l3lab/ntp-mathlib huggingface dataset:
theorem add_succ : ∀ m n : Num, m + succ n = succ (m + n)
  | 0, n => by 

Proof state:
α : Type u_1
n : Num
⊢ 0 + succ n = succ (0 + n)

Next tactic:
simp [zero_add]
theorem add_succ : ∀ m n : Num, m + succ n = succ (m + n)
  | 0, n => by simp [zero_add]
  | pos p, 0 => show pos (p + 1) = succ (pos p + 0) by 

Proof state:
α : Type u_1
p : PosNum
⊢ pos (p + 1) = succ (pos p + 0)

Next tactic:
rw [PosNum.add_one, add_zero, succ, succ']
@[simp]
theorem ofNat'_zero : Num.ofNat' 0 = 0 := by 

Proof state:
α : Type u_1
⊢ ofNat' 0 = 0

Next tactic:
simp [Num.ofNat']
@[simp]
theorem ofNat'_one : Num.ofNat' 1 = 1 := by 

Proof state:
α : Type u_1
⊢ ofNat' 1 = 1

Next tactic:
erw [ofNat'_bit true 0, cond, ofNat'_zero]
@[simp]
theorem ofNat'_one : Num.ofNat' 1 = 1 := by erw [ofNat'_bit true 0, cond, ofNat'_zero]; 

Proof state:
α : Type u_1
⊢ (match (motive := Bool → Num → Num) true with
      | true => Num.bit1
      | false => Num.bit0)
      0 =
    1

Next tactic:
rfl
theorem ofNat'_succ : ∀ {n}, ofNat' (n + 1) = ofNat' n + 1 :=
  @(Nat.binaryRec (by 

Proof state:
α : Type u_1
⊢ ofNat' (0 + 1) = ofNat' 0 + 1

Next tactic:
simp [zero_add]
theorem ofNat'_succ : ∀ {n}, ofNat' (n + 1) = ofNat' n + 1 :=
  @(Nat.binaryRec (by simp [zero_add]) fun b n ih => by
    

Proof state:
α : Type u_1
b : Bool
n : ℕ
ih : ofNat' (n + 1) = ofNat' n + 1
⊢ ofNat' (Nat.bit b n + 1) = ofNat' (Nat.bit b n) + 1

Next tactic:
cases b
theorem ofNat'_succ : ∀ {n}, ofNat' (n + 1) = ofNat' n + 1 :=
  @(Nat.binaryRec (by simp [zero_add]) fun b n ih => by
    cases b
    · 

Proof state:
case false
α : Type u_1
n : ℕ
ih : ofNat' (n + 1) = ofNat' n + 1
⊢ ofNat' (Nat.bit false n + 1) = ofNat' (Nat.bit false n) + 1

Next tactic:
erw [ofNat'_bit true n, ofNat'_bit]
theorem ofNat'_succ : ∀ {n}, ofNat' (n + 1) = ofNat' n + 1 :=
  @(Nat.binaryRec (by simp [zero_add]) fun b n ih => by
    cases b
    · erw [ofNat'_bit true n, ofNat'_bit]
      

Proof state:
case false
α : Type u_1
n : ℕ
ih : ofNat' (n + 1) = ofNat' n + 1
⊢ cond true Num.bit1 Num.bit0 (ofNat' n) = cond false Num.bit1 Num.bit0 (ofNat' n) + 1

Next tactic:
simp only [← bit1_of_bit1, ← bit0_of_bit0, cond, _root_.bit1]
theorem ofNat'_succ : ∀ {n}, ofNat' (n + 1) = ofNat' n + 1 :=
  @(Nat.binaryRec (by simp [zero_add]) fun b n ih => by
    cases b
    · erw [ofNat'_bit true n, ofNat'_bit]
      simp only [← bit1_of_bit1, ← bit0_of_bit0, cond, _root_.bit1]
    -- Porting note: `cc` was not ported yet so `exact Nat.add_left_comm n 1 1` is used.
    · 

Proof state:
case true
α : Type u_1
n : ℕ
ih : ofNat' (n + 1) = ofNat' n + 1
⊢ ofNat' (Nat.bit true n + 1) = ofNat' (Nat.bit true n) + 1

