Theorem: toZ_mono
File: Mathlib/Order/SuccPred/LinearLocallyFinite.lean

tactic: by_cases hi_max : IsMax i
state_before: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
⊢ toZ i0 i ≤ toZ i0 j
state_after: case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : IsMax i
⊢ toZ i0 i ≤ toZ i0 j

case neg
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
⊢ toZ i0 i ≤ toZ i0 j

tactic: by_cases hj_min : IsMin j
state_before: case neg
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
⊢ toZ i0 i ≤ toZ i0 j
state_after: case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : IsMin j
⊢ toZ i0 i ≤ toZ i0 j

case neg
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
⊢ toZ i0 i ≤ toZ i0 j

tactic: rcases le_or_lt i0 i with hi | hi <;> rcases le_or_lt i0 j with hj | hj
state_before: case neg
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
⊢ toZ i0 i ≤ toZ i0 j
state_after: case neg.inl.inl
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
⊢ toZ i0 i ≤ toZ i0 j

case neg.inl.inr
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : j < i0
⊢ toZ i0 i ≤ toZ i0 j

case neg.inr.inl
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : i0 ≤ j
⊢ toZ i0 i ≤ toZ i0 j

case neg.inr.inr
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
⊢ toZ i0 i ≤ toZ i0 j

tactic: rw [le_antisymm h_le (hi_max h_le)]
state_before: case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : IsMax i
⊢ toZ i0 i ≤ toZ i0 j
state_after: no goals

tactic: rw [le_antisymm h_le (hj_min h_le)]
state_before: case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : IsMin j
⊢ toZ i0 i ≤ toZ i0 j
state_after: no goals

tactic: let m := Nat.find (exists_succ_iterate_of_le h_le)
state_before: case neg.inl.inl
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
⊢ toZ i0 i ≤ toZ i0 j
state_after: case neg.inl.inl
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
⊢ toZ i0 i ≤ toZ i0 j

tactic: have hm : succ^[m] i = j := Nat.find_spec (exists_succ_iterate_of_le h_le)
state_before: case neg.inl.inl
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
⊢ toZ i0 i ≤ toZ i0 j
state_after: case neg.inl.inl
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
⊢ toZ i0 i ≤ toZ i0 j

tactic: have hj_eq : j = succ^[(toZ i0 i).toNat + m] i0 := by
  rw [← hm, add_comm]
  nth_rw 1 [← iterate_succ_toZ i hi]
  rw [Function.iterate_add]
  rfl
state_before: case neg.inl.inl
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
⊢ toZ i0 i ≤ toZ i0 j
state_after: case neg.inl.inl
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
⊢ toZ i0 i ≤ toZ i0 j

tactic: by_contra h
state_before: case neg.inl.inl
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
⊢ toZ i0 i ≤ toZ i0 j
state_after: case neg.inl.inl
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
⊢ False

tactic: by_cases hm0 : m = 0
state_before: case neg.inl.inl
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
⊢ False
state_after: case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : m = 0
⊢ False

case neg
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ False

tactic: refine hi_max (max_of_succ_le (le_trans ?_ (@le_of_toZ_le _ _ _ _ _ i0 j i ?_)))
state_before: case neg
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ False
state_after: case neg.refine_1
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ succ i ≤ j

case neg.refine_2
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ toZ i0 j ≤ toZ i0 i

tactic: rw [← hm, add_comm]
state_before: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
⊢ j = succ^[(toZ i0 i).toNat + m] i0
state_after: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
⊢ succ^[m] i = succ^[m + (toZ i0 i).toNat] i0

tactic: nth_rw 1 [← iterate_succ_toZ i hi]
state_before: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
⊢ succ^[m] i = succ^[m + (toZ i0 i).toNat] i0
state_after: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
⊢ succ^[m] (succ^[(toZ i0 i).toNat] i0) = succ^[m + (toZ i0 i).toNat] i0

tactic: rw [Function.iterate_add]
state_before: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
⊢ succ^[m] (succ^[(toZ i0 i).toNat] i0) = succ^[m + (toZ i0 i).toNat] i0
state_after: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
⊢ succ^[m] (succ^[(toZ i0 i).toNat] i0) = (succ^[m] ∘ succ^[(toZ i0 i).toNat]) i0

