awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.CharP.CharAndCard (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
p : ℕ
inst_1 : Fact (Nat.Prime p)
hR : ringChar R ≠ 0
hch : Eq.{u_1 + 1} (α := R) (↑(ringChar R) : R) 0
hp : Nat.Prime p
h₁ : IsUnit.{u_1} (M := R) (↑p : R)
q : ℕ
hq : ringChar R = p * q
a : R
ha : a * (↑p : R) = 1
h₃ : ¬ringChar R ∣ q
⊢ False
have inst : Fact True
Tactic: simp_all only [ne_eq]
Cleared hypotheses: inst_1, hR, hch, hp, h₁, hq, ha, h₃
-- Lean 4 proof step from Mathlib.Algebra.CharP.CharAndCard (hypothesis)
Goal:
R : Type u_1
inst_1 : CommRing R
p : ℕ
inst_2 : Fact (Nat.Prime p)
hR : ringChar R ≠ 0
hch : Eq.{u_1 + 1} (α := R) (↑(ringChar R) : R) 0
hp : Nat.Prime p
h₁ : IsUnit.{u_1} (M := R) (↑p : R)
q : ℕ
hq : ringChar R = p * q
a : R
ha : a * (↑p : R) = 1
h₃ : ¬ringChar R ∣ q
inst : Fact True
⊢ False
have hR₁ : ¬p * q = 0
Tactic: simp_all only [ne_eq]
Cleared hypotheses: inst_2, hR, hch, hp, h₁, hq, ha, h₃
-- Lean 4 proof step from Mathlib.Algebra.CharP.CharAndCard (hypothesis)
Goal:
R : Type u_1
inst_1 : CommRing R
p : ℕ
inst_2 : Fact (Nat.Prime p)
hR : ringChar R ≠ 0
hch : Eq.{u_1 + 1} (α := R) (↑(ringChar R) : R) 0
hp : Nat.Prime p
h₁ : IsUnit.{u_1} (M := R) (↑p : R)
q : ℕ
hq : ringChar R = p * q
a : R
ha : a * (↑p : R) = 1
h₃ : ¬ringChar R ∣ q
inst : Fact True
hR₁ : ¬p * q = 0
⊢ False
have hch₁ : Eq.{u_1 + 1} (α := R) (↑(p * q) : R) 0
Tactic: simp_all only [ne_eq]
Cleared hypotheses: inst_2, hR, hch, hp, h₁, hq, ha, h₃
-- Lean 4 proof step from Mathlib.Algebra.CharP.CharAndCard (hypothesis)
Goal:
R : Type u_1
inst_1 : CommRing R
p : ℕ
inst_2 : Fact (Nat.Prime p)
hR : ringChar R ≠ 0
hch : Eq.{u_1 + 1} (α := R) (↑(ringChar R) : R) 0
hp : Nat.Prime p
h₁ : IsUnit.{u_1} (M := R) (↑p : R)
q : ℕ
hq : ringChar R = p * q
a : R
ha : a * (↑p : R) = 1
h₃ : ¬ringChar R ∣ q
inst : Fact True
hR₁ : ¬p * q = 0
hch₁ : Eq.{u_1 + 1} (α := R) (↑(p * q) : R) 0
hp₁ : Nat.Prime p
h₂ : IsUnit.{u_1} (M := R) (↑p : R)
hq₁ : ringChar R = p * q
ha₁ : a * (↑p : R) = 1
⊢ False
have h₄ : ¬p * q ∣ q
Tactic: simp_all only [ne_eq]
Cleared hypotheses: inst_2, hR, hch, hp, h₁, hq, ha, h₃
-- Lean 4 proof step from Mathlib.Algebra.CharP.CharAndCard (goal)
Goal:
R : Type u_1
inst : CommRing R
p : ℕ
inst_1 : Fact (Nat.Prime p)
hR : ringChar R ≠ 0
hch : Eq.{u_1 + 1} (α := R) (↑(ringChar R) : R) 0
hp : Nat.Prime p
h : ¬p ∣ ringChar R
⊢ IsUnit.{u_1} (M := R) (↑p : R)
have mpr : ∀ (a b : ℤ) (hab : a * (↑p : ℤ) + b * (↑(ringChar R) : ℤ) = 1), IsUnit.{u_1} (M := R) (↑p : R)
Tactic: rcases (hp.coprime_iff_not_dvd.mpr h).isCoprime with ⟨a, b, hab⟩
-- Lean 4 proof step from Mathlib.Algebra.CharP.CharAndCard (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
p : ℕ
inst_1 : Fact (Nat.Prime p)
hR : ringChar R ≠ 0
