awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
n : ℕ
inst : NeZero n
S : Set ℕ
A : Type u
B : Type v
inst_1 : CommRing A
inst_2 : CommRing B
inst_3 : Algebra A B
h : ∃ s ∈ S, s ≠ 0 ∧ n ∣ s
H : IsCyclotomicExtension S A B
⊢ Eq.{v + 1} (α := Subalgebra A B) (adjoin A {b : B | ∃ n_1 ∈ S ∪ {n}, n_1 ≠ 0 ∧ b ^ n_1 = 1}) ⊤
have refine_2 : LE.le.{v} (α := Subalgebra A B) ⊤ (adjoin A {b : B | ∃ n_1 ∈ S ∪ {n}, n_1 ≠ 0 ∧ b ^ n_1 = 1})
Tactic: refine _root_.eq_top_iff.2 ?_
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
n : ℕ
inst : NeZero n
S : Set ℕ
A : Type u
B : Type v
inst_1 : CommRing A
inst_2 : CommRing B
inst_3 : Algebra A B
h : ∃ s ∈ S, s ≠ 0 ∧ n ∣ s
H : IsCyclotomicExtension S A B
⊢ LE.le.{v} (α := Subalgebra A B) ⊤ (adjoin A {b : B | ∃ n_1 ∈ S ∪ {n}, n_1 ≠ 0 ∧ b ^ n_1 = 1})
have refine_2 : LE.le.{v} (α := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})
  (adjoin A {b : B | ∃ n_1 ∈ S ∪ {n}, n_1 ≠ 0 ∧ b ^ n_1 = 1})
Tactic: rw [← ((iff_adjoin_eq_top S A B).1 H).2]
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
n : ℕ
inst : NeZero n
S : Set ℕ
A : Type u
B : Type v
inst_1 : CommRing A
inst_2 : CommRing B
inst_3 : Algebra A B
h : ∃ s ∈ S, s ≠ 0 ∧ n ∣ s
H : IsCyclotomicExtension S A B
⊢ LE.le.{v} (α := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})
    (adjoin A {b : B | ∃ n_1 ∈ S ∪ {n}, n_1 ≠ 0 ∧ b ^ n_1 = 1})
have refine_2 : ∀ (x : B) (hx : Membership.mem (γ := Set B) {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1} x),
  Membership.mem (γ := Set B) {b : B | ∃ n_1 ∈ S ∪ {n}, n_1 ≠ 0 ∧ b ^ n_1 = 1} x
Tactic: refine adjoin_mono fun x hx ↦ ?_
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
n : ℕ
inst : NeZero n
S : Set ℕ
A : Type u
B : Type v
inst_1 : CommRing A
inst_2 : CommRing B
inst_3 : Algebra A B
h : ∃ s ∈ S, s ≠ 0 ∧ n ∣ s
H : IsCyclotomicExtension S A B
x : B
hx : Membership.mem (γ := Set B) {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1} x
⊢ Membership.mem (γ := Set B) {b : B | ∃ n_1 ∈ S ∪ {n}, n_1 ≠ 0 ∧ b ^ n_1 = 1} x
have refine_2 : ∃ (n_1 : ℕ), (n_1 = n ∨ n_1 ∈ S) ∧ n_1 ≠ 0 ∧ x ^ n_1 = 1
Tactic: simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
n : ℕ
inst : NeZero n
S : Set ℕ
A : Type u
B : Type v
inst_1 : CommRing A
inst_2 : CommRing B
inst_3 : Algebra A B
h : ∃ s ∈ S, s ≠ 0 ∧ n ∣ s
H : IsCyclotomicExtension S A B
x : B
hx : ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1
⊢ ∃ (n_1 : ℕ), (n_1 = n ∨ n_1 ∈ S) ∧ n_1 ≠ 0 ∧ x ^ n_1 = 1
have refine_2 : ∀ (m : ℕ) (hm : m ∈ S) (hm' : m ≠ 0 ∧ x ^ m = 1), ∃ (n_1 : ℕ), (n_1 = n ∨ n_1 ∈ S) ∧ n_1 ≠ 0 ∧ x ^ n_1 = 1
