awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
iha : a = 0
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
e : a.fundamentalSequence = Sum.inl none
m' : ℕ
e' : m.natPred = m' + 1
⊢ m = (m' + 1).succPNat
have rw : m = m.natPred.succPNat
Tactic: rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
iha : a = 0
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
e : a.fundamentalSequence = Sum.inl none
m' : ℕ
e' : m.natPred = m' + 1
rw : m = m.natPred.succPNat
⊢ m = (m' + 1).succPNat
have rw₁ : (↑m : ℕ) = (↑m.natPred.succPNat : ℕ)
Tactic: rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
iha : a = 0
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
e : a.fundamentalSequence = Sum.inl none
m' : ℕ
e' : m.natPred = m' + 1
rw : m = m.natPred.succPNat
rw₁ : (↑m : ℕ) = (↑m.natPred.succPNat : ℕ)
⊢ m = (m' + 1).succPNat
have rw₂ : (↑m : ℕ) = m.natPred.succ
Tactic: rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
iha : a = 0
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
e : a.fundamentalSequence = Sum.inl none
m' : ℕ
e' : m.natPred = m' + 1
rw : m = m.natPred.succPNat
rw₁ : (↑m : ℕ) = (↑m.natPred.succPNat : ℕ)
rw₂ : (↑m : ℕ) = m.natPred.succ
rw₃ : (↑m : ℕ) = m.natPred + 1
⊢ m = (m' + 1).succPNat
have rw₄ : (↑m : ℕ) = (↑m : ℕ)
Tactic: rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
iha : a = 0
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
e : a.fundamentalSequence = Sum.inl none
m' : ℕ
e' : m.natPred = m' + 1
⊢ (a.oadd m b).repr = succ (zero.oadd m'.succPNat zero).repr ∧ ((a.oadd m b).NF → (zero.oadd m'.succPNat zero).NF)
have oadd : (a.oadd (m' + 1).succPNat b).repr = succ (zero.oadd m'.succPNat zero).repr ∧
  ((a.oadd (m' + 1).succPNat b).NF → (zero.oadd m'.succPNat zero).NF)
Tactic: rw [show m = (m' + 1).succPNat by
              rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one]]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
iha : a = 0
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
e : a.fundamentalSequence = Sum.inl none
m' : ℕ
e' : m.natPred = m' + 1
⊢ (a.oadd m b).repr = succ (zero.oadd m'.succPNat zero).repr ∧ ((a.oadd m b).NF → (zero.oadd m'.succPNat zero).NF)
have oadd : (a.oadd (m' + 1).succPNat b).repr = succ (zero.oadd m'.succPNat zero).repr ∧
  ((a.oadd (m' + 1).succPNat b).NF → (zero.oadd m'.succPNat zero).NF)
Tactic: try rw [show m = (m' + 1).succPNat by
              rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one]]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
a' : ONote
iha : a.repr = succ a'.repr ∧ (a.NF → a'.NF)
e : a.fundamentalSequence = Sum.inl (some a')
m' : ℕ
e' : m.natPred = m' + 1
⊢ m = (m' + 1).succPNat
have rw : m = m.natPred.succPNat
Tactic: rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
a' : ONote
iha : a.repr = succ a'.repr ∧ (a.NF → a'.NF)
e : a.fundamentalSequence = Sum.inl (some a')
m' : ℕ
e' : m.natPred = m' + 1
rw : m = m.natPred.succPNat