hch : Eq.{u_1 + 1} (α := R) (↑(ringChar R) : R) 0
hp : Nat.Prime p
h : ¬p ∣ ringChar R
a b : ℤ
hab : a * (↑p : ℤ) + b * (↑(ringChar R) : ℤ) = 1
⊢ IsUnit.{u_1} (M := R) (↑p : R)
have hab₁ : Eq.{u_1 + 1} (α := R) (↑(a * (↑p : ℤ) + b * (↑(ringChar R) : ℤ)) : R) (↑1 : R)
Tactic: apply_fun ((↑) : ℤ → R) at hab
Cleared hypotheses: hab
-- Lean 4 proof step from Mathlib.Algebra.CharP.CharAndCard (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
p : ℕ
inst_1 : Fact (Nat.Prime p)
hR : ringChar R ≠ 0
hch : Eq.{u_1 + 1} (α := R) (↑(ringChar R) : R) 0
hp : Nat.Prime p
h : ¬p ∣ ringChar R
a b : ℤ
hab : Eq.{u_1 + 1} (α := R) (↑(a * (↑p : ℤ) + b * (↑(ringChar R) : ℤ)) : R) (↑1 : R)
⊢ IsUnit.{u_1} (M := R) (↑p : R)
have hab₁ : HMul.hMul (α := R) (↑a : R) (↑p : R) + HMul.hMul (α := R) (↑b : R) (↑(ringChar R) : R) = 1
Tactic: push_cast at hab
Cleared hypotheses: hab
-- Lean 4 proof step from Mathlib.Algebra.CharP.CharAndCard (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
p : ℕ
inst_1 : Fact (Nat.Prime p)
hR : ringChar R ≠ 0
hch : Eq.{u_1 + 1} (α := R) (↑(ringChar R) : R) 0
hp : Nat.Prime p
h : ¬p ∣ ringChar R
a b : ℤ
hab : HMul.hMul (α := R) (↑a : R) (↑p : R) + HMul.hMul (α := R) (↑b : R) (↑(ringChar R) : R) = 1
⊢ IsUnit.{u_1} (M := R) (↑p : R)
have hab₁ : HMul.hMul (α := R) (↑p : R) (↑a : R) = 1
Tactic: rw [hch, mul_zero, add_zero, mul_comm] at hab
Cleared hypotheses: hab
-- Lean 4 proof step from Mathlib.Algebra.CharP.CharAndCard (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : Fintype R
p : ℕ
inst_2 : Fact (Nat.Prime p)
⊢ p ∣ ringChar R ↔ p ∣ Fintype.card R
have h : ∀ (h : p ∣ Fintype.card R), p ∣ ringChar R
Tactic: refine
    ⟨fun h =>
      h.trans <|
        Int.natCast_dvd_natCast.mp <|
          (CharP.intCast_eq_zero_iff R (ringChar R) (Fintype.card R)).mp <|
            mod_cast Nat.cast_card_eq_zero R,
      fun h => ?_⟩
-- Lean 4 proof step from Mathlib.Algebra.CharP.CharAndCard (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : Fintype R
p : ℕ
inst_2 : Fact (Nat.Prime p)
h : p ∣ Fintype.card R
⊢ p ∣ ringChar R
have by_contra : ∀ (h₀ : ¬p ∣ ringChar R), False
Tactic: by_contra h₀
-- Lean 4 proof step from Mathlib.Algebra.CharP.CharAndCard (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : Fintype R
p : ℕ
inst_2 : Fact (Nat.Prime p)
h : p ∣ Fintype.card R
h₀ : ¬p ∣ ringChar R
⊢ False
have rcases : ∀ (r : R) (hr : addOrderOf r = p), False
Tactic: rcases exists_prime_addOrderOf_dvd_card p h with ⟨r, hr⟩
-- Lean 4 proof step from Mathlib.Algebra.CharP.CharAndCard (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : Fintype R
p : ℕ
inst_2 : Fact (Nat.Prime p)
h : p ∣ Fintype.card R
h₀ : ¬p ∣ ringChar R
r : R
hr : addOrderOf r = p
⊢ False
have hr₁ : addOrderOf r • r = 0