Tactic: obtain ⟨m, hm, hm'⟩ := hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
n : ℕ
inst : NeZero n
S : Set ℕ
A : Type u
B : Type v
inst_1 : CommRing A
inst_2 : CommRing B
inst_3 : Algebra A B
h : ∃ s ∈ S, s ≠ 0 ∧ n ∣ s
⊢ IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B
have refine_1 : ∀ (H : IsCyclotomicExtension (S ∪ {n}) A B) (s : ℕ) (hs : s ∈ S), s ≠ 0 → ∃ (r : B), IsPrimitiveRoot r s
Tactic: refine
    ⟨fun H ↦ of_union_of_dvd A B h, fun H => (iff_adjoin_eq_top _ A _).2
      ⟨fun s hs ↦ ?_, ?_⟩⟩
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
n : ℕ
inst : NeZero n
S : Set ℕ
A : Type u
B : Type v
inst_1 : CommRing A
inst_2 : CommRing B
inst_3 : Algebra A B
h : ∃ s ∈ S, s ≠ 0 ∧ n ∣ s
refine_1 : ∀ (H : IsCyclotomicExtension (S ∪ {n}) A B) (s : ℕ) (hs : s ∈ S), s ≠ 0 → ∃ (r : B), IsPrimitiveRoot r s
⊢ IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B
have refine_2 : ∀ (H : IsCyclotomicExtension (S ∪ {n}) A B),
  Eq.{v + 1} (α := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) ⊤
Tactic: refine
    ⟨fun H ↦ of_union_of_dvd A B h, fun H => (iff_adjoin_eq_top _ A _).2
      ⟨fun s hs ↦ ?_, ?_⟩⟩
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
n : ℕ
inst : NeZero n
S : Set ℕ
A : Type u
B : Type v
inst_1 : CommRing A
inst_2 : CommRing B
inst_3 : Algebra A B
h : ∃ s ∈ S, s ≠ 0 ∧ n ∣ s
H : IsCyclotomicExtension (S ∪ {n}) A B
⊢ Eq.{v + 1} (α := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) ⊤
have refine_2 : LE.le.{v} (α := Subalgebra A B) (adjoin A {b : B | ∃ n_1 ∈ S ∪ {n}, n_1 ≠ 0 ∧ b ^ n_1 = 1})
  (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})
Tactic: rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
n : ℕ
inst : NeZero n
S : Set ℕ
A : Type u
B : Type v
inst_1 : CommRing A
inst_2 : CommRing B
inst_3 : Algebra A B
h : ∃ s ∈ S, s ≠ 0 ∧ n ∣ s
H : IsCyclotomicExtension (S ∪ {n}) A B
⊢ LE.le.{v} (α := Subalgebra A B) (adjoin A {b : B | ∃ n_1 ∈ S ∪ {n}, n_1 ≠ 0 ∧ b ^ n_1 = 1})
    (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})
have refine_2 : ∀ (x : B) (hx : Membership.mem (γ := Set B) {b : B | ∃ n_1 ∈ S ∪ {n}, n_1 ≠ 0 ∧ b ^ n_1 = 1} x),
  Membership.mem (γ := Set B) {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1} x
Tactic: refine adjoin_mono fun x hx => ?_
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (hypothesis)
Goal:
n : ℕ
inst : NeZero n
S : Set ℕ
A : Type u
B : Type v
inst_1 : CommRing A
inst_2 : CommRing B
inst_3 : Algebra A B
h : ∃ s ∈ S, s ≠ 0 ∧ n ∣ s
H : IsCyclotomicExtension (S ∪ {n}) A B
x : B
hx : Membership.mem (γ := Set B) {b : B | ∃ n_1 ∈ S ∪ {n}, n_1 ≠ 0 ∧ b ^ n_1 = 1} x
