awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Infinite K
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
⊢ False
have H₁ : Set.Infinite (α := K) Set.univ
Tactic: rw [← Set.infinite_univ_iff] at H
Cleared hypotheses: H, h
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
⊢ False
have a : Set.Finite (α := K) Set.univ
Tactic: apply H
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
this :
  Eq (α := Set K) Set.univ
    (⋃ (x : K),
      ⋃ (_ : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x), {y : K | x = y + y⁻¹ ∨ y = 0})
⊢ Set.Finite (α := K) Set.univ
have rw : Set.Finite (α := K)
  (⋃ (x : K),
    ⋃ (_ : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x), {y : K | x = y + y⁻¹ ∨ y = 0})
Tactic: rw [this]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
⊢ Set.Finite (α := K)
    (⋃ (x : K),
      ⋃ (_ : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x), {y : K | x = y + y⁻¹ ∨ y = 0})
have refine : ∀ (x : K),
  Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x →
    Set.Finite (α := K) {y : K | x = y + y⁻¹ ∨ y = 0}
Tactic: refine h.biUnion fun x _ => ?_
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H_1 : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
x : K
h_1 : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x
φ : K[X] := HPow.hPow (α := K[X]) (β := ℕ) X 2 - HMul.hMul (α := K[X]) (β := K[X]) ((C : (a : K) → K[X]) x : K[X]) X + 1
H : φ = 0
⊢ eval 0 φ = 0
have rw : Eq (α := K) (eval 0 0) 0
Tactic: rw [H, eval_zero]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H_1 : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
x : K
h_1 : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x
φ : K[X] := HPow.hPow (α := K[X]) (β := ℕ) X 2 - HMul.hMul (α := K[X]) (β := K[X]) ((C : (a : K) → K[X]) x : K[X]) X + 1
H : φ = 0
rw : Eq (α := K) (eval 0 0) 0
⊢ eval 0 φ = 0
have rw₁ : Eq (α := K) 0 0
Tactic: rw [H, eval_zero]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H_1 : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
x : K
h_1 : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x
φ : K[X] := HPow.hPow (α := K[X]) (β := ℕ) X 2 - HMul.hMul (α := K[X]) (β := K[X]) ((C : (a : K) → K[X]) x : K[X]) X + 1
H : φ = 0
⊢ False
have this : eval 0 φ = 0
Tactic: have : φ.eval 0 = 0 := by rw [H, eval_zero]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
x : K
h_1 : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x
φ : K[X] := HPow.hPow (α := K[X]) (β := ℕ) X 2 - HMul.hMul (α := K[X]) (β := K[X]) ((C : (a : K) → K[X]) x : K[X]) X + 1
⊢ Set.Finite (α := K) {y : K | x = y + y⁻¹ ∨ y = 0}
have hφ : φ ≠ 0
Tactic: have hφ : φ ≠ 0 := by
        intro H
        have : φ.eval 0 = 0 := by rw [H, eval_zero]
        simpa [φ, eval_X, eval_one, eval_pow, eval_sub, sub_zero, eval_add, eval_mul,
          mul_zero, sq, zero_add, one_ne_zero]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
x : K
h_1 : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x
φ : K[X] := HPow.hPow (α := K[X]) (β := ℕ) X 2 - HMul.hMul (α := K[X]) (β := K[X]) ((C : (a : K) → K[X]) x : K[X]) X + 1
hφ : φ ≠ 0
⊢ Set.Finite (α := K) {y : K | x = y + y⁻¹ ∨ y = 0}
have h₁ : {y : K | x = y + y⁻¹ ∨ y = 0} = (↑(φ.roots ∪ {0}).toFinset : Set K)
