l3lab/ntp-mathlib huggingface dataset:
instance : LocallyFiniteOrder ℤ
    where
  finsetIcc a b :=
    (Finset.range (b + 1 - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding a
  finsetIco a b := (Finset.range (b - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding a
  finsetIoc a b :=
    (Finset.range (b - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)
  finsetIoo a b :=
    (Finset.range (b - a - 1).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)
  finset_mem_Icc a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [lt_sub_iff_add_lt, Int.lt_add_one_iff, add_comm] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - a).toNat
      rw [← lt_add_one_iff] at hb
      rw [toNat_sub_of_le ha]
      exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ico a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      exact ⟨Int.le.intro a rfl, lt_sub_iff_add_lt'.mp h⟩
    · rintro ⟨ha, hb⟩
      use (x - a).toNat
      rw [toNat_sub_of_le ha]
      exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ioc a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [← add_one_le_iff, le_sub_iff_add_le', add_comm _ (1 : ℤ), ← add_assoc] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - (a + 1)).toNat
      

Proof state:
case h
a b x : ℤ
ha : a < x
hb : x ≤ b
⊢ ↑(toNat (x - (a + 1))) < b - a ∧ a + 1 + ↑(toNat (x - (a + 1))) = x

Next tactic:
rw [toNat_sub_of_le ha, ← add_one_le_iff, sub_add, add_sub_cancel]
instance : LocallyFiniteOrder ℤ
    where
  finsetIcc a b :=
    (Finset.range (b + 1 - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding a
  finsetIco a b := (Finset.range (b - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding a
  finsetIoc a b :=
    (Finset.range (b - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)
  finsetIoo a b :=
    (Finset.range (b - a - 1).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)
  finset_mem_Icc a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [lt_sub_iff_add_lt, Int.lt_add_one_iff, add_comm] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - a).toNat
      rw [← lt_add_one_iff] at hb
      rw [toNat_sub_of_le ha]
      exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ico a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      exact ⟨Int.le.intro a rfl, lt_sub_iff_add_lt'.mp h⟩
    · rintro ⟨ha, hb⟩
      use (x - a).toNat
      rw [toNat_sub_of_le ha]
      exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ioc a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [← add_one_le_iff, le_sub_iff_add_le', add_comm _ (1 : ℤ), ← add_assoc] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - (a + 1)).toNat
      rw [toNat_sub_of_le ha, ← add_one_le_iff, sub_add, add_sub_cancel]
      

Proof state:
case h
a b x : ℤ
ha : a < x
hb : x ≤ b
⊢ x - a ≤ b - a ∧ a + 1 + (x - (a + 1)) = x

Next tactic:
exact ⟨sub_le_sub_right hb _, add_sub_cancel'_right _ _⟩
instance : LocallyFiniteOrder ℤ
    where
  finsetIcc a b :=
    (Finset.range (b + 1 - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding a
  finsetIco a b := (Finset.range (b - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding a
  finsetIoc a b :=
    (Finset.range (b - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)
  finsetIoo a b :=
    (Finset.range (b - a - 1).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)
  finset_mem_Icc a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [lt_sub_iff_add_lt, Int.lt_add_one_iff, add_comm] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - a).toNat
      rw [← lt_add_one_iff] at hb
      rw [toNat_sub_of_le ha]
      exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ico a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      exact ⟨Int.le.intro a rfl, lt_sub_iff_add_lt'.mp h⟩
    · rintro ⟨ha, hb⟩
      use (x - a).toNat
      rw [toNat_sub_of_le ha]
      exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ioc a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [← add_one_le_iff, le_sub_iff_add_le', add_comm _ (1 : ℤ), ← add_assoc] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - (a + 1)).toNat
      rw [toNat_sub_of_le ha, ← add_one_le_iff, sub_add, add_sub_cancel]
      exact ⟨sub_le_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ioo a b x := by
    

Proof state:
a b x : ℤ
⊢ x ∈
      (fun a b =>
          map (Function.Embedding.trans Nat.castEmbedding (addLeftEmbedding (a + 1))) (range (toNat (b - a - 1))))
        a b ↔
    a < x ∧ x < b