⊢ m = (m' + 1).succPNat
have rw₁ : (↑m : ℕ) = (↑m.natPred.succPNat : ℕ)
Tactic: rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
a' : ONote
iha : a.repr = succ a'.repr ∧ (a.NF → a'.NF)
e : a.fundamentalSequence = Sum.inl (some a')
m' : ℕ
e' : m.natPred = m' + 1
rw : m = m.natPred.succPNat
rw₁ : (↑m : ℕ) = (↑m.natPred.succPNat : ℕ)
⊢ m = (m' + 1).succPNat
have rw₂ : (↑m : ℕ) = m.natPred.succ
Tactic: rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
a' : ONote
iha : a.repr = succ a'.repr ∧ (a.NF → a'.NF)
e : a.fundamentalSequence = Sum.inl (some a')
m' : ℕ
e' : m.natPred = m' + 1
rw : m = m.natPred.succPNat
rw₁ : (↑m : ℕ) = (↑m.natPred.succPNat : ℕ)
rw₂ : (↑m : ℕ) = m.natPred.succ
rw₃ : (↑m : ℕ) = m.natPred + 1
⊢ m = (m' + 1).succPNat
have rw₄ : (↑m : ℕ) = (↑m : ℕ)
Tactic: rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
a' : ONote
iha : a.repr = succ a'.repr ∧ (a.NF → a'.NF)
e : a.fundamentalSequence = Sum.inl (some a')
m' : ℕ
e' : m.natPred = m' + 1
⊢ IsSuccLimit (a.oadd m b).repr ∧
    (∀ (i : ℕ),
        a.oadd m'.succPNat (a'.oadd i.succPNat zero) < a.oadd m'.succPNat (a'.oadd (i + 1).succPNat zero) ∧
          a.oadd m'.succPNat (a'.oadd i.succPNat zero) < a.oadd m b ∧
            ((a.oadd m b).NF → (a.oadd m'.succPNat (a'.oadd i.succPNat zero)).NF)) ∧
      ∀ a_1 < (a.oadd m b).repr, ∃ (i : ℕ), a_1 < (a.oadd m'.succPNat (a'.oadd i.succPNat zero)).repr
have oadd : IsSuccLimit (a.oadd (m' + 1).succPNat b).repr ∧
  (∀ (i : ℕ),
      a.oadd m'.succPNat (a'.oadd i.succPNat zero) < a.oadd m'.succPNat (a'.oadd (i + 1).succPNat zero) ∧
        a.oadd m'.succPNat (a'.oadd i.succPNat zero) < a.oadd (m' + 1).succPNat b ∧
          ((a.oadd (m' + 1).succPNat b).NF → (a.oadd m'.succPNat (a'.oadd i.succPNat zero)).NF)) ∧
    ∀ a_1 < (a.oadd (m' + 1).succPNat b).repr, ∃ (i : ℕ), a_1 < (a.oadd m'.succPNat (a'.oadd i.succPNat zero)).repr
Tactic: rw [show m = (m' + 1).succPNat by
              rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one]]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
a' : ONote
iha : a.repr = succ a'.repr ∧ (a.NF → a'.NF)
e : a.fundamentalSequence = Sum.inl (some a')
m' : ℕ
e' : m.natPred = m' + 1
⊢ IsSuccLimit (a.oadd m b).repr ∧
    (∀ (i : ℕ),
        a.oadd m'.succPNat (a'.oadd i.succPNat zero) < a.oadd m'.succPNat (a'.oadd (i + 1).succPNat zero) ∧
          a.oadd m'.succPNat (a'.oadd i.succPNat zero) < a.oadd m b ∧
            ((a.oadd m b).NF → (a.oadd m'.succPNat (a'.oadd i.succPNat zero)).NF)) ∧
      ∀ a_1 < (a.oadd m b).repr, ∃ (i : ℕ), a_1 < (a.oadd m'.succPNat (a'.oadd i.succPNat zero)).repr
have oadd : IsSuccLimit (a.oadd (m' + 1).succPNat b).repr ∧
  (∀ (i : ℕ),
      a.oadd m'.succPNat (a'.oadd i.succPNat zero) < a.oadd m'.succPNat (a'.oadd (i + 1).succPNat zero) ∧
        a.oadd m'.succPNat (a'.oadd i.succPNat zero) < a.oadd (m' + 1).succPNat b ∧
          ((a.oadd (m' + 1).succPNat b).NF → (a.oadd m'.succPNat (a'.oadd i.succPNat zero)).NF)) ∧
