l3lab/ntp-mathlib huggingface dataset:
instance : LocallyFiniteOrder ℕ where
  finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
  finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
  finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
  finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
  finset_mem_Icc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le (Nat.lt_succ_of_le h).le, Nat.lt_succ_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 (succ_le_of_lt h), add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2)
  finset_mem_Ico a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le h]
    

Proof state:
case inr
a b x : ℕ
h : b < a
⊢ a ≤ x ∧ x < a + (b - a) ↔ a ≤ x ∧ x < b

Next tactic:
| inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2.le)
instance : LocallyFiniteOrder ℕ where
  finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
  finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
  finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
  finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
  finset_mem_Icc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le (Nat.lt_succ_of_le h).le, Nat.lt_succ_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 (succ_le_of_lt h), add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2)
  finset_mem_Ico a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le h]
    | inr h =>
      

Proof state:
case inr
a b x : ℕ
h : b < a
⊢ a ≤ x ∧ x < a + (b - a) ↔ a ≤ x ∧ x < b

Next tactic:
rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
instance : LocallyFiniteOrder ℕ where
  finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
  finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
  finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
  finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
  finset_mem_Icc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le (Nat.lt_succ_of_le h).le, Nat.lt_succ_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 (succ_le_of_lt h), add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2)
  finset_mem_Ico a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le h]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      

Proof state:
case inr
a b x : ℕ
h : b < a
⊢ a ≤ x ∧ x < a ↔ a ≤ x ∧ x < b

Next tactic:
exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2.le)
instance : LocallyFiniteOrder ℕ where
  finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
  finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
  finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
  finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
  finset_mem_Icc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le (Nat.lt_succ_of_le h).le, Nat.lt_succ_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 (succ_le_of_lt h), add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2)
  finset_mem_Ico a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le h]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2.le)
  finset_mem_Ioc a b x := by
    

Proof state:
a b x : ℕ
⊢ x ∈
      (fun a b => { val := ↑(List.range' (a + 1) (b - a)), nodup := (_ : List.Nodup (List.range' (a + 1) (b - a))) }) a
        b ↔
    a < x ∧ x ≤ b

Next tactic:
rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
instance : LocallyFiniteOrder ℕ where
  finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
  finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
  finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
  finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
  finset_mem_Icc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le (Nat.lt_succ_of_le h).le, Nat.lt_succ_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 (succ_le_of_lt h), add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2)
  finset_mem_Ico a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le h]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2.le)
  finset_mem_Ioc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    

Proof state:
a b x : ℕ
⊢ a + 1 ≤ x ∧ x < a + 1 + (b - a) ↔ a < x ∧ x ≤ b

Next tactic:
cases le_or_lt a b with
    | inl h =>
      rw [← succ_sub_succ, add_tsub_cancel_of_le (succ_le_succ h), Nat.lt_succ_iff, Nat.succ_le_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.le.trans hx.2)
instance : LocallyFiniteOrder ℕ where
  finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
  finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
  finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
  finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
  finset_mem_Icc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le (Nat.lt_succ_of_le h).le, Nat.lt_succ_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 (succ_le_of_lt h), add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2)
  finset_mem_Ico a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le h]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2.le)
  finset_mem_Ioc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    

Proof state:
a b x : ℕ
x✝ : a ≤ b ∨ b < a
⊢ a + 1 ≤ x ∧ x < a + 1 + (b - a) ↔ a < x ∧ x ≤ b

Next tactic:
cases le_or_lt a b with
    | inl h =>
      rw [← succ_sub_succ, add_tsub_cancel_of_le (succ_le_succ h), Nat.lt_succ_iff, Nat.succ_le_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.le.trans hx.2)
instance : LocallyFiniteOrder ℕ where
  finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
  finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
  finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
  finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
  finset_mem_Icc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le (Nat.lt_succ_of_le h).le, Nat.lt_succ_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 (succ_le_of_lt h), add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2)
  finset_mem_Ico a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le h]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2.le)
  finset_mem_Ioc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    

Proof state:
case inl
a b x : ℕ
h : a ≤ b
⊢ a + 1 ≤ x ∧ x < a + 1 + (b - a) ↔ a < x ∧ x ≤ b

