awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
L : Type u_3
inst_2 : Field L
inst_3 : Algebra K L
inst_4 : Finite L
this : Fintype L
m : ℕ
lt : m < Module.finrank K L
pos : 0 < m
eq : HPow.hPow (α := L →ₐ[K] L) (frobeniusAlgHom K L) m = 1
x : L
h : Membership.mem (γ := Multiset L) (val univ) x
⊢ x ∈ (HPow.hPow (α := L[X]) X (q ^ m) - X).roots
have simp_rw : HPow.hPow (α := L[X]) X (q ^ m) - X ≠ 0 ∧ x ^ q ^ m - x = 0
Tactic: simp_rw [mem_roots', IsRoot, eval_sub, eval_pow, eval_X]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (hypothesis)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
L : Type u_3
inst_2 : Field L
inst_3 : Algebra K L
inst_4 : Finite L
this : Fintype L
m : ℕ
lt : m < Module.finrank K L
pos : 0 < m
eq : HPow.hPow (α := L →ₐ[K] L) (frobeniusAlgHom K L) m = 1
x : L
h : Membership.mem (γ := Multiset L) (val univ) x
⊢ HPow.hPow (α := L[X]) X (q ^ m) - X ≠ 0 ∧ x ^ q ^ m - x = 0
have this₁ : Eq (α := L) ((HPow.hPow (α := L →ₐ[K] L) (frobeniusAlgHom K L) m : (a : L) → L) x : L) ((1 : (a : L) → L) x : L)
Tactic: have := DFunLike.congr_fun eq x
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (hypothesis)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
L : Type u_3
inst_2 : Field L
inst_3 : Algebra K L
inst_4 : Finite L
this_1 : Fintype L
m : ℕ
lt : m < Module.finrank K L
pos : 0 < m
eq : HPow.hPow (α := L →ₐ[K] L) (frobeniusAlgHom K L) m = 1
x : L
h : Membership.mem (γ := Multiset L) (val univ) x
this : Eq (α := L) ((HPow.hPow (α := L →ₐ[K] L) (frobeniusAlgHom K L) m : (a : L) → L) x : L) ((1 : (a : L) → L) x : L)
⊢ HPow.hPow (α := L[X]) X (q ^ m) - X ≠ 0 ∧ x ^ q ^ m - x = 0
have this₁ : (fun (x : L) ↦ x ^ q ^ m) x - x = 0
Tactic: rw [AlgHom.coe_pow, coe_frobeniusAlgHom, pow_iterate, AlgHom.one_apply, ← sub_eq_zero] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
L : Type u_3
inst_2 : Field L
inst_3 : Algebra K L
inst_4 : Finite L
this_1 : Fintype L
m : ℕ
lt : m < Module.finrank K L
pos : 0 < m
eq : HPow.hPow (α := L →ₐ[K] L) (frobeniusAlgHom K L) m = 1
x : L
h : Membership.mem (γ := Multiset L) (val univ) x
this : (fun (x : L) ↦ x ^ q ^ m) x - x = 0
⊢ HPow.hPow (α := L[X]) X (q ^ m) - X ≠ 0 ∧ x ^ q ^ m - x = 0
have refine : ∀ (h : HPow.hPow (α := L[X]) X (q ^ m) - X = 0), False
Tactic: refine ⟨fun h ↦ ?_, this⟩
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (hypothesis)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
L : Type u_3
inst_2 : Field L
inst_3 : Algebra K L
inst_4 : Finite L
this : Fintype L
m : ℕ
lt : m < Module.finrank K L
pos : 0 < m
eq : HPow.hPow (α := L →ₐ[K] L) (frobeniusAlgHom K L) m = 1
⊢ False
have this₁ : Finset.card.{u_3} (α := L) univ ≤ (HPow.hPow (α := L[X]) X (q ^ m) - X).natDegree
Tactic: have := card_le_degree_of_subset_roots (R := L) (p := X ^ q ^ m - X) (Z := univ) fun x _ ↦ by
      simp_rw [mem_roots', IsRoot, eval_sub, eval_pow, eval_X]
      have := DFunLike.congr_fun eq x
      rw [AlgHom.coe_pow, coe_frobeniusAlgHom, pow_iterate, AlgHom.one_apply, ← sub_eq_zero] at this
      refine ⟨fun h ↦ ?_, this⟩
      simpa [Fintype.one_lt_card.ne, pos.ne, eqComm] using congr_arg (coeff · 1) h
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
L : Type u_3