    ∀ a_1 < (a.oadd (m' + 1).succPNat b).repr, ∃ (i : ℕ), a_1 < (a.oadd m'.succPNat (a'.oadd i.succPNat zero)).repr
Tactic: try rw [show m = (m' + 1).succPNat by
              rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one]]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
f : ℕ → ONote
iha :
  IsSuccLimit a.repr ∧
    (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (a.NF → (f i).NF)) ∧ ∀ a_1 < a.repr, ∃ (i : ℕ), a_1 < (f i).repr
e : a.fundamentalSequence = Sum.inr f
m' : ℕ
e' : m.natPred = m' + 1
⊢ m = (m' + 1).succPNat
have rw : m = m.natPred.succPNat
Tactic: rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
f : ℕ → ONote
iha :
  IsSuccLimit a.repr ∧
    (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (a.NF → (f i).NF)) ∧ ∀ a_1 < a.repr, ∃ (i : ℕ), a_1 < (f i).repr
e : a.fundamentalSequence = Sum.inr f
m' : ℕ
e' : m.natPred = m' + 1
rw : m = m.natPred.succPNat
⊢ m = (m' + 1).succPNat
have rw₁ : (↑m : ℕ) = (↑m.natPred.succPNat : ℕ)
Tactic: rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
f : ℕ → ONote
iha :
  IsSuccLimit a.repr ∧
    (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (a.NF → (f i).NF)) ∧ ∀ a_1 < a.repr, ∃ (i : ℕ), a_1 < (f i).repr
e : a.fundamentalSequence = Sum.inr f
m' : ℕ
e' : m.natPred = m' + 1
rw : m = m.natPred.succPNat
rw₁ : (↑m : ℕ) = (↑m.natPred.succPNat : ℕ)
⊢ m = (m' + 1).succPNat
have rw₂ : (↑m : ℕ) = m.natPred.succ
Tactic: rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
f : ℕ → ONote
iha :
  IsSuccLimit a.repr ∧
    (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (a.NF → (f i).NF)) ∧ ∀ a_1 < a.repr, ∃ (i : ℕ), a_1 < (f i).repr
e : a.fundamentalSequence = Sum.inr f
m' : ℕ
e' : m.natPred = m' + 1
rw : m = m.natPred.succPNat
rw₁ : (↑m : ℕ) = (↑m.natPred.succPNat : ℕ)
rw₂ : (↑m : ℕ) = m.natPred.succ
rw₃ : (↑m : ℕ) = m.natPred + 1
⊢ m = (m' + 1).succPNat
have rw₄ : (↑m : ℕ) = (↑m : ℕ)
Tactic: rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
f : ℕ → ONote
iha :
  IsSuccLimit a.repr ∧
    (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (a.NF → (f i).NF)) ∧ ∀ a_1 < a.repr, ∃ (i : ℕ), a_1 < (f i).repr
e : a.fundamentalSequence = Sum.inr f
m' : ℕ
e' : m.natPred = m' + 1
⊢ IsSuccLimit (a.oadd m b).repr ∧
    (∀ (i : ℕ),
        a.oadd m'.succPNat ((f i).oadd 1 zero) < a.oadd m'.succPNat ((f (i + 1)).oadd 1 zero) ∧
          a.oadd m'.succPNat ((f i).oadd 1 zero) < a.oadd m b ∧
            ((a.oadd m b).NF → (a.oadd m'.succPNat ((f i).oadd 1 zero)).NF)) ∧
      ∀ a_1 < (a.oadd m b).repr, ∃ (i : ℕ), a_1 < (a.oadd m'.succPNat ((f i).oadd 1 zero)).repr
have oadd : IsSuccLimit (a.oadd (m' + 1).succPNat b).repr ∧
  (∀ (i : ℕ),
      a.oadd m'.succPNat ((f i).oadd 1 zero) < a.oadd m'.succPNat ((f (i + 1)).oadd 1 zero) ∧
        a.oadd m'.succPNat ((f i).oadd 1 zero) < a.oadd (m' + 1).succPNat b ∧
          ((a.oadd (m' + 1).succPNat b).NF → (a.oadd m'.succPNat ((f i).oadd 1 zero)).NF)) ∧
