awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (goal)
Goal:
k : Type u_1
V : Type u_2
P : Type u_3
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
inst_3 : AffineSpace V P
ι : Type u_4
inst_4 : Nontrivial k
p : ι → P
ha : AffineIndependent k p
s₁ s₂ : Set ι
h : Eq (α := Submodule k V) (vectorSpan k (p '' s₁)) (vectorSpan k (p '' s₂))
he : ¬s₁ = s₂
h₁ : ¬s₁.Subsingleton
h₂ : ¬s₂.Subsingleton
hi : (∃ i ∈ s₁, i ∉ s₂) ∨ ∃ i ∈ s₂, i ∉ s₁
⊢ False
have neg : ∀ (i : ι) (his₁ : i ∈ s₁) (his₂ : i ∉ s₂), False
Tactic: rcases hi with ⟨i, his₁, his₂⟩ | ⟨i, his₂, his₁⟩
Cleared hypotheses: hi
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (goal)
Goal:
k : Type u_1
V : Type u_2
P : Type u_3
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
inst_3 : AffineSpace V P
ι : Type u_4
inst_4 : Nontrivial k
p : ι → P
ha : AffineIndependent k p
s₁ s₂ : Set ι
h : Eq (α := Submodule k V) (vectorSpan k (p '' s₁)) (vectorSpan k (p '' s₂))
he : ¬s₁ = s₂
h₁ : ¬s₁.Subsingleton
h₂ : ¬s₂.Subsingleton
hi : (∃ i ∈ s₁, i ∉ s₂) ∨ ∃ i ∈ s₂, i ∉ s₁
neg : ∀ (i : ι) (his₁ : i ∈ s₁) (his₂ : i ∉ s₂), False
⊢ False
have neg₁ : ∀ (i : ι) (his₂ : i ∈ s₂) (his₁ : i ∉ s₁), False
Tactic: rcases hi with ⟨i, his₁, his₂⟩ | ⟨i, his₂, his₁⟩
Cleared hypotheses: hi
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (goal)
Goal:
k : Type u_1
V : Type u_2
P : Type u_3
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
inst_3 : AffineSpace V P
ι : Type u_4
inst_4 : Nontrivial k
p : ι → P
ha : AffineIndependent k p
s₁ s₂ : Set ι
h : s₁ = s₂ ∨ s₁.Subsingleton ∧ s₂.Subsingleton
⊢ Eq (α := Submodule k V) (vectorSpan k (p '' s₁)) (vectorSpan k (p '' s₂))
have mpr : Eq (α := Submodule k V) (vectorSpan k (p '' s₁)) (vectorSpan k (p '' s₁))
Tactic: rcases h with rfl | ⟨h₁, h₂⟩
Cleared hypotheses: s₂, h
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (goal)
Goal:
k : Type u_1
V : Type u_2
P : Type u_3
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
inst_3 : AffineSpace V P
ι : Type u_4
inst_4 : Nontrivial k
p : ι → P
ha : AffineIndependent k p
s₁ s₂ : Set ι
h : s₁ = s₂ ∨ s₁.Subsingleton ∧ s₂.Subsingleton
mpr : Eq (α := Submodule k V) (vectorSpan k (p '' s₁)) (vectorSpan k (p '' s₁))
⊢ Eq (α := Submodule k V) (vectorSpan k (p '' s₁)) (vectorSpan k (p '' s₂))
have mpr₁ : ∀ (h₁ : s₁.Subsingleton) (h₂ : s₂.Subsingleton),
  Eq (α := Submodule k V) (vectorSpan k (p '' s₁)) (vectorSpan k (p '' s₂))
Tactic: rcases h with rfl | ⟨h₁, h₂⟩
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (hypothesis)
Goal:
k : Type u_1
V : Type u_2
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
t : Finset V
h : ¬AffineIndependent k (P := V) (ι := ↥t) (Subtype.val (p := fun (x : V) ↦ x ∈ t))
⊢ ∃ (f : V → k), Eq (α := V) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := k) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
have h₁ : ¬∀ (w : ↥t → k),
    Eq.{u_1 + 1} (α := k) (∑ i : ↥t, w i) 0 →
      Eq.{u_2 + 1} (α := V) ((weightedVSub univ (Subtype.val (p := fun (x : V) ↦ x ∈ t)) : (a : ↥t → k) → V) w : V)
