awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (hypothesis)
Goal:
K : Type u_3
inst : Field K
a b c d : Fin 3 → K
hac : a ⬝ᵥ c = 0
hbc : b ⬝ᵥ c = 0
had : a ⬝ᵥ d = 0
hbd : b ⬝ᵥ d = 0
h :
  ¬(Eq (α := Fin 3 → K)
        ((((crossProduct : (a : Fin 3 → K) → (Fin 3 → K) →ₗ[K] Fin 3 → K) a : (Fin 3 → K) →ₗ[K] Fin 3 → K) :
              (a : Fin 3 → K) → Fin 3 → K)
            b :
          Fin 3 → K)
        (0 : Fin 3 → K) ∨
      Eq (α := Fin 3 → K)
        ((((crossProduct : (a : Fin 3 → K) → (Fin 3 → K) →ₗ[K] Fin 3 → K) c : (Fin 3 → K) →ₗ[K] Fin 3 → K) :
              (a : Fin 3 → K) → Fin 3 → K)
            d :
          Fin 3 → K)
        (0 : Fin 3 → K))
⊢ False
have h₁ : LinearIndependent K ![a, b] ∧ LinearIndependent K ![c, d]
Tactic: simp_rw [not_or, ← ne_eq, crossProduct_ne_zero_iff_linearIndependent] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (hypothesis)
Goal:
K : Type u_3
inst : Field K
a b c d : Fin 3 → K
hac : a ⬝ᵥ c = 0
hbc : b ⬝ᵥ c = 0
had : a ⬝ᵥ d = 0
hbd : b ⬝ᵥ d = 0
h : LinearIndependent K ![a, b] ∧ LinearIndependent K ![c, d]
⊢ False
have h₁ : LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Fin 3 → K)
    (row ((of : (a : Fin (Nat.succ 0).succ → Fin 3 → K) → Matrix (Fin (Nat.succ 0).succ) (Fin 3) K) ![a, b])) ∧
  LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Fin 3 → K)
    (row ((of : (a : Fin (Nat.succ 0).succ → Fin 3 → K) → Matrix (Fin (Nat.succ 0).succ) (Fin 3) K) ![c, d]))
Tactic: rw [← Matrix.of_row (![a, b]), ← Matrix.of_row (![c, d])] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
K : Type u_3
inst : Field K
a b c d : Fin 3 → K
hac : a ⬝ᵥ c = 0
hbc : b ⬝ᵥ c = 0
had : a ⬝ᵥ d = 0
hbd : b ⬝ᵥ d = 0
h :
  LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Fin 3 → K)
      (row ((of : (a : Fin (Nat.succ 0).succ → Fin 3 → K) → Matrix (Fin (Nat.succ 0).succ) (Fin 3) K) ![a, b])) ∧
    LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Fin 3 → K)
      (row ((of : (a : Fin (Nat.succ 0).succ → Fin 3 → K) → Matrix (Fin (Nat.succ 0).succ) (Fin 3) K) ![c, d]))
A : Matrix (Fin 2) (Fin 3) K := (of : (a : Fin 2 → Fin 3 → K) → Matrix (Fin 2) (Fin 3) K) ![a, b]
B : Matrix (Fin 2) (Fin 3) K := (of : (a : Fin 2 → Fin 3 → K) → Matrix (Fin 2) (Fin 3) K) ![c, d]
⊢ A * Bᵀ = 0
have a₁ : ∀ (i j : Fin 2), (A * Bᵀ) i j = 0 i j
Tactic: ext i j
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
K : Type u_3
inst : Field K
a b c d : Fin 3 → K
hac : a ⬝ᵥ c = 0
hbc : b ⬝ᵥ c = 0
had : a ⬝ᵥ d = 0
hbd : b ⬝ᵥ d = 0
h :
  LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Fin 3 → K)
      (row ((of : (a : Fin (Nat.succ 0).succ → Fin 3 → K) → Matrix (Fin (Nat.succ 0).succ) (Fin 3) K) ![a, b])) ∧
    LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Fin 3 → K)