Tactic: have hr₁ := addOrderOf_nsmul_eq_zero r
-- Lean 4 proof step from Mathlib.Algebra.CharP.CharAndCard (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : Fintype R
p : ℕ
inst_2 : Fact (Nat.Prime p)
h : p ∣ Fintype.card R
h₀ : ¬p ∣ ringChar R
r : R
hr : addOrderOf r = p
hr₁ : addOrderOf r • r = 0
⊢ False
have hr₂ : (↑p : R) * r = 0
Tactic: rw [hr, nsmul_eq_mul] at hr₁
Cleared hypotheses: hr₁
-- Lean 4 proof step from Mathlib.Algebra.CharP.CharAndCard (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : Fintype R
p : ℕ
inst_2 : Fact (Nat.Prime p)
h : p ∣ Fintype.card R
h₀ : ¬p ∣ ringChar R
r : R
hr : addOrderOf r = p
hr₁ : (↑p : R) * r = 0
⊢ False
have rcases : ∀ (u : R) (hu : u * (↑p : R) = 1), False
Tactic: rcases IsUnit.exists_left_inv ((isUnit_iff_not_dvd_char R p).mpr h₀) with ⟨u, hu⟩
-- Lean 4 proof step from Mathlib.Algebra.CharP.CharAndCard (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : Fintype R
p : ℕ
inst_2 : Fact (Nat.Prime p)
h : p ∣ Fintype.card R
h₀ : ¬p ∣ ringChar R
r : R
hr : addOrderOf r = p
hr₁ : (↑p : R) * r = 0
u : R
hu : u * (↑p : R) = 1
⊢ False
have hr₂ : u * ((↑p : R) * r) = u * 0
Tactic: apply_fun (· * ·) u at hr₁
Cleared hypotheses: hr₁
-- Lean 4 proof step from Mathlib.Algebra.CharP.CharAndCard (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : Fintype R
p : ℕ
inst_2 : Fact (Nat.Prime p)
h : p ∣ Fintype.card R
h₀ : ¬p ∣ ringChar R
r : R
hr : addOrderOf r = p
u : R
hu : u * (↑p : R) = 1
hr₁ : u * ((↑p : R) * r) = u * 0
⊢ False
have hr₂ : r = 0
Tactic: rw [mul_zero, ← mul_assoc, hu, one_mul] at hr₁
Cleared hypotheses: hr₁
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : AddMonoidWithOne R
p : ℕ
inst_1 : CharP R p
a : ℕ
inst_2 : a.AtLeastTwo
h : p ∣ a
⊢ Eq.{u_1 + 1} (α := R) (OfNat.ofNat a) 0
have rwa : ∀ (h : Eq.{u_1 + 1} (α := R) (↑a : R) 0), Eq.{u_1 + 1} (α := R) (OfNat.ofNat a) 0
Tactic: rwa [← CharP.cast_eq_zero_iff R p] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : AddMonoidWithOne R
p : ℕ
inst_1 : CharP R p
k a : ℕ
⊢ Eq.{u_1 + 1} (α := R) (↑(p * a) : R) 0
have rw : p ∣ p * a
Tactic: rw [CharP.cast_eq_zero_iff R p]
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : AddMonoidWithOne R
p : ℕ
inst_1 : CharP R p
k : ℕ
this : ∀ (a : ℕ), Eq.{u_1 + 1} (α := R) (↑(p * a) : R) 0
⊢ Eq.{u_1 + 1} (α := R) (↑k : R) (↑(k % p + p * (k / p)) : R)
have rw : Eq.{u_1 + 1} (α := R) (↑k : R) (↑k : R)
Tactic: rw [Nat.mod_add_div]
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : AddMonoidWithOne R
p : ℕ
inst_1 : CharP R p
a b : ℕ
inst_2 : IsLeftCancelAdd R
⊢ Eq.{u_1 + 1} (α := R) (↑a : R) (↑b : R) ↔ a % p = b % p
have inr : ∀
  (this :
    ∀ (R : Type u_1) [inst : AddMonoidWithOne R] (p : ℕ) [CharP R p] {a b : ℕ} [IsLeftCancelAdd R] (hle : a ≤ b),