          (0 : V) →
        ∀ (i : ↥t), w i = 0
Tactic: rw [affineIndependent_iff_of_fintype] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (hypothesis)
Goal:
k : Type u_1
V : Type u_2
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
t : Finset V
h :
  ¬∀ (w : ↥t → k),
      Eq.{u_1 + 1} (α := k) (∑ i : ↥t, w i) 0 →
        Eq.{u_2 + 1} (α := V) ((weightedVSub univ (Subtype.val (p := fun (x : V) ↦ x ∈ t)) : (a : ↥t → k) → V) w : V)
            (0 : V) →
          ∀ (i : ↥t), w i = 0
⊢ ∃ (f : V → k), Eq (α := V) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := k) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
have h₁ : ∃ (x : ↥t → k),
  Eq.{u_1 + 1} (α := k) (∑ i : ↥t, x i) 0 ∧
    Eq.{u_2 + 1} (α := V) ((weightedVSub univ (Subtype.val (p := fun (x : V) ↦ x ∈ t)) : (a : ↥t → k) → V) x : V)
        (0 : V) ∧
      ∃ (x_1 : ↥t), ¬x x_1 = 0
Tactic: simp only [exists_prop, not_forall] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (goal)
Goal:
k : Type u_1
V : Type u_2
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
t : Finset V
h :
  ∃ (x : ↥t → k),
    Eq.{u_1 + 1} (α := k) (∑ i : ↥t, x i) 0 ∧
      Eq.{u_2 + 1} (α := V) ((weightedVSub univ (Subtype.val (p := fun (x : V) ↦ x ∈ t)) : (a : ↥t → k) → V) x : V)
          (0 : V) ∧
        ∃ (x_1 : ↥t), ¬x x_1 = 0
⊢ ∃ (f : V → k), Eq (α := V) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := k) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
have obtain : ∀ (w : ↥t → k) (hw : Eq.{u_1 + 1} (α := k) (∑ i : ↥t, w i) 0)
  (hwt :
    Eq.{u_2 + 1} (α := V) ((weightedVSub univ (Subtype.val (p := fun (x : V) ↦ x ∈ t)) : (a : ↥t → k) → V) w : V)
      (0 : V))
  (i : ↥t) (hi : ¬w i = 0),
  ∃ (f : V → k), Eq (α := V) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := k) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
Tactic: obtain ⟨w, hw, hwt, i, hi⟩ := h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (hypothesis)
Goal:
k : Type u_1
V : Type u_2
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
t : Finset V
w : ↥t → k
hw : Eq.{u_1 + 1} (α := k) (∑ i : ↥t, w i) 0
hwt :
  Eq.{u_2 + 1} (α := V) ((weightedVSub univ (Subtype.val (p := fun (x : V) ↦ x ∈ t)) : (a : ↥t → k) → V) w : V) (0 : V)
i : ↥t
hi : ¬w i = 0
⊢ ∃ (f : V → k), Eq (α := V) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := k) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
have hwt₁ : Eq.{u_2 + 1} (α := V) (∑ x : ↥t, w x • (↑x : V)) 0
Tactic: simp only [Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero _ w ((↑) : t → V) hw 0,
      vsub_eq_sub, Finset.weightedVSubOfPoint_apply, sub_zero] at hwt
Cleared hypotheses: hwt
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (goal)
Goal:
k : Type u_1
V : Type u_2
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
t : Finset V
w : ↥t → k
hw : Eq.{u_1 + 1} (α := k) (∑ i : ↥t, w i) 0
i : ↥t
hi : ¬w i = 0
hwt : Eq.{u_2 + 1} (α := V) (∑ x : ↥t, w x • (↑x : V)) 0
f : (x : V) → x ∈ t → k := fun (x : V) (hx : x ∈ t) ↦ w (Subtype.mk (p := fun (x : V) ↦ x ∈ t) x hx)