Next tactic:
erw [show n.bit true + 1 = (n + 1).bit false by
        simp [Nat.bit, _root_.bit1, _root_.bit0]; exact Nat.add_left_comm n 1 1,
        ofNat'_bit, ofNat'_bit, ih]
theorem ofNat'_succ : ∀ {n}, ofNat' (n + 1) = ofNat' n + 1 :=
  @(Nat.binaryRec (by simp [zero_add]) fun b n ih => by
    cases b
    · erw [ofNat'_bit true n, ofNat'_bit]
      simp only [← bit1_of_bit1, ← bit0_of_bit0, cond, _root_.bit1]
    -- Porting note: `cc` was not ported yet so `exact Nat.add_left_comm n 1 1` is used.
    · erw [show n.bit true + 1 = (n + 1).bit false by
        

Proof state:
α : Type u_1
n : ℕ
ih : ofNat' (n + 1) = ofNat' n + 1
⊢ Nat.bit true n + 1 = Nat.bit false (n + 1)

Next tactic:
simp [Nat.bit, _root_.bit1, _root_.bit0]
theorem ofNat'_succ : ∀ {n}, ofNat' (n + 1) = ofNat' n + 1 :=
  @(Nat.binaryRec (by simp [zero_add]) fun b n ih => by
    cases b
    · erw [ofNat'_bit true n, ofNat'_bit]
      simp only [← bit1_of_bit1, ← bit0_of_bit0, cond, _root_.bit1]
    -- Porting note: `cc` was not ported yet so `exact Nat.add_left_comm n 1 1` is used.
    · erw [show n.bit true + 1 = (n + 1).bit false by
        simp [Nat.bit, _root_.bit1, _root_.bit0]; 

Proof state:
α : Type u_1
n : ℕ
ih : ofNat' (n + 1) = ofNat' n + 1
⊢ n + (1 + 1) = 1 + (n + 1)

Next tactic:
exact Nat.add_left_comm n 1 1
theorem ofNat'_succ : ∀ {n}, ofNat' (n + 1) = ofNat' n + 1 :=
  @(Nat.binaryRec (by simp [zero_add]) fun b n ih => by
    cases b
    · erw [ofNat'_bit true n, ofNat'_bit]
      simp only [← bit1_of_bit1, ← bit0_of_bit0, cond, _root_.bit1]
    -- Porting note: `cc` was not ported yet so `exact Nat.add_left_comm n 1 1` is used.
    · erw [show n.bit true + 1 = (n + 1).bit false by
        simp [Nat.bit, _root_.bit1, _root_.bit0]; exact Nat.add_left_comm n 1 1,
        ofNat'_bit, ofNat'_bit, ih]
      

Proof state:
case true
α : Type u_1
n : ℕ
ih : ofNat' (n + 1) = ofNat' n + 1
⊢ cond false Num.bit1 Num.bit0 (ofNat' n + 1) = cond true Num.bit1 Num.bit0 (ofNat' n) + 1

Next tactic:
simp only [cond, add_one, bit1_succ]
@[simp]
theorem add_ofNat' (m n) : Num.ofNat' (m + n) = Num.ofNat' m + Num.ofNat' n := by
  -- Porting note: `simp` fails to unify `ofNat' (n + 1)` with `ofNat' n.succ`
  

Proof state:
α : Type u_1
m n : ℕ
⊢ ofNat' (m + n) = ofNat' m + ofNat' n

Next tactic:
have : ∀ {n}, ofNat' n.succ = ofNat' n + 1 := ofNat'_succ
@[simp]
theorem add_ofNat' (m n) : Num.ofNat' (m + n) = Num.ofNat' m + Num.ofNat' n := by
  -- Porting note: `simp` fails to unify `ofNat' (n + 1)` with `ofNat' n.succ`
  have : ∀ {n}, ofNat' n.succ = ofNat' n + 1 := ofNat'_succ
  

Proof state:
α : Type u_1
m n : ℕ
this : ∀ {n : ℕ}, ofNat' (Nat.succ n) = ofNat' n + 1
⊢ ofNat' (m + n) = ofNat' m + ofNat' n

Next tactic:
induction n
@[simp]
theorem add_ofNat' (m n) : Num.ofNat' (m + n) = Num.ofNat' m + Num.ofNat' n := by
  -- Porting note: `simp` fails to unify `ofNat' (n + 1)` with `ofNat' n.succ`
  have : ∀ {n}, ofNat' n.succ = ofNat' n + 1 := ofNat'_succ
  induction n <;> 

