awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
m : ℕ
inst_1 : NeZero n
ψ : DirichletCharacter R m
h :
  Eq (α := DirichletCharacter R (n * m))
    ((changeLevel ⋯ : (a : DirichletCharacter R n) → DirichletCharacter R (n * m)) χ : DirichletCharacter R (n * m))
    ((changeLevel ⋯ : (a : DirichletCharacter R m) → DirichletCharacter R (n * m)) ψ : DirichletCharacter R (n * m))
x : (ZMod n)ˣ
hx : x ∈ (ZMod.unitsMap ⋯).ker
z : ℕ
hz₁ : (↑z : ZMod n) = (↑x : ZMod n)
hz₂ : Eq (α := ZMod m) (↑z : ZMod m) 1
hz₀ : z.gcd (n * m) = 1
this_1 :
  Eq (α := R)
    ((((changeLevel ⋯ : (a : DirichletCharacter R n) → DirichletCharacter R (n * m)) χ : DirichletCharacter R (n * m)) :
          (a : ZMod (n * m)) → R)
        (↑z : ZMod (n * m)) :
      R)
    ((χ : (a : ZMod n) → R) (↑z : ZMod n) : R)
this :
  Eq (α := R)
    ((((changeLevel ⋯ : (a : DirichletCharacter R m) → DirichletCharacter R (n * m)) ψ : DirichletCharacter R (n * m)) :
          (a : ZMod (n * m)) → R)
        (↑z : ZMod (n * m)) :
      R)
    ((ψ : (a : ZMod m) → R) (↑z : ZMod m) : R)
⊢ Eq (α := R)
    ((((changeLevel ⋯ : (a : DirichletCharacter R m) → DirichletCharacter R (n * m)) ψ : DirichletCharacter R (n * m)) :
          (a : ZMod (n * m)) → R)
        (↑z : ZMod (n * m)) :
      R)
    (1 : R)
have rw : Eq (α := R) ((ψ : (a : ZMod m) → R) (↑z : ZMod m) : R) (1 : R)
Tactic: rw [this, hz₂, map_one]
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
m : ℕ
inst_1 : NeZero n
ψ : DirichletCharacter R m
h :
  Eq (α := DirichletCharacter R (n * m))
    ((changeLevel ⋯ : (a : DirichletCharacter R n) → DirichletCharacter R (n * m)) χ : DirichletCharacter R (n * m))
    ((changeLevel ⋯ : (a : DirichletCharacter R m) → DirichletCharacter R (n * m)) ψ : DirichletCharacter R (n * m))
x : (ZMod n)ˣ
hx : x ∈ (ZMod.unitsMap ⋯).ker
z : ℕ
hz₁ : (↑z : ZMod n) = (↑x : ZMod n)
hz₂ : Eq (α := ZMod m) (↑z : ZMod m) 1
hz₀ : z.gcd (n * m) = 1
this_1 :
  Eq (α := R)
    ((((changeLevel ⋯ : (a : DirichletCharacter R n) → DirichletCharacter R (n * m)) χ : DirichletCharacter R (n * m)) :
          (a : ZMod (n * m)) → R)
        (↑z : ZMod (n * m)) :
      R)
    ((χ : (a : ZMod n) → R) (↑z : ZMod n) : R)
this :
  Eq (α := R)
    ((((changeLevel ⋯ : (a : DirichletCharacter R m) → DirichletCharacter R (n * m)) ψ : DirichletCharacter R (n * m)) :
          (a : ZMod (n * m)) → R)
        (↑z : ZMod (n * m)) :
      R)
    ((ψ : (a : ZMod m) → R) (↑z : ZMod m) : R)
rw : Eq (α := R) ((ψ : (a : ZMod m) → R) (↑z : ZMod m) : R) (1 : R)
⊢ Eq (α := R)
    ((((changeLevel ⋯ : (a : DirichletCharacter R m) → DirichletCharacter R (n * m)) ψ : DirichletCharacter R (n * m)) :
          (a : ZMod (n * m)) → R)
        (↑z : ZMod (n * m)) :
      R)
    (1 : R)
have rw₁ : Eq (α := R) ((ψ : (a : ZMod m) → R) 1 : R) (1 : R)
Tactic: rw [this, hz₂, map_one]