      (row ((of : (a : Fin (Nat.succ 0).succ → Fin 3 → K) → Matrix (Fin (Nat.succ 0).succ) (Fin 3) K) ![c, d]))
A : Matrix (Fin 2) (Fin 3) K := (of : (a : Fin 2 → Fin 3 → K) → Matrix (Fin 2) (Fin 3) K) ![a, b]
B : Matrix (Fin 2) (Fin 3) K := (of : (a : Fin 2 → Fin 3 → K) → Matrix (Fin 2) (Fin 3) K) ![c, d]
i j : Fin 2
⊢ (A * Bᵀ) i j = 0 i j
have a₁ : (A * Bᵀ) ((fun (i : Fin 2) ↦ i) (⟨0, sorry⟩ : Fin 2)) j = 0 ((fun (i : Fin 2) ↦ i) ⟨0, sorry⟩) j
Tactic: fin_cases i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
K : Type u_3
inst : Field K
a b c d : Fin 3 → K
hac : a ⬝ᵥ c = 0
hbc : b ⬝ᵥ c = 0
had : a ⬝ᵥ d = 0
hbd : b ⬝ᵥ d = 0
h :
  LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Fin 3 → K)
      (row ((of : (a : Fin (Nat.succ 0).succ → Fin 3 → K) → Matrix (Fin (Nat.succ 0).succ) (Fin 3) K) ![a, b])) ∧
    LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Fin 3 → K)
      (row ((of : (a : Fin (Nat.succ 0).succ → Fin 3 → K) → Matrix (Fin (Nat.succ 0).succ) (Fin 3) K) ![c, d]))
A : Matrix (Fin 2) (Fin 3) K := (of : (a : Fin 2 → Fin 3 → K) → Matrix (Fin 2) (Fin 3) K) ![a, b]
B : Matrix (Fin 2) (Fin 3) K := (of : (a : Fin 2 → Fin 3 → K) → Matrix (Fin 2) (Fin 3) K) ![c, d]
i j : Fin 2
a₁ : (A * Bᵀ) ((fun (i : Fin 2) ↦ i) (⟨0, ⋯⟩ : Fin 2)) j = 0 ((fun (i : Fin 2) ↦ i) ⟨0, ⋯⟩) j
⊢ (A * Bᵀ) i j = 0 i j
have a₂ : (A * Bᵀ) ((fun (i : Fin 2) ↦ i) (⟨1, sorry⟩ : Fin 2)) j = 0 ((fun (i : Fin 2) ↦ i) ⟨1, sorry⟩) j
Tactic: fin_cases i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
K : Type u_3
inst : Field K
a b c d : Fin 3 → K
hac : a ⬝ᵥ c = 0
hbc : b ⬝ᵥ c = 0
had : a ⬝ᵥ d = 0
hbd : b ⬝ᵥ d = 0
h :
  LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Fin 3 → K)
      (row ((of : (a : Fin (Nat.succ 0).succ → Fin 3 → K) → Matrix (Fin (Nat.succ 0).succ) (Fin 3) K) ![a, b])) ∧
    LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Fin 3 → K)
      (row ((of : (a : Fin (Nat.succ 0).succ → Fin 3 → K) → Matrix (Fin (Nat.succ 0).succ) (Fin 3) K) ![c, d]))
A : Matrix (Fin 2) (Fin 3) K := (of : (a : Fin 2 → Fin 3 → K) → Matrix (Fin 2) (Fin 3) K) ![a, b]
B : Matrix (Fin 2) (Fin 3) K := (of : (a : Fin 2 → Fin 3 → K) → Matrix (Fin 2) (Fin 3) K) ![c, d]
j : Fin 2
⊢ (A * Bᵀ) ((fun (i : Fin 2) ↦ i) (⟨0, ⋯⟩ : Fin 2)) j = 0 ((fun (i : Fin 2) ↦ i) ⟨0, ⋯⟩) j
have a₁ : (A * Bᵀ) ((fun (i : Fin 2) ↦ i) (⟨0, sorry⟩ : Fin 2)) ((fun (i : Fin 2) ↦ i) (⟨0, sorry⟩ : Fin 2)) =
  0 ((fun (i : Fin 2) ↦ i) ⟨0, sorry⟩) ((fun (i : Fin 2) ↦ i) ⟨0, sorry⟩)
Tactic: fin_cases j
Cleared hypotheses: j
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
K : Type u_3
inst : Field K
a b c d : Fin 3 → K
hac : a ⬝ᵥ c = 0
hbc : b ⬝ᵥ c = 0
had : a ⬝ᵥ d = 0
hbd : b ⬝ᵥ d = 0
h :
  LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Fin 3 → K)
      (row ((of : (a : Fin (Nat.succ 0).succ → Fin 3 → K) → Matrix (Fin (Nat.succ 0).succ) (Fin 3) K) ![a, b])) ∧
    LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Fin 3 → K)
      (row ((of : (a : Fin (Nat.succ 0).succ → Fin 3 → K) → Matrix (Fin (Nat.succ 0).succ) (Fin 3) K) ![c, d]))
A : Matrix (Fin 2) (Fin 3) K := (of : (a : Fin 2 → Fin 3 → K) → Matrix (Fin 2) (Fin 3) K) ![a, b]
B : Matrix (Fin 2) (Fin 3) K := (of : (a : Fin 2 → Fin 3 → K) → Matrix (Fin 2) (Fin 3) K) ![c, d]
j : Fin 2
a₁ :
  (A * Bᵀ) ((fun (i : Fin 2) ↦ i) (⟨0, ⋯⟩ : Fin 2)) ((fun (i : Fin 2) ↦ i) (⟨0, ⋯⟩ : Fin 2)) =
    0 ((fun (i : Fin 2) ↦ i) ⟨0, ⋯⟩) ((fun (i : Fin 2) ↦ i) ⟨0, ⋯⟩)
⊢ (A * Bᵀ) ((fun (i : Fin 2) ↦ i) (⟨0, ⋯⟩ : Fin 2)) j = 0 ((fun (i : Fin 2) ↦ i) ⟨0, ⋯⟩) j
have a₂ : (A * Bᵀ) ((fun (i : Fin 2) ↦ i) (⟨0, sorry⟩ : Fin 2)) ((fun (i : Fin 2) ↦ i) (⟨1, sorry⟩ : Fin 2)) =
  0 ((fun (i : Fin 2) ↦ i) ⟨0, sorry⟩) ((fun (i : Fin 2) ↦ i) ⟨1, sorry⟩)
Tactic: fin_cases j
Cleared hypotheses: j
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
K : Type u_3
inst : Field K
a b c d : Fin 3 → K
hac : a ⬝ᵥ c = 0
hbc : b ⬝ᵥ c = 0
had : a ⬝ᵥ d = 0
hbd : b ⬝ᵥ d = 0
h :
  LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Fin 3 → K)
      (row ((of : (a : Fin (Nat.succ 0).succ → Fin 3 → K) → Matrix (Fin (Nat.succ 0).succ) (Fin 3) K) ![a, b])) ∧
    LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Fin 3 → K)
      (row ((of : (a : Fin (Nat.succ 0).succ → Fin 3 → K) → Matrix (Fin (Nat.succ 0).succ) (Fin 3) K) ![c, d]))
A : Matrix (Fin 2) (Fin 3) K := (of : (a : Fin 2 → Fin 3 → K) → Matrix (Fin 2) (Fin 3) K) ![a, b]
B : Matrix (Fin 2) (Fin 3) K := (of : (a : Fin 2 → Fin 3 → K) → Matrix (Fin 2) (Fin 3) K) ![c, d]
j : Fin 2
⊢ (A * Bᵀ) ((fun (i : Fin 2) ↦ i) (⟨1, ⋯⟩ : Fin 2)) j = 0 ((fun (i : Fin 2) ↦ i) ⟨1, ⋯⟩) j
have a₁ : (A * Bᵀ) ((fun (i : Fin 2) ↦ i) (⟨1, sorry⟩ : Fin 2)) ((fun (i : Fin 2) ↦ i) (⟨0, sorry⟩ : Fin 2)) =
  0 ((fun (i : Fin 2) ↦ i) ⟨1, sorry⟩) ((fun (i : Fin 2) ↦ i) ⟨0, sorry⟩)
Tactic: fin_cases j
Cleared hypotheses: j
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
K : Type u_3
inst : Field K
a b c d : Fin 3 → K
hac : a ⬝ᵥ c = 0
hbc : b ⬝ᵥ c = 0
had : a ⬝ᵥ d = 0
hbd : b ⬝ᵥ d = 0
h :
  LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Fin 3 → K)
      (row ((of : (a : Fin (Nat.succ 0).succ → Fin 3 → K) → Matrix (Fin (Nat.succ 0).succ) (Fin 3) K) ![a, b])) ∧
    LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Fin 3 → K)
      (row ((of : (a : Fin (Nat.succ 0).succ → Fin 3 → K) → Matrix (Fin (Nat.succ 0).succ) (Fin 3) K) ![c, d]))
A : Matrix (Fin 2) (Fin 3) K := (of : (a : Fin 2 → Fin 3 → K) → Matrix (Fin 2) (Fin 3) K) ![a, b]
B : Matrix (Fin 2) (Fin 3) K := (of : (a : Fin 2 → Fin 3 → K) → Matrix (Fin 2) (Fin 3) K) ![c, d]
j : Fin 2
a₁ :
  (A * Bᵀ) ((fun (i : Fin 2) ↦ i) (⟨1, ⋯⟩ : Fin 2)) ((fun (i : Fin 2) ↦ i) (⟨0, ⋯⟩ : Fin 2)) =