inst_2 : Field L
inst_3 : Algebra K L
inst_4 : Finite L
this_1 : Fintype L
m : ℕ
lt : m < Module.finrank K L
pos : 0 < m
eq : HPow.hPow (α := L →ₐ[K] L) (frobeniusAlgHom K L) m = 1
this : Finset.card.{u_3} (α := L) univ ≤ (HPow.hPow (α := L[X]) X (q ^ m) - X).natDegree
⊢ False
have refine_2 : max (HPow.hPow (α := L[X]) X (q ^ m)).natDegree (natDegree.{u_3} (R := L) X) = q ^ m
Tactic: refine this.not_gt (((natDegree_sub_le ..).trans_eq ?_).trans_lt <|
      (Nat.pow_lt_pow_right Fintype.one_lt_card lt).trans_eq Module.card_eq_pow_finrank.symm)
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K' : Type u_3
inst : Field K'
p : ℕ
hp : 1 < p
⊢ degree.{u_3} (R := K') X < (HPow.hPow (α := K'[X]) X p).degree
have rw : LT.lt (α := WithBot ℕ) 1 (↑p : WithBot ℕ)
Tactic: rw [degree_X_pow, degree_X]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (hypothesis)
Goal:
K' : Type u_3
inst : Field K'
p : ℕ
hp : 1 < p
⊢ (HPow.hPow (α := K'[X]) X p - X).natDegree = p
have h1 : degree.{u_3} (R := K') X < (HPow.hPow (α := K'[X]) X p).degree
Tactic: have h1 : (X : K'[X]).degree < (X ^ p : K'[X]).degree := by
    rw [degree_X_pow, degree_X]
    exact mod_cast hp
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K' : Type u_3
inst : Field K'
p : ℕ
hp : 1 < p
h1 : degree.{u_3} (R := K') X < (HPow.hPow (α := K'[X]) X p).degree
⊢ (HPow.hPow (α := K'[X]) X p - X).natDegree = p
have rw : (HPow.hPow (α := K'[X]) X p).natDegree = p
Tactic: rw [natDegree_eq_of_degree_eq (degree_sub_eq_left_of_degree_lt h1), natDegree_X_pow]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K' : Type u_3
inst : Field K'
p : ℕ
hp : 1 < p
h1 : degree.{u_3} (R := K') X < (HPow.hPow (α := K'[X]) X p).degree
rw : (HPow.hPow (α := K'[X]) X p).natDegree = p
⊢ (HPow.hPow (α := K'[X]) X p - X).natDegree = p
have rw₁ : p = p
Tactic: rw [natDegree_eq_of_degree_eq (degree_sub_eq_left_of_degree_lt h1), natDegree_X_pow]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (hypothesis)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
⊢ (HPow.hPow (α := K[X]) X q - X).roots = val univ
have aux : HPow.hPow (α := K[X]) X q - X ≠ 0
Tactic: have aux : (X ^ q - X : K[X]) ≠ 0 := X_pow_card_sub_X_ne_zero K Fintype.one_lt_card
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
aux : HPow.hPow (α := K[X]) X q - X ≠ 0
⊢ (HPow.hPow (α := K[X]) X q - X).roots.toFinset = univ
have rw : ∀ (x : K), x ∈ (HPow.hPow (α := K[X]) X q - X).roots.toFinset
Tactic: rw [eq_univ_iff_forall]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
aux : HPow.hPow (α := K[X]) X q - X ≠ 0
x : K
⊢ x ∈ (HPow.hPow (α := K[X]) X q - X).roots.toFinset
have rw : x ∈ (HPow.hPow (α := K[X]) X q - X).roots
Tactic: rw [Multiset.mem_toFinset, mem_roots aux, IsRoot.def, eval_sub, eval_pow, eval_X,
        sub_eq_zero, pow_card]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
aux : HPow.hPow (α := K[X]) X q - X ≠ 0
x : K
rw : x ∈ (HPow.hPow (α := K[X]) X q - X).roots
⊢ x ∈ (HPow.hPow (α := K[X]) X q - X).roots.toFinset
have rw₁ : (HPow.hPow (α := K[X]) X q - X).IsRoot x
Tactic: rw [Multiset.mem_toFinset, mem_roots aux, IsRoot.def, eval_sub, eval_pow, eval_X,
        sub_eq_zero, pow_card]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