⊢ Membership.mem (γ := Set B) {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1} x
have hx₁ : ∃ (n_1 : ℕ), (n_1 = n ∨ n_1 ∈ S) ∧ n_1 ≠ 0 ∧ x ^ n_1 = 1
Tactic: simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
n : ℕ
inst : NeZero n
S : Set ℕ
A : Type u
B : Type v
inst_1 : CommRing A
inst_2 : CommRing B
inst_3 : Algebra A B
h : ∃ s ∈ S, s ≠ 0 ∧ n ∣ s
H : IsCyclotomicExtension (S ∪ {n}) A B
x : B
hx : ∃ (n_1 : ℕ), (n_1 = n ∨ n_1 ∈ S) ∧ n_1 ≠ 0 ∧ x ^ n_1 = 1
⊢ ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1
have refine_2 : ∀ (m : ℕ) (hxpow : m ≠ 0 ∧ x ^ m = 1) [NeZero m] (h : ∃ s ∈ S, s ≠ 0 ∧ m ∣ s) (H : IsCyclotomicExtension (S ∪ {m}) A B),
  ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1
Tactic: obtain ⟨m, rfl | hm, hxpow⟩ := hx
Cleared hypotheses: n, inst, h, H, hx
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
n : ℕ
inst : NeZero n
S : Set ℕ
A : Type u
B : Type v
inst_1 : CommRing A
inst_2 : CommRing B
inst_3 : Algebra A B
h : ∃ s ∈ S, s ≠ 0 ∧ n ∣ s
H : IsCyclotomicExtension (S ∪ {n}) A B
x : B
hx : ∃ (n_1 : ℕ), (n_1 = n ∨ n_1 ∈ S) ∧ n_1 ≠ 0 ∧ x ^ n_1 = 1
refine_2 :
  ∀ (m : ℕ) (hxpow : m ≠ 0 ∧ x ^ m = 1) [NeZero m] (h : ∃ s ∈ S, s ≠ 0 ∧ m ∣ s)
    (H : IsCyclotomicExtension (S ∪ {m}) A B), ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1
⊢ ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1
have refine_2₁ : ∀ (m : ℕ) (hxpow : m ≠ 0 ∧ x ^ m = 1) (hm : m ∈ S), ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1
Tactic: obtain ⟨m, rfl | hm, hxpow⟩ := hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
S : Set ℕ
A : Type u
B : Type v
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
x : B
m : ℕ
hxpow : m ≠ 0 ∧ x ^ m = 1
inst_3 : NeZero m
h : ∃ s ∈ S, s ≠ 0 ∧ m ∣ s
H : IsCyclotomicExtension (S ∪ {m}) A B
⊢ ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1
have refine_2 : ∀ (z : ℕ) (hy : m * z ∈ S) (hy' : m * z ≠ 0), ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1
Tactic: obtain ⟨y, ⟨hy, hy', ⟨z, rfl⟩⟩⟩ := h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
S : Set ℕ
A : Type u
B : Type v
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
⊢ IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B
have pos : ∀ (hS : ∃ s ∈ S, s ≠ 0), IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B
Tactic: by_cases hS : ∃ s ∈ S, s ≠ 0
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
S : Set ℕ
A : Type u
B : Type v
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
pos : ∀ (hS : ∃ s ∈ S, s ≠ 0), IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B