    0 ((fun (i : Fin 2) ↦ i) ⟨1, ⋯⟩) ((fun (i : Fin 2) ↦ i) ⟨0, ⋯⟩)
⊢ (A * Bᵀ) ((fun (i : Fin 2) ↦ i) (⟨1, ⋯⟩ : Fin 2)) j = 0 ((fun (i : Fin 2) ↦ i) ⟨1, ⋯⟩) j
have a₂ : (A * Bᵀ) ((fun (i : Fin 2) ↦ i) (⟨1, sorry⟩ : Fin 2)) ((fun (i : Fin 2) ↦ i) (⟨1, sorry⟩ : Fin 2)) =
  0 ((fun (i : Fin 2) ↦ i) ⟨1, sorry⟩) ((fun (i : Fin 2) ↦ i) ⟨1, sorry⟩)
Tactic: fin_cases j
Cleared hypotheses: j
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (hypothesis)
Goal:
K : Type u_3
inst : Field K
a b c d : Fin 3 → K
hac : a ⬝ᵥ c = 0
hbc : b ⬝ᵥ c = 0
had : a ⬝ᵥ d = 0
hbd : b ⬝ᵥ d = 0
h :
  LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Fin 3 → K)
      (row ((of : (a : Fin (Nat.succ 0).succ → Fin 3 → K) → Matrix (Fin (Nat.succ 0).succ) (Fin 3) K) ![a, b])) ∧
    LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Fin 3 → K)
      (row ((of : (a : Fin (Nat.succ 0).succ → Fin 3 → K) → Matrix (Fin (Nat.succ 0).succ) (Fin 3) K) ![c, d]))
A : Matrix (Fin 2) (Fin 3) K := (of : (a : Fin 2 → Fin 3 → K) → Matrix (Fin 2) (Fin 3) K) ![a, b]
B : Matrix (Fin 2) (Fin 3) K := (of : (a : Fin 2 → Fin 3 → K) → Matrix (Fin 2) (Fin 3) K) ![c, d]
⊢ False
have hAB : A * Bᵀ = 0
Tactic: have hAB : A * B.transpose = 0 := by
    ext i j
    fin_cases i <;> fin_cases j <;> assumption
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (hypothesis)
Goal:
K : Type u_3
inst : Field K
a b c d : Fin 3 → K
hac : a ⬝ᵥ c = 0
hbc : b ⬝ᵥ c = 0
had : a ⬝ᵥ d = 0
hbd : b ⬝ᵥ d = 0
h :
  LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Fin 3 → K)
      (row ((of : (a : Fin (Nat.succ 0).succ → Fin 3 → K) → Matrix (Fin (Nat.succ 0).succ) (Fin 3) K) ![a, b])) ∧
    LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Fin 3 → K)
      (row ((of : (a : Fin (Nat.succ 0).succ → Fin 3 → K) → Matrix (Fin (Nat.succ 0).succ) (Fin 3) K) ![c, d]))
A : Matrix (Fin 2) (Fin 3) K := (of : (a : Fin 2 → Fin 3 → K) → Matrix (Fin 2) (Fin 3) K) ![a, b]
B : Matrix (Fin 2) (Fin 3) K := (of : (a : Fin 2 → Fin 3 → K) → Matrix (Fin 2) (Fin 3) K) ![c, d]
hAB : A * Bᵀ = 0
⊢ False
have hAB₁ : A.rank + Bᵀ.rank ≤ Fintype.card (Fin 3)
Tactic: replace hAB := rank_add_rank_le_card_of_mul_eq_zero hAB
Cleared hypotheses: hAB
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (hypothesis)
Goal:
K : Type u_3
inst : Field K
a b c d : Fin 3 → K
hac : a ⬝ᵥ c = 0
hbc : b ⬝ᵥ c = 0
had : a ⬝ᵥ d = 0
hbd : b ⬝ᵥ d = 0
h :
  LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Fin 3 → K)
      (row ((of : (a : Fin (Nat.succ 0).succ → Fin 3 → K) → Matrix (Fin (Nat.succ 0).succ) (Fin 3) K) ![a, b])) ∧
    LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Fin 3 → K)
      (row ((of : (a : Fin (Nat.succ 0).succ → Fin 3 → K) → Matrix (Fin (Nat.succ 0).succ) (Fin 3) K) ![c, d]))