Next tactic:
| inl h =>
      rw [← succ_sub_succ, add_tsub_cancel_of_le (succ_le_succ h), Nat.lt_succ_iff, Nat.succ_le_iff]
instance : LocallyFiniteOrder ℕ where
  finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
  finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
  finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
  finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
  finset_mem_Icc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le (Nat.lt_succ_of_le h).le, Nat.lt_succ_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 (succ_le_of_lt h), add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2)
  finset_mem_Ico a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le h]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2.le)
  finset_mem_Ioc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h =>
      

Proof state:
case inl
a b x : ℕ
h : a ≤ b
⊢ a + 1 ≤ x ∧ x < a + 1 + (b - a) ↔ a < x ∧ x ≤ b

Next tactic:
rw [← succ_sub_succ, add_tsub_cancel_of_le (succ_le_succ h), Nat.lt_succ_iff, Nat.succ_le_iff]
instance : LocallyFiniteOrder ℕ where
  finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
  finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
  finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
  finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
  finset_mem_Icc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le (Nat.lt_succ_of_le h).le, Nat.lt_succ_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 (succ_le_of_lt h), add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2)
  finset_mem_Ico a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le h]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2.le)
  finset_mem_Ioc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h =>
      rw [← succ_sub_succ, add_tsub_cancel_of_le (succ_le_succ h), Nat.lt_succ_iff, Nat.succ_le_iff]
    

Proof state:
case inr
a b x : ℕ
h : b < a
⊢ a + 1 ≤ x ∧ x < a + 1 + (b - a) ↔ a < x ∧ x ≤ b

Next tactic:
| inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.le.trans hx.2)
instance : LocallyFiniteOrder ℕ where
  finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
  finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
  finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
  finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
  finset_mem_Icc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le (Nat.lt_succ_of_le h).le, Nat.lt_succ_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 (succ_le_of_lt h), add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2)
  finset_mem_Ico a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le h]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2.le)
  finset_mem_Ioc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h =>
      rw [← succ_sub_succ, add_tsub_cancel_of_le (succ_le_succ h), Nat.lt_succ_iff, Nat.succ_le_iff]
    | inr h =>
      

Proof state:
case inr
a b x : ℕ
h : b < a
⊢ a + 1 ≤ x ∧ x < a + 1 + (b - a) ↔ a < x ∧ x ≤ b

Next tactic:
rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
instance : LocallyFiniteOrder ℕ where
  finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
  finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
  finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
  finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
  finset_mem_Icc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le (Nat.lt_succ_of_le h).le, Nat.lt_succ_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 (succ_le_of_lt h), add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2)
  finset_mem_Ico a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le h]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2.le)
  finset_mem_Ioc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h =>
      rw [← succ_sub_succ, add_tsub_cancel_of_le (succ_le_succ h), Nat.lt_succ_iff, Nat.succ_le_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      

Proof state:
case inr
a b x : ℕ
h : b < a
⊢ a + 1 ≤ x ∧ x < a + 1 ↔ a < x ∧ x ≤ b

Next tactic:
exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.le.trans hx.2)
instance : LocallyFiniteOrder ℕ where
  finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
  finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
  finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
  finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
  finset_mem_Icc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le (Nat.lt_succ_of_le h).le, Nat.lt_succ_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 (succ_le_of_lt h), add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2)
  finset_mem_Ico a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le h]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2.le)
  finset_mem_Ioc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h =>
      rw [← succ_sub_succ, add_tsub_cancel_of_le (succ_le_succ h), Nat.lt_succ_iff, Nat.succ_le_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.le.trans hx.2)
  finset_mem_Ioo a b x := by
    

Proof state:
a b x : ℕ
⊢ x ∈
      (fun a b =>
          { val := ↑(List.range' (a + 1) (b - a - 1)), nodup := (_ : List.Nodup (List.range' (a + 1) (b - a - 1))) })
        a b ↔
    a < x ∧ x < b