⊢ IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B
have neg : ∀ (hS : ¬∃ s ∈ S, s ≠ 0), IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B
Tactic: by_cases hS : ∃ s ∈ S, s ≠ 0
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
S : Set ℕ
A : Type u
B : Type v
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
hS : ¬∃ s ∈ S, s ≠ 0
⊢ IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B
have neg : IsCyclotomicExtension ∅ A B ↔ IsCyclotomicExtension {1} A B
Tactic: rw [eq_self_sdiff_zero S, eq_self_sdiff_zero (S ∪ {1}), union_diff_distrib,
      show S \ {0} = ∅ by aesop, empty_union, show {1} \ {0} = {1} by simp]
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
S : Set ℕ
A : Type u
B : Type v
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
hS : ¬∃ s ∈ S, s ≠ 0
⊢ IsCyclotomicExtension ∅ A B ↔ IsCyclotomicExtension {1} A B
have neg : ∀ (H : IsCyclotomicExtension ∅ A B), IsCyclotomicExtension {1} A B
Tactic: refine ⟨fun H ↦ ?_, fun H ↦ ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
S : Set ℕ
A : Type u
B : Type v
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
hS : ¬∃ s ∈ S, s ≠ 0
neg : ∀ (H : IsCyclotomicExtension ∅ A B), IsCyclotomicExtension {1} A B
⊢ IsCyclotomicExtension ∅ A B ↔ IsCyclotomicExtension {1} A B
have neg₁ : ∀ (H : IsCyclotomicExtension {1} A B), IsCyclotomicExtension ∅ A B
Tactic: refine ⟨fun H ↦ ?_, fun H ↦ ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
S : Set ℕ
A : Type u
B : Type v
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
hS : ¬∃ s ∈ S, s ≠ 0
H : IsCyclotomicExtension ∅ A B
⊢ IsCyclotomicExtension {1} A B
have neg : Eq.{v + 1} (α := Subalgebra A B) (adjoin A {b : B | ∃ (n : ℕ), Membership.mem (γ := Set ℕ) {1} n ∧ n ≠ 0 ∧ b ^ n = 1}) ⊤
Tactic: refine (iff_adjoin_eq_top _ A _).2 ⟨fun s hs _ ↦ ⟨1, by simp [mem_singleton_iff.1 hs]⟩, ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
S : Set ℕ
A : Type u
B : Type v
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
hS : ¬∃ s ∈ S, s ≠ 0
H : IsCyclotomicExtension {1} A B
⊢ IsCyclotomicExtension ∅ A B
have neg : Eq.{v + 1} (α := Subalgebra A B) (adjoin A {b : B | ∃ (n : ℕ), Membership.mem (γ := Set ℕ) ∅ n ∧ n ≠ 0 ∧ b ^ n = 1}) ⊤
Tactic: refine (iff_adjoin_eq_top _ A _).2 ⟨fun s hs ↦ (notMem_empty s hs).elim, ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
A : Type u
B : Type v
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
h : Eq.{v + 1} (α := Subalgebra A B) ⊥ ⊤
⊢ IsCyclotomicExtension {1} A B
have h₁ : Eq (α := Set ℕ) {1} (({0} ∪ {1}) \ {0})
Tactic: convert!