      Eq.{u_1 + 1} (α := R) (↑a : R) (↑b : R) ↔ a % p = b % p)
  (hle : b < a), Eq.{u_1 + 1} (α := R) (↑a : R) (↑b : R) ↔ a % p = b % p
Tactic: wlog! hle : a ≤ b
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : AddMonoidWithOne R
p : ℕ
inst_1 : CharP R p
a b : ℕ
inst_2 : IsLeftCancelAdd R
inr :
  ∀
    (this :
      ∀ (R : Type u_1) [inst : AddMonoidWithOne R] (p : ℕ) [CharP R p] {a b : ℕ} [IsLeftCancelAdd R] (hle : a ≤ b),
        Eq.{u_1 + 1} (α := R) (↑a : R) (↑b : R) ↔ a % p = b % p)
    (hle : b < a), Eq.{u_1 + 1} (α := R) (↑a : R) (↑b : R) ↔ a % p = b % p
⊢ Eq.{u_1 + 1} (α := R) (↑a : R) (↑b : R) ↔ a % p = b % p
have wlog! : ∀ (R : Type u_1) [inst : AddMonoidWithOne R] (p : ℕ) [CharP R p] {a b : ℕ} [IsLeftCancelAdd R] (hle : a ≤ b),
  Eq.{u_1 + 1} (α := R) (↑a : R) (↑b : R) ↔ a % p = b % p
Tactic: wlog! hle : a ≤ b
Cleared hypotheses: inst_1, inst_2
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R_1 : Type u_1
inst : AddMonoidWithOne R_1
p_1 a_1 b_1 : ℕ
R : Type u_1
inst_1 : AddMonoidWithOne R
p : ℕ
inst_2 : CharP R p
a b : ℕ
inst_3 : IsLeftCancelAdd R
hle : a ≤ b
⊢ Eq.{u_1 + 1} (α := R) (↑a : R) (↑b : R) ↔ a % p = b % p
have obtain : ∀ (c : ℕ) (hle : a ≤ a + c), Eq.{u_1 + 1} (α := R) (↑a : R) (↑(a + c) : R) ↔ a % p = (a + c) % p
Tactic: obtain ⟨c, rfl⟩ := Nat.exists_eq_add_of_le hle
Cleared hypotheses: b, hle
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R_1 : Type u_1
inst : AddMonoidWithOne R_1
p_1 a_1 b : ℕ
R : Type u_1
inst_1 : AddMonoidWithOne R
p : ℕ
inst_2 : CharP R p
a : ℕ
inst_3 : IsLeftCancelAdd R
c : ℕ
hle : a ≤ a + c
⊢ Eq.{u_1 + 1} (α := R) (↑a : R) (↑(a + c) : R) ↔ a % p = (a + c) % p
have rw : p ∣ c ↔ a % p = (a + c) % p
Tactic: rw [Nat.cast_add, left_eq_add, CharP.cast_eq_zero_iff R p]
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R_1 : Type u_1
inst : AddMonoidWithOne R_1
p_1 a_1 b : ℕ
R : Type u_1
inst_1 : AddMonoidWithOne R
p : ℕ
inst_2 : CharP R p
a : ℕ
inst_3 : IsLeftCancelAdd R
c : ℕ
hle : a ≤ a + c
⊢ p ∣ c ↔ a % p = (a + c) % p
have mp : p ∣ c → a % p = (a + c) % p
Tactic: constructor
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R_1 : Type u_1
inst : AddMonoidWithOne R_1
p_1 a_1 b : ℕ
R : Type u_1
inst_1 : AddMonoidWithOne R
p : ℕ
inst_2 : CharP R p
a : ℕ
inst_3 : IsLeftCancelAdd R
c : ℕ
hle : a ≤ a + c
mp : p ∣ c → a % p = (a + c) % p
⊢ p ∣ c ↔ a % p = (a + c) % p
have mpr : a % p = (a + c) % p → p ∣ c
Tactic: constructor
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (hypothesis)
Goal:
R_1 : Type u_1
inst : AddMonoidWithOne R_1
p_1 a_1 b : ℕ
R : Type u_1
inst_1 : AddMonoidWithOne R
p : ℕ
inst_2 : CharP R p
a : ℕ
inst_3 : IsLeftCancelAdd R
c : ℕ
hle : a ≤ a + c
h : a % p = (a + c) % p
⊢ p ∣ c
have this : (a + c - a) % p = 0
Tactic: have := Nat.sub_mod_eq_zero_of_mod_eq h.symm
