l3lab/ntp-mathlib huggingface dataset:
/-- If for all `n ≠ 0 ∈ ℤ`, `n • A = A`, then `A` is divisible.
-/
noncomputable def divisibleByIntOfSMulTopEqTop
    (H : ∀ {n : ℤ} (_hn : n ≠ 0), n • (⊤ : AddSubgroup A) = ⊤) : DivisibleBy A ℤ where
  div a n :=
    if hn : n = 0 then 0 else (show a ∈ n • (⊤ : AddSubgroup A) by rw [H hn]; trivial).choose
  div_zero a := dif_pos rfl
  div_cancel a hn := by
    simp_rw [dif_neg hn]
    

Proof state:
A : Type u_1
inst✝ : AddCommGroup A
H : ∀ {n : ℤ}, n ≠ 0 → n • ⊤ = ⊤
n✝ : ℤ
a : A
hn : n✝ ≠ 0
⊢ n✝ • Exists.choose (_ : a ∈ n✝ • ⊤) = a

Next tactic:
generalize_proofs h1
/-- If for all `n ≠ 0 ∈ ℤ`, `n • A = A`, then `A` is divisible.
-/
noncomputable def divisibleByIntOfSMulTopEqTop
    (H : ∀ {n : ℤ} (_hn : n ≠ 0), n • (⊤ : AddSubgroup A) = ⊤) : DivisibleBy A ℤ where
  div a n :=
    if hn : n = 0 then 0 else (show a ∈ n • (⊤ : AddSubgroup A) by rw [H hn]; trivial).choose
  div_zero a := dif_pos rfl
  div_cancel a hn := by
    simp_rw [dif_neg hn]
    generalize_proofs h1
    

Proof state:
A : Type u_1
inst✝ : AddCommGroup A
H : ∀ {n : ℤ}, n ≠ 0 → n • ⊤ = ⊤
n✝ : ℤ
a : A
hn : n✝ ≠ 0
h1 : a ∈ n✝ • ⊤
⊢ n✝ • Exists.choose h1 = a

Next tactic:
exact h1.choose_spec.2
instance (priority := 100) divisibleByIntOfCharZero {𝕜} [DivisionRing 𝕜] [CharZero 𝕜] :
    DivisibleBy 𝕜 ℤ where
  div q n := q / n
  div_zero q := by 

Proof state:
𝕜 : Type ?u.13579
inst✝¹ : DivisionRing 𝕜
inst✝ : CharZero 𝕜
q : 𝕜
⊢ (fun q n => q / ↑n) q 0 = 0

Next tactic:
norm_num
instance (priority := 100) divisibleByIntOfCharZero {𝕜} [DivisionRing 𝕜] [CharZero 𝕜] :
    DivisibleBy 𝕜 ℤ where
  div q n := q / n
  div_zero q := by norm_num
  div_cancel {n} q hn := by
    

Proof state:
𝕜 : Type ?u.13579
inst✝¹ : DivisionRing 𝕜
inst✝ : CharZero 𝕜
n : ℤ
q : 𝕜
hn : n ≠ 0
⊢ n • (fun q n => q / ↑n) q n = q

Next tactic:
rw [zsmul_eq_mul, (Int.cast_commute n _).eq, div_mul_cancel q (Int.cast_ne_zero.mpr hn)]
open Int in
/-- A group is `ℤ`-rootable if it is `ℕ`-rootable.
-/
@[to_additive "An additive group is `ℤ`-divisible if it is `ℕ`-divisible."]
def rootableByIntOfRootableByNat [RootableBy A ℕ] : RootableBy A ℤ where
  root a z :=
    match z with
    | (n : ℕ) => RootableBy.root a n
    | -[n+1] => (RootableBy.root a (n + 1))⁻¹
  root_zero a := RootableBy.root_zero a
  root_cancel {n} a hn := by
    

Proof state:
A : Type u_1
inst✝¹ : Group A
inst✝ : RootableBy A ℕ
n : ℤ
a : A
hn : n ≠ 0
⊢ (fun a z =>
          match z with
          | ofNat n => RootableBy.root a n
          | -[n+1] => (RootableBy.root a (n + 1))⁻¹)
        a n ^
      n =
    a