    ∀ a_1 < (a.oadd (m' + 1).succPNat b).repr, ∃ (i : ℕ), a_1 < (a.oadd m'.succPNat ((f i).oadd 1 zero)).repr
Tactic: rw [show m = (m' + 1).succPNat by
              rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one]]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
f : ℕ → ONote
iha :
  IsSuccLimit a.repr ∧
    (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (a.NF → (f i).NF)) ∧ ∀ a_1 < a.repr, ∃ (i : ℕ), a_1 < (f i).repr
e : a.fundamentalSequence = Sum.inr f
m' : ℕ
e' : m.natPred = m' + 1
⊢ IsSuccLimit (a.oadd m b).repr ∧
    (∀ (i : ℕ),
        a.oadd m'.succPNat ((f i).oadd 1 zero) < a.oadd m'.succPNat ((f (i + 1)).oadd 1 zero) ∧
          a.oadd m'.succPNat ((f i).oadd 1 zero) < a.oadd m b ∧
            ((a.oadd m b).NF → (a.oadd m'.succPNat ((f i).oadd 1 zero)).NF)) ∧
      ∀ a_1 < (a.oadd m b).repr, ∃ (i : ℕ), a_1 < (a.oadd m'.succPNat ((f i).oadd 1 zero)).repr
have oadd : IsSuccLimit (a.oadd (m' + 1).succPNat b).repr ∧
  (∀ (i : ℕ),
      a.oadd m'.succPNat ((f i).oadd 1 zero) < a.oadd m'.succPNat ((f (i + 1)).oadd 1 zero) ∧
        a.oadd m'.succPNat ((f i).oadd 1 zero) < a.oadd (m' + 1).succPNat b ∧
          ((a.oadd (m' + 1).succPNat b).NF → (a.oadd m'.succPNat ((f i).oadd 1 zero)).NF)) ∧
    ∀ a_1 < (a.oadd (m' + 1).succPNat b).repr, ∃ (i : ℕ), a_1 < (a.oadd m'.succPNat ((f i).oadd 1 zero)).repr
Tactic: try rw [show m = (m' + 1).succPNat by
              rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one]]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
iha : a = 0
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
e : a.fundamentalSequence = Sum.inl none
e' : m.natPred = 0
⊢ (a.oadd 1 b).repr = succ zero.repr ∧ ((a.oadd 1 b).NF → zero.NF)
have oadd : Eq (α := Ordinal.{0}) 1 (succ 0) ∧ ((oadd 0 1 0).NF → NF 0)
Tactic: simp only [repr, iha, ihb, opow_lt_opow_iff_right one_lt_omega0, add_lt_add_iff_left,
        add_zero, lt_add_iff_pos_right, lt_def, mul_one, Nat.cast_zero,
        Nat.cast_succ, Nat.succPNat_coe, opow_succ, opow_zero, mul_add_one, PNat.one_coe,
        _root_.zero_add, zero_def]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
iha : a = 0
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
e : a.fundamentalSequence = Sum.inl none
m' : ℕ
e' : m.natPred = m' + 1
⊢ (a.oadd (m' + 1).succPNat b).repr = succ (zero.oadd m'.succPNat zero).repr ∧
    ((a.oadd (m' + 1).succPNat b).NF → (zero.oadd m'.succPNat zero).NF)
have oadd : HMul.hMul (α := Ordinal.{0}) 1 (↑m' : Ordinal.{0}) + 1 + 1 =
    succ (HMul.hMul (α := Ordinal.{0}) 1 (↑m' : Ordinal.{0}) + 1) ∧
  ((oadd 0 (m' + 1).succPNat 0).NF → (oadd 0 m'.succPNat 0).NF)
Tactic: simp only [repr, iha, ihb, opow_lt_opow_iff_right one_lt_omega0, add_lt_add_iff_left,
        add_zero, lt_add_iff_pos_right, lt_def, mul_one, Nat.cast_zero,
        Nat.cast_succ, Nat.succPNat_coe, opow_succ, opow_zero, mul_add_one, PNat.one_coe,
        _root_.zero_add, zero_def]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
