l3lab/ntp-mathlib huggingface dataset:
instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFiniteOrder ι] :
    IsSuccArchimedean ι where
  exists_succ_iterate_of_le := by
    intro i j hij
    rw [le_iff_lt_or_eq] at hij
    cases' hij with hij hij
    swap
    · refine' ⟨0, _⟩
      

Proof state:
case inr
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i j : ι
hij : i = j
⊢ succ^[0] i = j

Next tactic:
simpa only [Function.iterate_zero, id.def] using hij
instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFiniteOrder ι] :
    IsSuccArchimedean ι where
  exists_succ_iterate_of_le := by
    intro i j hij
    rw [le_iff_lt_or_eq] at hij
    cases' hij with hij hij
    swap
    · refine' ⟨0, _⟩
      simpa only [Function.iterate_zero, id.def] using hij
    

Proof state:
case inl
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i j : ι
hij : i < j
⊢ ∃ n, succ^[n] i = j

Next tactic:
by_contra! h
instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFiniteOrder ι] :
    IsSuccArchimedean ι where
  exists_succ_iterate_of_le := by
    intro i j hij
    rw [le_iff_lt_or_eq] at hij
    cases' hij with hij hij
    swap
    · refine' ⟨0, _⟩
      simpa only [Function.iterate_zero, id.def] using hij
    by_contra! h
    

Proof state:
case inl
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
⊢ False

Next tactic:
have h_lt : ∀ n, succ^[n] i < j := by
      intro n
      induction' n with n hn
      · simpa only [Function.iterate_zero, id.def] using hij
      · refine' lt_of_le_of_ne _ (h _)
        rw [Function.iterate_succ', Function.comp_apply]
        exact succ_le_of_lt hn
instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFiniteOrder ι] :
    IsSuccArchimedean ι where
  exists_succ_iterate_of_le := by
    intro i j hij
    rw [le_iff_lt_or_eq] at hij
    cases' hij with hij hij
    swap
    · refine' ⟨0, _⟩
      simpa only [Function.iterate_zero, id.def] using hij
    by_contra! h
    have h_lt : ∀ n, succ^[n] i < j := by
      

Proof state:
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
⊢ ∀ (n : ℕ), succ^[n] i < j

Next tactic:
intro n
instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFiniteOrder ι] :
    IsSuccArchimedean ι where
  exists_succ_iterate_of_le := by
    intro i j hij
    rw [le_iff_lt_or_eq] at hij
    cases' hij with hij hij
    swap
    · refine' ⟨0, _⟩
      simpa only [Function.iterate_zero, id.def] using hij
    by_contra! h
    have h_lt : ∀ n, succ^[n] i < j := by
      intro n
      

Proof state:
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
n : ℕ
⊢ succ^[n] i < j

Next tactic:
induction' n with n hn
instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFiniteOrder ι] :
    IsSuccArchimedean ι where
  exists_succ_iterate_of_le := by
    intro i j hij
    rw [le_iff_lt_or_eq] at hij
    cases' hij with hij hij
    swap
    · refine' ⟨0, _⟩
      simpa only [Function.iterate_zero, id.def] using hij
    by_contra! h
    have h_lt : ∀ n, succ^[n] i < j := by
      intro n
      induction' n with n hn
      · 

Proof state:
case zero
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
⊢ succ^[Nat.zero] i < j

Next tactic:
simpa only [Function.iterate_zero, id.def] using hij
instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFiniteOrder ι] :
    IsSuccArchimedean ι where
  exists_succ_iterate_of_le := by
    intro i j hij
    rw [le_iff_lt_or_eq] at hij
    cases' hij with hij hij
    swap
    · refine' ⟨0, _⟩
      simpa only [Function.iterate_zero, id.def] using hij
    by_contra! h
    have h_lt : ∀ n, succ^[n] i < j := by
      intro n
      induction' n with n hn
      · simpa only [Function.iterate_zero, id.def] using hij
      · 

Proof state:
case succ
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
n : ℕ
hn : succ^[n] i < j
⊢ succ^[Nat.succ n] i < j