Next tactic:
simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
instance : LocallyFiniteOrder ℤ
    where
  finsetIcc a b :=
    (Finset.range (b + 1 - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding a
  finsetIco a b := (Finset.range (b - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding a
  finsetIoc a b :=
    (Finset.range (b - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)
  finsetIoo a b :=
    (Finset.range (b - a - 1).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)
  finset_mem_Icc a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [lt_sub_iff_add_lt, Int.lt_add_one_iff, add_comm] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - a).toNat
      rw [← lt_add_one_iff] at hb
      rw [toNat_sub_of_le ha]
      exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ico a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      exact ⟨Int.le.intro a rfl, lt_sub_iff_add_lt'.mp h⟩
    · rintro ⟨ha, hb⟩
      use (x - a).toNat
      rw [toNat_sub_of_le ha]
      exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ioc a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [← add_one_le_iff, le_sub_iff_add_le', add_comm _ (1 : ℤ), ← add_assoc] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - (a + 1)).toNat
      rw [toNat_sub_of_le ha, ← add_one_le_iff, sub_add, add_sub_cancel]
      exact ⟨sub_le_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ioo a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    

Proof state:
a b x : ℤ
⊢ (∃ a_1, ↑a_1 < b - a - 1 ∧ a + 1 + ↑a_1 = x) ↔ a < x ∧ x < b

Next tactic:
constructor
instance : LocallyFiniteOrder ℤ
    where
  finsetIcc a b :=
    (Finset.range (b + 1 - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding a
  finsetIco a b := (Finset.range (b - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding a
  finsetIoc a b :=
    (Finset.range (b - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)
  finsetIoo a b :=
    (Finset.range (b - a - 1).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)
  finset_mem_Icc a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [lt_sub_iff_add_lt, Int.lt_add_one_iff, add_comm] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - a).toNat
      rw [← lt_add_one_iff] at hb
      rw [toNat_sub_of_le ha]
      exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ico a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      exact ⟨Int.le.intro a rfl, lt_sub_iff_add_lt'.mp h⟩
    · rintro ⟨ha, hb⟩
      use (x - a).toNat
      rw [toNat_sub_of_le ha]
      exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ioc a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [← add_one_le_iff, le_sub_iff_add_le', add_comm _ (1 : ℤ), ← add_assoc] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - (a + 1)).toNat
      rw [toNat_sub_of_le ha, ← add_one_le_iff, sub_add, add_sub_cancel]
      exact ⟨sub_le_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ioo a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · 

Proof state:
case mp
a b x : ℤ
⊢ (∃ a_1, ↑a_1 < b - a - 1 ∧ a + 1 + ↑a_1 = x) → a < x ∧ x < b

Next tactic:
rintro ⟨a, h, rfl⟩
instance : LocallyFiniteOrder ℤ
    where
  finsetIcc a b :=
    (Finset.range (b + 1 - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding a
  finsetIco a b := (Finset.range (b - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding a
  finsetIoc a b :=
    (Finset.range (b - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)
  finsetIoo a b :=
    (Finset.range (b - a - 1).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)
  finset_mem_Icc a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [lt_sub_iff_add_lt, Int.lt_add_one_iff, add_comm] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - a).toNat
      rw [← lt_add_one_iff] at hb
      rw [toNat_sub_of_le ha]
      exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ico a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      exact ⟨Int.le.intro a rfl, lt_sub_iff_add_lt'.mp h⟩
    · rintro ⟨ha, hb⟩
      use (x - a).toNat
      rw [toNat_sub_of_le ha]
      exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ioc a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [← add_one_le_iff, le_sub_iff_add_le', add_comm _ (1 : ℤ), ← add_assoc] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - (a + 1)).toNat
      rw [toNat_sub_of_le ha, ← add_one_le_iff, sub_add, add_sub_cancel]
      exact ⟨sub_le_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ioo a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      