aux : HPow.hPow (α := K[X]) X q - X ≠ 0
x : K
rw : x ∈ (HPow.hPow (α := K[X]) X q - X).roots
rw₁ : (HPow.hPow (α := K[X]) X q - X).IsRoot x
⊢ x ∈ (HPow.hPow (α := K[X]) X q - X).roots.toFinset
have rw₂ : eval x (HPow.hPow (α := K[X]) X q - X) = 0
Tactic: rw [Multiset.mem_toFinset, mem_roots aux, IsRoot.def, eval_sub, eval_pow, eval_X,
        sub_eq_zero, pow_card]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
aux : HPow.hPow (α := K[X]) X q - X ≠ 0
x : K
rw : x ∈ (HPow.hPow (α := K[X]) X q - X).roots
rw₁ : (HPow.hPow (α := K[X]) X q - X).IsRoot x
rw₂ : eval x (HPow.hPow (α := K[X]) X q - X) = 0
⊢ x ∈ (HPow.hPow (α := K[X]) X q - X).roots.toFinset
have rw₃ : eval x (HPow.hPow (α := K[X]) X q) - eval x X = 0
Tactic: rw [Multiset.mem_toFinset, mem_roots aux, IsRoot.def, eval_sub, eval_pow, eval_X,
        sub_eq_zero, pow_card]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
aux : HPow.hPow (α := K[X]) X q - X ≠ 0
x : K
rw : x ∈ (HPow.hPow (α := K[X]) X q - X).roots
rw₁ : (HPow.hPow (α := K[X]) X q - X).IsRoot x
rw₂ : eval x (HPow.hPow (α := K[X]) X q - X) = 0
rw₃ : eval x (HPow.hPow (α := K[X]) X q) - eval x X = 0
⊢ x ∈ (HPow.hPow (α := K[X]) X q - X).roots.toFinset
have rw₄ : eval x X ^ q - eval x X = 0
Tactic: rw [Multiset.mem_toFinset, mem_roots aux, IsRoot.def, eval_sub, eval_pow, eval_X,
        sub_eq_zero, pow_card]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
aux : HPow.hPow (α := K[X]) X q - X ≠ 0
x : K
rw : x ∈ (HPow.hPow (α := K[X]) X q - X).roots
rw₁ : (HPow.hPow (α := K[X]) X q - X).IsRoot x
rw₂ : eval x (HPow.hPow (α := K[X]) X q - X) = 0
rw₃ : eval x (HPow.hPow (α := K[X]) X q) - eval x X = 0
rw₄ : eval x X ^ q - eval x X = 0
⊢ x ∈ (HPow.hPow (α := K[X]) X q - X).roots.toFinset
have rw₅ : x ^ q - x = 0
Tactic: rw [Multiset.mem_toFinset, mem_roots aux, IsRoot.def, eval_sub, eval_pow, eval_X,
        sub_eq_zero, pow_card]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
aux : HPow.hPow (α := K[X]) X q - X ≠ 0
x : K
rw : x ∈ (HPow.hPow (α := K[X]) X q - X).roots
rw₁ : (HPow.hPow (α := K[X]) X q - X).IsRoot x
rw₂ : eval x (HPow.hPow (α := K[X]) X q - X) = 0
rw₃ : eval x (HPow.hPow (α := K[X]) X q) - eval x X = 0
rw₄ : eval x X ^ q - eval x X = 0
rw₅ : x ^ q - x = 0
⊢ x ∈ (HPow.hPow (α := K[X]) X q - X).roots.toFinset
have rw₆ : x ^ q = x
Tactic: rw [Multiset.mem_toFinset, mem_roots aux, IsRoot.def, eval_sub, eval_pow, eval_X,
        sub_eq_zero, pow_card]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
aux : HPow.hPow (α := K[X]) X q - X ≠ 0
x : K
rw : x ∈ (HPow.hPow (α := K[X]) X q - X).roots
rw₁ : (HPow.hPow (α := K[X]) X q - X).IsRoot x
rw₂ : eval x (HPow.hPow (α := K[X]) X q - X) = 0
rw₃ : eval x (HPow.hPow (α := K[X]) X q) - eval x X = 0
rw₄ : eval x X ^ q - eval x X = 0
rw₅ : x ^ q - x = 0
rw₆ : x ^ q = x
⊢ x ∈ (HPow.hPow (α := K[X]) X q - X).roots.toFinset
have rw₇ : x = x
Tactic: rw [Multiset.mem_toFinset, mem_roots aux, IsRoot.def, eval_sub, eval_pow, eval_X,
        sub_eq_zero, pow_card]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (hypothesis)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
aux : HPow.hPow (α := K[X]) X q - X ≠ 0
⊢ (HPow.hPow (α := K[X]) X q - X).roots = val univ