A : Matrix (Fin 2) (Fin 3) K := (of : (a : Fin 2 → Fin 3 → K) → Matrix (Fin 2) (Fin 3) K) ![a, b]
B : Matrix (Fin 2) (Fin 3) K := (of : (a : Fin 2 → Fin 3 → K) → Matrix (Fin 2) (Fin 3) K) ![c, d]
hAB : A.rank + Bᵀ.rank ≤ Fintype.card (Fin 3)
⊢ False
have hAB₁ : (Nat.succ 0).succ + (Nat.succ 0).succ ≤ 3
Tactic: rw [rank_transpose, h.1.rank_matrix, h.2.rank_matrix, Fintype.card_fin, Fintype.card_fin] at hAB
Cleared hypotheses: hAB
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
K : Type u_3
inst : Field K
a : Fin 3 → K
ha : a ≠ 0
b : Fin 3 → K
hb : b ≠ 0
c : Fin 3 → K
hc : c ≠ 0
hac : orthogonal (Projectivization.mk K a ha) (Projectivization.mk K c hc)
hbc : orthogonal (Projectivization.mk K b hb) (Projectivization.mk K c hc)
d : Fin 3 → K
hd : d ≠ 0
had : orthogonal (Projectivization.mk K a ha) (Projectivization.mk K d hd)
hbd : orthogonal (Projectivization.mk K b hb) (Projectivization.mk K d hd)
⊢ Eq (α := ℙ K (Fin 3 → K)) (Projectivization.mk K a ha) (Projectivization.mk K b hb) ∨
    Eq (α := ℙ K (Fin 3 → K)) (Projectivization.mk K c hc) (Projectivization.mk K d hd)
have h : Eq (α := Fin 3 → K)
    ((((crossProduct : (a : Fin 3 → K) → (Fin 3 → K) →ₗ[K] Fin 3 → K) a : (Fin 3 → K) →ₗ[K] Fin 3 → K) :
          (a : Fin 3 → K) → Fin 3 → K)
        b :
      Fin 3 → K)
    (0 : Fin 3 → K) ∨
  Eq (α := Fin 3 → K)
    ((((crossProduct : (a : Fin 3 → K) → (Fin 3 → K) →ₗ[K] Fin 3 → K) c : (Fin 3 → K) →ₗ[K] Fin 3 → K) :
          (a : Fin 3 → K) → Fin 3 → K)
        d :
      Fin 3 → K)
    (0 : Fin 3 → K)
Tactic: rw [mk_eq_mk_iff_crossProduct_eq_zero, mk_eq_mk_iff_crossProduct_eq_zero]
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
K : Type u_3
inst : Field K
d : ℙ K (Fin 3 → K)
a : Fin 3 → K
ha : a ≠ 0
had : orthogonal (Projectivization.mk K a ha) d
b : Fin 3 → K
hb : b ≠ 0
hbd : orthogonal (Projectivization.mk K b hb) d
c : Fin 3 → K
hc : c ≠ 0
hac : orthogonal (Projectivization.mk K a ha) (Projectivization.mk K c hc)
hbc : orthogonal (Projectivization.mk K b hb) (Projectivization.mk K c hc)
⊢ Eq (α := ℙ K (Fin 3 → K)) (Projectivization.mk K a ha) (Projectivization.mk K b hb) ∨ Projectivization.mk K c hc = d
have h : ∀ (d : Fin 3 → K) (hd : d ≠ 0) (had : orthogonal (Projectivization.mk K a ha) (Projectivization.mk K d hd))
  (hbd : orthogonal (Projectivization.mk K b hb) (Projectivization.mk K d hd)),
  Eq (α := ℙ K (Fin 3 → K)) (Projectivization.mk K a ha) (Projectivization.mk K b hb) ∨
    Eq (α := ℙ K (Fin 3 → K)) (Projectivization.mk K c hc) (Projectivization.mk K d hd)
Tactic: induction d with | h d hd =>
  rw [mk_eq_mk_iff_crossProduct_eq_zero, mk_eq_mk_iff_crossProduct_eq_zero]
  exact crossProduct_eq_zero_of_dotProduct_eq_zero hac hbc had hbd
Cleared hypotheses: d, had, hbd
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
K : Type u_3
inst : Field K