Proof state:
case mp.intro.intro
a✝ b : ℤ
a : ℕ
h : ↑a < b - a✝ - 1
⊢ a✝ < a✝ + 1 + ↑a ∧ a✝ + 1 + ↑a < b

Next tactic:
rw [sub_sub, lt_sub_iff_add_lt'] at h
instance : LocallyFiniteOrder ℤ
    where
  finsetIcc a b :=
    (Finset.range (b + 1 - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding a
  finsetIco a b := (Finset.range (b - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding a
  finsetIoc a b :=
    (Finset.range (b - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)
  finsetIoo a b :=
    (Finset.range (b - a - 1).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)
  finset_mem_Icc a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [lt_sub_iff_add_lt, Int.lt_add_one_iff, add_comm] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - a).toNat
      rw [← lt_add_one_iff] at hb
      rw [toNat_sub_of_le ha]
      exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ico a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      exact ⟨Int.le.intro a rfl, lt_sub_iff_add_lt'.mp h⟩
    · rintro ⟨ha, hb⟩
      use (x - a).toNat
      rw [toNat_sub_of_le ha]
      exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ioc a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [← add_one_le_iff, le_sub_iff_add_le', add_comm _ (1 : ℤ), ← add_assoc] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - (a + 1)).toNat
      rw [toNat_sub_of_le ha, ← add_one_le_iff, sub_add, add_sub_cancel]
      exact ⟨sub_le_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ioo a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [sub_sub, lt_sub_iff_add_lt'] at h
      

Proof state:
case mp.intro.intro
a✝ b : ℤ
a : ℕ
h : a✝ + 1 + ↑a < b
⊢ a✝ < a✝ + 1 + ↑a ∧ a✝ + 1 + ↑a < b

Next tactic:
exact ⟨Int.le.intro a rfl, h⟩
instance : LocallyFiniteOrder ℤ
    where
  finsetIcc a b :=
    (Finset.range (b + 1 - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding a
  finsetIco a b := (Finset.range (b - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding a
  finsetIoc a b :=
    (Finset.range (b - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)
  finsetIoo a b :=
    (Finset.range (b - a - 1).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)
  finset_mem_Icc a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [lt_sub_iff_add_lt, Int.lt_add_one_iff, add_comm] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - a).toNat
      rw [← lt_add_one_iff] at hb
      rw [toNat_sub_of_le ha]
      exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ico a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      exact ⟨Int.le.intro a rfl, lt_sub_iff_add_lt'.mp h⟩
    · rintro ⟨ha, hb⟩
      use (x - a).toNat
      rw [toNat_sub_of_le ha]
      exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ioc a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [← add_one_le_iff, le_sub_iff_add_le', add_comm _ (1 : ℤ), ← add_assoc] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - (a + 1)).toNat
      rw [toNat_sub_of_le ha, ← add_one_le_iff, sub_add, add_sub_cancel]
      exact ⟨sub_le_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ioo a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [sub_sub, lt_sub_iff_add_lt'] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · 

Proof state:
case mpr
a b x : ℤ
⊢ a < x ∧ x < b → ∃ a_2, ↑a_2 < b - a - 1 ∧ a + 1 + ↑a_2 = x

Next tactic:
rintro ⟨ha, hb⟩
instance : LocallyFiniteOrder ℤ
    where
  finsetIcc a b :=
    (Finset.range (b + 1 - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding a
  finsetIco a b := (Finset.range (b - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding a
  finsetIoc a b :=
    (Finset.range (b - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)
  finsetIoo a b :=
    (Finset.range (b - a - 1).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)
  finset_mem_Icc a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [lt_sub_iff_add_lt, Int.lt_add_one_iff, add_comm] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - a).toNat
      rw [← lt_add_one_iff] at hb
      rw [toNat_sub_of_le ha]
      exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ico a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      exact ⟨Int.le.intro a rfl, lt_sub_iff_add_lt'.mp h⟩
    · rintro ⟨ha, hb⟩
      use (x - a).toNat
      rw [toNat_sub_of_le ha]
      exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ioc a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [← add_one_le_iff, le_sub_iff_add_le', add_comm _ (1 : ℤ), ← add_assoc] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - (a + 1)).toNat
      rw [toNat_sub_of_le ha, ← add_one_le_iff, sub_add, add_sub_cancel]
      exact ⟨sub_le_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ioo a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [sub_sub, lt_sub_iff_add_lt'] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      