have this : (HPow.hPow (α := K[X]) X q - X).roots.toFinset = univ
Tactic: have : (roots (X ^ q - X : K[X])).toFinset = Finset.univ := by
      rw [eq_univ_iff_forall]
      intro x
      rw [Multiset.mem_toFinset, mem_roots aux, IsRoot.def, eval_sub, eval_pow, eval_X,
        sub_eq_zero, pow_card]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
aux : HPow.hPow (α := K[X]) X q - X ≠ 0
this : (HPow.hPow (α := K[X]) X q - X).roots.toFinset = univ
⊢ (HPow.hPow (α := K[X]) X q - X).roots = val univ
have rw : (HPow.hPow (α := K[X]) X q - X).roots.Nodup
Tactic: rw [← this, Multiset.toFinset_val, eq_comm, Multiset.dedup_eq_self]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
aux : HPow.hPow (α := K[X]) X q - X ≠ 0
this : (HPow.hPow (α := K[X]) X q - X).roots.toFinset = univ
⊢ (HPow.hPow (α := K[X]) X q - X).roots.Nodup
have hsep : (HPow.hPow (α := K[X]) X q - X).Separable
Tactic: apply nodup_roots
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
aux : HPow.hPow (α := K[X]) X q - X ≠ 0
this : (HPow.hPow (α := K[X]) X q - X).roots.toFinset = univ
⊢ IsCoprime (HPow.hPow (α := K[X]) X q - X) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q - X))
have h : Eq (α := K[X]) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q - X) : K[X]) (-1 : K[X])
Tactic: convert! isCoprime_one_right.neg_right (R := K[X]) using 1
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
aux : HPow.hPow (α := K[X]) X q - X ≠ 0
this : (HPow.hPow (α := K[X]) X q - X).roots.toFinset = univ
⊢ Eq (α := K[X]) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q - X) : K[X]) (-1 : K[X])
have h : HSub.hSub (α := K[X]) (β := K[X]) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q) : K[X])
    ((derivative : (a : K[X]) → K[X]) X : K[X]) =
  -1
Tactic: rw [derivative_sub, derivative_X, derivative_X_pow, Nat.cast_card_eq_zero K, C_0,
      zero_mul, zero_sub]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
aux : HPow.hPow (α := K[X]) X q - X ≠ 0
this : (HPow.hPow (α := K[X]) X q - X).roots.toFinset = univ
h :
  HSub.hSub (α := K[X]) (β := K[X]) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q) : K[X])
      ((derivative : (a : K[X]) → K[X]) X : K[X]) =
    -1
⊢ Eq (α := K[X]) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q - X) : K[X]) (-1 : K[X])
have h₁ : HSub.hSub (α := K[X]) (β := K[X]) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q) : K[X]) (1 : K[X]) = -1
Tactic: rw [derivative_sub, derivative_X, derivative_X_pow, Nat.cast_card_eq_zero K, C_0,
      zero_mul, zero_sub]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
aux : HPow.hPow (α := K[X]) X q - X ≠ 0
this : (HPow.hPow (α := K[X]) X q - X).roots.toFinset = univ
h :
  HSub.hSub (α := K[X]) (β := K[X]) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q) : K[X])
      ((derivative : (a : K[X]) → K[X]) X : K[X]) =
    -1
h₁ :
  HSub.hSub (α := K[X]) (β := K[X]) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q) : K[X]) (1 : K[X]) =
    -1
⊢ Eq (α := K[X]) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q - X) : K[X]) (-1 : K[X])
have h₂ : (C : (a : K) → K[X]) (↑q : K) * HPow.hPow (α := K[X]) X (q - 1) - 1 = -1