Next tactic:
induction n
open Int in
/-- A group is `ℤ`-rootable if it is `ℕ`-rootable.
-/
@[to_additive "An additive group is `ℤ`-divisible if it is `ℕ`-divisible."]
def rootableByIntOfRootableByNat [RootableBy A ℕ] : RootableBy A ℤ where
  root a z :=
    match z with
    | (n : ℕ) => RootableBy.root a n
    | -[n+1] => (RootableBy.root a (n + 1))⁻¹
  root_zero a := RootableBy.root_zero a
  root_cancel {n} a hn := by
    induction n
    · 

Proof state:
case ofNat
A : Type u_1
inst✝¹ : Group A
inst✝ : RootableBy A ℕ
a : A
a✝ : ℕ
hn : ofNat a✝ ≠ 0
⊢ (fun a z =>
          match z with
          | ofNat n => RootableBy.root a n
          | -[n+1] => (RootableBy.root a (n + 1))⁻¹)
        a (ofNat a✝) ^
      ofNat a✝ =
    a

Next tactic:
change RootableBy.root a _ ^ _ = a
open Int in
/-- A group is `ℤ`-rootable if it is `ℕ`-rootable.
-/
@[to_additive "An additive group is `ℤ`-divisible if it is `ℕ`-divisible."]
def rootableByIntOfRootableByNat [RootableBy A ℕ] : RootableBy A ℤ where
  root a z :=
    match z with
    | (n : ℕ) => RootableBy.root a n
    | -[n+1] => (RootableBy.root a (n + 1))⁻¹
  root_zero a := RootableBy.root_zero a
  root_cancel {n} a hn := by
    induction n
    · change RootableBy.root a _ ^ _ = a
      

Proof state:
case ofNat
A : Type u_1
inst✝¹ : Group A
inst✝ : RootableBy A ℕ
a : A
a✝ : ℕ
hn : ofNat a✝ ≠ 0
⊢ RootableBy.root a a✝ ^ ofNat a✝ = a

Next tactic:
norm_num
open Int in
/-- A group is `ℤ`-rootable if it is `ℕ`-rootable.
-/
@[to_additive "An additive group is `ℤ`-divisible if it is `ℕ`-divisible."]
def rootableByIntOfRootableByNat [RootableBy A ℕ] : RootableBy A ℤ where
  root a z :=
    match z with
    | (n : ℕ) => RootableBy.root a n
    | -[n+1] => (RootableBy.root a (n + 1))⁻¹
  root_zero a := RootableBy.root_zero a
  root_cancel {n} a hn := by
    induction n
    · change RootableBy.root a _ ^ _ = a
      norm_num
      

Proof state:
case ofNat
A : Type u_1
inst✝¹ : Group A
inst✝ : RootableBy A ℕ
a : A
a✝ : ℕ
hn : ofNat a✝ ≠ 0
⊢ RootableBy.root a a✝ ^ a✝ = a

Next tactic:
rw [RootableBy.root_cancel]
open Int in
/-- A group is `ℤ`-rootable if it is `ℕ`-rootable.
-/
@[to_additive "An additive group is `ℤ`-divisible if it is `ℕ`-divisible."]
def rootableByIntOfRootableByNat [RootableBy A ℕ] : RootableBy A ℤ where
  root a z :=
    match z with
    | (n : ℕ) => RootableBy.root a n
    | -[n+1] => (RootableBy.root a (n + 1))⁻¹
  root_zero a := RootableBy.root_zero a
  root_cancel {n} a hn := by
    induction n
    · change RootableBy.root a _ ^ _ = a
      norm_num
      rw [RootableBy.root_cancel]
      

Proof state:
case ofNat.a
A : Type u_1
inst✝¹ : Group A
inst✝ : RootableBy A ℕ
a : A
a✝ : ℕ
hn : ofNat a✝ ≠ 0
⊢ a✝ ≠ 0

Next tactic:
rw [Int.ofNat_eq_coe] at hn
open Int in
/-- A group is `ℤ`-rootable if it is `ℕ`-rootable.
-/
@[to_additive "An additive group is `ℤ`-divisible if it is `ℕ`-divisible."]
def rootableByIntOfRootableByNat [RootableBy A ℕ] : RootableBy A ℤ where
  root a z :=
    match z with
    | (n : ℕ) => RootableBy.root a n
    | -[n+1] => (RootableBy.root a (n + 1))⁻¹
  root_zero a := RootableBy.root_zero a
  root_cancel {n} a hn := by
    induction n
    · change RootableBy.root a _ ^ _ = a
      norm_num
      rw [RootableBy.root_cancel]
      rw [Int.ofNat_eq_coe] at hn
      