Tactic: convert! (φ.roots ∪ {0}).toFinset.finite_toSet using 1
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
x : K
h_1 : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x
φ : K[X] := HPow.hPow (α := K[X]) (β := ℕ) X 2 - HMul.hMul (α := K[X]) (β := K[X]) ((C : (a : K) → K[X]) x : K[X]) X + 1
hφ : φ ≠ 0
⊢ {y : K | x = y + y⁻¹ ∨ y = 0} = (↑(φ.roots ∪ {0}).toFinset : Set K)
have h₁ : ∀ (y : K), Membership.mem (γ := Set K) {y : K | x = y + y⁻¹ ∨ y = 0} y ↔ y ∈ (↑(φ.roots ∪ {0}).toFinset : Set K)
Tactic: ext1 y
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
x : K
h_1 : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x
φ : K[X] := HPow.hPow (α := K[X]) (β := ℕ) X 2 - HMul.hMul (α := K[X]) (β := K[X]) ((C : (a : K) → K[X]) x : K[X]) X + 1
hφ : φ ≠ 0
y : K
⊢ Membership.mem (γ := Set K) {y : K | x = y + y⁻¹ ∨ y = 0} y ↔ y ∈ (↑(φ.roots ∪ {0}).toFinset : Set K)
have h₁ : x = y + y⁻¹ ∨ y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0 ∨ y = 0
Tactic: simp only [φ, Multiset.mem_toFinset, Set.mem_setOf_eq, Finset.mem_coe, Multiset.mem_union,
          mem_roots hφ, IsRoot, eval_add, eval_sub, eval_pow, eval_mul, eval_X, eval_C, eval_one,
          Multiset.mem_singleton]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
x : K
h_1 : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x
φ : K[X] := HPow.hPow (α := K[X]) (β := ℕ) X 2 - HMul.hMul (α := K[X]) (β := K[X]) ((C : (a : K) → K[X]) x : K[X]) X + 1
hφ : φ ≠ 0
y : K
⊢ x = y + y⁻¹ ∨ y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0 ∨ y = 0
have pos : ∀ (hy : y = 0), x = y + y⁻¹ ∨ y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0 ∨ y = 0
Tactic: by_cases hy : y = 0
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
x : K
h_1 : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x
φ : K[X] := HPow.hPow (α := K[X]) (β := ℕ) X 2 - HMul.hMul (α := K[X]) (β := K[X]) ((C : (a : K) → K[X]) x : K[X]) X + 1
hφ : φ ≠ 0
y : K
pos : ∀ (hy : y = 0), x = y + y⁻¹ ∨ y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0 ∨ y = 0
⊢ x = y + y⁻¹ ∨ y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0 ∨ y = 0
have neg : ∀ (hy : ¬y = 0), x = y + y⁻¹ ∨ y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0 ∨ y = 0
Tactic: by_cases hy : y = 0
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
x : K
h_1 : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x
φ : K[X] := HPow.hPow (α := K[X]) (β := ℕ) X 2 - HMul.hMul (α := K[X]) (β := K[X]) ((C : (a : K) → K[X]) x : K[X]) X + 1
hφ : φ ≠ 0
y : K
hy : ¬y = 0
⊢ x = y + y⁻¹ ∨ y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0 ∨ y = 0
have apply : x = y + y⁻¹ ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0
Tactic: apply or_congr _ Iff.rfl
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
x : K
h_1 : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x
φ : K[X] := HPow.hPow (α := K[X]) (β := ℕ) X 2 - HMul.hMul (α := K[X]) (β := K[X]) ((C : (a : K) → K[X]) x : K[X]) X + 1
hφ : φ ≠ 0
y : K
hy : ¬y = 0
⊢ x = y + y⁻¹ ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0
have rw : y * y + 1 - x * y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0
Tactic: rw [← mul_left_inj' hy, eq_comm, ← sub_eq_zero, add_mul, inv_mul_cancel₀ hy]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