    eq_self_sdiff_zero _ A B ▸ (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (hypothesis)
Goal:
S : Set ℕ
A : Type u
B : Type v
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
C : Type u_1
inst_3 : CommRing C
inst_4 : Algebra A C
h : IsCyclotomicExtension S A B
f : B ≃ₐ[A] C
this : Algebra B C := (↑f : B →ₐ[A] C).toAlgebra
⊢ IsCyclotomicExtension S A C
have this₁ : IsCyclotomicExtension {1} B C
Tactic: haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (hypothesis)
Goal:
S : Set ℕ
A : Type u
B : Type v
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
C : Type u_1
inst_3 : CommRing C
inst_4 : Algebra A C
h : IsCyclotomicExtension S A B
f : B ≃ₐ[A] C
this_1 : Algebra B C := (↑f : B →ₐ[A] C).toAlgebra
this : IsCyclotomicExtension {1} B C
⊢ IsCyclotomicExtension S A C
have this₁ : IsScalarTower A B C
Tactic: haveI : IsScalarTower A B C := IsScalarTower.of_algHom f.toAlgHom
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
S : Set ℕ
A : Type u
B : Type v
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
inst_3 : IsCyclotomicExtension S A B
⊢ Algebra.IsIntegral A B
have rw : Algebra.IsIntegral.{u, v} A ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})
Tactic: rw [← (Subalgebra.equivOfEq _ _ ((IsCyclotomicExtension.iff_adjoin_eq_top S A B).1 ‹_›).2
    |>.trans Subalgebra.topEquiv).isIntegral_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
S : Set ℕ
A : Type u
B : Type v
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
h : ∀ n ∈ S, n ≠ 0 → ∃ (r : B), IsPrimitiveRoot r n
n : ℕ
hn1 : n ∈ S
hn2 : n ≠ 0
⊢ ∃ (r : ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})), IsPrimitiveRoot r n
have obtain : ∀ (r : B) (hr1 : r ^ n = 1) (hr2 : ∀ (l : ℕ), r ^ l = 1 → n ∣ l),
  ∃ (r : ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})), IsPrimitiveRoot r n
Tactic: obtain ⟨r, hr1, hr2⟩ := h n hn1 hn2
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
S : Set ℕ
A : Type u
B : Type v
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
h : ∀ n ∈ S, n ≠ 0 → ∃ (r : B), IsPrimitiveRoot r n
x_1 x : B
hx : Membership.mem (γ := Set B) {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1} x
⊢ Membership.mem (γ := Subalgebra A ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}))
    (adjoin A {b : ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})
    (Subtype.mk (p := fun (x : B) ↦
      Membership.mem (γ := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) x) x ⋯)
have mem : ∀ (n : ℕ) (hn1 : n ∈ S) (hn2 : n ≠ 0) (hx : x ^ n = 1),
  Membership.mem (γ := Subalgebra A ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}))
    (adjoin A {b : ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})
    (Subtype.mk (p := fun (x : B) ↦
      Membership.mem (γ := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) x) x sorry)
Tactic: obtain ⟨n, hn1, hn2, hx⟩ := hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
S : Set ℕ
A : Type u
B : Type v
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
h : ∀ n ∈ S, n ≠ 0 → ∃ (r : B), IsPrimitiveRoot r n
x : B
hx : Membership.mem (γ := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) x
⊢ Membership.mem (γ := Subalgebra A ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}))
    (adjoin A {b : ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})
    (Subtype.mk (p := fun (x : B) ↦
      Membership.mem (γ := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) x) x hx)
have mem : ∀ (x : B) (hx : Membership.mem (γ := Set B) {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1} x),
  Membership.mem (γ := Subalgebra A ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}))
    (adjoin A {b : ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})
    (Subtype.mk (p := fun (x : B) ↦
      Membership.mem (γ := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) x) x sorry)
Tactic: induction hx using adjoin_induction with
    | mem x hx =>
      obtain ⟨n, hn1, hn2, hx⟩ := hx
      exact subset_adjoin ⟨n, hn1, hn2, Subtype.val_injective hx⟩