Tactic: rw [derivative_sub, derivative_X, derivative_X_pow, Nat.cast_card_eq_zero K, C_0,
      zero_mul, zero_sub]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
aux : HPow.hPow (α := K[X]) X q - X ≠ 0
this : (HPow.hPow (α := K[X]) X q - X).roots.toFinset = univ
h :
  HSub.hSub (α := K[X]) (β := K[X]) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q) : K[X])
      ((derivative : (a : K[X]) → K[X]) X : K[X]) =
    -1
h₁ :
  HSub.hSub (α := K[X]) (β := K[X]) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q) : K[X]) (1 : K[X]) =
    -1
h₂ : (C : (a : K) → K[X]) (↑q : K) * HPow.hPow (α := K[X]) X (q - 1) - 1 = -1
⊢ Eq (α := K[X]) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q - X) : K[X]) (-1 : K[X])
have h₃ : (C : (a : K) → K[X]) 0 * HPow.hPow (α := K[X]) X (q - 1) - 1 = -1
Tactic: rw [derivative_sub, derivative_X, derivative_X_pow, Nat.cast_card_eq_zero K, C_0,
      zero_mul, zero_sub]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
aux : HPow.hPow (α := K[X]) X q - X ≠ 0
this : (HPow.hPow (α := K[X]) X q - X).roots.toFinset = univ
h :
  HSub.hSub (α := K[X]) (β := K[X]) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q) : K[X])
      ((derivative : (a : K[X]) → K[X]) X : K[X]) =
    -1
h₁ :
  HSub.hSub (α := K[X]) (β := K[X]) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q) : K[X]) (1 : K[X]) =
    -1
h₂ : (C : (a : K) → K[X]) (↑q : K) * HPow.hPow (α := K[X]) X (q - 1) - 1 = -1
h₃ : (C : (a : K) → K[X]) 0 * HPow.hPow (α := K[X]) X (q - 1) - 1 = -1
⊢ Eq (α := K[X]) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q - X) : K[X]) (-1 : K[X])
have h₄ : 0 * HPow.hPow (α := K[X]) X (q - 1) - 1 = -1
Tactic: rw [derivative_sub, derivative_X, derivative_X_pow, Nat.cast_card_eq_zero K, C_0,
      zero_mul, zero_sub]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
aux : HPow.hPow (α := K[X]) X q - X ≠ 0
this : (HPow.hPow (α := K[X]) X q - X).roots.toFinset = univ
h :
  HSub.hSub (α := K[X]) (β := K[X]) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q) : K[X])
      ((derivative : (a : K[X]) → K[X]) X : K[X]) =
    -1
h₁ :
  HSub.hSub (α := K[X]) (β := K[X]) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q) : K[X]) (1 : K[X]) =
    -1
h₂ : (C : (a : K) → K[X]) (↑q : K) * HPow.hPow (α := K[X]) X (q - 1) - 1 = -1
h₃ : (C : (a : K) → K[X]) 0 * HPow.hPow (α := K[X]) X (q - 1) - 1 = -1
h₄ : 0 * HPow.hPow (α := K[X]) X (q - 1) - 1 = -1
⊢ Eq (α := K[X]) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q - X) : K[X]) (-1 : K[X])
have h₅ : HSub.hSub (α := K[X]) 0 1 = -1
Tactic: rw [derivative_sub, derivative_X, derivative_X_pow, Nat.cast_card_eq_zero K, C_0,
      zero_mul, zero_sub]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
aux : HPow.hPow (α := K[X]) X q - X ≠ 0
this : (HPow.hPow (α := K[X]) X q - X).roots.toFinset = univ
h :
  HSub.hSub (α := K[X]) (β := K[X]) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q) : K[X])
      ((derivative : (a : K[X]) → K[X]) X : K[X]) =
    -1
h₁ :
  HSub.hSub (α := K[X]) (β := K[X]) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q) : K[X]) (1 : K[X]) =
    -1
h₂ : (C : (a : K) → K[X]) (↑q : K) * HPow.hPow (α := K[X]) X (q - 1) - 1 = -1
h₃ : (C : (a : K) → K[X]) 0 * HPow.hPow (α := K[X]) X (q - 1) - 1 = -1