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
m : ℕ
inst_1 : NeZero n
ψ : DirichletCharacter R m
h :
  Eq (α := DirichletCharacter R (n * m))
    ((changeLevel ⋯ : (a : DirichletCharacter R n) → DirichletCharacter R (n * m)) χ : DirichletCharacter R (n * m))
    ((changeLevel ⋯ : (a : DirichletCharacter R m) → DirichletCharacter R (n * m)) ψ : DirichletCharacter R (n * m))
x : (ZMod n)ˣ
hx : x ∈ (ZMod.unitsMap ⋯).ker
z : ℕ
hz₁ : (↑z : ZMod n) = (↑x : ZMod n)
hz₂ : Eq (α := ZMod m) (↑z : ZMod m) 1
hz₀ : z.gcd (n * m) = 1
this_1 :
  Eq (α := R)
    ((((changeLevel ⋯ : (a : DirichletCharacter R n) → DirichletCharacter R (n * m)) χ : DirichletCharacter R (n * m)) :
          (a : ZMod (n * m)) → R)
        (↑z : ZMod (n * m)) :
      R)
    ((χ : (a : ZMod n) → R) (↑z : ZMod n) : R)
this :
  Eq (α := R)
    ((((changeLevel ⋯ : (a : DirichletCharacter R m) → DirichletCharacter R (n * m)) ψ : DirichletCharacter R (n * m)) :
          (a : ZMod (n * m)) → R)
        (↑z : ZMod (n * m)) :
      R)
    ((ψ : (a : ZMod m) → R) (↑z : ZMod m) : R)
rw : Eq (α := R) ((ψ : (a : ZMod m) → R) (↑z : ZMod m) : R) (1 : R)
rw₁ : Eq (α := R) ((ψ : (a : ZMod m) → R) 1 : R) (1 : R)
⊢ Eq (α := R)
    ((((changeLevel ⋯ : (a : DirichletCharacter R m) → DirichletCharacter R (n * m)) ψ : DirichletCharacter R (n * m)) :
          (a : ZMod (n * m)) → R)
        (↑z : ZMod (n * m)) :
      R)
    (1 : R)
have rw₂ : Eq.{u_1 + 1} (α := R) 1 1
Tactic: rw [this, hz₂, map_one]
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
χ : DirichletCharacter R 1
⊢ χ = 1
have h : ∀ (a : (ZMod 1)ˣ), Eq (α := R) ((χ : (a : ZMod 1) → R) (↑a : ZMod 1) : R) ((1 : (a : ZMod 1) → R) (↑a : ZMod 1) : R)
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
hn : n = 1
⊢ χ = 1
have subst : ∀ (χ : DirichletCharacter R 1), χ = 1
Tactic: subst hn
Cleared hypotheses: n, χ, hn
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
⊢ Subsingleton.{u_1 + 1} (DirichletCharacter R 1)
have refine : ∀ (χ χ' : DirichletCharacter R 1), χ = χ'
Tactic: refine subsingleton_iff.mpr (fun χ χ' ↦ ?_)
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
h : χ = 1
⊢ χ = (changeLevel ⋯ : (a : DirichletCharacter R 1) → DirichletCharacter R n) 1
have rw : Eq.{u_1 + 1} (α := DirichletCharacter R n) 1 ((changeLevel sorry : (a : DirichletCharacter R 1) → DirichletCharacter R n) 1)
Tactic: rw [h, changeLevel_one]
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
h : χ = 1
rw :
  Eq.{u_1 + 1} (α := DirichletCharacter R n) 1
    ((changeLevel ⋯ : (a : DirichletCharacter R 1) → DirichletCharacter R n) 1)
⊢ χ = (changeLevel ⋯ : (a : DirichletCharacter R 1) → DirichletCharacter R n) 1
have rw₁ : Eq.{u_1 + 1} (α := DirichletCharacter R n) 1 1
Tactic: rw [h, changeLevel_one]
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
⊢ χ.FactorsThrough 1 ↔ χ = 1