Proof state:
case zero
α : Type u_1
m : ℕ
this : ∀ {n : ℕ}, ofNat' (Nat.succ n) = ofNat' n + 1
⊢ ofNat' (m + Nat.zero) = ofNat' m + ofNat' Nat.zero

Next tactic:
simp [Nat.add_zero, this, add_zero, Nat.add_succ, add_one, add_succ, *]
@[simp]
theorem add_ofNat' (m n) : Num.ofNat' (m + n) = Num.ofNat' m + Num.ofNat' n := by
  -- Porting note: `simp` fails to unify `ofNat' (n + 1)` with `ofNat' n.succ`
  have : ∀ {n}, ofNat' n.succ = ofNat' n + 1 := ofNat'_succ
  induction n <;> 

Proof state:
case succ
α : Type u_1
m : ℕ
this : ∀ {n : ℕ}, ofNat' (Nat.succ n) = ofNat' n + 1
n✝ : ℕ
n_ih✝ : ofNat' (m + n✝) = ofNat' m + ofNat' n✝
⊢ ofNat' (m + Nat.succ n✝) = ofNat' m + ofNat' (Nat.succ n✝)

Next tactic:
simp [Nat.add_zero, this, add_zero, Nat.add_succ, add_one, add_succ, *]
theorem cmp_to_nat : ∀ m n, (Ordering.casesOn (cmp m n) ((m : ℕ) < n) (m = n) ((n : ℕ) < m) : Prop)
  | 0, 0 => rfl
  | 0, pos b => to_nat_pos _
  | pos a, 0 => to_nat_pos _
  | pos a, pos b => by
    

Proof state:
α : Type u_1
a b : PosNum
⊢ Ordering.casesOn (cmp (pos a) (pos b)) (↑(pos a) < ↑(pos b)) (pos a = pos b) (↑(pos b) < ↑(pos a))

Next tactic:
have := PosNum.cmp_to_nat a b
theorem cmp_to_nat : ∀ m n, (Ordering.casesOn (cmp m n) ((m : ℕ) < n) (m = n) ((n : ℕ) < m) : Prop)
  | 0, 0 => rfl
  | 0, pos b => to_nat_pos _
  | pos a, 0 => to_nat_pos _
  | pos a, pos b => by
    have := PosNum.cmp_to_nat a b; 

Proof state:
α : Type u_1
a b : PosNum
this : Ordering.casesOn (PosNum.cmp a b) (↑a < ↑b) (a = b) (↑b < ↑a)
⊢ Ordering.casesOn (cmp (pos a) (pos b)) (↑(pos a) < ↑(pos b)) (pos a = pos b) (↑(pos b) < ↑(pos a))

Next tactic:
revert this
theorem cmp_to_nat : ∀ m n, (Ordering.casesOn (cmp m n) ((m : ℕ) < n) (m = n) ((n : ℕ) < m) : Prop)
  | 0, 0 => rfl
  | 0, pos b => to_nat_pos _
  | pos a, 0 => to_nat_pos _
  | pos a, pos b => by
    have := PosNum.cmp_to_nat a b; revert this; 

Proof state:
α : Type u_1
a b : PosNum
⊢ Ordering.casesOn (PosNum.cmp a b) (↑a < ↑b) (a = b) (↑b < ↑a) →
    Ordering.casesOn (cmp (pos a) (pos b)) (↑(pos a) < ↑(pos b)) (pos a = pos b) (↑(pos b) < ↑(pos a))

Next tactic:
dsimp [cmp]
theorem cmp_to_nat : ∀ m n, (Ordering.casesOn (cmp m n) ((m : ℕ) < n) (m = n) ((n : ℕ) < m) : Prop)
  | 0, 0 => rfl
  | 0, pos b => to_nat_pos _
  | pos a, 0 => to_nat_pos _
  | pos a, pos b => by
    have := PosNum.cmp_to_nat a b; revert this; dsimp [cmp]; 

Proof state:
α : Type u_1
a b : PosNum
⊢ Ordering.rec (↑a < ↑b) (a = b) (↑b < ↑a) (PosNum.cmp a b) →
    Ordering.rec (↑a < ↑b) (pos a = pos b) (↑b < ↑a) (PosNum.cmp a b)