    | algebraMap x => exact Subalgebra.algebraMap_mem _ x
    | add x y hx hy ihx ihy => exact Subalgebra.add_mem _ ihx ihy
    | mul x y hx hy ihx ihy => exact Subalgebra.mul_mem _ ihx ihy
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
S : Set ℕ
A : Type u
B : Type v
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
h : ∀ n ∈ S, n ≠ 0 → ∃ (r : B), IsPrimitiveRoot r n
x : B
hx : Membership.mem (γ := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) x
mem :
  ∀ (x : B) (hx : Membership.mem (γ := Set B) {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1} x),
    Membership.mem (γ := Subalgebra A ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}))
      (adjoin A {b : ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})
      (Subtype.mk (p := fun (x : B) ↦
        Membership.mem (γ := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) x) x ⋯)
⊢ Membership.mem (γ := Subalgebra A ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}))
    (adjoin A {b : ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})
    (Subtype.mk (p := fun (x : B) ↦
      Membership.mem (γ := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) x) x hx)
have algebraMap : ∀ (x : A),
  Membership.mem.{v, v} (α := ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})) (γ :=
    Subalgebra A ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}))
    (adjoin A {b : ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})
    (Subtype.mk (p := fun (x : B) ↦
        Membership.mem (γ := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) x)
        ((algebraMap A B : (a : A) → B) x) sorry :
      ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}))
Tactic: induction hx using adjoin_induction with
    | mem x hx =>
      obtain ⟨n, hn1, hn2, hx⟩ := hx
      exact subset_adjoin ⟨n, hn1, hn2, Subtype.val_injective hx⟩
    | algebraMap x => exact Subalgebra.algebraMap_mem _ x
    | add x y hx hy ihx ihy => exact Subalgebra.add_mem _ ihx ihy
    | mul x y hx hy ihx ihy => exact Subalgebra.mul_mem _ ihx ihy
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
S : Set ℕ
A : Type u
B : Type v
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
h : ∀ n ∈ S, n ≠ 0 → ∃ (r : B), IsPrimitiveRoot r n
x : B
hx : Membership.mem (γ := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) x
mem :
  ∀ (x : B) (hx : Membership.mem (γ := Set B) {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1} x),
    Membership.mem (γ := Subalgebra A ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}))
      (adjoin A {b : ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})
      (Subtype.mk (p := fun (x : B) ↦
        Membership.mem (γ := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) x) x ⋯)
algebraMap :
  ∀ (x : A),
    Membership.mem.{v, v} (α := ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})) (γ :=
      Subalgebra A ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}))
      (adjoin A {b : ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})
      (Subtype.mk (p := fun (x : B) ↦
          Membership.mem (γ := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) x)
          ((Algebra.algebraMap A B : (a : A) → B) x) ⋯ :
        ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}))
⊢ Membership.mem (γ := Subalgebra A ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}))
    (adjoin A {b : ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})
    (Subtype.mk (p := fun (x : B) ↦
      Membership.mem (γ := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) x) x hx)
have add : ∀ (x y : B) (hx : Membership.mem (γ := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) x)
  (hy : Membership.mem (γ := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) y)
  (ihx :
    Membership.mem (γ := Subalgebra A ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}))
      (adjoin A {b : ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})
      (Subtype.mk (p := fun (x : B) ↦
        Membership.mem (γ := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) x) x hx))