have refine : χ.FactorsThrough 1 →
  ∀ (w : 1 ∣ n) (χ₀ : DirichletCharacter R 1)
    (hχ₀ : χ = (changeLevel w : (a : DirichletCharacter R 1) → DirichletCharacter R n) χ₀), χ = 1
Tactic: refine ⟨fun ⟨_, χ₀, hχ₀⟩ ↦ ?_,
          fun h ↦ ⟨one_dvd n, 1, by rw [h, changeLevel_one]⟩⟩
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
h : χ.FactorsThrough 1
w : 1 ∣ n
χ₀ : DirichletCharacter R 1
hχ₀ : χ = (changeLevel w : (a : DirichletCharacter R 1) → DirichletCharacter R n) χ₀
⊢ χ = 1
have rwa : ∀ (hχ₀ : χ = (changeLevel w : (a : DirichletCharacter R 1) → DirichletCharacter R n) 1), χ = 1
Tactic: rwa [level_one χ₀, changeLevel_one] at hχ₀
Cleared hypotheses: hχ₀
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
h : χ.FactorsThrough 1
w : 1 ∣ n
χ₀ : DirichletCharacter R 1
hχ₀ : χ = (changeLevel w : (a : DirichletCharacter R 1) → DirichletCharacter R n) χ₀
rwa : ∀ (hχ₀ : χ = (changeLevel w : (a : DirichletCharacter R 1) → DirichletCharacter R n) 1), χ = 1
⊢ χ = 1
have rwa₁ : ∀ (hχ₀ : χ = 1), χ = 1
Tactic: rwa [level_one χ₀, changeLevel_one] at hχ₀
Cleared hypotheses: hχ₀
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
inst_1 : NeZero n
this : FactorsThrough.{u_1} (R := R) (n := n) 1 1
⊢ conductor.{u_1} (R := R) (n := n) 1 = 1
have h : conductor.{u_1} (R := R) (n := n) 1 ≤ 1
Tactic: have h : conductor (1 : DirichletCharacter R n) ≤ 1 :=
      Nat.sInf_le <| (mem_conductorSet_iff _).mpr this
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
inst_1 : NeZero n
⊢ conductor.{u_1} (R := R) (n := n) 1 = 1
have suffices : FactorsThrough.{u_1} (R := R) (n := n) 1 1
Tactic: suffices FactorsThrough (1 : DirichletCharacter R n) 1 by
    have h : conductor (1 : DirichletCharacter R n) ≤ 1 :=
      Nat.sInf_le <| (mem_conductorSet_iff _).mpr this
    exact Nat.le_antisymm h (Nat.pos_of_ne_zero <| conductor_ne_zero _)
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
inst_1 : NeZero n
⊢ χ = 1 ↔ χ.conductor = 1
have refine : ∀ (hχ : χ.conductor = 1), χ = 1
Tactic: refine ⟨fun h ↦ h ▸ conductor_one, fun hχ ↦ ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
inst_1 : NeZero n
hχ : χ.conductor = 1
⊢ χ = 1
have obtain : ∀ (h' : χ.conductor ∣ n) (χ₀ : DirichletCharacter R χ.conductor)
  (h : χ = (changeLevel h' : (a : DirichletCharacter R χ.conductor) → DirichletCharacter R n) χ₀), χ = 1
Tactic: obtain ⟨h', χ₀, h⟩ := factorsThrough_conductor χ
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
⊢ χ.conductor = 0 ↔ n = 0
have refine_1 : χ.conductor = 0 → n = 0
Tactic: refine ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
refine_1 : χ.conductor = 0 → n = 0
⊢ χ.conductor = 0 ↔ n = 0
have refine_2 : n = 0 → χ.conductor = 0
Tactic: refine ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
⊢ χ.conductor = 0 → n = 0
have refine_1 : n ≠ 0 → χ.conductor ≠ 0
Tactic: contrapose!