Next tactic:
cases PosNum.cmp a b
theorem cmp_to_nat : ∀ m n, (Ordering.casesOn (cmp m n) ((m : ℕ) < n) (m = n) ((n : ℕ) < m) : Prop)
  | 0, 0 => rfl
  | 0, pos b => to_nat_pos _
  | pos a, 0 => to_nat_pos _
  | pos a, pos b => by
    have := PosNum.cmp_to_nat a b; revert this; dsimp [cmp]; cases PosNum.cmp a b
    

Proof state:
case lt
α : Type u_1
a b : PosNum
⊢ Ordering.rec (↑a < ↑b) (a = b) (↑b < ↑a) Ordering.lt → Ordering.rec (↑a < ↑b) (pos a = pos b) (↑b < ↑a) Ordering.lt
case eq
α : Type u_1
a b : PosNum
⊢ Ordering.rec (↑a < ↑b) (a = b) (↑b < ↑a) Ordering.eq → Ordering.rec (↑a < ↑b) (pos a = pos b) (↑b < ↑a) Ordering.eq
case gt
α : Type u_1
a b : PosNum
⊢ Ordering.rec (↑a < ↑b) (a = b) (↑b < ↑a) Ordering.gt → Ordering.rec (↑a < ↑b) (pos a = pos b) (↑b < ↑a) Ordering.gt

Next tactic:
exacts [id, congr_arg pos, id]
@[norm_cast]
theorem lt_to_nat {m n : Num} : (m : ℕ) < n ↔ m < n :=
  show (m : ℕ) < n ↔ cmp m n = Ordering.lt from
    match cmp m n, cmp_to_nat m n with
    | Ordering.lt, h => by 

Proof state:
α : Type u_1
m n : Num
h : Ordering.casesOn Ordering.lt (↑m < ↑n) (m = n) (↑n < ↑m)
⊢ ↑m < ↑n ↔ Ordering.lt = Ordering.lt

Next tactic:
simp at h
@[norm_cast]
theorem lt_to_nat {m n : Num} : (m : ℕ) < n ↔ m < n :=
  show (m : ℕ) < n ↔ cmp m n = Ordering.lt from
    match cmp m n, cmp_to_nat m n with
    | Ordering.lt, h => by simp at h; 

Proof state:
α : Type u_1
m n : Num
h : ↑m < ↑n
⊢ ↑m < ↑n ↔ Ordering.lt = Ordering.lt

Next tactic:
simp [h]
@[norm_cast]
theorem lt_to_nat {m n : Num} : (m : ℕ) < n ↔ m < n :=
  show (m : ℕ) < n ↔ cmp m n = Ordering.lt from
    match cmp m n, cmp_to_nat m n with
    | Ordering.lt, h => by simp at h; simp [h]
    | Ordering.eq, h => by 

Proof state:
α : Type u_1
m n : Num
h : Ordering.casesOn Ordering.eq (↑m < ↑n) (m = n) (↑n < ↑m)
⊢ ↑m < ↑n ↔ Ordering.eq = Ordering.lt

Next tactic:
simp at h
@[norm_cast]
theorem lt_to_nat {m n : Num} : (m : ℕ) < n ↔ m < n :=
  show (m : ℕ) < n ↔ cmp m n = Ordering.lt from
    match cmp m n, cmp_to_nat m n with
    | Ordering.lt, h => by simp at h; simp [h]
    | Ordering.eq, h => by simp at h; 

Proof state:
α : Type u_1
m n : Num
h : m = n
⊢ ↑m < ↑n ↔ Ordering.eq = Ordering.lt

Next tactic:
simp [h, lt_irrefl]
@[norm_cast]
theorem lt_to_nat {m n : Num} : (m : ℕ) < n ↔ m < n :=
  show (m : ℕ) < n ↔ cmp m n = Ordering.lt from
    match cmp m n, cmp_to_nat m n with
    | Ordering.lt, h => by simp at h; simp [h]
    | Ordering.eq, h => by simp at h; simp [h, lt_irrefl]
    | Ordering.gt, h => by 

Proof state:
α : Type u_1
m n : Num
h : Ordering.casesOn Ordering.gt (↑m < ↑n) (m = n) (↑n < ↑m)
⊢ ↑m < ↑n ↔ Ordering.gt = Ordering.lt