x : K
h_1 : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x
φ : K[X] := HPow.hPow (α := K[X]) (β := ℕ) X 2 - HMul.hMul (α := K[X]) (β := K[X]) ((C : (a : K) → K[X]) x : K[X]) X + 1
hφ : φ ≠ 0
y : K
hy : ¬y = 0
⊢ y * y + 1 - x * y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0
have a : y * y + 1 - x * y = HPow.hPow (β := ℕ) y 2 - x * y + 1
Tactic: apply eq_iff_eq_cancel_right.mpr
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
x : K
h_1 : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x
φ : K[X] := HPow.hPow (α := K[X]) (β := ℕ) X 2 - HMul.hMul (α := K[X]) (β := K[X]) ((C : (a : K) → K[X]) x : K[X]) X + 1
hφ : φ ≠ 0
⊢ Set.Finite (α := K) {y : K | x = y + y⁻¹ ∨ y = 0}
have h₁ : {y : K | x = y + y⁻¹ ∨ y = 0} = (↑(φ.roots ∪ {0}).toFinset : Set K)
Tactic: classical
        convert! (φ.roots ∪ {0}).toFinset.finite_toSet using 1
        ext1 y
        simp only [φ, Multiset.mem_toFinset, Set.mem_setOf_eq, Finset.mem_coe, Multiset.mem_union,
          mem_roots hφ, IsRoot, eval_add, eval_sub, eval_pow, eval_mul, eval_X, eval_C, eval_one,
          Multiset.mem_singleton]
        by_cases hy : y = 0
        · simp only [hy, or_true]
        apply or_congr _ Iff.rfl
        rw [← mul_left_inj' hy, eq_comm, ← sub_eq_zero, add_mul, inv_mul_cancel₀ hy]
        apply eq_iff_eq_cancel_right.mpr
        ring
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
x : K
h_1 : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x
φ : K[X] := HPow.hPow (α := K[X]) (β := ℕ) X 2 - HMul.hMul (α := K[X]) (β := K[X]) ((C : (a : K) → K[X]) x : K[X]) X + 1
hφ : φ ≠ 0
h₁ : {y : K | x = y + y⁻¹ ∨ y = 0} = (↑(φ.roots ∪ {0}).toFinset : Set K)
⊢ Set.Finite (α := K) {y : K | x = y + y⁻¹ ∨ y = 0}
have h₂ : ∀ (y : K), Membership.mem (γ := Set K) {y : K | x = y + y⁻¹ ∨ y = 0} y ↔ y ∈ (↑(φ.roots ∪ {0}).toFinset : Set K)
Tactic: classical
        convert! (φ.roots ∪ {0}).toFinset.finite_toSet using 1
        ext1 y
        simp only [φ, Multiset.mem_toFinset, Set.mem_setOf_eq, Finset.mem_coe, Multiset.mem_union,
          mem_roots hφ, IsRoot, eval_add, eval_sub, eval_pow, eval_mul, eval_X, eval_C, eval_one,
          Multiset.mem_singleton]
        by_cases hy : y = 0
        · simp only [hy, or_true]
        apply or_congr _ Iff.rfl
        rw [← mul_left_inj' hy, eq_comm, ← sub_eq_zero, add_mul, inv_mul_cancel₀ hy]
        apply eq_iff_eq_cancel_right.mpr
        ring
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
x : K
h_1 : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x
φ : K[X] := HPow.hPow (α := K[X]) (β := ℕ) X 2 - HMul.hMul (α := K[X]) (β := K[X]) ((C : (a : K) → K[X]) x : K[X]) X + 1
hφ : φ ≠ 0
h₁ : {y : K | x = y + y⁻¹ ∨ y = 0} = (↑(φ.roots ∪ {0}).toFinset : Set K)
h₂ : ∀ (y : K), Membership.mem (γ := Set K) {y : K | x = y + y⁻¹ ∨ y = 0} y ↔ y ∈ (↑(φ.roots ∪ {0}).toFinset : Set K)
⊢ Set.Finite (α := K) {y : K | x = y + y⁻¹ ∨ y = 0}
have h₃ : ∀ (y : K), x = y + y⁻¹ ∨ y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0 ∨ y = 0
Tactic: classical
        convert! (φ.roots ∪ {0}).toFinset.finite_toSet using 1
        ext1 y
        simp only [φ, Multiset.mem_toFinset, Set.mem_setOf_eq, Finset.mem_coe, Multiset.mem_union,