  (ihy :
    Membership.mem (γ := Subalgebra A ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}))
      (adjoin A {b : ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})
      (Subtype.mk (p := fun (x : B) ↦
        Membership.mem (γ := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) x) y hy)),
  Membership.mem (γ := Subalgebra A ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}))
    (adjoin A {b : ↥(adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})
    (Subtype.mk (p := fun (x : B) ↦
      Membership.mem (γ := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) x) (x + y) sorry)
Tactic: induction hx using adjoin_induction with
    | mem x hx =>
      obtain ⟨n, hn1, hn2, hx⟩ := hx
      exact subset_adjoin ⟨n, hn1, hn2, Subtype.val_injective hx⟩
    | algebraMap x => exact Subalgebra.algebraMap_mem _ x
    | add x y hx hy ihx ihy => exact Subalgebra.add_mem _ ihx ihy
    | mul x y hx hy ihx ihy => exact Subalgebra.mul_mem _ ihx ihy
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
S : Set ℕ
A : Type u
B : Type v
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
inst_3 : IsCyclotomicExtension S A B
inst_4 : IsDomain B
f g : B ≃ₐ[A] B
H : ∀ n ∈ S, n ≠ 0 → ∃ (r : B), IsPrimitiveRoot r n ∧ Eq (α := B) ((f : (a : B) → B) r : B) ((g : (a : B) → B) r : B)
⊢ f = g
have h : ∀ (x : B), Eq (α := B) ((f : (a : B) → B) x : B) ((g : (a : B) → B) x : B)
Tactic: ext x
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (hypothesis)
Goal:
S : Set ℕ
A : Type u
B : Type v
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
inst_3 : IsCyclotomicExtension S A B
inst_4 : IsDomain B
f g : B ≃ₐ[A] B
H : ∀ n ∈ S, n ≠ 0 → ∃ (r : B), IsPrimitiveRoot r n ∧ Eq (α := B) ((f : (a : B) → B) r : B) ((g : (a : B) → B) r : B)
x : B
⊢ Eq (α := B) ((f : (a : B) → B) x : B) ((g : (a : B) → B) x : B)
have hx : Membership.mem (γ := Subalgebra A B) (adjoin A {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}) x
Tactic: have hx := ‹IsCyclotomicExtension S A B›.adjoin_roots x
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
S : Set ℕ
A : Type u
B : Type v
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
inst_3 : IsCyclotomicExtension S A B
inst_4 : IsDomain B
f g : B ≃ₐ[A] B
H : ∀ n ∈ S, n ≠ 0 → ∃ (r : B), IsPrimitiveRoot r n ∧ Eq (α := B) ((f : (a : B) → B) r : B) ((g : (a : B) → B) r : B)
x y : B
hy : Membership.mem (γ := Set B) {b : B | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1} y
⊢ Eq (α := B) ((f : (a : B) → B) y : B) ((g : (a : B) → B) y : B)
have h : ∀ (n : ℕ) (hn : n ∈ S) (h1 : n ≠ 0) (h2 : y ^ n = 1), Eq (α := B) ((f : (a : B) → B) y : B) ((g : (a : B) → B) y : B)
Tactic: obtain ⟨n, hn, h1, h2⟩ := hy
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (goal)
Goal:
S : Set ℕ
A : Type u
B : Type v
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
inst_3 : IsCyclotomicExtension S A B
inst_4 : IsDomain B
f g : B ≃ₐ[A] B
H : ∀ n ∈ S, n ≠ 0 → ∃ (r : B), IsPrimitiveRoot r n ∧ Eq (α := B) ((f : (a : B) → B) r : B) ((g : (a : B) → B) r : B)
x y : B
n : ℕ
hn : n ∈ S
h1 : n ≠ 0
h2 : y ^ n = 1
⊢ Eq (α := B) ((f : (a : B) → B) y : B) ((g : (a : B) → B) y : B)
have h : ∀ (r : B) (hr1 : IsPrimitiveRoot r n) (hr2 : Eq (α := B) ((f : (a : B) → B) r : B) ((g : (a : B) → B) r : B)),
  Eq (α := B) ((f : (a : B) → B) y : B) ((g : (a : B) → B) y : B)
Tactic: obtain ⟨r, hr1, hr2⟩ := H n hn h1
-- Lean 4 proof step from Mathlib.NumberTheory.Cyclotomic.Basic (hypothesis)
Goal:
S : Set ℕ
A : Type u
B : Type v
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
inst_3 : IsCyclotomicExtension S A B
inst_4 : IsDomain B
f g : B ≃ₐ[A] B
H : ∀ n ∈ S, n ≠ 0 → ∃ (r : B), IsPrimitiveRoot r n ∧ Eq (α := B) ((f : (a : B) → B) r : B) ((g : (a : B) → B) r : B)
x y : B
n : ℕ
hn : n ∈ S
h1 : n ≠ 0
h2 : y ^ n = 1
r : B
hr1 : IsPrimitiveRoot r n
hr2 : Eq (α := B) ((f : (a : B) → B) r : B) ((g : (a : B) → B) r : B)
⊢ Eq (α := B) ((f : (a : B) → B) y : B) ((g : (a : B) → B) y : B)
have this : NeZero n
Tactic: have := NeZero.mk h1