Next tactic:
simp [not_lt_of_gt h]
@[norm_cast]
theorem le_to_nat {m n : Num} : (m : ℕ) ≤ n ↔ m ≤ n := by
  

Proof state:
α : Type u_1
m n : Num
⊢ ↑m ≤ ↑n ↔ m ≤ n

Next tactic:
rw [← not_lt]
@[norm_cast]
theorem le_to_nat {m n : Num} : (m : ℕ) ≤ n ↔ m ≤ n := by
  rw [← not_lt]; 

Proof state:
α : Type u_1
m n : Num
⊢ ¬↑n < ↑m ↔ m ≤ n

Next tactic:
exact not_congr lt_to_nat
@[simp]
theorem of_to_nat' : ∀ n : PosNum, Num.ofNat' (n : ℕ) = Num.pos n
  | 1 => by 

Proof state:
⊢ Num.ofNat' ↑1 = Num.pos 1

Next tactic:
erw [@Num.ofNat'_bit true 0, Num.ofNat'_zero]
@[simp]
theorem of_to_nat' : ∀ n : PosNum, Num.ofNat' (n : ℕ) = Num.pos n
  | 1 => by erw [@Num.ofNat'_bit true 0, Num.ofNat'_zero]; 

Proof state:
⊢ cond true Num.bit1 Num.bit0 0 = Num.pos 1

Next tactic:
rfl
@[simp]
theorem of_to_nat' : ∀ n : PosNum, Num.ofNat' (n : ℕ) = Num.pos n
  | 1 => by erw [@Num.ofNat'_bit true 0, Num.ofNat'_zero]; rfl
  | bit0 p => by 

Proof state:
p : PosNum
⊢ Num.ofNat' ↑(bit0 p) = Num.pos (bit0 p)

Next tactic:
erw [@Num.ofNat'_bit false, of_to_nat' p]
@[simp]
theorem of_to_nat' : ∀ n : PosNum, Num.ofNat' (n : ℕ) = Num.pos n
  | 1 => by erw [@Num.ofNat'_bit true 0, Num.ofNat'_zero]; rfl
  | bit0 p => by erw [@Num.ofNat'_bit false, of_to_nat' p]; 

Proof state:
p : PosNum
⊢ cond false Num.bit1 Num.bit0 (Num.pos p) = Num.pos (bit0 p)

Next tactic:
rfl
@[simp]
theorem of_to_nat' : ∀ n : PosNum, Num.ofNat' (n : ℕ) = Num.pos n
  | 1 => by erw [@Num.ofNat'_bit true 0, Num.ofNat'_zero]; rfl
  | bit0 p => by erw [@Num.ofNat'_bit false, of_to_nat' p]; rfl
  | bit1 p => by 

Proof state:
p : PosNum
⊢ Num.ofNat' ↑(bit1 p) = Num.pos (bit1 p)

Next tactic:
erw [@Num.ofNat'_bit true, of_to_nat' p]
@[simp]
theorem of_to_nat' : ∀ n : PosNum, Num.ofNat' (n : ℕ) = Num.pos n
  | 1 => by erw [@Num.ofNat'_bit true 0, Num.ofNat'_zero]; rfl
  | bit0 p => by erw [@Num.ofNat'_bit false, of_to_nat' p]; rfl
  | bit1 p => by erw [@Num.ofNat'_bit true, of_to_nat' p]; 

Proof state:
p : PosNum
⊢ cond true Num.bit1 Num.bit0 (Num.pos p) = Num.pos (bit1 p)

Next tactic:
rfl
instance addMonoid : AddMonoid Num where
  add := (· + ·)
  zero := 0
  zero_add := zero_add
  add_zero := add_zero
  add_assoc := by 

Proof state:
⊢ ∀ (a b c : Num), a + b + c = a + (b + c)

Next tactic:
transfer
instance commSemiring : CommSemiring Num := by
  

Proof state:
⊢ CommSemiring Num

Next tactic:
refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. }
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    

Proof state:
case refine'_1
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a b : Num), a + b = b + a

Next tactic:
try { intros; rfl }
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    try 

Proof state:
case refine'_1
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a b : Num), a + b = b + a

Next tactic:
{ intros; rfl }
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    try { 