⊢ ∃ x ∈ t, Ne.{u_1 + 1} (α := k) ((fun (x : V) ↦ (if hx : x ∈ t then f x hx else 0 : k)) x) 0
have h : (↑i : V) ∈ t ∧ Ne.{u_1 + 1} (α := k) ((fun (x : V) ↦ (if hx : x ∈ t then f x hx else 0 : k)) (↑i : V)) 0
Tactic: use i
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (goal)
Goal:
k : Type u_1
V : Type u_2
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
t : Finset V
w : ↥t → k
hw : Eq.{u_1 + 1} (α := k) (∑ i : ↥t, w i) 0
i : ↥t
hi : ¬w i = 0
hwt : Eq.{u_2 + 1} (α := V) (∑ x : ↥t, w x • (↑x : V)) 0
f : (x : V) → x ∈ t → k := fun (x : V) (hx : x ∈ t) ↦ w (Subtype.mk (p := fun (x : V) ↦ x ∈ t) x hx)
⊢ ∃ (f : V → k), Eq (α := V) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := k) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
have refine_1 : Eq (α := V) (∑ e ∈ t, HSMul.hSMul.{u_1, u_2, u_2} (α := k) ((fun (x : V) ↦ (if hx : x ∈ t then f x hx else 0 : k)) e) e)
  0
Tactic: refine ⟨fun x => if hx : x ∈ t then f x hx else (0 : k), ?_, ?_, by use i; simp [f, hi]⟩
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (goal)
Goal:
k : Type u_1
V : Type u_2
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
t : Finset V
w : ↥t → k
hw : Eq.{u_1 + 1} (α := k) (∑ i : ↥t, w i) 0
i : ↥t
hi : ¬w i = 0
hwt : Eq.{u_2 + 1} (α := V) (∑ x : ↥t, w x • (↑x : V)) 0
f : (x : V) → x ∈ t → k := fun (x : V) (hx : x ∈ t) ↦ w (Subtype.mk (p := fun (x : V) ↦ x ∈ t) x hx)
refine_1 :
  Eq (α := V)
    (∑ e ∈ t, HSMul.hSMul.{u_1, u_2, u_2} (α := k) ((fun (x : V) ↦ (if hx : x ∈ t then f x hx else 0 : k)) e) e) 0
⊢ ∃ (f : V → k), Eq (α := V) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := k) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
have refine_2 : Eq.{u_1 + 1} (α := k) (∑ e ∈ t, (fun (x : V) ↦ (if hx : x ∈ t then f x hx else 0 : k)) e) 0
Tactic: refine ⟨fun x => if hx : x ∈ t then f x hx else (0 : k), ?_, ?_, by use i; simp [f, hi]⟩
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (goal)
Goal:
k : Type u_1
V : Type u_2
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
t : Finset V
w : ↥t → k
hw : Eq.{u_1 + 1} (α := k) (∑ i : ↥t, w i) 0
i : ↥t
hi : ¬w i = 0
hwt : Eq.{u_2 + 1} (α := V) (∑ x : ↥t, w x • (↑x : V)) 0
f : (x : V) → x ∈ t → k := fun (x : V) (hx : x ∈ t) ↦ w (Subtype.mk (p := fun (x : V) ↦ x ∈ t) x hx)
this : Eq (α := V) (∑ e ∈ t, if hx : e ∈ t then f e hx • e else 0) 0
⊢ Eq (α := V)
    (∑ e ∈ t, HSMul.hSMul.{u_1, u_2, u_2} (α := k) ((fun (x : V) ↦ (if hx : x ∈ t then f x hx else 0 : k)) e) e) 0
have h : ∀ x ∈ t,
  HSMul.hSMul.{u_1, u_2, u_2} (α := k) ((fun (x : V) ↦ (if hx : x ∈ t then f x hx else 0 : k)) x) x =
    if hx : x ∈ t then f x hx • x else 0
Tactic: convert! this
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (goal)
Goal:
k : Type u_1
V : Type u_2
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
t : Finset V
w : ↥t → k
hw : Eq.{u_1 + 1} (α := k) (∑ i : ↥t, w i) 0
i : ↥t
hi : ¬w i = 0
hwt : Eq.{u_2 + 1} (α := V) (∑ x : ↥t, w x • (↑x : V)) 0