Proof state:
case ofNat.a
A : Type u_1
inst✝¹ : Group A
inst✝ : RootableBy A ℕ
a : A
a✝ : ℕ
hn : ↑a✝ ≠ 0
⊢ a✝ ≠ 0

Next tactic:
exact mod_cast hn
open Int in
/-- A group is `ℤ`-rootable if it is `ℕ`-rootable.
-/
@[to_additive "An additive group is `ℤ`-divisible if it is `ℕ`-divisible."]
def rootableByIntOfRootableByNat [RootableBy A ℕ] : RootableBy A ℤ where
  root a z :=
    match z with
    | (n : ℕ) => RootableBy.root a n
    | -[n+1] => (RootableBy.root a (n + 1))⁻¹
  root_zero a := RootableBy.root_zero a
  root_cancel {n} a hn := by
    induction n
    · change RootableBy.root a _ ^ _ = a
      norm_num
      rw [RootableBy.root_cancel]
      rw [Int.ofNat_eq_coe] at hn
      exact mod_cast hn
    · 

Proof state:
case negSucc
A : Type u_1
inst✝¹ : Group A
inst✝ : RootableBy A ℕ
a : A
a✝ : ℕ
hn : -[a✝+1] ≠ 0
⊢ (fun a z =>
          match z with
          | ofNat n => RootableBy.root a n
          | -[n+1] => (RootableBy.root a (n + 1))⁻¹)
        a -[a✝+1] ^
      -[a✝+1] =
    a

Next tactic:
change (RootableBy.root a _)⁻¹ ^ _ = a
open Int in
/-- A group is `ℤ`-rootable if it is `ℕ`-rootable.
-/
@[to_additive "An additive group is `ℤ`-divisible if it is `ℕ`-divisible."]
def rootableByIntOfRootableByNat [RootableBy A ℕ] : RootableBy A ℤ where
  root a z :=
    match z with
    | (n : ℕ) => RootableBy.root a n
    | -[n+1] => (RootableBy.root a (n + 1))⁻¹
  root_zero a := RootableBy.root_zero a
  root_cancel {n} a hn := by
    induction n
    · change RootableBy.root a _ ^ _ = a
      norm_num
      rw [RootableBy.root_cancel]
      rw [Int.ofNat_eq_coe] at hn
      exact mod_cast hn
    · change (RootableBy.root a _)⁻¹ ^ _ = a
      

Proof state:
case negSucc
A : Type u_1
inst✝¹ : Group A
inst✝ : RootableBy A ℕ
a : A
a✝ : ℕ
hn : -[a✝+1] ≠ 0
⊢ (RootableBy.root a (a✝ + 1))⁻¹ ^ -[a✝+1] = a

Next tactic:
norm_num
open Int in
/-- A group is `ℤ`-rootable if it is `ℕ`-rootable.
-/
@[to_additive "An additive group is `ℤ`-divisible if it is `ℕ`-divisible."]
def rootableByIntOfRootableByNat [RootableBy A ℕ] : RootableBy A ℤ where
  root a z :=
    match z with
    | (n : ℕ) => RootableBy.root a n
    | -[n+1] => (RootableBy.root a (n + 1))⁻¹
  root_zero a := RootableBy.root_zero a
  root_cancel {n} a hn := by
    induction n
    · change RootableBy.root a _ ^ _ = a
      norm_num
      rw [RootableBy.root_cancel]
      rw [Int.ofNat_eq_coe] at hn
      exact mod_cast hn
    · change (RootableBy.root a _)⁻¹ ^ _ = a
      norm_num
      

Proof state:
case negSucc
A : Type u_1
inst✝¹ : Group A
inst✝ : RootableBy A ℕ
a : A
a✝ : ℕ
hn : -[a✝+1] ≠ 0
⊢ RootableBy.root a (a✝ + 1) ^ (a✝ + 1) = a

Next tactic:
rw [RootableBy.root_cancel]
open Int in
/-- A group is `ℤ`-rootable if it is `ℕ`-rootable.
-/
@[to_additive "An additive group is `ℤ`-divisible if it is `ℕ`-divisible."]
def rootableByIntOfRootableByNat [RootableBy A ℕ] : RootableBy A ℤ where
  root a z :=
    match z with
    | (n : ℕ) => RootableBy.root a n
    | -[n+1] => (RootableBy.root a (n + 1))⁻¹
  root_zero a := RootableBy.root_zero a
  root_cancel {n} a hn := by
    induction n
    · change RootableBy.root a _ ^ _ = a
      norm_num
      rw [RootableBy.root_cancel]
      rw [Int.ofNat_eq_coe] at hn
      exact mod_cast hn
    · change (RootableBy.root a _)⁻¹ ^ _ = a
      norm_num
      rw [RootableBy.root_cancel]
      

