Theorem: Ordinal.CNF_snd_lt
File: Mathlib/SetTheory/Ordinal/CantorNormalForm.lean

tactic: refine CNFRec b ?_ (fun o ho IH ↦ ?_) o
state_before: b o : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
⊢ x ∈ CNF b o → x.2 < b
state_after: case refine_1
b o : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
⊢ x ∈ CNF b 0 → x.2 < b

case refine_2
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
⊢ x ∈ CNF b o → x.2 < b

tactic: simp only [CNF_zero, not_mem_nil, IsEmpty.forall_iff]
state_before: case refine_1
b o : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
⊢ x ∈ CNF b 0 → x.2 < b
state_after: no goals

tactic: rw [CNF_ne_zero ho]
state_before: case refine_2
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
⊢ x ∈ CNF b o → x.2 < b
state_after: case refine_2
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
⊢ x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o) → x.2 < b

tactic: intro h
state_before: case refine_2
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
⊢ x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o) → x.2 < b
state_after: case refine_2
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
h : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)
⊢ x.2 < b

tactic: cases' (mem_cons.mp h) with h h
state_before: case refine_2
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
h : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)
⊢ x.2 < b
state_after: case refine_2.inl
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
h✝ : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)
h : x = (log b o, o / b ^ log b o)
⊢ x.2 < b

case refine_2.inr
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
h✝ : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)
h : x ∈ CNF b (o % b ^ log b o)
⊢ x.2 < b

tactic: rw [h]
state_before: case refine_2.inl
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
h✝ : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)
h : x = (log b o, o / b ^ log b o)
⊢ x.2 < b
state_after: case refine_2.inl
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
h✝ : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)
h : x = (log b o, o / b ^ log b o)
⊢ (log b o, o / b ^ log b o).2 < b

tactic: simpa only using div_opow_log_lt o hb
state_before: case refine_2.inl
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
h✝ : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)
h : x = (log b o, o / b ^ log b o)
⊢ (log b o, o / b ^ log b o).2 < b
state_after: no goals

tactic: exact IH h
state_before: case refine_2.inr
b o✝ : Ordinal.{u}
hb : 1 < b
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b
h✝ : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)
h : x ∈ CNF b (o % b ^ log b o)
⊢ x.2 < b
state_after: no goals