          mem_roots hφ, IsRoot, eval_add, eval_sub, eval_pow, eval_mul, eval_X, eval_C, eval_one,
          Multiset.mem_singleton]
        by_cases hy : y = 0
        · simp only [hy, or_true]
        apply or_congr _ Iff.rfl
        rw [← mul_left_inj' hy, eq_comm, ← sub_eq_zero, add_mul, inv_mul_cancel₀ hy]
        apply eq_iff_eq_cancel_right.mpr
        ring
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
x : K
h_1 : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x
φ : K[X] := HPow.hPow (α := K[X]) (β := ℕ) X 2 - HMul.hMul (α := K[X]) (β := K[X]) ((C : (a : K) → K[X]) x : K[X]) X + 1
hφ : φ ≠ 0
h₁ : {y : K | x = y + y⁻¹ ∨ y = 0} = (↑(φ.roots ∪ {0}).toFinset : Set K)
h₂ : ∀ (y : K), Membership.mem (γ := Set K) {y : K | x = y + y⁻¹ ∨ y = 0} y ↔ y ∈ (↑(φ.roots ∪ {0}).toFinset : Set K)
h₃ : ∀ (y : K), x = y + y⁻¹ ∨ y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0 ∨ y = 0
⊢ Set.Finite (α := K) {y : K | x = y + y⁻¹ ∨ y = 0}
have pos : ∀ (y : K) (hy : y = 0), x = y + y⁻¹ ∨ y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0 ∨ y = 0
Tactic: classical
        convert! (φ.roots ∪ {0}).toFinset.finite_toSet using 1
        ext1 y
        simp only [φ, Multiset.mem_toFinset, Set.mem_setOf_eq, Finset.mem_coe, Multiset.mem_union,
          mem_roots hφ, IsRoot, eval_add, eval_sub, eval_pow, eval_mul, eval_X, eval_C, eval_one,
          Multiset.mem_singleton]
        by_cases hy : y = 0
        · simp only [hy, or_true]
        apply or_congr _ Iff.rfl
        rw [← mul_left_inj' hy, eq_comm, ← sub_eq_zero, add_mul, inv_mul_cancel₀ hy]
        apply eq_iff_eq_cancel_right.mpr
        ring
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
x : K
h_1 : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x
φ : K[X] := HPow.hPow (α := K[X]) (β := ℕ) X 2 - HMul.hMul (α := K[X]) (β := K[X]) ((C : (a : K) → K[X]) x : K[X]) X + 1
hφ : φ ≠ 0
h₁ : {y : K | x = y + y⁻¹ ∨ y = 0} = (↑(φ.roots ∪ {0}).toFinset : Set K)
h₂ : ∀ (y : K), Membership.mem (γ := Set K) {y : K | x = y + y⁻¹ ∨ y = 0} y ↔ y ∈ (↑(φ.roots ∪ {0}).toFinset : Set K)
h₃ : ∀ (y : K), x = y + y⁻¹ ∨ y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0 ∨ y = 0
pos : ∀ (y : K) (hy : y = 0), x = y + y⁻¹ ∨ y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0 ∨ y = 0
⊢ Set.Finite (α := K) {y : K | x = y + y⁻¹ ∨ y = 0}
have neg : ∀ (y : K) (hy : ¬y = 0), x = y + y⁻¹ ∨ y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0 ∨ y = 0
Tactic: classical
        convert! (φ.roots ∪ {0}).toFinset.finite_toSet using 1
        ext1 y
        simp only [φ, Multiset.mem_toFinset, Set.mem_setOf_eq, Finset.mem_coe, Multiset.mem_union,
          mem_roots hφ, IsRoot, eval_add, eval_sub, eval_pow, eval_mul, eval_X, eval_C, eval_one,
          Multiset.mem_singleton]
        by_cases hy : y = 0
        · simp only [hy, or_true]
        apply or_congr _ Iff.rfl
        rw [← mul_left_inj' hy, eq_comm, ← sub_eq_zero, add_mul, inv_mul_cancel₀ hy]
        apply eq_iff_eq_cancel_right.mpr
        ring
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
x : K
h_1 : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x
φ : K[X] := HPow.hPow (α := K[X]) (β := ℕ) X 2 - HMul.hMul (α := K[X]) (β := K[X]) ((C : (a : K) → K[X]) x : K[X]) X + 1
hφ : φ ≠ 0