Next tactic:
rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1, ← tsub_add_eq_tsub_tsub]
instance : LocallyFiniteOrder ℕ where
  finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
  finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
  finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
  finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
  finset_mem_Icc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le (Nat.lt_succ_of_le h).le, Nat.lt_succ_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 (succ_le_of_lt h), add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2)
  finset_mem_Ico a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le h]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2.le)
  finset_mem_Ioc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h =>
      rw [← succ_sub_succ, add_tsub_cancel_of_le (succ_le_succ h), Nat.lt_succ_iff, Nat.succ_le_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.le.trans hx.2)
  finset_mem_Ioo a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1, ← tsub_add_eq_tsub_tsub]
    

Proof state:
a b x : ℕ
⊢ a + 1 ≤ x ∧ x < a + 1 + (b - (a + 1)) ↔ a < x ∧ x < b

Next tactic:
cases le_or_lt (a + 1) b with
    | inl h => rw [add_tsub_cancel_of_le h, Nat.succ_le_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2)
instance : LocallyFiniteOrder ℕ where
  finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
  finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
  finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
  finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
  finset_mem_Icc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le (Nat.lt_succ_of_le h).le, Nat.lt_succ_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 (succ_le_of_lt h), add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2)
  finset_mem_Ico a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le h]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2.le)
  finset_mem_Ioc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h =>
      rw [← succ_sub_succ, add_tsub_cancel_of_le (succ_le_succ h), Nat.lt_succ_iff, Nat.succ_le_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.le.trans hx.2)
  finset_mem_Ioo a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1, ← tsub_add_eq_tsub_tsub]
    

Proof state:
a b x : ℕ
x✝ : a + 1 ≤ b ∨ b < a + 1
⊢ a + 1 ≤ x ∧ x < a + 1 + (b - (a + 1)) ↔ a < x ∧ x < b

Next tactic:
cases le_or_lt (a + 1) b with
    | inl h => rw [add_tsub_cancel_of_le h, Nat.succ_le_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2)
instance : LocallyFiniteOrder ℕ where
  finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
  finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
  finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
  finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
  finset_mem_Icc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le (Nat.lt_succ_of_le h).le, Nat.lt_succ_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 (succ_le_of_lt h), add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2)
  finset_mem_Ico a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le h]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2.le)
  finset_mem_Ioc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h =>
      rw [← succ_sub_succ, add_tsub_cancel_of_le (succ_le_succ h), Nat.lt_succ_iff, Nat.succ_le_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.le.trans hx.2)
  finset_mem_Ioo a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1, ← tsub_add_eq_tsub_tsub]
    cases le_or_lt (a + 1) b with
    

Proof state:
case inl
a b x : ℕ
h : a + 1 ≤ b
⊢ a + 1 ≤ x ∧ x < a + 1 + (b - (a + 1)) ↔ a < x ∧ x < b

Next tactic:
| inl h => rw [add_tsub_cancel_of_le h, Nat.succ_le_iff]
instance : LocallyFiniteOrder ℕ where
  finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
  finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
  finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
  finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
  finset_mem_Icc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le (Nat.lt_succ_of_le h).le, Nat.lt_succ_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 (succ_le_of_lt h), add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2)
  finset_mem_Ico a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h => rw [add_tsub_cancel_of_le h]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.trans hx.2.le)
  finset_mem_Ioc a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]
    cases le_or_lt a b with
    | inl h =>
      rw [← succ_sub_succ, add_tsub_cancel_of_le (succ_le_succ h), Nat.lt_succ_iff, Nat.succ_le_iff]
    | inr h =>
      rw [tsub_eq_zero_iff_le.2 h.le, add_zero]
      exact iff_of_false (fun hx => hx.2.not_le hx.1) fun hx => h.not_le (hx.1.le.trans hx.2)
  finset_mem_Ioo a b x := by
    rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1, ← tsub_add_eq_tsub_tsub]
    cases le_or_lt (a + 1) b with
    | inl h => 

Proof state:
case inl
a b x : ℕ
h : a + 1 ≤ b
⊢ a + 1 ≤ x ∧ x < a + 1 + (b - (a + 1)) ↔ a < x ∧ x < b

Next tactic:
rw [add_tsub_cancel_of_le h, Nat.succ_le_iff]