c d : ℙ K (Fin 3 → K)
a : Fin 3 → K
ha : a ≠ 0
hac : orthogonal (Projectivization.mk K a ha) c
had : orthogonal (Projectivization.mk K a ha) d
b : Fin 3 → K
hb : b ≠ 0
hbc : orthogonal (Projectivization.mk K b hb) c
hbd : orthogonal (Projectivization.mk K b hb) d
⊢ Eq (α := ℙ K (Fin 3 → K)) (Projectivization.mk K a ha) (Projectivization.mk K b hb) ∨ c = d
have h : ∀ (c : Fin 3 → K) (hc : c ≠ 0) (hac : orthogonal (Projectivization.mk K a ha) (Projectivization.mk K c hc))
  (hbc : orthogonal (Projectivization.mk K b hb) (Projectivization.mk K c hc)),
  Eq (α := ℙ K (Fin 3 → K)) (Projectivization.mk K a ha) (Projectivization.mk K b hb) ∨ Projectivization.mk K c hc = d
Tactic: induction c with | h c hc =>
  induction d with | h d hd =>
  rw [mk_eq_mk_iff_crossProduct_eq_zero, mk_eq_mk_iff_crossProduct_eq_zero]
  exact crossProduct_eq_zero_of_dotProduct_eq_zero hac hbc had hbd
Cleared hypotheses: c, hac, hbc
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
K : Type u_3
inst : Field K
b c d : ℙ K (Fin 3 → K)
hbc : b.orthogonal c
hbd : b.orthogonal d
a : Fin 3 → K
ha : a ≠ 0
hac : orthogonal (Projectivization.mk K a ha) c
had : orthogonal (Projectivization.mk K a ha) d
⊢ Projectivization.mk K a ha = b ∨ c = d
have h : ∀ (b : Fin 3 → K) (hb : b ≠ 0) (hbc : orthogonal (Projectivization.mk K b hb) c)
  (hbd : orthogonal (Projectivization.mk K b hb) d),
  Eq (α := ℙ K (Fin 3 → K)) (Projectivization.mk K a ha) (Projectivization.mk K b hb) ∨ c = d
Tactic: induction b with | h b hb =>
  induction c with | h c hc =>
  induction d with | h d hd =>
  rw [mk_eq_mk_iff_crossProduct_eq_zero, mk_eq_mk_iff_crossProduct_eq_zero]
  exact crossProduct_eq_zero_of_dotProduct_eq_zero hac hbc had hbd
Cleared hypotheses: b, hbc, hbd
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
K : Type u_3
inst : Field K
a b c d : ℙ K (Fin 3 → K)
hac : a.orthogonal c
hbc : b.orthogonal c
had : a.orthogonal d
hbd : b.orthogonal d
⊢ a = b ∨ c = d
have h : ∀ (a : Fin 3 → K) (ha : a ≠ 0) (hac : orthogonal (Projectivization.mk K a ha) c)
  (had : orthogonal (Projectivization.mk K a ha) d), Projectivization.mk K a ha = b ∨ c = d
Tactic: induction a with | h a ha =>
  induction b with | h b hb =>
  induction c with | h c hc =>
  induction d with | h d hd =>
  rw [mk_eq_mk_iff_crossProduct_eq_zero, mk_eq_mk_iff_crossProduct_eq_zero]
  exact crossProduct_eq_zero_of_dotProduct_eq_zero hac hbc had hbd
Cleared hypotheses: a, hac, had
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
K : Type u_3
inst_1 : Field K
inst_2 : DecidableEq K
⊢ ∃ (p₁ : ℙ K (Fin 3 → K)) (p₂ : ℙ K (Fin 3 → K)) (p₃ : ℙ K (Fin 3 → K)) (l₁ : ℙ K (Fin 3 → K)) (l₂ : ℙ K (Fin 3 → K))
    (l₃ : ℙ K (Fin 3 → K)), p₁ ∉ l₂ ∧ p₁ ∉ l₃ ∧ p₂ ∉ l₁ ∧ p₂ ∈ l₂ ∧ p₂ ∈ l₃ ∧ p₃ ∉ l₁ ∧ p₃ ∈ l₂ ∧ p₃ ∉ l₃