Next tactic:
refine' lt_of_le_of_ne _ (h _)
instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFiniteOrder ι] :
    IsSuccArchimedean ι where
  exists_succ_iterate_of_le := by
    intro i j hij
    rw [le_iff_lt_or_eq] at hij
    cases' hij with hij hij
    swap
    · refine' ⟨0, _⟩
      simpa only [Function.iterate_zero, id.def] using hij
    by_contra! h
    have h_lt : ∀ n, succ^[n] i < j := by
      intro n
      induction' n with n hn
      · simpa only [Function.iterate_zero, id.def] using hij
      · refine' lt_of_le_of_ne _ (h _)
        

Proof state:
case succ
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
n : ℕ
hn : succ^[n] i < j
⊢ succ^[Nat.succ n] i ≤ j

Next tactic:
rw [Function.iterate_succ', Function.comp_apply]
instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFiniteOrder ι] :
    IsSuccArchimedean ι where
  exists_succ_iterate_of_le := by
    intro i j hij
    rw [le_iff_lt_or_eq] at hij
    cases' hij with hij hij
    swap
    · refine' ⟨0, _⟩
      simpa only [Function.iterate_zero, id.def] using hij
    by_contra! h
    have h_lt : ∀ n, succ^[n] i < j := by
      intro n
      induction' n with n hn
      · simpa only [Function.iterate_zero, id.def] using hij
      · refine' lt_of_le_of_ne _ (h _)
        rw [Function.iterate_succ', Function.comp_apply]
        

Proof state:
case succ
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
n : ℕ
hn : succ^[n] i < j
⊢ succ (succ^[n] i) ≤ j

Next tactic:
exact succ_le_of_lt hn
instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFiniteOrder ι] :
    IsSuccArchimedean ι where
  exists_succ_iterate_of_le := by
    intro i j hij
    rw [le_iff_lt_or_eq] at hij
    cases' hij with hij hij
    swap
    · refine' ⟨0, _⟩
      simpa only [Function.iterate_zero, id.def] using hij
    by_contra! h
    have h_lt : ∀ n, succ^[n] i < j := by
      intro n
      induction' n with n hn
      · simpa only [Function.iterate_zero, id.def] using hij
      · refine' lt_of_le_of_ne _ (h _)
        rw [Function.iterate_succ', Function.comp_apply]
        exact succ_le_of_lt hn
    

Proof state:
case inl
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
h_lt : ∀ (n : ℕ), succ^[n] i < j
⊢ False

Next tactic:
have h_mem : ∀ n, succ^[n] i ∈ Finset.Icc i j :=
      fun n ↦ Finset.mem_Icc.mpr ⟨le_succ_iterate n i, (h_lt n).le⟩
instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFiniteOrder ι] :
    IsSuccArchimedean ι where
  exists_succ_iterate_of_le := by
    intro i j hij
    rw [le_iff_lt_or_eq] at hij
    cases' hij with hij hij
    swap
    · refine' ⟨0, _⟩
      simpa only [Function.iterate_zero, id.def] using hij
    by_contra! h
    have h_lt : ∀ n, succ^[n] i < j := by
      intro n
      induction' n with n hn
      · simpa only [Function.iterate_zero, id.def] using hij
      · refine' lt_of_le_of_ne _ (h _)
        rw [Function.iterate_succ', Function.comp_apply]
        exact succ_le_of_lt hn
    have h_mem : ∀ n, succ^[n] i ∈ Finset.Icc i j :=
      fun n ↦ Finset.mem_Icc.mpr ⟨le_succ_iterate n i, (h_lt n).le⟩
    

Proof state:
case inl
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
h_lt : ∀ (n : ℕ), succ^[n] i < j
h_mem : ∀ (n : ℕ), succ^[n] i ∈ Finset.Icc i j
⊢ False