h₄ : 0 * HPow.hPow (α := K[X]) X (q - 1) - 1 = -1
h₅ : HSub.hSub (α := K[X]) 0 1 = -1
⊢ Eq (α := K[X]) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q - X) : K[X]) (-1 : K[X])
have h₆ : Eq.{u_1 + 1} (α := K[X]) (-1) (-1)
Tactic: rw [derivative_sub, derivative_X, derivative_X_pow, Nat.cast_card_eq_zero K, C_0,
      zero_mul, zero_sub]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
⊢ (HPow.hPow (α := K[X]) X q - X).roots = val univ
have have : ∀ (aux : HPow.hPow (α := K[X]) X q - X ≠ 0), (HPow.hPow (α := K[X]) X q - X).roots = val univ
Tactic: classical
    have aux : (X ^ q - X : K[X]) ≠ 0 := X_pow_card_sub_X_ne_zero K Fintype.one_lt_card
    have : (roots (X ^ q - X : K[X])).toFinset = Finset.univ := by
      rw [eq_univ_iff_forall]
      intro x
      rw [Multiset.mem_toFinset, mem_roots aux, IsRoot.def, eval_sub, eval_pow, eval_X,
        sub_eq_zero, pow_card]
    rw [← this, Multiset.toFinset_val, eq_comm, Multiset.dedup_eq_self]
    apply nodup_roots
    rw [separable_def]
    convert! isCoprime_one_right.neg_right (R := K[X]) using 1
    rw [derivative_sub, derivative_X, derivative_X_pow, Nat.cast_card_eq_zero K, C_0,
      zero_mul, zero_sub]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
have have₁ have₂ : ∀ (aux : HPow.hPow (α := K[X]) X q - X ≠ 0), (HPow.hPow (α := K[X]) X q - X).roots = val univ
⊢ (HPow.hPow (α := K[X]) X q - X).roots = val univ
have have₃ : ∀ (aux : HPow.hPow (α := K[X]) X q - X ≠ 0) (this : (HPow.hPow (α := K[X]) X q - X).roots.toFinset = univ),
  (HPow.hPow (α := K[X]) X q - X).roots = val univ
Tactic: classical
    have aux : (X ^ q - X : K[X]) ≠ 0 := X_pow_card_sub_X_ne_zero K Fintype.one_lt_card
    have : (roots (X ^ q - X : K[X])).toFinset = Finset.univ := by
      rw [eq_univ_iff_forall]
      intro x
      rw [Multiset.mem_toFinset, mem_roots aux, IsRoot.def, eval_sub, eval_pow, eval_X,
        sub_eq_zero, pow_card]
    rw [← this, Multiset.toFinset_val, eq_comm, Multiset.dedup_eq_self]
    apply nodup_roots
    rw [separable_def]
    convert! isCoprime_one_right.neg_right (R := K[X]) using 1
    rw [derivative_sub, derivative_X, derivative_X_pow, Nat.cast_card_eq_zero K, C_0,
      zero_mul, zero_sub]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
have have₁ have₂ : ∀ (aux : HPow.hPow (α := K[X]) X q - X ≠ 0), (HPow.hPow (α := K[X]) X q - X).roots = val univ
have₃ :
  ∀ (aux : HPow.hPow (α := K[X]) X q - X ≠ 0) (this : (HPow.hPow (α := K[X]) X q - X).roots.toFinset = univ),
    (HPow.hPow (α := K[X]) X q - X).roots = val univ
⊢ (HPow.hPow (α := K[X]) X q - X).roots = val univ
have have₄ : ∀ (aux : HPow.hPow (α := K[X]) X q - X ≠ 0) (this : (HPow.hPow (α := K[X]) X q - X).roots.toFinset = univ),
  (HPow.hPow (α := K[X]) X q - X).roots.Nodup
Tactic: classical
    have aux : (X ^ q - X : K[X]) ≠ 0 := X_pow_card_sub_X_ne_zero K Fintype.one_lt_card
    have : (roots (X ^ q - X : K[X])).toFinset = Finset.univ := by
      rw [eq_univ_iff_forall]
      intro x
      rw [Multiset.mem_toFinset, mem_roots aux, IsRoot.def, eval_sub, eval_pow, eval_X,
        sub_eq_zero, pow_card]
    rw [← this, Multiset.toFinset_val, eq_comm, Multiset.dedup_eq_self]