h₁ : {y : K | x = y + y⁻¹ ∨ y = 0} = (↑(φ.roots ∪ {0}).toFinset : Set K)
h₂ : ∀ (y : K), Membership.mem (γ := Set K) {y : K | x = y + y⁻¹ ∨ y = 0} y ↔ y ∈ (↑(φ.roots ∪ {0}).toFinset : Set K)
h₃ : ∀ (y : K), x = y + y⁻¹ ∨ y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0 ∨ y = 0
pos : ∀ (y : K) (hy : y = 0), x = y + y⁻¹ ∨ y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0 ∨ y = 0
neg neg₁ : ∀ (y : K) (hy : ¬y = 0), x = y + y⁻¹ ∨ y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0 ∨ y = 0
⊢ Set.Finite (α := K) {y : K | x = y + y⁻¹ ∨ y = 0}
have convert! : ∀ (y : K) (hy : ¬y = 0), x = y + y⁻¹ ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0
Tactic: classical
        convert! (φ.roots ∪ {0}).toFinset.finite_toSet using 1
        ext1 y
        simp only [φ, Multiset.mem_toFinset, Set.mem_setOf_eq, Finset.mem_coe, Multiset.mem_union,
          mem_roots hφ, IsRoot, eval_add, eval_sub, eval_pow, eval_mul, eval_X, eval_C, eval_one,
          Multiset.mem_singleton]
        by_cases hy : y = 0
        · simp only [hy, or_true]
        apply or_congr _ Iff.rfl
        rw [← mul_left_inj' hy, eq_comm, ← sub_eq_zero, add_mul, inv_mul_cancel₀ hy]
        apply eq_iff_eq_cancel_right.mpr
        ring
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
x : K
h_1 : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x
φ : K[X] := HPow.hPow (α := K[X]) (β := ℕ) X 2 - HMul.hMul (α := K[X]) (β := K[X]) ((C : (a : K) → K[X]) x : K[X]) X + 1
hφ : φ ≠ 0
h₁ : {y : K | x = y + y⁻¹ ∨ y = 0} = (↑(φ.roots ∪ {0}).toFinset : Set K)
h₂ : ∀ (y : K), Membership.mem (γ := Set K) {y : K | x = y + y⁻¹ ∨ y = 0} y ↔ y ∈ (↑(φ.roots ∪ {0}).toFinset : Set K)
h₃ : ∀ (y : K), x = y + y⁻¹ ∨ y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0 ∨ y = 0
pos : ∀ (y : K) (hy : y = 0), x = y + y⁻¹ ∨ y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0 ∨ y = 0
neg neg₁ : ∀ (y : K) (hy : ¬y = 0), x = y + y⁻¹ ∨ y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0 ∨ y = 0
convert! : ∀ (y : K) (hy : ¬y = 0), x = y + y⁻¹ ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0
⊢ Set.Finite (α := K) {y : K | x = y + y⁻¹ ∨ y = 0}
have convert!₁ : ∀ (y : K) (hy : ¬y = 0), y * y + 1 - x * y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0
Tactic: classical
        convert! (φ.roots ∪ {0}).toFinset.finite_toSet using 1
        ext1 y
        simp only [φ, Multiset.mem_toFinset, Set.mem_setOf_eq, Finset.mem_coe, Multiset.mem_union,
          mem_roots hφ, IsRoot, eval_add, eval_sub, eval_pow, eval_mul, eval_X, eval_C, eval_one,
          Multiset.mem_singleton]
        by_cases hy : y = 0
        · simp only [hy, or_true]
        apply or_congr _ Iff.rfl
        rw [← mul_left_inj' hy, eq_comm, ← sub_eq_zero, add_mul, inv_mul_cancel₀ hy]
        apply eq_iff_eq_cancel_right.mpr
        ring
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
x : K
h_1 : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x
φ : K[X] := HPow.hPow (α := K[X]) (β := ℕ) X 2 - HMul.hMul (α := K[X]) (β := K[X]) ((C : (a : K) → K[X]) x : K[X]) X + 1
hφ : φ ≠ 0
h₁ : {y : K | x = y + y⁻¹ ∨ y = 0} = (↑(φ.roots ∪ {0}).toFinset : Set K)
h₂ : ∀ (y : K), Membership.mem (γ := Set K) {y : K | x = y + y⁻¹ ∨ y = 0} y ↔ y ∈ (↑(φ.roots ∪ {0}).toFinset : Set K)
