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

tactic: cases' n with n n
state_before: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
n : ℕ
hn : ¬IsMin (pred^[n] i0)
⊢ toZ i0 (pred^[n] i0) = -↑n
state_after: case zero
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
hn : ¬IsMin (pred^[0] i0)
⊢ toZ i0 (pred^[0] i0) = -↑0

case succ
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)

tactic: have : pred^[n.succ] i0 < i0 := by
  refine lt_of_le_of_ne (pred_iterate_le _ _) fun h_pred_iterate_eq ↦ hn ?_
  have h_pred_eq_pred : pred^[n.succ] i0 = pred^[0] i0 := by
    rwa [Function.iterate_zero, id]
  exact isMin_iterate_pred_of_eq_of_ne h_pred_eq_pred (Nat.succ_ne_zero n)
state_before: case succ
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)
state_after: case succ
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)

tactic: let m := (-toZ i0 (pred^[n.succ] i0)).toNat
state_before: case succ
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)
state_after: case succ
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
m : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat
⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)

tactic: have h_eq : pred^[m] i0 = pred^[n.succ] i0 := iterate_pred_toZ _ this
state_before: case succ
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
m : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat
⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)
state_after: case succ
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
m : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat
h_eq : pred^[m] i0 = pred^[n.succ] i0
⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)

tactic: by_cases hmn : m = n + 1
state_before: case succ
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
m : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat
h_eq : pred^[m] i0 = pred^[n.succ] i0
⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)
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 : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
m : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat
h_eq : pred^[m] i0 = pred^[n.succ] i0
hmn : m = n + 1
⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)

case neg
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
m : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat
h_eq : pred^[m] i0 = pred^[n.succ] i0
hmn : ¬m = n + 1
⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)

tactic: simp only [Nat.zero_eq, Function.iterate_zero, id, toZ_of_eq, Nat.cast_zero, neg_zero]
state_before: case zero
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
hn : ¬IsMin (pred^[0] i0)
⊢ toZ i0 (pred^[0] i0) = -↑0
state_after: case zero
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
hn : ¬IsMin (pred^[0] i0)
⊢ 0 = -↑0

tactic: rfl
state_before: case zero
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
hn : ¬IsMin (pred^[0] i0)
⊢ 0 = -↑0
state_after: no goals

tactic: refine lt_of_le_of_ne (pred_iterate_le _ _) fun h_pred_iterate_eq ↦ hn ?_
state_before: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
⊢ pred^[n.succ] i0 < i0
state_after: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
h_pred_iterate_eq : pred^[n.succ] i0 = i0
⊢ IsMin (pred^[n + 1] i0)

tactic: have h_pred_eq_pred : pred^[n.succ] i0 = pred^[0] i0 := by
  rwa [Function.iterate_zero, id]
state_before: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
h_pred_iterate_eq : pred^[n.succ] i0 = i0
⊢ IsMin (pred^[n + 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 : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
h_pred_iterate_eq : pred^[n.succ] i0 = i0
h_pred_eq_pred : pred^[n.succ] i0 = pred^[0] i0
⊢ IsMin (pred^[n + 1] i0)

tactic: exact isMin_iterate_pred_of_eq_of_ne h_pred_eq_pred (Nat.succ_ne_zero n)
state_before: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
h_pred_iterate_eq : pred^[n.succ] i0 = i0
h_pred_eq_pred : pred^[n.succ] i0 = pred^[0] i0
⊢ IsMin (pred^[n + 1] i0)
state_after: no goals

tactic: rwa [Function.iterate_zero, id]
state_before: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : IsSuccArchimedean ι
inst✝ : PredOrder ι
i0 i : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
h_pred_iterate_eq : pred^[n.succ] i0 = i0
⊢ pred^[n.succ] i0 = pred^[0] i0
state_after: no goals

tactic: nth_rw 2 [← hmn]
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 : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
m : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat
h_eq : pred^[m] i0 = pred^[n.succ] i0
hmn : m = n + 1
⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)
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 : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
m : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat
h_eq : pred^[m] i0 = pred^[n.succ] i0
hmn : m = n + 1
⊢ toZ i0 (pred^[n + 1] i0) = -↑m

tactic: rw [Int.toNat_eq_max, toZ_of_lt this, max_eq_left, neg_neg]
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 : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
m : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat
h_eq : pred^[m] i0 = pred^[n.succ] i0
hmn : m = n + 1
⊢ toZ i0 (pred^[n + 1] i0) = -↑m
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 : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
m : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat
h_eq : pred^[m] i0 = pred^[n.succ] i0
hmn : m = n + 1
⊢ 0 ≤ - -↑(Nat.find ⋯)

tactic: rw [neg_neg]
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 : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
m : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat
h_eq : pred^[m] i0 = pred^[n.succ] i0
hmn : m = n + 1
⊢ 0 ≤ - -↑(Nat.find ⋯)
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 : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
m : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat
h_eq : pred^[m] i0 = pred^[n.succ] i0
hmn : m = n + 1
⊢ 0 ≤ ↑(Nat.find ⋯)

tactic: exact Int.natCast_nonneg _
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 : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
m : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat
h_eq : pred^[m] i0 = pred^[n.succ] i0
hmn : m = n + 1
⊢ 0 ≤ ↑(Nat.find ⋯)
state_after: no goals

tactic: suffices IsMin (pred^[n.succ] i0) from absurd this hn
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 : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
m : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat
h_eq : pred^[m] i0 = pred^[n.succ] i0
hmn : ¬m = n + 1
⊢ toZ i0 (pred^[n + 1] i0) = -↑(n + 1)
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 : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
m : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat
h_eq : pred^[m] i0 = pred^[n.succ] i0
hmn : ¬m = n + 1
⊢ IsMin (pred^[n.succ] i0)

tactic: exact isMin_iterate_pred_of_eq_of_ne h_eq.symm (Ne.symm hmn)
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 : ι
n : ℕ
hn : ¬IsMin (pred^[n + 1] i0)
this : pred^[n.succ] i0 < i0
m : ℕ := (-toZ i0 (pred^[n.succ] i0)).toNat
h_eq : pred^[m] i0 = pred^[n.succ] i0
hmn : ¬m = n + 1
⊢ IsMin (pred^[n.succ] i0)
state_after: no goals