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : AddMonoidWithOne R
p : ℕ
inst_1 : CharP R p
a b : ℕ
inst_2 : CharZero R
x : ℕ
⊢ Eq.{u_1 + 1} (α := R) (↑x : R) 0 ↔ 0 ∣ x
have rw : Eq.{u_1 + 1} (α := R) (↑x : R) 0 ↔ x = 0
Tactic: rw [zero_dvd_iff, ← Nat.cast_zero, Nat.cast_inj]
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : AddMonoidWithOne R
p : ℕ
inst_1 : CharP R p
a b : ℕ
inst_2 : CharZero R
x : ℕ
rw : Eq.{u_1 + 1} (α := R) (↑x : R) 0 ↔ x = 0
⊢ Eq.{u_1 + 1} (α := R) (↑x : R) 0 ↔ 0 ∣ x
have rw₁ : Eq.{u_1 + 1} (α := R) (↑x : R) (↑0 : R) ↔ x = 0
Tactic: rw [zero_dvd_iff, ← Nat.cast_zero, Nat.cast_inj]
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : AddMonoidWithOne R
p : ℕ
inst_1 : CharP R p
a b : ℕ
inst_2 : CharZero R
x : ℕ
rw : Eq.{u_1 + 1} (α := R) (↑x : R) 0 ↔ x = 0
rw₁ : Eq.{u_1 + 1} (α := R) (↑x : R) (↑0 : R) ↔ x = 0
⊢ Eq.{u_1 + 1} (α := R) (↑x : R) 0 ↔ 0 ∣ x
have rw₂ : x = 0 ↔ x = 0
Tactic: rw [zero_dvd_iff, ← Nat.cast_zero, Nat.cast_inj]
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : AddGroupWithOne R
p : ℕ
inst_1 : CharP R p
a : ℤ
⊢ Eq.{u_1 + 1} (α := R) (↑a : R) 0 ↔ (↑p : ℤ) ∣ a
have inl : ∀ (h : a < 0), Eq.{u_1 + 1} (α := R) (↑a : R) 0 ↔ (↑p : ℤ) ∣ a
Tactic: rcases lt_trichotomy a 0 with (h | rfl | h)
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : AddGroupWithOne R
p : ℕ
inst_1 : CharP R p
a : ℤ
inl : ∀ (h : a < 0), Eq.{u_1 + 1} (α := R) (↑a : R) 0 ↔ (↑p : ℤ) ∣ a
⊢ Eq.{u_1 + 1} (α := R) (↑a : R) 0 ↔ (↑p : ℤ) ∣ a
have inr : Eq.{u_1 + 1} (α := R) (↑0 : R) 0 ↔ Dvd.dvd (α := ℤ) (↑p : ℤ) 0
Tactic: rcases lt_trichotomy a 0 with (h | rfl | h)
Cleared hypotheses: a
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : AddGroupWithOne R
p : ℕ
inst_1 : CharP R p
a : ℤ
inl : ∀ (h : a < 0), Eq.{u_1 + 1} (α := R) (↑a : R) 0 ↔ (↑p : ℤ) ∣ a
inr : Eq.{u_1 + 1} (α := R) (↑0 : R) 0 ↔ Dvd.dvd (α := ℤ) (↑p : ℤ) 0
⊢ Eq.{u_1 + 1} (α := R) (↑a : R) 0 ↔ (↑p : ℤ) ∣ a
have inr₁ : ∀ (h : 0 < a), Eq.{u_1 + 1} (α := R) (↑a : R) 0 ↔ (↑p : ℤ) ∣ a
Tactic: rcases lt_trichotomy a 0 with (h | rfl | h)
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : AddGroupWithOne R
p : ℕ
inst_1 : CharP R p
a : ℤ
h : a < 0
⊢ Eq.{u_1 + 1} (α := R) (↑a : R) 0 ↔ (↑p : ℤ) ∣ a
have inl : Eq.{u_1 + 1} (α := R) (↑(-a) : R) 0 ↔ (↑p : ℤ) ∣ -a
Tactic: rw [← neg_eq_zero, ← Int.cast_neg, ← Int.dvd_neg]
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : AddGroupWithOne R
p : ℕ
inst_1 : CharP R p
a : ℤ
h : a < 0
⊢ Eq.{u_1 + 1} (α := R) (↑(-a) : R) 0 ↔ (↑p : ℤ) ∣ -a
have inl : ∀ (b : ℕ), Eq.{u_1 + 1} (α := R) (↑(↑b : ℤ) : R) 0 ↔ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑b : ℤ)
Tactic: lift -a to ℕ using Int.neg_nonneg.mpr (le_of_lt h) with b
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : AddGroupWithOne R