h₃ : ∀ (y : K), x = y + y⁻¹ ∨ y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0 ∨ y = 0
pos : ∀ (y : K) (hy : y = 0), x = y + y⁻¹ ∨ y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0 ∨ y = 0
neg neg₁ : ∀ (y : K) (hy : ¬y = 0), x = y + y⁻¹ ∨ y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0 ∨ y = 0
convert! : ∀ (y : K) (hy : ¬y = 0), x = y + y⁻¹ ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0
convert!₁ : ∀ (y : K) (hy : ¬y = 0), y * y + 1 - x * y = 0 ↔ HPow.hPow (β := ℕ) y 2 - x * y + 1 = 0
⊢ Set.Finite (α := K) {y : K | x = y + y⁻¹ ∨ y = 0}
have a : ∀ (y : K) (hy : ¬y = 0), y * y + 1 - x * y = HPow.hPow (β := ℕ) y 2 - x * y + 1
Tactic: classical
        convert! (φ.roots ∪ {0}).toFinset.finite_toSet using 1
        ext1 y
        simp only [φ, Multiset.mem_toFinset, Set.mem_setOf_eq, Finset.mem_coe, Multiset.mem_union,
          mem_roots hφ, IsRoot, eval_add, eval_sub, eval_pow, eval_mul, eval_X, eval_C, eval_one,
          Multiset.mem_singleton]
        by_cases hy : y = 0
        · simp only [hy, or_true]
        apply or_congr _ Iff.rfl
        rw [← mul_left_inj' hy, eq_comm, ← sub_eq_zero, add_mul, inv_mul_cancel₀ hy]
        apply eq_iff_eq_cancel_right.mpr
        ring
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
⊢ Set.Finite (α := K) Set.univ
have a : Eq (α := Set K) Set.univ
  (⋃ (x : K),
    ⋃ (_ : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x), {y : K | x = y + y⁻¹ ∨ y = 0})
Tactic: suffices (Set.univ : Set K) =
        ⋃ x ∈ { x : K | ∃ y : K, x = y + y⁻¹ ∧ y ≠ 0 }, { y | x = y + y⁻¹ ∨ y = 0 }  by
      rw [this]
      clear this
      refine h.biUnion fun x _ => ?_
      -- The following quadratic polynomial has as solutions the `y` for which `x = y + y⁻¹`.
      let φ : K[X] := X ^ 2 - C x * X + 1
      have hφ : φ ≠ 0 := by
        intro H
        have : φ.eval 0 = 0 := by rw [H, eval_zero]
        simpa [φ, eval_X, eval_one, eval_pow, eval_sub, sub_zero, eval_add, eval_mul,
          mul_zero, sq, zero_add, one_ne_zero]
      classical
        convert! (φ.roots ∪ {0}).toFinset.finite_toSet using 1
        ext1 y
        simp only [φ, Multiset.mem_toFinset, Set.mem_setOf_eq, Finset.mem_coe, Multiset.mem_union,
          mem_roots hφ, IsRoot, eval_add, eval_sub, eval_pow, eval_mul, eval_X, eval_C, eval_one,
          Multiset.mem_singleton]
        by_cases hy : y = 0
        · simp only [hy, or_true]
        apply or_congr _ Iff.rfl
        rw [← mul_left_inj' hy, eq_comm, ← sub_eq_zero, add_mul, inv_mul_cancel₀ hy]
        apply eq_iff_eq_cancel_right.mpr
        ring
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Dickson (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
K : Type
w : Field K
w_1 : CharP K p
H : Set.Infinite (α := K) Set.univ
h : Set.Finite (α := K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0}
⊢ Eq (α := Set K) Set.univ
    (⋃ (x : K),
      ⋃ (_ : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x), {y : K | x = y + y⁻¹ ∨ y = 0})
have apply : ∀ (x : K),
  Membership.mem (γ := Set K)
    (⋃ (x : K),
      ⋃ (_ : Membership.mem (γ := Set K) {x : K | ∃ (y : K), x = y + y⁻¹ ∧ y ≠ 0} x), {y : K | x = y + y⁻¹ ∨ y = 0})
    x
Tactic: apply (Set.eq_univ_of_forall _).symm