a' : ONote
iha : a.repr = succ a'.repr ∧ (a.NF → a'.NF)
e : a.fundamentalSequence = Sum.inl (some a')
e' : m.natPred = 0
⊢ IsSuccLimit (a.oadd 1 b).repr ∧
    (∀ (i : ℕ),
        a'.oadd i.succPNat zero < a'.oadd (i + 1).succPNat zero ∧
          a'.oadd i.succPNat zero < a.oadd 1 b ∧ ((a.oadd 1 b).NF → (a'.oadd i.succPNat zero).NF)) ∧
      ∀ a_1 < (a.oadd 1 b).repr, ∃ (i : ℕ), a_1 < (a'.oadd i.succPNat zero).repr
have oadd : IsSuccLimit (ω ^ a'.repr * ω) ∧
  (∀ (i : ℕ),
      0 < ω ^ a'.repr ∧
        ω ^ a'.repr * (↑i : Ordinal.{0}) + ω ^ a'.repr < ω ^ a'.repr * ω ∧
          ((a.oadd 1 0).NF → (a'.oadd i.succPNat 0).NF)) ∧
    ∀ a < ω ^ a'.repr * ω, ∃ (i : ℕ), a < ω ^ a'.repr * (↑i : Ordinal.{0}) + ω ^ a'.repr
Tactic: simp only [repr, iha, ihb, opow_lt_opow_iff_right one_lt_omega0, add_lt_add_iff_left,
        add_zero, lt_add_iff_pos_right, lt_def, mul_one, Nat.cast_zero,
        Nat.cast_succ, Nat.succPNat_coe, opow_succ, opow_zero, mul_add_one, PNat.one_coe,
        _root_.zero_add, zero_def]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
a' : ONote
iha : a.repr = succ a'.repr ∧ (a.NF → a'.NF)
e : a.fundamentalSequence = Sum.inl (some a')
m' : ℕ
e' : m.natPred = m' + 1
⊢ IsSuccLimit (a.oadd (m' + 1).succPNat b).repr ∧
    (∀ (i : ℕ),
        a.oadd m'.succPNat (a'.oadd i.succPNat zero) < a.oadd m'.succPNat (a'.oadd (i + 1).succPNat zero) ∧
          a.oadd m'.succPNat (a'.oadd i.succPNat zero) < a.oadd (m' + 1).succPNat b ∧
            ((a.oadd (m' + 1).succPNat b).NF → (a.oadd m'.succPNat (a'.oadd i.succPNat zero)).NF)) ∧
      ∀ a_1 < (a.oadd (m' + 1).succPNat b).repr, ∃ (i : ℕ), a_1 < (a.oadd m'.succPNat (a'.oadd i.succPNat zero)).repr
have oadd : IsSuccLimit (ω ^ a'.repr * ω * (↑m' : Ordinal.{0}) + ω ^ a'.repr * ω + ω ^ a'.repr * ω) ∧
  (∀ (i : ℕ),
      0 < ω ^ a'.repr ∧
        ω ^ a'.repr * (↑i : Ordinal.{0}) + ω ^ a'.repr < ω ^ a'.repr * ω ∧
          ((a.oadd (m' + 1).succPNat 0).NF → (a.oadd m'.succPNat (a'.oadd i.succPNat 0)).NF)) ∧
    ∀ a < ω ^ a'.repr * ω * (↑m' : Ordinal.{0}) + ω ^ a'.repr * ω + ω ^ a'.repr * ω,
      ∃ (i : ℕ),
        a < ω ^ a'.repr * ω * (↑m' : Ordinal.{0}) + ω ^ a'.repr * ω + (ω ^ a'.repr * (↑i : Ordinal.{0}) + ω ^ a'.repr)
Tactic: simp only [repr, iha, ihb, opow_lt_opow_iff_right one_lt_omega0, add_lt_add_iff_left,
        add_zero, lt_add_iff_pos_right, lt_def, mul_one, Nat.cast_zero,
        Nat.cast_succ, Nat.succPNat_coe, opow_succ, opow_zero, mul_add_one, PNat.one_coe,
        _root_.zero_add, zero_def]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
f : ℕ → ONote
iha :
  IsSuccLimit a.repr ∧
    (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (a.NF → (f i).NF)) ∧ ∀ a_1 < a.repr, ∃ (i : ℕ), a_1 < (f i).repr
e : a.fundamentalSequence = Sum.inr f
e' : m.natPred = 0
⊢ IsSuccLimit (a.oadd 1 b).repr ∧
    (∀ (i : ℕ),
        (f i).oadd 1 zero < (f (i + 1)).oadd 1 zero ∧