p : ℕ
inst_1 : CharP R p
a : ℤ
h : a < 0
b : ℕ
⊢ Eq.{u_1 + 1} (α := R) (↑(↑b : ℤ) : R) 0 ↔ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑b : ℤ)
have inl : Eq.{u_1 + 1} (α := R) (↑b : R) 0 ↔ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑b : ℤ)
Tactic: rw [Int.cast_natCast, CharP.cast_eq_zero_iff R p, Int.natCast_dvd_natCast]
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : AddGroupWithOne R
p : ℕ
inst_1 : CharP R p
a : ℤ
h : a < 0
b : ℕ
inl : Eq.{u_1 + 1} (α := R) (↑b : R) 0 ↔ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑b : ℤ)
⊢ Eq.{u_1 + 1} (α := R) (↑(↑b : ℤ) : R) 0 ↔ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑b : ℤ)
have inl₁ : p ∣ b ↔ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑b : ℤ)
Tactic: rw [Int.cast_natCast, CharP.cast_eq_zero_iff R p, Int.natCast_dvd_natCast]
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : AddGroupWithOne R
p : ℕ
inst_1 : CharP R p
a : ℤ
h : a < 0
b : ℕ
inl : Eq.{u_1 + 1} (α := R) (↑b : R) 0 ↔ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑b : ℤ)
inl₁ : p ∣ b ↔ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑b : ℤ)
⊢ Eq.{u_1 + 1} (α := R) (↑(↑b : ℤ) : R) 0 ↔ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑b : ℤ)
have inl₂ : p ∣ b ↔ p ∣ b
Tactic: rw [Int.cast_natCast, CharP.cast_eq_zero_iff R p, Int.natCast_dvd_natCast]
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : AddGroupWithOne R
p : ℕ
inst_1 : CharP R p
a : ℤ
h : 0 < a
⊢ Eq.{u_1 + 1} (α := R) (↑a : R) 0 ↔ (↑p : ℤ) ∣ a
have inr : ∀ (b : ℕ) (h : LT.lt (α := ℤ) 0 (↑b : ℤ)), Eq.{u_1 + 1} (α := R) (↑(↑b : ℤ) : R) 0 ↔ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑b : ℤ)
Tactic: lift a to ℕ using le_of_lt h with b
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : AddGroupWithOne R
p : ℕ
inst_1 : CharP R p
a : ℤ
b : ℕ
h : LT.lt (α := ℤ) 0 (↑b : ℤ)
⊢ Eq.{u_1 + 1} (α := R) (↑(↑b : ℤ) : R) 0 ↔ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑b : ℤ)
have inr : Eq.{u_1 + 1} (α := R) (↑b : R) 0 ↔ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑b : ℤ)
Tactic: rw [Int.cast_natCast, CharP.cast_eq_zero_iff R p, Int.natCast_dvd_natCast]
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : AddGroupWithOne R
p : ℕ
inst_1 : CharP R p
a : ℤ
b : ℕ
h : LT.lt (α := ℤ) 0 (↑b : ℤ)
inr : Eq.{u_1 + 1} (α := R) (↑b : R) 0 ↔ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑b : ℤ)
⊢ Eq.{u_1 + 1} (α := R) (↑(↑b : ℤ) : R) 0 ↔ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑b : ℤ)
have inr₁ : p ∣ b ↔ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑b : ℤ)
Tactic: rw [Int.cast_natCast, CharP.cast_eq_zero_iff R p, Int.natCast_dvd_natCast]
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : AddGroupWithOne R
p : ℕ
inst_1 : CharP R p
a : ℤ