Proof state:
case mpr.intro
a b x : ℤ
ha : a < x
hb : x < b
⊢ ∃ a_1, ↑a_1 < b - a - 1 ∧ a + 1 + ↑a_1 = x

Next tactic:
use (x - (a + 1)).toNat
instance : LocallyFiniteOrder ℤ
    where
  finsetIcc a b :=
    (Finset.range (b + 1 - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding a
  finsetIco a b := (Finset.range (b - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding a
  finsetIoc a b :=
    (Finset.range (b - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)
  finsetIoo a b :=
    (Finset.range (b - a - 1).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)
  finset_mem_Icc a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [lt_sub_iff_add_lt, Int.lt_add_one_iff, add_comm] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - a).toNat
      rw [← lt_add_one_iff] at hb
      rw [toNat_sub_of_le ha]
      exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ico a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      exact ⟨Int.le.intro a rfl, lt_sub_iff_add_lt'.mp h⟩
    · rintro ⟨ha, hb⟩
      use (x - a).toNat
      rw [toNat_sub_of_le ha]
      exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ioc a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [← add_one_le_iff, le_sub_iff_add_le', add_comm _ (1 : ℤ), ← add_assoc] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - (a + 1)).toNat
      rw [toNat_sub_of_le ha, ← add_one_le_iff, sub_add, add_sub_cancel]
      exact ⟨sub_le_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ioo a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [sub_sub, lt_sub_iff_add_lt'] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - (a + 1)).toNat
      

Proof state:
case h
a b x : ℤ
ha : a < x
hb : x < b
⊢ ↑(toNat (x - (a + 1))) < b - a - 1 ∧ a + 1 + ↑(toNat (x - (a + 1))) = x

Next tactic:
rw [toNat_sub_of_le ha, sub_sub]
instance : LocallyFiniteOrder ℤ
    where
  finsetIcc a b :=
    (Finset.range (b + 1 - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding a
  finsetIco a b := (Finset.range (b - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding a
  finsetIoc a b :=
    (Finset.range (b - a).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)
  finsetIoo a b :=
    (Finset.range (b - a - 1).toNat).map <| Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)
  finset_mem_Icc a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [lt_sub_iff_add_lt, Int.lt_add_one_iff, add_comm] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - a).toNat
      rw [← lt_add_one_iff] at hb
      rw [toNat_sub_of_le ha]
      exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ico a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      exact ⟨Int.le.intro a rfl, lt_sub_iff_add_lt'.mp h⟩
    · rintro ⟨ha, hb⟩
      use (x - a).toNat
      rw [toNat_sub_of_le ha]
      exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ioc a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [← add_one_le_iff, le_sub_iff_add_le', add_comm _ (1 : ℤ), ← add_assoc] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - (a + 1)).toNat
      rw [toNat_sub_of_le ha, ← add_one_le_iff, sub_add, add_sub_cancel]
      exact ⟨sub_le_sub_right hb _, add_sub_cancel'_right _ _⟩
  finset_mem_Ioo a b x := by
    simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply,
      Nat.castEmbedding_apply, addLeftEmbedding_apply]
    constructor
    · rintro ⟨a, h, rfl⟩
      rw [sub_sub, lt_sub_iff_add_lt'] at h
      exact ⟨Int.le.intro a rfl, h⟩
    · rintro ⟨ha, hb⟩
      use (x - (a + 1)).toNat
      rw [toNat_sub_of_le ha, sub_sub]
      

Proof state:
case h
a b x : ℤ
ha : a < x
hb : x < b
⊢ x - (a + 1) < b - (a + 1) ∧ a + 1 + (x - (a + 1)) = x

Next tactic:
exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