Proof state:
case negSucc.a
A : Type u_1
inst✝¹ : Group A
inst✝ : RootableBy A ℕ
a : A
a✝ : ℕ
hn : -[a✝+1] ≠ 0
⊢ a✝ + 1 ≠ 0

Next tactic:
norm_num
/-- A group is `ℕ`-rootable if it is `ℤ`-rootable
-/
@[to_additive "An additive group is `ℕ`-divisible if it `ℤ`-divisible."]
def rootableByNatOfRootableByInt [RootableBy A ℤ] : RootableBy A ℕ where
  root a n := RootableBy.root a (n : ℤ)
  root_zero a := RootableBy.root_zero a
  root_cancel {n} a hn := by
    -- Porting note: replaced `norm_num`
    

Proof state:
A : Type u_1
inst✝¹ : Group A
inst✝ : RootableBy A ℤ
n : ℕ
a : A
hn : n ≠ 0
⊢ (fun a n => RootableBy.root a ↑n) a n ^ n = a

Next tactic:
simpa only [zpow_coe_nat] using RootableBy.root_cancel a (show (n : ℤ) ≠ 0 from mod_cast hn)
/--
If `f : A → B` is a surjective homomorphism and `A` is `α`-rootable, then `B` is also `α`-rootable.
-/
@[to_additive
      "If `f : A → B` is a surjective homomorphism and `A` is `α`-divisible, then `B` is also
      `α`-divisible."]
noncomputable def Function.Surjective.rootableBy (hf : Function.Surjective f)
    (hpow : ∀ (a : A) (n : α), f (a ^ n) = f a ^ n) : RootableBy B α :=
  rootableByOfPowLeftSurj _ _ fun {n} hn x =>
    let ⟨y, hy⟩ := hf x
    ⟨f <| RootableBy.root y n,
      (by 

Proof state:
A : Type u_1
B : Type u_2
α : Type u_3
inst✝⁵ : Zero α
inst✝⁴ : Monoid A
inst✝³ : Monoid B
inst✝² : Pow A α
inst✝¹ : Pow B α
inst✝ : RootableBy A α
f : A → B
hf : Surjective f
hpow : ∀ (a : A) (n : α), f (a ^ n) = f a ^ n
n : α
hn : n ≠ 0
x : B
y : A
hy : f y = x
⊢ f (RootableBy.root y n) ^ n = x

Next tactic:
rw [← hpow (RootableBy.root y n) n, RootableBy.root_cancel _ hn, hy]
@[to_additive]
instance Interval.mulOneClass [OrderedCommMonoid α] : MulOneClass (Interval α) where
  mul := (· * ·)
  one := 1
  one_mul s :=
    (Option.map₂_coe_left _ _ _).trans <| by
      

Proof state:
ι : Type u_1
α : Type u_2
inst✝ : OrderedCommMonoid α
s : Interval α
⊢ Option.map (fun b => NonemptyInterval.pure 1 * b) s = s

Next tactic:
simp_rw [NonemptyInterval.pure_one, one_mul, ← id_def, Option.map_id, id]
@[to_additive]
instance Interval.mulOneClass [OrderedCommMonoid α] : MulOneClass (Interval α) where
  mul := (· * ·)
  one := 1
  one_mul s :=
    (Option.map₂_coe_left _ _ _).trans <| by
      simp_rw [NonemptyInterval.pure_one, one_mul, ← id_def, Option.map_id, id]
  mul_one s :=
    (Option.map₂_coe_right _ _ _).trans <| by
      

Proof state:
ι : Type u_1
α : Type u_2
inst✝ : OrderedCommMonoid α
s : Interval α
⊢ Option.map (fun a => a * NonemptyInterval.pure 1) s = s

Next tactic:
simp_rw [NonemptyInterval.pure_one, mul_one, ← id_def, Option.map_id, id]
@[to_additive]
protected theorem mul_eq_one_iff : s * t = 1 ↔ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1 := by
  