b : ℕ
h : LT.lt (α := ℤ) 0 (↑b : ℤ)
inr : Eq.{u_1 + 1} (α := R) (↑b : R) 0 ↔ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑b : ℤ)
inr₁ : p ∣ b ↔ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑b : ℤ)
⊢ Eq.{u_1 + 1} (α := R) (↑(↑b : ℤ) : R) 0 ↔ Dvd.dvd (α := ℤ) (↑p : ℤ) (↑b : ℤ)
have inr₂ : p ∣ b ↔ p ∣ b
Tactic: rw [Int.cast_natCast, CharP.cast_eq_zero_iff R p, Int.natCast_dvd_natCast]
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : NonAssocSemiring R
this : DecidableEq R := Classical.decEq R
H : ∀ (p : ℕ), Eq.{u_1 + 1} (α := R) (↑p : R) 0 → p = 0
x : ℕ
⊢ Eq.{u_1 + 1} (α := R) (↑x : R) 0 ↔ 0 ∣ x
have rw : Eq.{u_1 + 1} (α := R) (↑x : R) 0 ↔ x = 0
Tactic: rw [zero_dvd_iff]
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : NonAssocSemiring R
this : DecidableEq R := Classical.decEq R
H : ¬∀ (p : ℕ), Eq.{u_1 + 1} (α := R) (↑p : R) 0 → p = 0
x : ℕ
H1 : Eq.{u_1 + 1} (α := R) (↑x : R) 0
H2 : ¬x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) ⋯ = 0
⊢ Eq.{u_1 + 1} (α := R) (↑(x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) ⋯) : R) 0
have rwa : ∀
  (H1 :
    Eq.{u_1 + 1} (α := R)
      (↑(x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) sorry +
            Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) sorry *
              (x / Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) sorry)) :
        R)
      0)
  (H2 : ¬x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) sorry = 0),
  Eq.{u_1 + 1} (α := R) (↑(x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) sorry) : R) 0
Tactic: rwa [← Nat.mod_add_div x (Nat.find (not_forall.1 H)), Nat.cast_add,
                      Nat.cast_mul,
                      of_not_not (not_not_of_not_imp <| Nat.find_spec (not_forall.1 H)),
                      zero_mul, add_zero] at H1
Cleared hypotheses: H1, H2
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : NonAssocSemiring R
this : DecidableEq R := Classical.decEq R
H : ¬∀ (p : ℕ), Eq.{u_1 + 1} (α := R) (↑p : R) 0 → p = 0
x : ℕ
H1 : Eq.{u_1 + 1} (α := R) (↑x : R) 0
H2 : ¬x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) ⋯ = 0
rwa :
  ∀
    (H1 :
      Eq.{u_1 + 1} (α := R)
        (↑(x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) ⋯ +
              Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) ⋯ *
                (x / Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) ⋯)) :
          R)
        0)
    (H2 : ¬x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) ⋯ = 0),
    Eq.{u_1 + 1} (α := R) (↑(x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) ⋯) : R) 0
⊢ Eq.{u_1 + 1} (α := R) (↑(x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) ⋯) : R) 0
have rwa₁ : ∀
  (H1 :
    HAdd.hAdd (α := R) (↑(x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) sorry) : R)