f : (x : V) → x ∈ t → k := fun (x : V) (hx : x ∈ t) ↦ w (Subtype.mk (p := fun (x : V) ↦ x ∈ t) x hx)
this : Eq (α := V) (∑ e ∈ t, if hx : e ∈ t then f e hx • e else 0) 0
x : V
h : x ∈ t
⊢ HSMul.hSMul.{u_1, u_2, u_2} (α := k) ((fun (x : V) ↦ (if hx : x ∈ t then f x hx else 0 : k)) x) x =
    if hx : x ∈ t then f x hx • x else 0
have pos : ∀ (hx : x ∈ t),
  HSMul.hSMul.{u_1, u_2, u_2} (α := k) ((fun (x : V) ↦ (if hx : x ∈ t then f x hx else 0 : k)) x) x =
    if hx : x ∈ t then f x hx • x else 0
Tactic: by_cases hx : x ∈ t
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (goal)
Goal:
k : Type u_1
V : Type u_2
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
t : Finset V
w : ↥t → k
hw : Eq.{u_1 + 1} (α := k) (∑ i : ↥t, w i) 0
i : ↥t
hi : ¬w i = 0
hwt : Eq.{u_2 + 1} (α := V) (∑ x : ↥t, w x • (↑x : V)) 0
f : (x : V) → x ∈ t → k := fun (x : V) (hx : x ∈ t) ↦ w (Subtype.mk (p := fun (x : V) ↦ x ∈ t) x hx)
this : Eq (α := V) (∑ e ∈ t, if hx : e ∈ t then f e hx • e else 0) 0
x : V
h : x ∈ t
pos :
  ∀ (hx : x ∈ t),
    HSMul.hSMul.{u_1, u_2, u_2} (α := k) ((fun (x : V) ↦ (if hx : x ∈ t then f x hx else 0 : k)) x) x =
      if hx : x ∈ t then f x hx • x else 0
⊢ HSMul.hSMul.{u_1, u_2, u_2} (α := k) ((fun (x : V) ↦ (if hx : x ∈ t then f x hx else 0 : k)) x) x =
    if hx : x ∈ t then f x hx • x else 0
have neg : ∀ (hx : x ∉ t),
  HSMul.hSMul.{u_1, u_2, u_2} (α := k) ((fun (x : V) ↦ (if hx : x ∈ t then f x hx else 0 : k)) x) x =
    if hx : x ∈ t then f x hx • x else 0
Tactic: by_cases hx : x ∈ t
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (goal)
Goal:
k : Type u_1
V : Type u_2
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
t : Finset V
w : ↥t → k
hw : Eq.{u_1 + 1} (α := k) (∑ i : ↥t, w i) 0
i : ↥t
hi : ¬w i = 0
hwt : Eq.{u_2 + 1} (α := V) (∑ x : ↥t, w x • (↑x : V)) 0
f : (x : V) → x ∈ t → k := fun (x : V) (hx : x ∈ t) ↦ w (Subtype.mk (p := fun (x : V) ↦ x ∈ t) x hx)
⊢ Eq (α := V)
    (∑ e ∈ t, HSMul.hSMul.{u_1, u_2, u_2} (α := k) ((fun (x : V) ↦ (if hx : x ∈ t then f x hx else 0 : k)) e) e) 0
have refine_1 : Eq (α := V) (∑ e ∈ t, if hx : e ∈ t then f e hx • e else 0) 0
Tactic: suffices (∑ e ∈ t, dite (e ∈ t) (fun hx => f e hx • e) fun _ => 0) = 0 by
        convert! this
        rename V => x
        by_cases hx : x ∈ t <;> simp [hx]
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (goal)
Goal:
k : Type u_1
V : Type u_2
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
t : Finset V
w : ↥t → k
hw : Eq.{u_1 + 1} (α := k) (∑ i : ↥t, w i) 0
i : ↥t
hi : ¬w i = 0
hwt : Eq.{u_2 + 1} (α := V) (∑ x : ↥t, w x • (↑x : V)) 0
f : (x : V) → x ∈ t → k := fun (x : V) (hx : x ∈ t) ↦ w (Subtype.mk (p := fun (x : V) ↦ x ∈ t) x hx)
⊢ Eq (α := V)
    (∑ e ∈ t, HSMul.hSMul.{u_1, u_2, u_2} (α := k) ((fun (x : V) ↦ (if hx : x ∈ t then f x hx else 0 : k)) e) e) 0
have refine_1 : Eq (α := V) (∑ e ∈ t, if hx : e ∈ t then f e hx • e else 0) 0
Tactic: on_goal 1 =>
      suffices (∑ e ∈ t, dite (e ∈ t) (fun hx => f e hx • e) fun _ => 0) = 0 by
        convert! this
        rename V => x