Proof state:
ι : Type u_1
α : Type u_2
inst✝ : OrderedCommGroup α
s t : NonemptyInterval α
⊢ s * t = 1 ↔ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1

Next tactic:
refine' ⟨fun h => _, _⟩
@[to_additive]
protected theorem mul_eq_one_iff : s * t = 1 ↔ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1 := by
  refine' ⟨fun h => _, _⟩
  · 

Proof state:
case refine'_1
ι : Type u_1
α : Type u_2
inst✝ : OrderedCommGroup α
s t : NonemptyInterval α
h : s * t = 1
⊢ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1

Next tactic:
rw [ext_iff, Prod.ext_iff] at h
@[to_additive]
protected theorem mul_eq_one_iff : s * t = 1 ↔ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1 := by
  refine' ⟨fun h => _, _⟩
  · rw [ext_iff, Prod.ext_iff] at h
    

Proof state:
case refine'_1
ι : Type u_1
α : Type u_2
inst✝ : OrderedCommGroup α
s t : NonemptyInterval α
h : (s * t).toProd.1 = 1.toProd.1 ∧ (s * t).toProd.2 = 1.toProd.2
⊢ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1

Next tactic:
have := (mul_le_mul_iff_of_ge s.fst_le_snd t.fst_le_snd).1 (h.2.trans h.1.symm).le
@[to_additive]
protected theorem mul_eq_one_iff : s * t = 1 ↔ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1 := by
  refine' ⟨fun h => _, _⟩
  · rw [ext_iff, Prod.ext_iff] at h
    have := (mul_le_mul_iff_of_ge s.fst_le_snd t.fst_le_snd).1 (h.2.trans h.1.symm).le
    

Proof state:
case refine'_1
ι : Type u_1
α : Type u_2
inst✝ : OrderedCommGroup α
s t : NonemptyInterval α
h : (s * t).toProd.1 = 1.toProd.1 ∧ (s * t).toProd.2 = 1.toProd.2
this : s.toProd.1 = s.toProd.2 ∧ t.toProd.1 = t.toProd.2
⊢ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1

Next tactic:
refine' ⟨s.fst, t.fst, _, _, h.1⟩
@[to_additive]
protected theorem mul_eq_one_iff : s * t = 1 ↔ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1 := by
  refine' ⟨fun h => _, _⟩
  · rw [ext_iff, Prod.ext_iff] at h
    have := (mul_le_mul_iff_of_ge s.fst_le_snd t.fst_le_snd).1 (h.2.trans h.1.symm).le
    refine' ⟨s.fst, t.fst, _, _, h.1⟩ <;> 

Proof state:
case refine'_1.refine'_1
ι : Type u_1
α : Type u_2
inst✝ : OrderedCommGroup α
s t : NonemptyInterval α
h : (s * t).toProd.1 = 1.toProd.1 ∧ (s * t).toProd.2 = 1.toProd.2
this : s.toProd.1 = s.toProd.2 ∧ t.toProd.1 = t.toProd.2
⊢ s = pure s.toProd.1

Next tactic:
apply NonemptyInterval.ext
@[to_additive]
protected theorem mul_eq_one_iff : s * t = 1 ↔ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1 := by
  refine' ⟨fun h => _, _⟩
  · rw [ext_iff, Prod.ext_iff] at h
    have := (mul_le_mul_iff_of_ge s.fst_le_snd t.fst_le_snd).1 (h.2.trans h.1.symm).le
    refine' ⟨s.fst, t.fst, _, _, h.1⟩ <;> 

Proof state:
case refine'_1.refine'_2
ι : Type u_1
α : Type u_2
inst✝ : OrderedCommGroup α
s t : NonemptyInterval α
h : (s * t).toProd.1 = 1.toProd.1 ∧ (s * t).toProd.2 = 1.toProd.2
this : s.toProd.1 = s.toProd.2 ∧ t.toProd.1 = t.toProd.2
⊢ t = pure t.toProd.1

Next tactic:
apply NonemptyInterval.ext
@[to_additive]
protected theorem mul_eq_one_iff : s * t = 1 ↔ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1 := by
  refine' ⟨fun h => _, _⟩
  · rw [ext_iff, Prod.ext_iff] at h
    have := (mul_le_mul_iff_of_ge s.fst_le_snd t.fst_le_snd).1 (h.2.trans h.1.symm).le
    refine' ⟨s.fst, t.fst, _, _, h.1⟩ <;> apply NonemptyInterval.ext <;> 