Next tactic:
obtain ⟨n, m, hnm, h_eq⟩ : ∃ n m, n < m ∧ succ^[n] i = succ^[m] i := by
      let f : ℕ → Finset.Icc i j := fun n ↦ ⟨succ^[n] i, h_mem n⟩
      obtain ⟨n, m, hnm_ne, hfnm⟩ : ∃ n m, n ≠ m ∧ f n = f m
      exact Finite.exists_ne_map_eq_of_infinite f
      have hnm_eq : succ^[n] i = succ^[m] i := by simpa only [Subtype.mk_eq_mk] using hfnm
      rcases le_total n m with h_le | h_le
      · exact ⟨n, m, lt_of_le_of_ne h_le hnm_ne, hnm_eq⟩
      · exact ⟨m, n, lt_of_le_of_ne h_le hnm_ne.symm, hnm_eq.symm⟩
instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFiniteOrder ι] :
    IsSuccArchimedean ι where
  exists_succ_iterate_of_le := by
    intro i j hij
    rw [le_iff_lt_or_eq] at hij
    cases' hij with hij hij
    swap
    · refine' ⟨0, _⟩
      simpa only [Function.iterate_zero, id.def] using hij
    by_contra! h
    have h_lt : ∀ n, succ^[n] i < j := by
      intro n
      induction' n with n hn
      · simpa only [Function.iterate_zero, id.def] using hij
      · refine' lt_of_le_of_ne _ (h _)
        rw [Function.iterate_succ', Function.comp_apply]
        exact succ_le_of_lt hn
    have h_mem : ∀ n, succ^[n] i ∈ Finset.Icc i j :=
      fun n ↦ Finset.mem_Icc.mpr ⟨le_succ_iterate n i, (h_lt n).le⟩
    obtain ⟨n, m, hnm, h_eq⟩ : ∃ n m, n < m ∧ succ^[n] i = succ^[m] i := by
      

Proof state:
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
h_lt : ∀ (n : ℕ), succ^[n] i < j
h_mem : ∀ (n : ℕ), succ^[n] i ∈ Finset.Icc i j
⊢ ∃ n m, n < m ∧ succ^[n] i = succ^[m] i

Next tactic:
let f : ℕ → Finset.Icc i j := fun n ↦ ⟨succ^[n] i, h_mem n⟩
instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFiniteOrder ι] :
    IsSuccArchimedean ι where
  exists_succ_iterate_of_le := by
    intro i j hij
    rw [le_iff_lt_or_eq] at hij
    cases' hij with hij hij
    swap
    · refine' ⟨0, _⟩
      simpa only [Function.iterate_zero, id.def] using hij
    by_contra! h
    have h_lt : ∀ n, succ^[n] i < j := by
      intro n
      induction' n with n hn
      · simpa only [Function.iterate_zero, id.def] using hij
      · refine' lt_of_le_of_ne _ (h _)
        rw [Function.iterate_succ', Function.comp_apply]
        exact succ_le_of_lt hn
    have h_mem : ∀ n, succ^[n] i ∈ Finset.Icc i j :=
      fun n ↦ Finset.mem_Icc.mpr ⟨le_succ_iterate n i, (h_lt n).le⟩
    obtain ⟨n, m, hnm, h_eq⟩ : ∃ n m, n < m ∧ succ^[n] i = succ^[m] i := by
      let f : ℕ → Finset.Icc i j := fun n ↦ ⟨succ^[n] i, h_mem n⟩
      

Proof state:
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
h_lt : ∀ (n : ℕ), succ^[n] i < j
h_mem : ∀ (n : ℕ), succ^[n] i ∈ Finset.Icc i j
f : ℕ → { x // x ∈ Finset.Icc i j } := fun n => { val := succ^[n] i, property := (_ : succ^[n] i ∈ Finset.Icc i j) }
⊢ ∃ n m, n < m ∧ succ^[n] i = succ^[m] i

Next tactic:
obtain ⟨n, m, hnm_ne, hfnm⟩ : ∃ n m, n ≠ m ∧ f n = f m
instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFiniteOrder ι] :
    IsSuccArchimedean ι where
  exists_succ_iterate_of_le := by
    intro i j hij
    rw [le_iff_lt_or_eq] at hij
    cases' hij with hij hij
    swap
    · refine' ⟨0, _⟩
      simpa only [Function.iterate_zero, id.def] using hij
    by_contra! h
    have h_lt : ∀ n, succ^[n] i < j := by
      intro n
      induction' n with n hn
      · simpa only [Function.iterate_zero, id.def] using hij
      · refine' lt_of_le_of_ne _ (h _)
        rw [Function.iterate_succ', Function.comp_apply]
        exact succ_le_of_lt hn
    have h_mem : ∀ n, succ^[n] i ∈ Finset.Icc i j :=
      fun n ↦ Finset.mem_Icc.mpr ⟨le_succ_iterate n i, (h_lt n).le⟩
    obtain ⟨n, m, hnm, h_eq⟩ : ∃ n m, n < m ∧ succ^[n] i = succ^[m] i := by
      let f : ℕ → Finset.Icc i j := fun n ↦ ⟨succ^[n] i, h_mem n⟩
      obtain ⟨n, m, hnm_ne, hfnm⟩ : ∃ n m, n ≠ m ∧ f n = f m
      