          (f i).oadd 1 zero < a.oadd 1 b ∧ ((a.oadd 1 b).NF → ((f i).oadd 1 zero).NF)) ∧
      ∀ a_1 < (a.oadd 1 b).repr, ∃ (i : ℕ), a_1 < ((f i).oadd 1 zero).repr
have oadd : IsSuccLimit (ω ^ a.repr) ∧
  (∀ (i : ℕ), (f i).repr < (f (i + 1)).repr ∧ (f i).repr < a.repr ∧ ((a.oadd 1 0).NF → ((f i).oadd 1 0).NF)) ∧
    ∀ a_1 < ω ^ a.repr, ∃ (i : ℕ), a_1 < ω ^ (f i).repr
Tactic: simp only [repr, iha, ihb, opow_lt_opow_iff_right one_lt_omega0, add_lt_add_iff_left,
        add_zero, lt_add_iff_pos_right, lt_def, mul_one, Nat.cast_zero,
        Nat.cast_succ, Nat.succPNat_coe, opow_succ, opow_zero, mul_add_one, PNat.one_coe,
        _root_.zero_add, zero_def]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
f : ℕ → ONote
iha :
  IsSuccLimit a.repr ∧
    (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (a.NF → (f i).NF)) ∧ ∀ a_1 < a.repr, ∃ (i : ℕ), a_1 < (f i).repr
e : a.fundamentalSequence = Sum.inr f
m' : ℕ
e' : m.natPred = m' + 1
⊢ IsSuccLimit (a.oadd (m' + 1).succPNat b).repr ∧
    (∀ (i : ℕ),
        a.oadd m'.succPNat ((f i).oadd 1 zero) < a.oadd m'.succPNat ((f (i + 1)).oadd 1 zero) ∧
          a.oadd m'.succPNat ((f i).oadd 1 zero) < a.oadd (m' + 1).succPNat b ∧
            ((a.oadd (m' + 1).succPNat b).NF → (a.oadd m'.succPNat ((f i).oadd 1 zero)).NF)) ∧
      ∀ a_1 < (a.oadd (m' + 1).succPNat b).repr, ∃ (i : ℕ), a_1 < (a.oadd m'.succPNat ((f i).oadd 1 zero)).repr
have oadd : IsSuccLimit (ω ^ a.repr * (↑m' : Ordinal.{0}) + ω ^ a.repr + ω ^ a.repr) ∧
  (∀ (i : ℕ),
      (f i).repr < (f (i + 1)).repr ∧
        (f i).repr < a.repr ∧ ((a.oadd (m' + 1).succPNat 0).NF → (a.oadd m'.succPNat ((f i).oadd 1 0)).NF)) ∧
    ∀ a_1 < ω ^ a.repr * (↑m' : Ordinal.{0}) + ω ^ a.repr + ω ^ a.repr,
      ∃ (i : ℕ), a_1 < ω ^ a.repr * (↑m' : Ordinal.{0}) + ω ^ a.repr + ω ^ (f i).repr
Tactic: simp only [repr, iha, ihb, opow_lt_opow_iff_right one_lt_omega0, add_lt_add_iff_left,
        add_zero, lt_add_iff_pos_right, lt_def, mul_one, Nat.cast_zero,
        Nat.cast_succ, Nat.succPNat_coe, opow_succ, opow_zero, mul_add_one, PNat.one_coe,
        _root_.zero_add, zero_def]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
iha : a = 0
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
e : a.fundamentalSequence = Sum.inl none
e' : m.natPred = 0
⊢ Eq (α := Ordinal.{0}) 1 (succ 0) ∧ ((oadd 0 1 0).NF → NF 0)
have oadd : Eq (α := Ordinal.{0}) 1 (succ 0)
Tactic: constructor
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
iha : a = 0
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
e : a.fundamentalSequence = Sum.inl none
e' : m.natPred = 0
oadd : Eq (α := Ordinal.{0}) 1 (succ 0)
⊢ Eq (α := Ordinal.{0}) 1 (succ 0) ∧ ((ONote.oadd 0 1 0).NF → NF 0)
have oadd₁ : (ONote.oadd 0 1 0).NF → NF 0
Tactic: constructor
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (hypothesis)
Goal:
a : ONote
m : ℕ+
b : ONote
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
a' : ONote