        by_cases hx : x ∈ t <;> simp [hx]
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (goal)
Goal:
k : Type u_1
V : Type u_2
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
t : Finset V
h : ¬AffineIndependent k (P := V) (ι := ↥t) (Subtype.val (p := fun (x : V) ↦ x ∈ t))
⊢ ∃ (f : V → k), Eq (α := V) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := k) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
have rw : ∀
  (h :
    ¬∀ (w : ↥t → k),
        Eq.{u_1 + 1} (α := k) (∑ i : ↥t, w i) 0 →
          Eq.{u_2 + 1} (α := V) ((weightedVSub univ (Subtype.val (p := fun (x : V) ↦ x ∈ t)) : (a : ↥t → k) → V) w : V)
              (0 : V) →
            ∀ (i : ↥t), w i = 0),
  ∃ (f : V → k), Eq (α := V) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := k) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
Tactic: classical
    rw [affineIndependent_iff_of_fintype] at h
    simp only [exists_prop, not_forall] at h
    obtain ⟨w, hw, hwt, i, hi⟩ := h
    simp only [Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero _ w ((↑) : t → V) hw 0,
      vsub_eq_sub, Finset.weightedVSubOfPoint_apply, sub_zero] at hwt
    let f : ∀ x : V, x ∈ t → k := fun x hx => w ⟨x, hx⟩
    refine ⟨fun x => if hx : x ∈ t then f x hx else (0 : k), ?_, ?_, by use i; simp [f, hi]⟩
    on_goal 1 =>
      suffices (∑ e ∈ t, dite (e ∈ t) (fun hx => f e hx • e) fun _ => 0) = 0 by
        convert! this
        rename V => x
        by_cases hx : x ∈ t <;> simp [hx]
    all_goals
      simp only [f, Finset.sum_dite_of_true fun _ h => h, Finset.mk_coe, hwt, hw]
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (goal)
Goal:
k : Type u_1
V : Type u_2
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
t : Finset V
h : ¬AffineIndependent k (P := V) (ι := ↥t) (Subtype.val (p := fun (x : V) ↦ x ∈ t))
rw rw₁ rw₂ :
  ∀
    (h :
      ¬∀ (w : ↥t → k),
          Eq.{u_1 + 1} (α := k) (∑ i : ↥t, w i) 0 →
            Eq.{u_2 + 1} (α := V)
                ((weightedVSub univ (Subtype.val (p := fun (x : V) ↦ x ∈ t)) : (a : ↥t → k) → V) w : V) (0 : V) →
              ∀ (i : ↥t), w i = 0),
    ∃ (f : V → k), Eq (α := V) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := k) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
⊢ ∃ (f : V → k), Eq (α := V) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := k) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
have rw₃ : ∀
  (h :
    ∃ (x : ↥t → k),
      Eq.{u_1 + 1} (α := k) (∑ i : ↥t, x i) 0 ∧
        Eq.{u_2 + 1} (α := V) ((weightedVSub univ (Subtype.val (p := fun (x : V) ↦ x ∈ t)) : (a : ↥t → k) → V) x : V)
            (0 : V) ∧
          ∃ (x_1 : ↥t), ¬x x_1 = 0),
  ∃ (f : V → k), Eq (α := V) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := k) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
Tactic: classical
    rw [affineIndependent_iff_of_fintype] at h
    simp only [exists_prop, not_forall] at h
    obtain ⟨w, hw, hwt, i, hi⟩ := h
    simp only [Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero _ w ((↑) : t → V) hw 0,