Proof state:
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
h_lt : ∀ (n : ℕ), succ^[n] i < j
h_mem : ∀ (n : ℕ), succ^[n] i ∈ Finset.Icc i j
f : ℕ → { x // x ∈ Finset.Icc i j } := fun n => { val := succ^[n] i, property := (_ : succ^[n] i ∈ Finset.Icc i j) }
⊢ ∃ n m, n ≠ m ∧ f n = f m
case intro.intro.intro
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
h_lt : ∀ (n : ℕ), succ^[n] i < j
h_mem : ∀ (n : ℕ), succ^[n] i ∈ Finset.Icc i j
f : ℕ → { x // x ∈ Finset.Icc i j } := fun n => { val := succ^[n] i, property := (_ : succ^[n] i ∈ Finset.Icc i j) }
n m : ℕ
hnm_ne : n ≠ m
hfnm : f n = f m
⊢ ∃ n m, n < m ∧ succ^[n] i = succ^[m] i

Next tactic:
exact Finite.exists_ne_map_eq_of_infinite f
instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFiniteOrder ι] :
    IsSuccArchimedean ι where
  exists_succ_iterate_of_le := by
    intro i j hij
    rw [le_iff_lt_or_eq] at hij
    cases' hij with hij hij
    swap
    · refine' ⟨0, _⟩
      simpa only [Function.iterate_zero, id.def] using hij
    by_contra! h
    have h_lt : ∀ n, succ^[n] i < j := by
      intro n
      induction' n with n hn
      · simpa only [Function.iterate_zero, id.def] using hij
      · refine' lt_of_le_of_ne _ (h _)
        rw [Function.iterate_succ', Function.comp_apply]
        exact succ_le_of_lt hn
    have h_mem : ∀ n, succ^[n] i ∈ Finset.Icc i j :=
      fun n ↦ Finset.mem_Icc.mpr ⟨le_succ_iterate n i, (h_lt n).le⟩
    obtain ⟨n, m, hnm, h_eq⟩ : ∃ n m, n < m ∧ succ^[n] i = succ^[m] i := by
      let f : ℕ → Finset.Icc i j := fun n ↦ ⟨succ^[n] i, h_mem n⟩
      obtain ⟨n, m, hnm_ne, hfnm⟩ : ∃ n m, n ≠ m ∧ f n = f m
      exact Finite.exists_ne_map_eq_of_infinite f
      

Proof state:
case intro.intro.intro
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
h_lt : ∀ (n : ℕ), succ^[n] i < j
h_mem : ∀ (n : ℕ), succ^[n] i ∈ Finset.Icc i j
f : ℕ → { x // x ∈ Finset.Icc i j } := fun n => { val := succ^[n] i, property := (_ : succ^[n] i ∈ Finset.Icc i j) }
n m : ℕ
hnm_ne : n ≠ m
hfnm : f n = f m
⊢ ∃ n m, n < m ∧ succ^[n] i = succ^[m] i

Next tactic:
have hnm_eq : succ^[n] i = succ^[m] i := by simpa only [Subtype.mk_eq_mk] using hfnm
instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFiniteOrder ι] :
    IsSuccArchimedean ι where
  exists_succ_iterate_of_le := by
    intro i j hij
    rw [le_iff_lt_or_eq] at hij
    cases' hij with hij hij
    swap
    · refine' ⟨0, _⟩
      simpa only [Function.iterate_zero, id.def] using hij
    by_contra! h
    have h_lt : ∀ n, succ^[n] i < j := by
      intro n
      induction' n with n hn
      · simpa only [Function.iterate_zero, id.def] using hij
      · refine' lt_of_le_of_ne _ (h _)
        rw [Function.iterate_succ', Function.comp_apply]
        exact succ_le_of_lt hn
    have h_mem : ∀ n, succ^[n] i ∈ Finset.Icc i j :=
      fun n ↦ Finset.mem_Icc.mpr ⟨le_succ_iterate n i, (h_lt n).le⟩
    obtain ⟨n, m, hnm, h_eq⟩ : ∃ n m, n < m ∧ succ^[n] i = succ^[m] i := by
      let f : ℕ → Finset.Icc i j := fun n ↦ ⟨succ^[n] i, h_mem n⟩
      obtain ⟨n, m, hnm_ne, hfnm⟩ : ∃ n m, n ≠ m ∧ f n = f m
      exact Finite.exists_ne_map_eq_of_infinite f
      have hnm_eq : succ^[n] i = succ^[m] i := by 