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
d : ℕ
hd : d ∈ χ.conductorSet
⊢ χ.conductor ≤
    conductor.{u_1} (R := R) (n := d)
      (Classical.choose (p := fun (x : DirichletCharacter R d) ↦
        χ = (changeLevel ⋯ : (a : DirichletCharacter R d) → DirichletCharacter R n) x) ⋯)
have refine : χ =
  (changeLevel sorry : (a : DirichletCharacter R (Classical.choose sorry).conductor) → DirichletCharacter R n)
    (Exists.choose (p :=
      fun
        (χ₀ :
          DirichletCharacter R
            (conductor.{u_1} (R := R) (n := d)
              (Classical.choose (p := fun (x : DirichletCharacter R d) ↦
                χ = (changeLevel sorry : (a : DirichletCharacter R d) → DirichletCharacter R n) x) sorry))) ↦
      Eq (α := DirichletCharacter R d)
        (Classical.choose (p := fun (x : DirichletCharacter R d) ↦
            χ = (changeLevel sorry : (a : DirichletCharacter R d) → DirichletCharacter R n) x) sorry :
          DirichletCharacter R d)
        ((changeLevel sorry : (a : DirichletCharacter R (Classical.choose sorry).conductor) → DirichletCharacter R d) χ₀ :
          DirichletCharacter R d))
      sorry)
Tactic: refine Nat.sInf_le <| (mem_conductorSet_iff χ).mpr <|
    ⟨dvd_trans (conductor_dvd_level _) hd.1,
     (factorsThrough_conductor (Classical.choose hd.2)).2.choose, ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
d : ℕ
hd : d ∈ χ.conductorSet
⊢ χ =
    (changeLevel ⋯ : (a : DirichletCharacter R (Classical.choose ⋯).conductor) → DirichletCharacter R n)
      (Exists.choose (p :=
        fun
          (χ₀ :
            DirichletCharacter R
              (conductor.{u_1} (R := R) (n := d)
                (Classical.choose (p := fun (x : DirichletCharacter R d) ↦
                  χ = (changeLevel ⋯ : (a : DirichletCharacter R d) → DirichletCharacter R n) x) ⋯))) ↦
        Eq (α := DirichletCharacter R d)
          (Classical.choose (p := fun (x : DirichletCharacter R d) ↦
              χ = (changeLevel ⋯ : (a : DirichletCharacter R d) → DirichletCharacter R n) x) ⋯ :
            DirichletCharacter R d)
          ((changeLevel ⋯ : (a : DirichletCharacter R (Classical.choose ⋯).conductor) → DirichletCharacter R d) χ₀ :
            DirichletCharacter R d))
        ⋯)
have rw : χ =
  (changeLevel sorry : (a : DirichletCharacter R d) → DirichletCharacter R n)
    (Classical.choose (p := fun (x : DirichletCharacter R d) ↦
      χ = (changeLevel sorry : (a : DirichletCharacter R d) → DirichletCharacter R n) x) sorry)
Tactic: rw [changeLevel_trans _ (conductor_dvd_level _) hd.dvd,
      ← (factorsThrough_conductor (Classical.choose hd.2)).2.choose_spec]
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
⊢ conductor.{u_1} (R := R) (n := 1) 1 ∣ n
have rw : 1 ∣ n
Tactic: rw [(isPrimitive_def _).mp isPrimitive_one_level_one]
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
⊢ χ.primitiveCharacter.IsPrimitive
have pos : ∀ (h : χ.conductor = 0), χ.primitiveCharacter.IsPrimitive
Tactic: by_cases h : χ.conductor = 0
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