    apply nodup_roots
    rw [separable_def]
    convert! isCoprime_one_right.neg_right (R := K[X]) using 1
    rw [derivative_sub, derivative_X, derivative_X_pow, Nat.cast_card_eq_zero K, C_0,
      zero_mul, zero_sub]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
have have₁ have₂ : ∀ (aux : HPow.hPow (α := K[X]) X q - X ≠ 0), (HPow.hPow (α := K[X]) X q - X).roots = val univ
have₃ :
  ∀ (aux : HPow.hPow (α := K[X]) X q - X ≠ 0) (this : (HPow.hPow (α := K[X]) X q - X).roots.toFinset = univ),
    (HPow.hPow (α := K[X]) X q - X).roots = val univ
have₄ :
  ∀ (aux : HPow.hPow (α := K[X]) X q - X ≠ 0) (this : (HPow.hPow (α := K[X]) X q - X).roots.toFinset = univ),
    (HPow.hPow (α := K[X]) X q - X).roots.Nodup
⊢ (HPow.hPow (α := K[X]) X q - X).roots = val univ
have hsep : ∀ (aux : HPow.hPow (α := K[X]) X q - X ≠ 0) (this : (HPow.hPow (α := K[X]) X q - X).roots.toFinset = univ),
  (HPow.hPow (α := K[X]) X q - X).Separable
Tactic: classical
    have aux : (X ^ q - X : K[X]) ≠ 0 := X_pow_card_sub_X_ne_zero K Fintype.one_lt_card
    have : (roots (X ^ q - X : K[X])).toFinset = Finset.univ := by
      rw [eq_univ_iff_forall]
      intro x
      rw [Multiset.mem_toFinset, mem_roots aux, IsRoot.def, eval_sub, eval_pow, eval_X,
        sub_eq_zero, pow_card]
    rw [← this, Multiset.toFinset_val, eq_comm, Multiset.dedup_eq_self]
    apply nodup_roots
    rw [separable_def]
    convert! isCoprime_one_right.neg_right (R := K[X]) using 1
    rw [derivative_sub, derivative_X, derivative_X_pow, Nat.cast_card_eq_zero K, C_0,
      zero_mul, zero_sub]
-- Lean 4 proof step from Mathlib.FieldTheory.Finite.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : Fintype K
have have₁ have₂ : ∀ (aux : HPow.hPow (α := K[X]) X q - X ≠ 0), (HPow.hPow (α := K[X]) X q - X).roots = val univ
have₃ :
  ∀ (aux : HPow.hPow (α := K[X]) X q - X ≠ 0) (this : (HPow.hPow (α := K[X]) X q - X).roots.toFinset = univ),
    (HPow.hPow (α := K[X]) X q - X).roots = val univ
have₄ :
  ∀ (aux : HPow.hPow (α := K[X]) X q - X ≠ 0) (this : (HPow.hPow (α := K[X]) X q - X).roots.toFinset = univ),
    (HPow.hPow (α := K[X]) X q - X).roots.Nodup
hsep :
  ∀ (aux : HPow.hPow (α := K[X]) X q - X ≠ 0) (this : (HPow.hPow (α := K[X]) X q - X).roots.toFinset = univ),
    (HPow.hPow (α := K[X]) X q - X).Separable
⊢ (HPow.hPow (α := K[X]) X q - X).roots = val univ
have hsep₁ : ∀ (aux : HPow.hPow (α := K[X]) X q - X ≠ 0) (this : (HPow.hPow (α := K[X]) X q - X).roots.toFinset = univ),
  IsCoprime (HPow.hPow (α := K[X]) X q - X) ((derivative : (a : K[X]) → K[X]) (HPow.hPow (α := K[X]) X q - X))
Tactic: classical
    have aux : (X ^ q - X : K[X]) ≠ 0 := X_pow_card_sub_X_ne_zero K Fintype.one_lt_card
    have : (roots (X ^ q - X : K[X])).toFinset = Finset.univ := by
      rw [eq_univ_iff_forall]
      intro x
      rw [Multiset.mem_toFinset, mem_roots aux, IsRoot.def, eval_sub, eval_pow, eval_X,
        sub_eq_zero, pow_card]
    rw [← this, Multiset.toFinset_val, eq_comm, Multiset.dedup_eq_self]
    apply nodup_roots
    rw [separable_def]
    convert! isCoprime_one_right.neg_right (R := K[X]) using 1
    rw [derivative_sub, derivative_X, derivative_X_pow, Nat.cast_card_eq_zero K, C_0,
      zero_mul, zero_sub]