Proof state:
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
h_lt : ∀ (n : ℕ), succ^[n] i < j
h_mem : ∀ (n : ℕ), succ^[n] i ∈ Finset.Icc i j
f : ℕ → { x // x ∈ Finset.Icc i j } := fun n => { val := succ^[n] i, property := (_ : succ^[n] i ∈ Finset.Icc i j) }
n m : ℕ
hnm_ne : n ≠ m
hfnm : f n = f m
⊢ succ^[n] i = succ^[m] i

Next tactic:
simpa only [Subtype.mk_eq_mk] using hfnm
instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFiniteOrder ι] :
    IsSuccArchimedean ι where
  exists_succ_iterate_of_le := by
    intro i j hij
    rw [le_iff_lt_or_eq] at hij
    cases' hij with hij hij
    swap
    · refine' ⟨0, _⟩
      simpa only [Function.iterate_zero, id.def] using hij
    by_contra! h
    have h_lt : ∀ n, succ^[n] i < j := by
      intro n
      induction' n with n hn
      · simpa only [Function.iterate_zero, id.def] using hij
      · refine' lt_of_le_of_ne _ (h _)
        rw [Function.iterate_succ', Function.comp_apply]
        exact succ_le_of_lt hn
    have h_mem : ∀ n, succ^[n] i ∈ Finset.Icc i j :=
      fun n ↦ Finset.mem_Icc.mpr ⟨le_succ_iterate n i, (h_lt n).le⟩
    obtain ⟨n, m, hnm, h_eq⟩ : ∃ n m, n < m ∧ succ^[n] i = succ^[m] i := by
      let f : ℕ → Finset.Icc i j := fun n ↦ ⟨succ^[n] i, h_mem n⟩
      obtain ⟨n, m, hnm_ne, hfnm⟩ : ∃ n m, n ≠ m ∧ f n = f m
      exact Finite.exists_ne_map_eq_of_infinite f
      have hnm_eq : succ^[n] i = succ^[m] i := by simpa only [Subtype.mk_eq_mk] using hfnm
      

Proof state:
case intro.intro.intro
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
h_lt : ∀ (n : ℕ), succ^[n] i < j
h_mem : ∀ (n : ℕ), succ^[n] i ∈ Finset.Icc i j
f : ℕ → { x // x ∈ Finset.Icc i j } := fun n => { val := succ^[n] i, property := (_ : succ^[n] i ∈ Finset.Icc i j) }
n m : ℕ
hnm_ne : n ≠ m
hfnm : f n = f m
hnm_eq : succ^[n] i = succ^[m] i
⊢ ∃ n m, n < m ∧ succ^[n] i = succ^[m] i

Next tactic:
rcases le_total n m with h_le | h_le
instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFiniteOrder ι] :
    IsSuccArchimedean ι where
  exists_succ_iterate_of_le := by
    intro i j hij
    rw [le_iff_lt_or_eq] at hij
    cases' hij with hij hij
    swap
    · refine' ⟨0, _⟩
      simpa only [Function.iterate_zero, id.def] using hij
    by_contra! h
    have h_lt : ∀ n, succ^[n] i < j := by
      intro n
      induction' n with n hn
      · simpa only [Function.iterate_zero, id.def] using hij
      · refine' lt_of_le_of_ne _ (h _)
        rw [Function.iterate_succ', Function.comp_apply]
        exact succ_le_of_lt hn
    have h_mem : ∀ n, succ^[n] i ∈ Finset.Icc i j :=
      fun n ↦ Finset.mem_Icc.mpr ⟨le_succ_iterate n i, (h_lt n).le⟩
    obtain ⟨n, m, hnm, h_eq⟩ : ∃ n m, n < m ∧ succ^[n] i = succ^[m] i := by
      let f : ℕ → Finset.Icc i j := fun n ↦ ⟨succ^[n] i, h_mem n⟩
      obtain ⟨n, m, hnm_ne, hfnm⟩ : ∃ n m, n ≠ m ∧ f n = f m
      exact Finite.exists_ne_map_eq_of_infinite f
      have hnm_eq : succ^[n] i = succ^[m] i := by simpa only [Subtype.mk_eq_mk] using hfnm
      rcases le_total n m with h_le | h_le
      · 