tactic: rfl
state_before: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
⊢ succ^[m] (succ^[(toZ i0 i).toNat] i0) = (succ^[m] ∘ succ^[(toZ i0 i).toNat]) i0
state_after: no goals

tactic: rw [hm0, Function.iterate_zero, id] at hm
state_before: case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : m = 0
⊢ False
state_after: case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : m = 0
⊢ False

tactic: rw [hm] at h
state_before: case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : m = 0
⊢ False
state_after: case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
h : ¬toZ i0 j ≤ toZ i0 j
hm0 : m = 0
⊢ False

tactic: exact h (le_of_eq rfl)
state_before: case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
h : ¬toZ i0 j ≤ toZ i0 j
hm0 : m = 0
⊢ False
state_after: no goals

tactic: have h_succ_le : succ^[(toZ i0 i).toNat + 1] i0 ≤ j := by
  rw [hj_eq]
  refine Monotone.monotone_iterate_of_le_map succ_mono (le_succ i0) (add_le_add_left ?_ _)
  exact Nat.one_le_iff_ne_zero.mpr hm0
state_before: case neg.refine_1
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ succ i ≤ j
state_after: case neg.refine_1
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
h_succ_le : succ^[(toZ i0 i).toNat + 1] i0 ≤ j
⊢ succ i ≤ j

tactic: rwa [Function.iterate_succ', Function.comp_apply, iterate_succ_toZ i hi] at h_succ_le
state_before: case neg.refine_1
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
h_succ_le : succ^[(toZ i0 i).toNat + 1] i0 ≤ j
⊢ succ i ≤ j
state_after: no goals

tactic: rw [hj_eq]
state_before: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ succ^[(toZ i0 i).toNat + 1] i0 ≤ j
state_after: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ succ^[(toZ i0 i).toNat + 1] i0 ≤ succ^[(toZ i0 i).toNat + m] i0

tactic: refine Monotone.monotone_iterate_of_le_map succ_mono (le_succ i0) (add_le_add_left ?_ _)
state_before: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ succ^[(toZ i0 i).toNat + 1] i0 ≤ succ^[(toZ i0 i).toNat + m] i0
state_after: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ 1 ≤ m

tactic: exact Nat.one_le_iff_ne_zero.mpr hm0
state_before: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ 1 ≤ m
state_after: no goals

tactic: exact le_of_not_le h
state_before: case neg.refine_2
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : i0 ≤ j
m : ℕ := Nat.find ⋯
hm : succ^[m] i = j
hj_eq : j = succ^[(toZ i0 i).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ toZ i0 j ≤ toZ i0 i
state_after: no goals

tactic: exact absurd h_le (not_le.mpr (hj.trans_le hi))
state_before: case neg.inl.inr
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i0 ≤ i
hj : j < i0
⊢ toZ i0 i ≤ toZ i0 j
state_after: no goals

tactic: exact (toZ_neg hi).le.trans (toZ_nonneg hj)
state_before: case neg.inr.inl
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : i0 ≤ j
⊢ toZ i0 i ≤ toZ i0 j
state_after: no goals

tactic: let m := Nat.find (exists_pred_iterate_of_le h_le)
state_before: case neg.inr.inr
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
⊢ toZ i0 i ≤ toZ i0 j
state_after: case neg.inr.inr
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
⊢ toZ i0 i ≤ toZ i0 j

tactic: have hm : pred^[m] j = i := Nat.find_spec (exists_pred_iterate_of_le h_le)
state_before: case neg.inr.inr
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
⊢ toZ i0 i ≤ toZ i0 j
state_after: case neg.inr.inr
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
⊢ toZ i0 i ≤ toZ i0 j

tactic: have hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0 := by
  rw [← hm, add_comm]
  nth_rw 1 [← iterate_pred_toZ j hj]
  rw [Function.iterate_add]
  rfl
state_before: case neg.inr.inr
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
⊢ toZ i0 i ≤ toZ i0 j
state_after: case neg.inr.inr
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
⊢ toZ i0 i ≤ toZ i0 j

tactic: by_contra h
state_before: case neg.inr.inr
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
⊢ toZ i0 i ≤ toZ i0 j
state_after: case neg.inr.inr
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
⊢ False