have refine_1 : Ne.{u_3 + 1} (α := Fin 3 → K) ![0, 1, 1] 0
Tactic: refine ⟨mk K ![0, 1, 1] ?_, mk K ![1, 0, 0] ?_, mk K ![1, 0, 1] ?_, mk K ![1, 0, 0] ?_,
        mk K ![0, 1, 0] ?_, mk K ![0, 0, 1] ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
K : Type u_3
inst_1 : Field K
inst_2 : DecidableEq K
refine_1 : Ne.{u_3 + 1} (α := Fin 3 → K) ![0, 1, 1] 0
⊢ ∃ (p₁ : ℙ K (Fin 3 → K)) (p₂ : ℙ K (Fin 3 → K)) (p₃ : ℙ K (Fin 3 → K)) (l₁ : ℙ K (Fin 3 → K)) (l₂ : ℙ K (Fin 3 → K))
    (l₃ : ℙ K (Fin 3 → K)), p₁ ∉ l₂ ∧ p₁ ∉ l₃ ∧ p₂ ∉ l₁ ∧ p₂ ∈ l₂ ∧ p₂ ∈ l₃ ∧ p₃ ∉ l₁ ∧ p₃ ∈ l₂ ∧ p₃ ∉ l₃
have refine_2 : Ne.{u_3 + 1} (α := Fin 3 → K) ![1, 0, 0] 0
Tactic: refine ⟨mk K ![0, 1, 1] ?_, mk K ![1, 0, 0] ?_, mk K ![1, 0, 1] ?_, mk K ![1, 0, 0] ?_,
        mk K ![0, 1, 0] ?_, mk K ![0, 0, 1] ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
K : Type u_3
inst_1 : Field K
inst_2 : DecidableEq K
refine_1 : Ne.{u_3 + 1} (α := Fin 3 → K) ![0, 1, 1] 0
refine_2 : Ne.{u_3 + 1} (α := Fin 3 → K) ![1, 0, 0] 0
⊢ ∃ (p₁ : ℙ K (Fin 3 → K)) (p₂ : ℙ K (Fin 3 → K)) (p₃ : ℙ K (Fin 3 → K)) (l₁ : ℙ K (Fin 3 → K)) (l₂ : ℙ K (Fin 3 → K))
    (l₃ : ℙ K (Fin 3 → K)), p₁ ∉ l₂ ∧ p₁ ∉ l₃ ∧ p₂ ∉ l₁ ∧ p₂ ∈ l₂ ∧ p₂ ∈ l₃ ∧ p₃ ∉ l₁ ∧ p₃ ∈ l₂ ∧ p₃ ∉ l₃
have refine_3 : Ne.{u_3 + 1} (α := Fin 3 → K) ![1, 0, 1] 0
Tactic: refine ⟨mk K ![0, 1, 1] ?_, mk K ![1, 0, 0] ?_, mk K ![1, 0, 1] ?_, mk K ![1, 0, 0] ?_,
        mk K ![0, 1, 0] ?_, mk K ![0, 0, 1] ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
K : Type u_3
inst_1 : Field K
inst_2 : DecidableEq K
refine_1 : Ne.{u_3 + 1} (α := Fin 3 → K) ![0, 1, 1] 0
refine_2 : Ne.{u_3 + 1} (α := Fin 3 → K) ![1, 0, 0] 0
refine_3 : Ne.{u_3 + 1} (α := Fin 3 → K) ![1, 0, 1] 0
refine_4 : Ne.{u_3 + 1} (α := Fin 3 → K) ![1, 0, 0] 0
⊢ ∃ (p₁ : ℙ K (Fin 3 → K)) (p₂ : ℙ K (Fin 3 → K)) (p₃ : ℙ K (Fin 3 → K)) (l₁ : ℙ K (Fin 3 → K)) (l₂ : ℙ K (Fin 3 → K))
    (l₃ : ℙ K (Fin 3 → K)), p₁ ∉ l₂ ∧ p₁ ∉ l₃ ∧ p₂ ∉ l₁ ∧ p₂ ∈ l₂ ∧ p₂ ∈ l₃ ∧ p₃ ∉ l₁ ∧ p₃ ∈ l₂ ∧ p₃ ∉ l₃
have refine_5 : Ne.{u_3 + 1} (α := Fin 3 → K) ![0, 1, 0] 0
Tactic: refine ⟨mk K ![0, 1, 1] ?_, mk K ![1, 0, 0] ?_, mk K ![1, 0, 1] ?_, mk K ![1, 0, 0] ?_,
        mk K ![0, 1, 0] ?_, mk K ![0, 0, 1] ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
K : Type u_3
inst_1 : Field K
inst_2 : DecidableEq K
refine_1 : Ne.{u_3 + 1} (α := Fin 3 → K) ![0, 1, 1] 0
refine_2 : Ne.{u_3 + 1} (α := Fin 3 → K) ![1, 0, 0] 0
refine_3 : Ne.{u_3 + 1} (α := Fin 3 → K) ![1, 0, 1] 0
refine_4 : Ne.{u_3 + 1} (α := Fin 3 → K) ![1, 0, 0] 0
refine_5 : Ne.{u_3 + 1} (α := Fin 3 → K) ![0, 1, 0] 0
⊢ ∃ (p₁ : ℙ K (Fin 3 → K)) (p₂ : ℙ K (Fin 3 → K)) (p₃ : ℙ K (Fin 3 → K)) (l₁ : ℙ K (Fin 3 → K)) (l₂ : ℙ K (Fin 3 → K))
    (l₃ : ℙ K (Fin 3 → K)), p₁ ∉ l₂ ∧ p₁ ∉ l₃ ∧ p₂ ∉ l₁ ∧ p₂ ∈ l₂ ∧ p₂ ∈ l₃ ∧ p₃ ∉ l₁ ∧ p₃ ∈ l₂ ∧ p₃ ∉ l₃
have refine_6 : Ne.{u_3 + 1} (α := Fin 3 → K) ![0, 0, 1] 0