      vsub_eq_sub, Finset.weightedVSubOfPoint_apply, sub_zero] at hwt
    let f : ∀ x : V, x ∈ t → k := fun x hx => w ⟨x, hx⟩
    refine ⟨fun x => if hx : x ∈ t then f x hx else (0 : k), ?_, ?_, by use i; simp [f, hi]⟩
    on_goal 1 =>
      suffices (∑ e ∈ t, dite (e ∈ t) (fun hx => f e hx • e) fun _ => 0) = 0 by
        convert! this
        rename V => x
        by_cases hx : x ∈ t <;> simp [hx]
    all_goals
      simp only [f, Finset.sum_dite_of_true fun _ h => h, Finset.mk_coe, hwt, hw]
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (goal)
Goal:
k : Type u_1
V : Type u_2
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
t : Finset V
h : ¬AffineIndependent k (P := V) (ι := ↥t) (Subtype.val (p := fun (x : V) ↦ x ∈ t))
rw rw₁ rw₂ :
  ∀
    (h :
      ¬∀ (w : ↥t → k),
          Eq.{u_1 + 1} (α := k) (∑ i : ↥t, w i) 0 →
            Eq.{u_2 + 1} (α := V)
                ((weightedVSub univ (Subtype.val (p := fun (x : V) ↦ x ∈ t)) : (a : ↥t → k) → V) w : V) (0 : V) →
              ∀ (i : ↥t), w i = 0),
    ∃ (f : V → k), Eq (α := V) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := k) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
rw₃ :
  ∀
    (h :
      ∃ (x : ↥t → k),
        Eq.{u_1 + 1} (α := k) (∑ i : ↥t, x i) 0 ∧
          Eq.{u_2 + 1} (α := V) ((weightedVSub univ (Subtype.val (p := fun (x : V) ↦ x ∈ t)) : (a : ↥t → k) → V) x : V)
              (0 : V) ∧
            ∃ (x_1 : ↥t), ¬x x_1 = 0),
    ∃ (f : V → k), Eq (α := V) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := k) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
⊢ ∃ (f : V → k), Eq (α := V) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := k) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
have rw₄ : ∀ (w : ↥t → k) (hw : Eq.{u_1 + 1} (α := k) (∑ i : ↥t, w i) 0)
  (hwt :
    Eq.{u_2 + 1} (α := V) ((weightedVSub univ (Subtype.val (p := fun (x : V) ↦ x ∈ t)) : (a : ↥t → k) → V) w : V)
      (0 : V))
  (i : ↥t) (hi : ¬w i = 0),
  ∃ (f : V → k), Eq (α := V) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := k) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
Tactic: classical
    rw [affineIndependent_iff_of_fintype] at h
    simp only [exists_prop, not_forall] at h
    obtain ⟨w, hw, hwt, i, hi⟩ := h
    simp only [Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero _ w ((↑) : t → V) hw 0,
      vsub_eq_sub, Finset.weightedVSubOfPoint_apply, sub_zero] at hwt
    let f : ∀ x : V, x ∈ t → k := fun x hx => w ⟨x, hx⟩
    refine ⟨fun x => if hx : x ∈ t then f x hx else (0 : k), ?_, ?_, by use i; simp [f, hi]⟩
    on_goal 1 =>
      suffices (∑ e ∈ t, dite (e ∈ t) (fun hx => f e hx • e) fun _ => 0) = 0 by
        convert! this
        rename V => x
        by_cases hx : x ∈ t <;> simp [hx]
    all_goals
      simp only [f, Finset.sum_dite_of_true fun _ h => h, Finset.mk_coe, hwt, hw]
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (goal)
Goal:
k : Type u_1
V : Type u_2
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