Proof state:
case refine'_1.refine'_1.h
ι : Type u_1
α : Type u_2
inst✝ : OrderedCommGroup α
s t : NonemptyInterval α
h : (s * t).toProd.1 = 1.toProd.1 ∧ (s * t).toProd.2 = 1.toProd.2
this : s.toProd.1 = s.toProd.2 ∧ t.toProd.1 = t.toProd.2
⊢ s.toProd = (pure s.toProd.1).toProd

Next tactic:
dsimp [pure]
@[to_additive]
protected theorem mul_eq_one_iff : s * t = 1 ↔ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1 := by
  refine' ⟨fun h => _, _⟩
  · rw [ext_iff, Prod.ext_iff] at h
    have := (mul_le_mul_iff_of_ge s.fst_le_snd t.fst_le_snd).1 (h.2.trans h.1.symm).le
    refine' ⟨s.fst, t.fst, _, _, h.1⟩ <;> apply NonemptyInterval.ext <;> 

Proof state:
case refine'_1.refine'_2.h
ι : Type u_1
α : Type u_2
inst✝ : OrderedCommGroup α
s t : NonemptyInterval α
h : (s * t).toProd.1 = 1.toProd.1 ∧ (s * t).toProd.2 = 1.toProd.2
this : s.toProd.1 = s.toProd.2 ∧ t.toProd.1 = t.toProd.2
⊢ t.toProd = (pure t.toProd.1).toProd

Next tactic:
dsimp [pure]
@[to_additive]
protected theorem mul_eq_one_iff : s * t = 1 ↔ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1 := by
  refine' ⟨fun h => _, _⟩
  · rw [ext_iff, Prod.ext_iff] at h
    have := (mul_le_mul_iff_of_ge s.fst_le_snd t.fst_le_snd).1 (h.2.trans h.1.symm).le
    refine' ⟨s.fst, t.fst, _, _, h.1⟩ <;> apply NonemptyInterval.ext <;> dsimp [pure]
    · 

Proof state:
case refine'_1.refine'_1.h
ι : Type u_1
α : Type u_2
inst✝ : OrderedCommGroup α
s t : NonemptyInterval α
h : (s * t).toProd.1 = 1.toProd.1 ∧ (s * t).toProd.2 = 1.toProd.2
this : s.toProd.1 = s.toProd.2 ∧ t.toProd.1 = t.toProd.2
⊢ s.toProd = (s.toProd.1, s.toProd.1)

Next tactic:
nth_rw 2 [this.1]
@[to_additive]
protected theorem mul_eq_one_iff : s * t = 1 ↔ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1 := by
  refine' ⟨fun h => _, _⟩
  · rw [ext_iff, Prod.ext_iff] at h
    have := (mul_le_mul_iff_of_ge s.fst_le_snd t.fst_le_snd).1 (h.2.trans h.1.symm).le
    refine' ⟨s.fst, t.fst, _, _, h.1⟩ <;> apply NonemptyInterval.ext <;> dsimp [pure]
    · nth_rw 2 [this.1]
    · 

Proof state:
case refine'_1.refine'_2.h
ι : Type u_1
α : Type u_2
inst✝ : OrderedCommGroup α
s t : NonemptyInterval α
h : (s * t).toProd.1 = 1.toProd.1 ∧ (s * t).toProd.2 = 1.toProd.2
this : s.toProd.1 = s.toProd.2 ∧ t.toProd.1 = t.toProd.2
⊢ t.toProd = (t.toProd.1, t.toProd.1)

Next tactic:
nth_rw 2 [this.2]
@[to_additive]
protected theorem mul_eq_one_iff : s * t = 1 ↔ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1 := by
  refine' ⟨fun h => _, _⟩
  · rw [ext_iff, Prod.ext_iff] at h
    have := (mul_le_mul_iff_of_ge s.fst_le_snd t.fst_le_snd).1 (h.2.trans h.1.symm).le
    refine' ⟨s.fst, t.fst, _, _, h.1⟩ <;> apply NonemptyInterval.ext <;> dsimp [pure]
    · nth_rw 2 [this.1]
    · nth_rw 2 [this.2]
  · 

Proof state:
case refine'_2
ι : Type u_1
α : Type u_2
inst✝ : OrderedCommGroup α
s t : NonemptyInterval α
⊢ (∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1) → s * t = 1