Proof state:
case intro.intro.intro.inl
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
h_lt : ∀ (n : ℕ), succ^[n] i < j
h_mem : ∀ (n : ℕ), succ^[n] i ∈ Finset.Icc i j
f : ℕ → { x // x ∈ Finset.Icc i j } := fun n => { val := succ^[n] i, property := (_ : succ^[n] i ∈ Finset.Icc i j) }
n m : ℕ
hnm_ne : n ≠ m
hfnm : f n = f m
hnm_eq : succ^[n] i = succ^[m] i
h_le : n ≤ m
⊢ ∃ n m, n < m ∧ succ^[n] i = succ^[m] i

Next tactic:
exact ⟨n, m, lt_of_le_of_ne h_le hnm_ne, hnm_eq⟩
instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFiniteOrder ι] :
    IsSuccArchimedean ι where
  exists_succ_iterate_of_le := by
    intro i j hij
    rw [le_iff_lt_or_eq] at hij
    cases' hij with hij hij
    swap
    · refine' ⟨0, _⟩
      simpa only [Function.iterate_zero, id.def] using hij
    by_contra! h
    have h_lt : ∀ n, succ^[n] i < j := by
      intro n
      induction' n with n hn
      · simpa only [Function.iterate_zero, id.def] using hij
      · refine' lt_of_le_of_ne _ (h _)
        rw [Function.iterate_succ', Function.comp_apply]
        exact succ_le_of_lt hn
    have h_mem : ∀ n, succ^[n] i ∈ Finset.Icc i j :=
      fun n ↦ Finset.mem_Icc.mpr ⟨le_succ_iterate n i, (h_lt n).le⟩
    obtain ⟨n, m, hnm, h_eq⟩ : ∃ n m, n < m ∧ succ^[n] i = succ^[m] i := by
      let f : ℕ → Finset.Icc i j := fun n ↦ ⟨succ^[n] i, h_mem n⟩
      obtain ⟨n, m, hnm_ne, hfnm⟩ : ∃ n m, n ≠ m ∧ f n = f m
      exact Finite.exists_ne_map_eq_of_infinite f
      have hnm_eq : succ^[n] i = succ^[m] i := by simpa only [Subtype.mk_eq_mk] using hfnm
      rcases le_total n m with h_le | h_le
      · exact ⟨n, m, lt_of_le_of_ne h_le hnm_ne, hnm_eq⟩
      · 

Proof state:
case intro.intro.intro.inr
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
h_lt : ∀ (n : ℕ), succ^[n] i < j
h_mem : ∀ (n : ℕ), succ^[n] i ∈ Finset.Icc i j
f : ℕ → { x // x ∈ Finset.Icc i j } := fun n => { val := succ^[n] i, property := (_ : succ^[n] i ∈ Finset.Icc i j) }
n m : ℕ
hnm_ne : n ≠ m
hfnm : f n = f m
hnm_eq : succ^[n] i = succ^[m] i
h_le : m ≤ n
⊢ ∃ n m, n < m ∧ succ^[n] i = succ^[m] i

Next tactic:
exact ⟨m, n, lt_of_le_of_ne h_le hnm_ne.symm, hnm_eq.symm⟩
instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFiniteOrder ι] :
    IsSuccArchimedean ι where
  exists_succ_iterate_of_le := by
    intro i j hij
    rw [le_iff_lt_or_eq] at hij
    cases' hij with hij hij
    swap
    · refine' ⟨0, _⟩
      simpa only [Function.iterate_zero, id.def] using hij
    by_contra! h
    have h_lt : ∀ n, succ^[n] i < j := by
      intro n
      induction' n with n hn
      · simpa only [Function.iterate_zero, id.def] using hij
      · refine' lt_of_le_of_ne _ (h _)
        rw [Function.iterate_succ', Function.comp_apply]
        exact succ_le_of_lt hn
    have h_mem : ∀ n, succ^[n] i ∈ Finset.Icc i j :=
      fun n ↦ Finset.mem_Icc.mpr ⟨le_succ_iterate n i, (h_lt n).le⟩
    obtain ⟨n, m, hnm, h_eq⟩ : ∃ n m, n < m ∧ succ^[n] i = succ^[m] i := by
      let f : ℕ → Finset.Icc i j := fun n ↦ ⟨succ^[n] i, h_mem n⟩
      obtain ⟨n, m, hnm_ne, hfnm⟩ : ∃ n m, n ≠ m ∧ f n = f m
      exact Finite.exists_ne_map_eq_of_infinite f
      have hnm_eq : succ^[n] i = succ^[m] i := by simpa only [Subtype.mk_eq_mk] using hfnm
      rcases le_total n m with h_le | h_le
      · exact ⟨n, m, lt_of_le_of_ne h_le hnm_ne, hnm_eq⟩
      · exact ⟨m, n, lt_of_le_of_ne h_le hnm_ne.symm, hnm_eq.symm⟩
    