t : Finset V
h : ¬AffineIndependent k (P := V) (ι := ↥t) (Subtype.val (p := fun (x : V) ↦ x ∈ t))
rw rw₁ rw₂ :
  ∀
    (h :
      ¬∀ (w : ↥t → k),
          Eq.{u_1 + 1} (α := k) (∑ i : ↥t, w i) 0 →
            Eq.{u_2 + 1} (α := V)
                ((weightedVSub univ (Subtype.val (p := fun (x : V) ↦ x ∈ t)) : (a : ↥t → k) → V) w : V) (0 : V) →
              ∀ (i : ↥t), w i = 0),
    ∃ (f : V → k), Eq (α := V) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := k) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
rw₃ :
  ∀
    (h :
      ∃ (x : ↥t → k),
        Eq.{u_1 + 1} (α := k) (∑ i : ↥t, x i) 0 ∧
          Eq.{u_2 + 1} (α := V) ((weightedVSub univ (Subtype.val (p := fun (x : V) ↦ x ∈ t)) : (a : ↥t → k) → V) x : V)
              (0 : V) ∧
            ∃ (x_1 : ↥t), ¬x x_1 = 0),
    ∃ (f : V → k), Eq (α := V) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := k) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
rw₄ :
  ∀ (w : ↥t → k) (hw : Eq.{u_1 + 1} (α := k) (∑ i : ↥t, w i) 0)
    (hwt :
      Eq.{u_2 + 1} (α := V) ((weightedVSub univ (Subtype.val (p := fun (x : V) ↦ x ∈ t)) : (a : ↥t → k) → V) w : V)
        (0 : V))
    (i : ↥t) (hi : ¬w i = 0),
    ∃ (f : V → k), Eq (α := V) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := k) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
⊢ ∃ (f : V → k), Eq (α := V) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := k) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
have rw₅ : ∀ (w : ↥t → k) (hw : Eq.{u_1 + 1} (α := k) (∑ i : ↥t, w i) 0) (i : ↥t) (hi : ¬w i = 0)
  (hwt : Eq.{u_2 + 1} (α := V) (∑ x : ↥t, w x • (↑x : V)) 0),
  ∃ (f : V → k), Eq (α := V) (∑ e ∈ t, f e • e) 0 ∧ Eq.{u_1 + 1} (α := k) (∑ e ∈ t, f e) 0 ∧ ∃ x ∈ t, f x ≠ 0
Tactic: classical
    rw [affineIndependent_iff_of_fintype] at h
    simp only [exists_prop, not_forall] at h
    obtain ⟨w, hw, hwt, i, hi⟩ := h
    simp only [Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero _ w ((↑) : t → V) hw 0,
      vsub_eq_sub, Finset.weightedVSubOfPoint_apply, sub_zero] at hwt
    let f : ∀ x : V, x ∈ t → k := fun x hx => w ⟨x, hx⟩
    refine ⟨fun x => if hx : x ∈ t then f x hx else (0 : k), ?_, ?_, by use i; simp [f, hi]⟩
    on_goal 1 =>
      suffices (∑ e ∈ t, dite (e ∈ t) (fun hx => f e hx • e) fun _ => 0) = 0 by
        convert! this
        rename V => x
        by_cases hx : x ∈ t <;> simp [hx]
    all_goals
      simp only [f, Finset.sum_dite_of_true fun _ h => h, Finset.mk_coe, hwt, hw]
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (goal)
Goal:
k : Type u_1
V : Type u_2
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
ι : Type u_4
s : Finset ι
w₁ w₂ : ι → k
p : ι → V
hp : AffineIndependent k p
hw : Eq.{u_1 + 1} (α := k) (∑ i ∈ s, w₁ i) (∑ i ∈ s, w₂ i)
hwp : Eq.{u_2 + 1} (α := V) (∑ i ∈ s, w₁ i • p i) (∑ i ∈ s, w₂ i • p i)
⊢ ∀ i ∈ s, w₁ i = w₂ i
have refine_1 : ∀ (i : ι) (hi : i ∈ s), Eq.{u_1 + 1} (α := k) (∑ i ∈ s, (w₁ - w₂) i) 0
Tactic: refine fun i hi ↦ sub_eq_zero.1 (hp.eq_zero_of_sum_eq_zero (w := w₁ - w₂) ?_ ?_ _ hi)