Next tactic:
rintro ⟨b, c, rfl, rfl, h⟩
@[to_additive]
protected theorem mul_eq_one_iff : s * t = 1 ↔ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1 := by
  refine' ⟨fun h => _, _⟩
  · rw [ext_iff, Prod.ext_iff] at h
    have := (mul_le_mul_iff_of_ge s.fst_le_snd t.fst_le_snd).1 (h.2.trans h.1.symm).le
    refine' ⟨s.fst, t.fst, _, _, h.1⟩ <;> apply NonemptyInterval.ext <;> dsimp [pure]
    · nth_rw 2 [this.1]
    · nth_rw 2 [this.2]
  · rintro ⟨b, c, rfl, rfl, h⟩
    

Proof state:
case refine'_2.intro.intro.intro.intro
ι : Type u_1
α : Type u_2
inst✝ : OrderedCommGroup α
b c : α
h : b * c = 1
⊢ pure b * pure c = 1

Next tactic:
rw [pure_mul_pure, h, pure_one]
instance subtractionCommMonoid {α : Type u} [OrderedAddCommGroup α] :
    SubtractionCommMonoid (NonemptyInterval α) :=
  { NonemptyInterval.addCommMonoid with
    neg := Neg.neg
    sub := Sub.sub
    sub_eq_add_neg := fun s t => by
      

Proof state:
ι : Type u_1
α✝ : Type u_2
inst✝¹ : OrderedCommGroup α✝
s✝ t✝ : NonemptyInterval α✝
α : Type u
inst✝ : OrderedAddCommGroup α
src✝ : AddCommMonoid (NonemptyInterval α) := addCommMonoid
s t : NonemptyInterval α
⊢ s - t = s + -t

Next tactic:
refine NonemptyInterval.ext _ _ (Prod.ext ?_ ?_)
instance subtractionCommMonoid {α : Type u} [OrderedAddCommGroup α] :
    SubtractionCommMonoid (NonemptyInterval α) :=
  { NonemptyInterval.addCommMonoid with
    neg := Neg.neg
    sub := Sub.sub
    sub_eq_add_neg := fun s t => by
      refine NonemptyInterval.ext _ _ (Prod.ext ?_ ?_) <;>
      

Proof state:
case refine_1
ι : Type u_1
α✝ : Type u_2
inst✝¹ : OrderedCommGroup α✝
s✝ t✝ : NonemptyInterval α✝
α : Type u
inst✝ : OrderedAddCommGroup α
src✝ : AddCommMonoid (NonemptyInterval α) := addCommMonoid
s t : NonemptyInterval α
⊢ (s - t).toProd.1 = (s + -t).toProd.1

Next tactic:
exact sub_eq_add_neg _ _
instance subtractionCommMonoid {α : Type u} [OrderedAddCommGroup α] :
    SubtractionCommMonoid (NonemptyInterval α) :=
  { NonemptyInterval.addCommMonoid with
    neg := Neg.neg
    sub := Sub.sub
    sub_eq_add_neg := fun s t => by
      refine NonemptyInterval.ext _ _ (Prod.ext ?_ ?_) <;>
      

Proof state:
case refine_2
ι : Type u_1
α✝ : Type u_2
inst✝¹ : OrderedCommGroup α✝
s✝ t✝ : NonemptyInterval α✝
α : Type u
inst✝ : OrderedAddCommGroup α
src✝ : AddCommMonoid (NonemptyInterval α) := addCommMonoid
s t : NonemptyInterval α
⊢ (s - t).toProd.2 = (s + -t).toProd.2

Next tactic:
exact sub_eq_add_neg _ _
instance subtractionCommMonoid {α : Type u} [OrderedAddCommGroup α] :
    SubtractionCommMonoid (NonemptyInterval α) :=
  { NonemptyInterval.addCommMonoid with
    neg := Neg.neg
    sub := Sub.sub
    sub_eq_add_neg := fun s t => by
      refine NonemptyInterval.ext _ _ (Prod.ext ?_ ?_) <;>
      exact sub_eq_add_neg _ _
    neg_neg := fun s => by 

Proof state:
ι : Type u_1
α✝ : Type u_2
inst✝¹ : OrderedCommGroup α✝
s✝ t : NonemptyInterval α✝
α : Type u
inst✝ : OrderedAddCommGroup α
src✝ : AddCommMonoid (NonemptyInterval α) := addCommMonoid
s : NonemptyInterval α
⊢ - -s = s