pos : ∀ (h : χ.conductor = 0), χ.primitiveCharacter.IsPrimitive
⊢ χ.primitiveCharacter.IsPrimitive
have neg : ∀ (h : ¬χ.conductor = 0), χ.primitiveCharacter.IsPrimitive
Tactic: by_cases h : χ.conductor = 0
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
inst_1 : NeZero n
⊢ Eq.{u_1 + 1} (α := DirichletCharacter R (conductor.{u_1} (R := R) (n := n) 1)) (primitiveCharacter 1) 1
have this : NeZero (conductor.{u_1} (R := R) (n := n) 1)
Tactic: have : NeZero (conductor (1 : DirichletCharacter R n)) :=
    ⟨@conductor_one R _ n _ ▸ Nat.one_ne_zero⟩
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
inst_1 : NeZero n
this : NeZero (conductor.{u_1} (R := R) (n := n) 1)
⊢ Eq.{u_1 + 1} (α := DirichletCharacter R (conductor.{u_1} (R := R) (n := n) 1)) (primitiveCharacter 1) 1
have rw : conductor.{u_1} (R := R) (n := conductor.{u_1} (R := R) (n := n) 1) (primitiveCharacter 1) = 1
Tactic: rw [eq_one_iff_conductor_eq_one,
    (isPrimitive_def _).1 (1 : DirichletCharacter R n).primitiveCharacter_isPrimitive,
    conductor_one]
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
inst_1 : NeZero n
this : NeZero (conductor.{u_1} (R := R) (n := n) 1)
rw : conductor.{u_1} (R := R) (n := conductor.{u_1} (R := R) (n := n) 1) (primitiveCharacter 1) = 1
⊢ Eq.{u_1 + 1} (α := DirichletCharacter R (conductor.{u_1} (R := R) (n := n) 1)) (primitiveCharacter 1) 1
have rw₁ : conductor.{u_1} (R := R) (n := n) 1 = 1
Tactic: rw [eq_one_iff_conductor_eq_one,
    (isPrimitive_def _).1 (1 : DirichletCharacter R n).primitiveCharacter_isPrimitive,
    conductor_one]
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
inst_1 : NeZero n
this : NeZero (conductor.{u_1} (R := R) (n := n) 1)
rw : conductor.{u_1} (R := R) (n := conductor.{u_1} (R := R) (n := n) 1) (primitiveCharacter 1) = 1
rw₁ : conductor.{u_1} (R := R) (n := n) 1 = 1
⊢ Eq.{u_1 + 1} (α := DirichletCharacter R (conductor.{u_1} (R := R) (n := n) 1)) (primitiveCharacter 1) 1
have rw₂ : Eq (α := ℕ) 1 1
Tactic: rw [eq_one_iff_conductor_eq_one,
    (isPrimitive_def _).1 (1 : DirichletCharacter R n).primitiveCharacter_isPrimitive,
    conductor_one]
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
d : ℕ
inst_1 : NeZero n
hd : d ∈ χ.conductorSet
⊢ d ≠ 0
have contrapose : ∀ (hd : d = 0), d ∉ χ.conductorSet
Tactic: contrapose hd
Cleared hypotheses: hd
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
d : ℕ
inst_1 : NeZero n
hd : d ∈ χ.conductorSet
⊢ χ.conductor ∣ d
have this : NeZero d
Tactic: have : NeZero d := ⟨by
    contrapose hd
    exact hd ▸ zero_ne_mem_conductorSet χ⟩
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
d : ℕ
inst_1 : NeZero n
hd : d ∈ χ.conductorSet
this_1 : NeZero d
this : d.gcd χ.conductor ∈ χ.conductorSet
⊢ χ.conductor ∣ d
have this₁ : χ.conductor ≤ d.gcd χ.conductor
Tactic: have : χ.conductor ≤ d.gcd χ.conductor := Nat.sInf_le this
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
d : ℕ
inst_1 : NeZero n
hd : d ∈ χ.conductorSet
this_1 : NeZero d
this_2 : d.gcd χ.conductor ∈ χ.conductorSet