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (goal)
Goal:
k : Type u_1
V : Type u_2
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
ι : Type u_4
s : Finset ι
w₁ w₂ : ι → k
p : ι → V
hp : AffineIndependent k p
hw : Eq.{u_1 + 1} (α := k) (∑ i ∈ s, w₁ i) (∑ i ∈ s, w₂ i)
hwp : Eq.{u_2 + 1} (α := V) (∑ i ∈ s, w₁ i • p i) (∑ i ∈ s, w₂ i • p i)
refine_1 : ∀ (i : ι) (hi : i ∈ s), Eq.{u_1 + 1} (α := k) (∑ i ∈ s, (w₁ - w₂) i) 0
⊢ ∀ i ∈ s, w₁ i = w₂ i
have refine_2 : ∀ (i : ι) (hi : i ∈ s), Eq.{u_2 + 1} (α := V) (∑ i ∈ s, (w₁ - w₂) i • p i) 0
Tactic: refine fun i hi ↦ sub_eq_zero.1 (hp.eq_zero_of_sum_eq_zero (w := w₁ - w₂) ?_ ?_ _ hi)
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (hypothesis)
Goal:
k : Type u_1
V : Type u_2
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
s : Finset V
hp : AffineIndependent k (P := V) (ι := ↥s) (Subtype.val (p := fun (x : V) ↦ x ∈ s))
w : V → k
hw₀ : Eq.{u_1 + 1} (α := k) (∑ x ∈ s, w x) 0
hw₁ : Eq (α := V) (∑ x ∈ s, w x • x) 0
⊢ ∀ x ∈ s, w x = 0
have hw₂ : Eq.{u_1 + 1} (α := k) (∑ x ∈ s.attach, w (↑x : V)) 0
Tactic: rw [← sum_attach] at hw₀ hw₁
Cleared hypotheses: hw₀, hw₁
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (hypothesis)
Goal:
k : Type u_1
V : Type u_2
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
s : Finset V
hp : AffineIndependent k (P := V) (ι := ↥s) (Subtype.val (p := fun (x : V) ↦ x ∈ s))
w : V → k
hw₀ : Eq.{u_1 + 1} (α := k) (∑ x ∈ s, w x) 0
hw₁ : Eq (α := V) (∑ x ∈ s, w x • x) 0
hw₂ : Eq.{u_1 + 1} (α := k) (∑ x ∈ s.attach, w (↑x : V)) 0
⊢ ∀ x ∈ s, w x = 0
have hw₃ : Eq (α := V) (∑ x ∈ s.attach, w (↑x : V) • (↑x : V)) 0
Tactic: rw [← sum_attach] at hw₀ hw₁
Cleared hypotheses: hw₀, hw₁
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (goal)
Goal:
k : Type u_1
V : Type u_2
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
s : Finset V
hp : AffineIndependent k (P := V) (ι := ↥s) (Subtype.val (p := fun (x : V) ↦ x ∈ s))
w₁ w₂ : V → k
hw : Eq.{u_1 + 1} (α := k) (∑ i ∈ s, w₁ i) (∑ i ∈ s, w₂ i)
hwp : Eq (α := V) (∑ i ∈ s, w₁ i • i) (∑ i ∈ s, w₂ i • i)
⊢ ∀ i ∈ s, w₁ i = w₂ i
have refine_1 : ∀ (i : V) (hi : i ∈ s), Eq.{u_1 + 1} (α := k) (∑ x ∈ s, (w₁ - w₂) x) 0
Tactic: refine fun i hi => sub_eq_zero.1 (hp.eq_zero_of_sum_eq_zero_subtype (w := w₁ - w₂) ?_ ?_ _ hi)
-- Lean 4 proof step from Mathlib.LinearAlgebra.AffineSpace.Independent (goal)
Goal:
k : Type u_1
V : Type u_2
inst : Ring k
inst_1 : AddCommGroup V
inst_2 : Module k V
s : Finset V
hp : AffineIndependent k (P := V) (ι := ↥s) (Subtype.val (p := fun (x : V) ↦ x ∈ s))
w₁ w₂ : V → k
hw : Eq.{u_1 + 1} (α := k) (∑ i ∈ s, w₁ i) (∑ i ∈ s, w₂ i)
hwp : Eq (α := V) (∑ i ∈ s, w₁ i • i) (∑ i ∈ s, w₂ i • i)
refine_1 : ∀ (i : V) (hi : i ∈ s), Eq.{u_1 + 1} (α := k) (∑ x ∈ s, (w₁ - w₂) x) 0
⊢ ∀ i ∈ s, w₁ i = w₂ i
have refine_2 : ∀ (i : V) (hi : i ∈ s), Eq (α := V) (∑ x ∈ s, (w₁ - w₂) x • x) 0
Tactic: refine fun i hi => sub_eq_zero.1 (hp.eq_zero_of_sum_eq_zero_subtype (w := w₁ - w₂) ?_ ?_ _ hi)