Next tactic:
apply NonemptyInterval.ext
instance subtractionCommMonoid {α : Type u} [OrderedAddCommGroup α] :
    SubtractionCommMonoid (NonemptyInterval α) :=
  { NonemptyInterval.addCommMonoid with
    neg := Neg.neg
    sub := Sub.sub
    sub_eq_add_neg := fun s t => by
      refine NonemptyInterval.ext _ _ (Prod.ext ?_ ?_) <;>
      exact sub_eq_add_neg _ _
    neg_neg := fun s => by apply NonemptyInterval.ext; 

Proof state:
case h
ι : Type u_1
α✝ : Type u_2
inst✝¹ : OrderedCommGroup α✝
s✝ t : NonemptyInterval α✝
α : Type u
inst✝ : OrderedAddCommGroup α
src✝ : AddCommMonoid (NonemptyInterval α) := addCommMonoid
s : NonemptyInterval α
⊢ (- -s).toProd = s.toProd

Next tactic:
exact neg_neg _
instance subtractionCommMonoid {α : Type u} [OrderedAddCommGroup α] :
    SubtractionCommMonoid (NonemptyInterval α) :=
  { NonemptyInterval.addCommMonoid with
    neg := Neg.neg
    sub := Sub.sub
    sub_eq_add_neg := fun s t => by
      refine NonemptyInterval.ext _ _ (Prod.ext ?_ ?_) <;>
      exact sub_eq_add_neg _ _
    neg_neg := fun s => by apply NonemptyInterval.ext; exact neg_neg _
    neg_add_rev := fun s t => by
      

Proof state:
ι : Type u_1
α✝ : Type u_2
inst✝¹ : OrderedCommGroup α✝
s✝ t✝ : NonemptyInterval α✝
α : Type u
inst✝ : OrderedAddCommGroup α
src✝ : AddCommMonoid (NonemptyInterval α) := addCommMonoid
s t : NonemptyInterval α
⊢ -(s + t) = -t + -s

Next tactic:
refine NonemptyInterval.ext _ _ (Prod.ext ?_ ?_)
instance subtractionCommMonoid {α : Type u} [OrderedAddCommGroup α] :
    SubtractionCommMonoid (NonemptyInterval α) :=
  { NonemptyInterval.addCommMonoid with
    neg := Neg.neg
    sub := Sub.sub
    sub_eq_add_neg := fun s t => by
      refine NonemptyInterval.ext _ _ (Prod.ext ?_ ?_) <;>
      exact sub_eq_add_neg _ _
    neg_neg := fun s => by apply NonemptyInterval.ext; exact neg_neg _
    neg_add_rev := fun s t => by
      refine NonemptyInterval.ext _ _ (Prod.ext ?_ ?_) <;>
      

Proof state:
case refine_1
ι : Type u_1
α✝ : Type u_2
inst✝¹ : OrderedCommGroup α✝
s✝ t✝ : NonemptyInterval α✝
α : Type u
inst✝ : OrderedAddCommGroup α
src✝ : AddCommMonoid (NonemptyInterval α) := addCommMonoid
s t : NonemptyInterval α
⊢ (-(s + t)).toProd.1 = (-t + -s).toProd.1

Next tactic:
exact neg_add_rev _ _
instance subtractionCommMonoid {α : Type u} [OrderedAddCommGroup α] :
    SubtractionCommMonoid (NonemptyInterval α) :=
  { NonemptyInterval.addCommMonoid with
    neg := Neg.neg
    sub := Sub.sub
    sub_eq_add_neg := fun s t => by
      refine NonemptyInterval.ext _ _ (Prod.ext ?_ ?_) <;>
      exact sub_eq_add_neg _ _
    neg_neg := fun s => by apply NonemptyInterval.ext; exact neg_neg _
    neg_add_rev := fun s t => by
      refine NonemptyInterval.ext _ _ (Prod.ext ?_ ?_) <;>
      

Proof state:
case refine_2
ι : Type u_1
α✝ : Type u_2
inst✝¹ : OrderedCommGroup α✝
s✝ t✝ : NonemptyInterval α✝
α : Type u
inst✝ : OrderedAddCommGroup α
src✝ : AddCommMonoid (NonemptyInterval α) := addCommMonoid
s t : NonemptyInterval α
⊢ (-(s + t)).toProd.2 = (-t + -s).toProd.2

Next tactic:
exact neg_add_rev _ _