iha : a.repr = succ a'.repr ∧ (a.NF → a'.NF)
e : a.fundamentalSequence = Sum.inl (some a')
e' : m.natPred = 0
⊢ IsSuccLimit (ω ^ a'.repr * ω) ∧
    (∀ (i : ℕ),
        0 < ω ^ a'.repr ∧
          ω ^ a'.repr * (↑i : Ordinal.{0}) + ω ^ a'.repr < ω ^ a'.repr * ω ∧
            ((a.oadd 1 0).NF → (a'.oadd i.succPNat 0).NF)) ∧
      ∀ a < ω ^ a'.repr * ω, ∃ (i : ℕ), a < ω ^ a'.repr * (↑i : Ordinal.{0}) + ω ^ a'.repr
have this : 0 < ω ^ a'.repr
Tactic: have := opow_pos (repr a') omega0_pos
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
a' : ONote
iha : a.repr = succ a'.repr ∧ (a.NF → a'.NF)
e : a.fundamentalSequence = Sum.inl (some a')
e' : m.natPred = 0
this : 0 < ω ^ a'.repr
⊢ IsSuccLimit (ω ^ a'.repr * ω) ∧
    (∀ (i : ℕ),
        0 < ω ^ a'.repr ∧
          ω ^ a'.repr * (↑i : Ordinal.{0}) + ω ^ a'.repr < ω ^ a'.repr * ω ∧
            ((a.oadd 1 0).NF → (a'.oadd i.succPNat 0).NF)) ∧
      ∀ a < ω ^ a'.repr * ω, ∃ (i : ℕ), a < ω ^ a'.repr * (↑i : Ordinal.{0}) + ω ^ a'.repr
have oadd : ∀ (i : ℕ), ω ^ a'.repr * (↑i : Ordinal.{0}) + ω ^ a'.repr < ω ^ a'.repr * ω
Tactic: refine
        ⟨isSuccLimit_mul_right this isSuccLimit_omega0, fun i =>
          ⟨this, ?_, fun H => @NF.oadd_zero _ _ (iha.2 H.fst)⟩, exists_lt_mul_omega0'⟩
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
a' : ONote
iha : a.repr = succ a'.repr ∧ (a.NF → a'.NF)
e : a.fundamentalSequence = Sum.inl (some a')
e' : m.natPred = 0
this : 0 < ω ^ a'.repr
i : ℕ
⊢ ω ^ a'.repr * (↑i : Ordinal.{0}) + ω ^ a'.repr < ω ^ a'.repr * ω
have oadd : ω ^ a'.repr * (↑(i + 1) : Ordinal.{0}) < ω ^ a'.repr * ω
Tactic: rw [← mul_add_one, ← Nat.cast_add_one]
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (goal)
Goal:
a : ONote
m : ℕ+
b : ONote
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
a' : ONote
iha : a.repr = succ a'.repr ∧ (a.NF → a'.NF)
e : a.fundamentalSequence = Sum.inl (some a')
e' : m.natPred = 0
this : 0 < ω ^ a'.repr
i : ℕ
⊢ ω ^ a'.repr * (↑(i + 1) : Ordinal.{0}) < ω ^ a'.repr * ω
have oadd : (↑(i + 1) : Ordinal.{0}) < ω
Tactic: gcongr
-- Lean 4 proof step from Mathlib.SetTheory.Ordinal.Notation (hypothesis)
Goal:
a : ONote
m : ℕ+
b : ONote
ihb : b = 0
e_1 : b.fundamentalSequence = Sum.inl none
a' : ONote
iha : a.repr = succ a'.repr ∧ (a.NF → a'.NF)
e : a.fundamentalSequence = Sum.inl (some a')
m' : ℕ
e' : m.natPred = m' + 1
⊢ IsSuccLimit (ω ^ a'.repr * ω * (↑m' : Ordinal.{0}) + ω ^ a'.repr * ω + ω ^ a'.repr * ω) ∧
    (∀ (i : ℕ),
        0 < ω ^ a'.repr ∧
          ω ^ a'.repr * (↑i : Ordinal.{0}) + ω ^ a'.repr < ω ^ a'.repr * ω ∧
            ((a.oadd (m' + 1).succPNat 0).NF → (a.oadd m'.succPNat (a'.oadd i.succPNat 0)).NF)) ∧
      ∀ a < ω ^ a'.repr * ω * (↑m' : Ordinal.{0}) + ω ^ a'.repr * ω + ω ^ a'.repr * ω,
        ∃ (i : ℕ),
          a < ω ^ a'.repr * ω * (↑m' : Ordinal.{0}) + ω ^ a'.repr * ω + (ω ^ a'.repr * (↑i : Ordinal.{0}) + ω ^ a'.repr)
have this : 0 < ω ^ a'.repr
Tactic: have := opow_pos (repr a') omega0_pos