this : χ.conductor ≤ d.gcd χ.conductor
⊢ χ.conductor ∣ d
have contrapose! : ∀ (this : ¬χ.conductor ∣ d), d.gcd χ.conductor < χ.conductor
Tactic: contrapose! this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
d : ℕ
inst_1 : NeZero n
hd : d ∈ χ.conductorSet
this_1 : NeZero d
this_2 : d.gcd χ.conductor ∈ χ.conductorSet
this : ¬χ.conductor ∣ d
⊢ d.gcd χ.conductor < χ.conductor
have refine : d.gcd χ.conductor ≤ χ.conductor
Tactic: refine Nat.lt_of_le_of_ne ?_ (Nat.gcd_eq_right_iff_dvd.not.mpr this)
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
d : ℕ
inst_1 : NeZero n
hd : d ∈ χ.conductorSet
this : NeZero d
⊢ χ.conductor ∣ d
have suffices : d.gcd χ.conductor ∈ χ.conductorSet
Tactic: suffices d.gcd χ.conductor ∈ χ.conductorSet by
    have : χ.conductor ≤ d.gcd χ.conductor := Nat.sInf_le this
    contrapose! this
    refine Nat.lt_of_le_of_ne ?_ (Nat.gcd_eq_right_iff_dvd.not.mpr this)
    exact Nat.gcd_le_right _ <| Nat.pos_of_ne_zero <| conductor_ne_zero χ
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
d : ℕ
inst_1 : NeZero n
hd : d ∈ χ.conductorSet
this : NeZero d
⊢ d.gcd χ.conductor ∈ χ.conductorSet
have obtain : ∀ (hd : d ∣ n) (χ₀ : DirichletCharacter R d)
  (hχ₀ : χ = (changeLevel hd : (a : DirichletCharacter R d) → DirichletCharacter R n) χ₀),
  d.gcd χ.conductor ∈ χ.conductorSet
Tactic: obtain ⟨hd, χ₀, hχ₀⟩ := hd
Cleared hypotheses: hd
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
d : ℕ
inst_1 : NeZero n
this_1 : NeZero d
hd : d ∣ n
χ₀ : DirichletCharacter R d
hχ₀ : χ = (changeLevel hd : (a : DirichletCharacter R d) → DirichletCharacter R n) χ₀
this :
  Eq (α := DirichletCharacter R (d * χ.conductor))
    ((changeLevel ⋯ : (a : DirichletCharacter R d) → DirichletCharacter R (d * χ.conductor)) χ₀ :
      DirichletCharacter R (d * χ.conductor))
    ((changeLevel ⋯ : (a : DirichletCharacter R χ.conductor) → DirichletCharacter R (d * χ.conductor))
        χ.primitiveCharacter :
      DirichletCharacter R (d * χ.conductor))
⊢ d.gcd χ.conductor ∈ χ.conductorSet
have obtain : ∀ (w : d.gcd χ.conductor ∣ d) (χ₁ : DirichletCharacter R (d.gcd χ.conductor))
  (hχ₁ : χ₀ = (changeLevel w : (a : DirichletCharacter R (d.gcd χ.conductor)) → DirichletCharacter R d) χ₁),
  d.gcd χ.conductor ∈ χ.conductorSet
Tactic: obtain ⟨_, χ₁, hχ₁⟩ := factorsThrough_gcd χ₀ χ.primitiveCharacter this
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
d : ℕ
inst_1 : NeZero n
this_1 : NeZero d
hd : d ∣ n
χ₀ : DirichletCharacter R d
hχ₀ : χ = (changeLevel hd : (a : DirichletCharacter R d) → DirichletCharacter R n) χ₀
this :
  Eq (α := DirichletCharacter R (d * χ.conductor))
    ((changeLevel ⋯ : (a : DirichletCharacter R d) → DirichletCharacter R (d * χ.conductor)) χ₀ :
      DirichletCharacter R (d * χ.conductor))
    ((changeLevel ⋯ : (a : DirichletCharacter R χ.conductor) → DirichletCharacter R (d * χ.conductor))
        χ.primitiveCharacter :
      DirichletCharacter R (d * χ.conductor))
w : d.gcd χ.conductor ∣ d
χ₁ : DirichletCharacter R (d.gcd χ.conductor)