Tactic: refine ⟨mk K ![0, 1, 1] ?_, mk K ![1, 0, 0] ?_, mk K ![1, 0, 1] ?_, mk K ![1, 0, 0] ?_,
        mk K ![0, 1, 0] ?_, mk K ![0, 0, 1] ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
K : Type u_3
inst_1 : Field K
inst_2 : DecidableEq K
refine_1 : Ne.{u_3 + 1} (α := Fin 3 → K) ![0, 1, 1] 0
refine_2 : Ne.{u_3 + 1} (α := Fin 3 → K) ![1, 0, 0] 0
refine_3 : Ne.{u_3 + 1} (α := Fin 3 → K) ![1, 0, 1] 0
refine_4 : Ne.{u_3 + 1} (α := Fin 3 → K) ![1, 0, 0] 0
refine_5 : Ne.{u_3 + 1} (α := Fin 3 → K) ![0, 1, 0] 0
refine_6 : Ne.{u_3 + 1} (α := Fin 3 → K) ![0, 0, 1] 0
⊢ ∃ (p₁ : ℙ K (Fin 3 → K)) (p₂ : ℙ K (Fin 3 → K)) (p₃ : ℙ K (Fin 3 → K)) (l₁ : ℙ K (Fin 3 → K)) (l₂ : ℙ K (Fin 3 → K))
    (l₃ : ℙ K (Fin 3 → K)), p₁ ∉ l₂ ∧ p₁ ∉ l₃ ∧ p₂ ∉ l₁ ∧ p₂ ∈ l₂ ∧ p₂ ∈ l₃ ∧ p₃ ∉ l₁ ∧ p₃ ∈ l₂ ∧ p₃ ∉ l₃
have refine_7 : Projectivization.mk K ![0, 1, 1] sorry ∉ Projectivization.mk K ![0, 1, 0] sorry ∧
  Projectivization.mk K ![0, 1, 1] sorry ∉ Projectivization.mk K ![0, 0, 1] sorry ∧
    Projectivization.mk K ![1, 0, 0] sorry ∉ Projectivization.mk K ![1, 0, 0] sorry ∧
      Membership.mem (γ := ℙ K (Fin 3 → K)) (Projectivization.mk K ![0, 1, 0] sorry) (Projectivization.mk K ![1, 0, 0] sorry) ∧
        Membership.mem (γ := ℙ K (Fin 3 → K)) (Projectivization.mk K ![0, 0, 1] sorry)
            (Projectivization.mk K ![1, 0, 0] sorry) ∧
          Projectivization.mk K ![1, 0, 1] sorry ∉ Projectivization.mk K ![1, 0, 0] sorry ∧
            Membership.mem (γ := ℙ K (Fin 3 → K)) (Projectivization.mk K ![0, 1, 0] sorry)
                (Projectivization.mk K ![1, 0, 1] sorry) ∧
              Projectivization.mk K ![1, 0, 1] sorry ∉ Projectivization.mk K ![0, 0, 1] sorry
Tactic: refine ⟨mk K ![0, 1, 1] ?_, mk K ![1, 0, 0] ?_, mk K ![1, 0, 1] ?_, mk K ![1, 0, 0] ?_,
        mk K ![0, 1, 0] ?_, mk K ![0, 0, 1] ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Combinatorics.Derangements.Basic (goal)
Goal:
α : Type u_1
β : Type u_2
p : α → Prop
inst : DecidablePred p
⊢ (↑(derangements (Subtype p)) : Type u_1) ≃
    { f : { f : Perm α // ∀ (a : α), ¬p a → Membership.mem (γ := Set α) (fixedPoints (⇑f : (a : α) → α)) a } //
      ∀ (a : α), Membership.mem (γ := Set α) (fixedPoints (⇑(↑f : Perm α) : (a : α) → α)) a → ¬p a }
have refine_1 : ∀ (f : Perm (Subtype p)) (hf : Membership.mem (γ := Set (Perm (Subtype p))) (derangements (Subtype p)) f) (a : α)
  (hfa :
    Membership.mem (γ := Set α)
      (fixedPoints
        (⇑(↑((Perm.subtypeEquivSubtypePerm p :
                    (a : Perm (Subtype p)) → { f : Perm α // ∀ (a : α), ¬p a → (f : (a : α) → α) a = a })
                  f) :
              Perm α) :
          (a : α) → α))
      a)
  (ha : p a), False
Tactic: refine (Perm.subtypeEquivSubtypePerm p).subtypeEquiv fun f => ⟨fun hf a hfa ha => ?_, ?_⟩