tactic: by_cases hm0 : m = 0
state_before: case neg.inr.inr
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
⊢ False
state_after: case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : m = 0
⊢ False

case neg
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ False

tactic: refine hj_min (min_of_le_pred ?_)
state_before: case neg
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ False
state_after: case neg
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ j ≤ pred j

tactic: refine (@le_of_toZ_le _ _ _ _ _ i0 j i ?_).trans ?_
state_before: case neg
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ j ≤ pred j
state_after: case neg.refine_1
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ toZ i0 j ≤ toZ i0 i

case neg.refine_2
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ i ≤ pred j

tactic: rw [← hm, add_comm]
state_before: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
⊢ i = pred^[(-toZ i0 j).toNat + m] i0
state_after: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
⊢ pred^[m] j = pred^[m + (-toZ i0 j).toNat] i0

tactic: nth_rw 1 [← iterate_pred_toZ j hj]
state_before: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
⊢ pred^[m] j = pred^[m + (-toZ i0 j).toNat] i0
state_after: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
⊢ pred^[m] (pred^[(-toZ i0 j).toNat] i0) = pred^[m + (-toZ i0 j).toNat] i0

tactic: rw [Function.iterate_add]
state_before: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
⊢ pred^[m] (pred^[(-toZ i0 j).toNat] i0) = pred^[m + (-toZ i0 j).toNat] i0
state_after: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
⊢ pred^[m] (pred^[(-toZ i0 j).toNat] i0) = (pred^[m] ∘ pred^[(-toZ i0 j).toNat]) i0

tactic: rfl
state_before: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
⊢ pred^[m] (pred^[(-toZ i0 j).toNat] i0) = (pred^[m] ∘ pred^[(-toZ i0 j).toNat]) i0
state_after: no goals

tactic: rw [hm0, Function.iterate_zero, id] at hm
state_before: case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : m = 0
⊢ False
state_after: case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : m = 0
⊢ False

tactic: rw [hm] at h
state_before: case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : m = 0
⊢ False
state_after: case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 i
hm0 : m = 0
⊢ False

tactic: exact h (le_of_eq rfl)
state_before: case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 i
hm0 : m = 0
⊢ False
state_after: no goals

tactic: exact le_of_not_le h
state_before: case neg.refine_1
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ toZ i0 j ≤ toZ i0 i
state_after: no goals

tactic: have h_le_pred : i ≤ pred^[(-toZ i0 j).toNat + 1] i0 := by
  rw [hj_eq]
  refine Monotone.antitone_iterate_of_map_le pred_mono (pred_le i0) (add_le_add_left ?_ _)
  exact Nat.one_le_iff_ne_zero.mpr hm0
state_before: case neg.refine_2
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ i ≤ pred j
state_after: case neg.refine_2
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
h_le_pred : i ≤ pred^[(-toZ i0 j).toNat + 1] i0
⊢ i ≤ pred j

tactic: rwa [Function.iterate_succ', Function.comp_apply, iterate_pred_toZ j hj] at h_le_pred
state_before: case neg.refine_2
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
h_le_pred : i ≤ pred^[(-toZ i0 j).toNat + 1] i0
⊢ i ≤ pred j
state_after: no goals

tactic: rw [hj_eq]
state_before: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ i ≤ pred^[(-toZ i0 j).toNat + 1] i0
state_after: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ pred^[(-toZ i0 j).toNat + m] i0 ≤ pred^[(-toZ i0 j).toNat + 1] i0

tactic: refine Monotone.antitone_iterate_of_map_le pred_mono (pred_le i0) (add_le_add_left ?_ _)
state_before: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ pred^[(-toZ i0 j).toNat + m] i0 ≤ pred^[(-toZ i0 j).toNat + 1] i0
state_after: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ 1 ≤ m

tactic: exact Nat.one_le_iff_ne_zero.mpr hm0
state_before: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i✝ i j : ι
h_le : i ≤ j
hi_max : ¬IsMax i
hj_min : ¬IsMin j
hi : i < i0
hj : j < i0
m : ℕ := Nat.find ⋯
hm : pred^[m] j = i
hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0
h : ¬toZ i0 i ≤ toZ i0 j
hm0 : ¬m = 0
⊢ 1 ≤ m
state_after: no goals