hχ₁ : χ₀ = (changeLevel w : (a : DirichletCharacter R (d.gcd χ.conductor)) → DirichletCharacter R d) χ₁
⊢ d.gcd χ.conductor ∈ χ.conductorSet
have refine : χ = (changeLevel sorry : (a : DirichletCharacter R (d.gcd χ.conductor)) → DirichletCharacter R n) χ₁
Tactic: refine ⟨Nat.dvd_trans (d.gcd_dvd_left χ.conductor) hd, χ₁, ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
d : ℕ
inst_1 : NeZero n
this_1 : NeZero d
hd : d ∣ n
χ₀ : DirichletCharacter R d
hχ₀ : χ = (changeLevel hd : (a : DirichletCharacter R d) → DirichletCharacter R n) χ₀
this :
  Eq (α := DirichletCharacter R (d * χ.conductor))
    ((changeLevel ⋯ : (a : DirichletCharacter R d) → DirichletCharacter R (d * χ.conductor)) χ₀ :
      DirichletCharacter R (d * χ.conductor))
    ((changeLevel ⋯ : (a : DirichletCharacter R χ.conductor) → DirichletCharacter R (d * χ.conductor))
        χ.primitiveCharacter :
      DirichletCharacter R (d * χ.conductor))
w : d.gcd χ.conductor ∣ d
χ₁ : DirichletCharacter R (d.gcd χ.conductor)
hχ₁ : χ₀ = (changeLevel w : (a : DirichletCharacter R (d.gcd χ.conductor)) → DirichletCharacter R d) χ₁
⊢ χ = (changeLevel ⋯ : (a : DirichletCharacter R (d.gcd χ.conductor)) → DirichletCharacter R n) χ₁
have rw : χ =
  (changeLevel hd : (a : DirichletCharacter R d) → DirichletCharacter R n)
    ((changeLevel sorry : (a : DirichletCharacter R (d.gcd χ.conductor)) → DirichletCharacter R d) χ₁)
Tactic: rw [changeLevel_trans _ (d.gcd_dvd_left χ.conductor), ← hχ₁, hχ₀]
-- Lean 4 proof step from Mathlib.NumberTheory.DirichletCharacter.Basic (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
d : ℕ
inst_1 : NeZero n
this_1 : NeZero d
hd : d ∣ n
χ₀ : DirichletCharacter R d
hχ₀ : χ = (changeLevel hd : (a : DirichletCharacter R d) → DirichletCharacter R n) χ₀
this :
  Eq (α := DirichletCharacter R (d * χ.conductor))
    ((changeLevel ⋯ : (a : DirichletCharacter R d) → DirichletCharacter R (d * χ.conductor)) χ₀ :
      DirichletCharacter R (d * χ.conductor))
    ((changeLevel ⋯ : (a : DirichletCharacter R χ.conductor) → DirichletCharacter R (d * χ.conductor))
        χ.primitiveCharacter :
      DirichletCharacter R (d * χ.conductor))
w : d.gcd χ.conductor ∣ d
χ₁ : DirichletCharacter R (d.gcd χ.conductor)
hχ₁ : χ₀ = (changeLevel w : (a : DirichletCharacter R (d.gcd χ.conductor)) → DirichletCharacter R d) χ₁
rw :
  χ =
    (changeLevel hd : (a : DirichletCharacter R d) → DirichletCharacter R n)
      ((changeLevel ⋯ : (a : DirichletCharacter R (d.gcd χ.conductor)) → DirichletCharacter R d) χ₁)
rw₁ : d ∣ n
rw₂ : χ = (changeLevel hd : (a : DirichletCharacter R d) → DirichletCharacter R n) χ₀
rw₃ : d ∣ n
⊢ χ = (changeLevel ⋯ : (a : DirichletCharacter R (d.gcd χ.conductor)) → DirichletCharacter R n) χ₁
have rw₄ : Eq (α := DirichletCharacter R n)
  ((changeLevel hd : (a : DirichletCharacter R d) → DirichletCharacter R n) χ₀ : DirichletCharacter R n)
  ((changeLevel hd : (a : DirichletCharacter R d) → DirichletCharacter R n) χ₀ : DirichletCharacter R n)
Tactic: rw [changeLevel_trans _ (d.gcd_dvd_left χ.conductor), ← hχ₁, hχ₀]