_iUnion]

Premises used: Set.mem_iUnion, exists_prop
-- Module: Mathlib.Analysis.Convex.Caratheodory
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
⊢ ∃ (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:
⟨Caratheodory.minCardFinsetOfMemConvexHull hx,
      Caratheodory.minCardFinsetOfMemConvexHull_subseteq hx,
      Caratheodory.affineIndependent_minCardFinsetOfMemConvexHull hx,
      Caratheodory.mem_minCardFinsetOfMemConvexHull hx⟩

Premises used: Caratheodory.mem_minCardFinsetOfMemConvexHull, Caratheodory.minCardFinsetOfMemConvexHull, Caratheodory.affineIndependent_minCardFinsetOfMemConvexHull, And.intro, Exists.intro, Caratheodory.minCardFinsetOfMemConvexHull_subseteq
-- Module: Mathlib.Analysis.Convex.Caratheodory
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
⊢ ∃ (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:
exact ⟨Caratheodory.minCardFinsetOfMemConvexHull hx,
      Caratheodory.minCardFinsetOfMemConvexHull_subseteq hx,
      Caratheodory.affineIndependent_minCardFinsetOfMemConvexHull hx,
      Caratheodory.mem_minCardFinsetOfMemConvexHull hx⟩
-- Module: Mathlib.Analysis.Convex.Caratheodory
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)
    (⋃ (i : Finset 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:
Set.iUnion_subset _

Premises used: Set.iUnion_subset
-- Module: Mathlib.Analysis.Convex.Caratheodory
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
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)
⊢ 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:
convert! Set.iUnion_subset _

Premises used: Set.iUnion_subset
-- Module: Mathlib.Analysis.Convex.Caratheodory
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)

Tactic:
Set.iUnion_subset _

Premises used: Set.iUnion_subset
-- Module: Mathlib.Analysis.Convex.Caratheodory
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
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)
⊢ 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 _

Premises used: Set.iUnion_subset
-- Module: Mathlib.Analysis.Convex.Caratheodory
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)

Tactic:
Set.iUnion_subset _

Premises used: Set.iUnion_subset
-- Module: Mathlib.Analysis.Convex.Caratheodory
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
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)
⊢ 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 _

Premises used: Set.iUnion_subset
-- Module: Mathlib.Analysis.Convex.Caratheodory
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
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)
⊢ 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:
iterate 3 convert! Set.iUnion_subset _; intro

Premises used: Set.iUnion_subset
-- Module: Mathlib.Analysis.Convex.Caratheodory
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
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:
convexHull_mono ‹_›

Premises used: convexHull_mono
-- Module: Mathlib.Analysis.Convex.Caratheodory
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
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:
exact convexHull_mono ‹_›

Premises used: convexHull_mono
-- Module: Mathlib.Analysis.Convex.Caratheodory
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

Tactic:
rw [convexHull_eq_union] at hx

Premises used: convexHull_eq_union
-- Module: Mathlib.Analysis.Convex.Caratheodory
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
⊢ ∃ (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

Premises used: Set.mem_iUnion, exists_prop
-- Module: Mathlib.Analysis.Convex.Caratheodory
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
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
⊢ ∃ (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
-- Module: Mathlib.Analysis.Convex.Caratheodory
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
⊢ ∃ (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

Premises used: Set.mem_setOf_eq, Finset.convexHull_eq
-- Module: Mathlib.Analysis.Convex.Caratheodory
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
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
⊢ ∃ (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
-- Module: Mathlib.Analysis.Convex.Caratheodory
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

Tactic:
⟨t', t'.fintypeCoeSort, ((↑) : t' → E), w ∘ ((↑) : t' → E), ?_, ?_, ?_, ?_, ?_⟩

Premises used: Function.comp, Membership.mem, Subtype, Finset.fintypeCoeSort, And.intro, Exists.intro, Subtype.val
-- Module: Mathlib.Analysis.Convex.Caratheodory
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
refine_5 : ∑ i : ↥t', (w ∘ Subtype.val (p := fun (x : E) ↦ x ∈ t')) i • (↑i : 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:
refine ⟨t', t'.fintypeCoeSort, ((↑) : t' → E), w ∘ ((↑) : t' → E), ?_, ?_, ?_, ?_, ?_⟩
-- Module: Mathlib.Analysis.Convex.Caratheodory
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 : {x : E | x ∈ t'} ⊆ s
⊢ Set.range (ι := ↥t') (Subtype.val (p := fun (x : E) ↦ x ∈ t')) ⊆ s

Tactic:
rw [Subtype.range_coe_subtype]

Premises used: Subtype.range_coe_subtype
-- Module: Mathlib.Analysis.Convex.Caratheodory
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}
⊢ {x : E | x ∈ t'} ⊆ s

Tactic:
Subset.trans (Finset.filter_subset _ t) ht₁

Premises used: Set.Subset.trans, Finset.filter_subset
-- Module: Mathlib.Analysis.Convex.Caratheodory
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}
⊢ {x : E | x ∈ t'} ⊆ s

Tactic:
exact Subset.trans (Finset.filter_subset _ t) ht₁
-- Module: Mathlib.Analysis.Convex.Caratheodory
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}
⊢ AffineIndependent 𝕜 (P := E) (ι := ↥t') (Subtype.val (p := fun (x : E) ↦ x ∈ t'))

Tactic:
ht₂.comp_embedding ⟨_, inclusion_injective (Finset.filter_subset (fun i => w i ≠ 0) t)⟩

Premises used: Set.Mem, AffineIndependent.comp_embedding, setOf, Function.Embedding.mk, Finset.filter_subset, Ne, Set.inclusion_injective, OfNat.ofNat
-- Module: Mathlib.Analysis.Convex.Caratheodory
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}
⊢ AffineIndependent 𝕜 (P := E) (ι := ↥t') (Subtype.val (p := fun (x : E) ↦ x ∈ t'))

Tactic:
exact ht₂.comp_embedding ⟨_, inclusion_injective (Finset.filter_subset (fun i => w i ≠ 0) t)⟩

Premises used: AffineIndependent.comp_embedding, Function.Embedding.mk, Finset.filter_subset, Ne, Set.inclusion_injective, OfNat.ofNat