Proof state:
case refine'_1
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a b : Num), a + b = b + a

Next tactic:
intros
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    try { intros; 

Proof state:
case refine'_1
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
a✝ b✝ : Num
⊢ a✝ + b✝ = b✝ + a✝

Next tactic:
rfl
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    

Proof state:
case refine'_2
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a b c : Num), a * (b + c) = a * b + a * c

Next tactic:
try { intros; rfl }
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    try 

Proof state:
case refine'_2
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a b c : Num), a * (b + c) = a * b + a * c

Next tactic:
{ intros; rfl }
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    try { 

Proof state:
case refine'_2
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a b c : Num), a * (b + c) = a * b + a * c

Next tactic:
intros
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    try { intros; 

Proof state:
case refine'_2
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
a✝ b✝ c✝ : Num
⊢ a✝ * (b✝ + c✝) = a✝ * b✝ + a✝ * c✝

Next tactic:
rfl
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    

Proof state:
case refine'_3
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a b c : Num), (a + b) * c = a * c + b * c

Next tactic:
try { intros; rfl }
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    try 

Proof state:
case refine'_3
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a b c : Num), (a + b) * c = a * c + b * c

Next tactic:
{ intros; rfl }
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    try { 

Proof state:
case refine'_3
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a b c : Num), (a + b) * c = a * c + b * c

Next tactic:
intros
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    try { intros; 

Proof state:
case refine'_3
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
a✝ b✝ c✝ : Num
⊢ (a✝ + b✝) * c✝ = a✝ * c✝ + b✝ * c✝

Next tactic:
rfl
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    

Proof state:
case refine'_4
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a : Num), 0 * a = 0

Next tactic:
try { intros; rfl }
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    try 

Proof state:
case refine'_4
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a : Num), 0 * a = 0

Next tactic:
{ intros; rfl }
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    try { 

Proof state:
case refine'_4
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a : Num), 0 * a = 0

Next tactic:
intros
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    try { intros; 

Proof state:
case refine'_4
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
a✝ : Num
⊢ 0 * a✝ = 0

Next tactic:
rfl
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    

Proof state:
case refine'_5
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a : Num), a * 0 = 0

Next tactic:
try { intros; rfl }
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    try 

Proof state:
case refine'_5
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a : Num), a * 0 = 0

Next tactic:
{ intros; rfl }
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    try { 

Proof state:
case refine'_5
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a : Num), a * 0 = 0

Next tactic:
intros
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    try { intros; 

Proof state:
case refine'_5
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
a✝ : Num
⊢ a✝ * 0 = 0

Next tactic:
rfl
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    

Proof state:
case refine'_6
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a b c : Num), a * b * c = a * (b * c)

Next tactic:
try { intros; rfl }
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    try 

Proof state:
case refine'_6
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a b c : Num), a * b * c = a * (b * c)

Next tactic:
{ intros; rfl }
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    try { 

Proof state:
case refine'_6
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a b c : Num), a * b * c = a * (b * c)

Next tactic:
intros
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    try { intros; 

Proof state:
case refine'_6
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
a✝ b✝ c✝ : Num
⊢ a✝ * b✝ * c✝ = a✝ * (b✝ * c✝)

Next tactic:
rfl
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    

Proof state:
case refine'_7
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a : Num), 1 * a = a

Next tactic:
try { intros; rfl }
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    try 

Proof state:
case refine'_7
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a : Num), 1 * a = a

Next tactic:
{ intros; rfl }
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    try { 

Proof state:
case refine'_7
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a : Num), 1 * a = a

Next tactic:
intros
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    try { intros; 

Proof state:
case refine'_7
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
a✝ : Num
⊢ 1 * a✝ = a✝

Next tactic:
rfl
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    

Proof state:
case refine'_8
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a : Num), a * 1 = a

Next tactic:
try { intros; rfl }
instance commSemiring : CommSemiring Num := by
  refine'
    { Num.addMonoid,
      Num.addMonoidWithOne with
      mul := (· * ·)
      npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩, .. } <;>
    try 

Proof state:
case refine'_8
src✝¹ : AddMonoid Num := addMonoid
src✝ : AddMonoidWithOne Num := addMonoidWithOne
⊢ ∀ (a : Num), a * 1 = a

Next tactic:
{ intros; rfl }