        (↑(Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) sorry *
              (x / Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) sorry)) :
          R) =
      0)
  (H2 : ¬x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) sorry = 0),
  Eq.{u_1 + 1} (α := R) (↑(x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) sorry) : R) 0
Tactic: rwa [← Nat.mod_add_div x (Nat.find (not_forall.1 H)), Nat.cast_add,
                      Nat.cast_mul,
                      of_not_not (not_not_of_not_imp <| Nat.find_spec (not_forall.1 H)),
                      zero_mul, add_zero] at H1
Cleared hypotheses: H1, H2
-- Lean 4 proof step from Mathlib.Algebra.CharP.Defs (goal)
Goal:
R : Type u_1
inst : NonAssocSemiring R
this : DecidableEq R := Classical.decEq R
H : ¬∀ (p : ℕ), Eq.{u_1 + 1} (α := R) (↑p : R) 0 → p = 0
x : ℕ
H1 : Eq.{u_1 + 1} (α := R) (↑x : R) 0
H2 : ¬x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) ⋯ = 0
rwa :
  ∀
    (H1 :
      Eq.{u_1 + 1} (α := R)
        (↑(x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) ⋯ +
              Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) ⋯ *
                (x / Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) ⋯)) :
          R)
        0)
    (H2 : ¬x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) ⋯ = 0),
    Eq.{u_1 + 1} (α := R) (↑(x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) ⋯) : R) 0
rwa₁ :
  ∀
    (H1 :
      HAdd.hAdd (α := R) (↑(x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) ⋯) : R)
          (↑(Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) ⋯ *
                (x / Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) ⋯)) :
            R) =
        0)
    (H2 : ¬x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) ⋯ = 0),
    Eq.{u_1 + 1} (α := R) (↑(x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) ⋯) : R) 0
⊢ Eq.{u_1 + 1} (α := R) (↑(x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) ⋯) : R) 0
have rwa₂ : ∀
  (H1 :
    (↑(x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) sorry) : R) +
        HMul.hMul (α := R) (↑(Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) sorry) : R)
          (↑(x / Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) sorry) : R) =
      0)
  (H2 : ¬x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) sorry = 0),
  Eq.{u_1 + 1} (α := R) (↑(x % Nat.find (p := fun (n : ℕ) ↦ ¬(Eq.{u_1 + 1} (α := R) (↑n : R) 0 → n = 0)) sorry) : R) 0
Tactic: rwa [← Nat.mod_add_div x (Nat.find (not_forall.1 H)), Nat.cast_add,
                      Nat.cast_mul,
                      of_not_not (not_not_of_not_imp <| Nat.find_spec (not_forall.1 H)),
                      zero_mul, add_zero] at H1
Cleared hypotheses: H1, H2