Proof state:
case inl.intro.intro.intro
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
h_lt : ∀ (n : ℕ), succ^[n] i < j
h_mem : ∀ (n : ℕ), succ^[n] i ∈ Finset.Icc i j
n m : ℕ
hnm : n < m
h_eq : succ^[n] i = succ^[m] i
⊢ False

Next tactic:
have h_max : IsMax (succ^[n] i) := isMax_iterate_succ_of_eq_of_ne h_eq hnm.ne
instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFiniteOrder ι] :
    IsSuccArchimedean ι where
  exists_succ_iterate_of_le := by
    intro i j hij
    rw [le_iff_lt_or_eq] at hij
    cases' hij with hij hij
    swap
    · refine' ⟨0, _⟩
      simpa only [Function.iterate_zero, id.def] using hij
    by_contra! h
    have h_lt : ∀ n, succ^[n] i < j := by
      intro n
      induction' n with n hn
      · simpa only [Function.iterate_zero, id.def] using hij
      · refine' lt_of_le_of_ne _ (h _)
        rw [Function.iterate_succ', Function.comp_apply]
        exact succ_le_of_lt hn
    have h_mem : ∀ n, succ^[n] i ∈ Finset.Icc i j :=
      fun n ↦ Finset.mem_Icc.mpr ⟨le_succ_iterate n i, (h_lt n).le⟩
    obtain ⟨n, m, hnm, h_eq⟩ : ∃ n m, n < m ∧ succ^[n] i = succ^[m] i := by
      let f : ℕ → Finset.Icc i j := fun n ↦ ⟨succ^[n] i, h_mem n⟩
      obtain ⟨n, m, hnm_ne, hfnm⟩ : ∃ n m, n ≠ m ∧ f n = f m
      exact Finite.exists_ne_map_eq_of_infinite f
      have hnm_eq : succ^[n] i = succ^[m] i := by simpa only [Subtype.mk_eq_mk] using hfnm
      rcases le_total n m with h_le | h_le
      · exact ⟨n, m, lt_of_le_of_ne h_le hnm_ne, hnm_eq⟩
      · exact ⟨m, n, lt_of_le_of_ne h_le hnm_ne.symm, hnm_eq.symm⟩
    have h_max : IsMax (succ^[n] i) := isMax_iterate_succ_of_eq_of_ne h_eq hnm.ne
    

Proof state:
case inl.intro.intro.intro
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i j : ι
hij : i < j
h : ∀ (n : ℕ), succ^[n] i ≠ j
h_lt : ∀ (n : ℕ), succ^[n] i < j
h_mem : ∀ (n : ℕ), succ^[n] i ∈ Finset.Icc i j
n m : ℕ
hnm : n < m
h_eq : succ^[n] i = succ^[m] i
h_max : IsMax (succ^[n] i)
⊢ False

Next tactic:
exact not_le.mpr (h_lt n) (h_max (h_lt n).le)
instance (priority := 100) LinearOrder.isPredArchimedean_of_isSuccArchimedean [SuccOrder ι]
    [PredOrder ι] [IsSuccArchimedean ι] : IsPredArchimedean ι where
  exists_pred_iterate_of_le := by
    

Proof state:
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : SuccOrder ι
inst✝¹ : PredOrder ι
inst✝ : IsSuccArchimedean ι
⊢ ∀ {a b : ι}, a ≤ b → ∃ n, pred^[n] b = a

Next tactic:
intro i j hij