l3lab/ntp-mathlib huggingface dataset:
/-- Induction principle to build a sequence, by adding one point at a time satisfying a given
relation with respect to all the previously chosen points.

More precisely, Assume that, for any finite set `s`, one can find another point satisfying
some relation `r` with respect to all the points in `s`. Then one may construct a
function `f : ℕ → α` such that `r (f m) (f n)` holds whenever `m < n`.
We also ensure that all constructed points satisfy a given predicate `P`. -/
theorem exists_seq_of_forall_finset_exists {α : Type*} (P : α → Prop) (r : α → α → Prop)
    (h : ∀ s : Finset α, (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y) :
    ∃ f : ℕ → α, (∀ n, P (f n)) ∧ ∀ m n, m < n → r (f m) (f n) := by
  classical
    have : Nonempty α := by
      

Proof state:
α✝ : Type u_1
β : Type u_2
γ : Type u_3
α : Type u_4
P : α → Prop
r : α → α → Prop
h : ∀ (s : Finset α), (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y
⊢ Nonempty α

Next tactic:
rcases h ∅ (by simp) with ⟨y, _⟩
/-- Induction principle to build a sequence, by adding one point at a time satisfying a given
relation with respect to all the previously chosen points.

More precisely, Assume that, for any finite set `s`, one can find another point satisfying
some relation `r` with respect to all the points in `s`. Then one may construct a
function `f : ℕ → α` such that `r (f m) (f n)` holds whenever `m < n`.
We also ensure that all constructed points satisfy a given predicate `P`. -/
theorem exists_seq_of_forall_finset_exists {α : Type*} (P : α → Prop) (r : α → α → Prop)
    (h : ∀ s : Finset α, (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y) :
    ∃ f : ℕ → α, (∀ n, P (f n)) ∧ ∀ m n, m < n → r (f m) (f n) := by
  classical
    have : Nonempty α := by
      rcases h ∅ (by 

Proof state:
α✝ : Type u_1
β : Type u_2
γ : Type u_3
α : Type u_4
P : α → Prop
r : α → α → Prop
h : ∀ (s : Finset α), (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y
⊢ ∀ x ∈ ∅, P x

Next tactic:
simp
/-- Induction principle to build a sequence, by adding one point at a time satisfying a given
relation with respect to all the previously chosen points.

More precisely, Assume that, for any finite set `s`, one can find another point satisfying
some relation `r` with respect to all the points in `s`. Then one may construct a
function `f : ℕ → α` such that `r (f m) (f n)` holds whenever `m < n`.
We also ensure that all constructed points satisfy a given predicate `P`. -/
theorem exists_seq_of_forall_finset_exists {α : Type*} (P : α → Prop) (r : α → α → Prop)
    (h : ∀ s : Finset α, (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y) :
    ∃ f : ℕ → α, (∀ n, P (f n)) ∧ ∀ m n, m < n → r (f m) (f n) := by
  classical
    have : Nonempty α := by
      rcases h ∅ (by simp) with ⟨y, _⟩
      

Proof state:
case intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
α : Type u_4
P : α → Prop
r : α → α → Prop
h : ∀ (s : Finset α), (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y
y : α
h✝ : P y ∧ ∀ x ∈ ∅, r x y
⊢ Nonempty α

Next tactic:
exact ⟨y⟩
/-- Induction principle to build a sequence, by adding one point at a time satisfying a given
relation with respect to all the previously chosen points.

More precisely, Assume that, for any finite set `s`, one can find another point satisfying
some relation `r` with respect to all the points in `s`. Then one may construct a
function `f : ℕ → α` such that `r (f m) (f n)` holds whenever `m < n`.
We also ensure that all constructed points satisfy a given predicate `P`. -/
theorem exists_seq_of_forall_finset_exists {α : Type*} (P : α → Prop) (r : α → α → Prop)
    (h : ∀ s : Finset α, (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y) :
    ∃ f : ℕ → α, (∀ n, P (f n)) ∧ ∀ m n, m < n → r (f m) (f n) := by
  classical
    have : Nonempty α := by
      rcases h ∅ (by simp) with ⟨y, _⟩
      exact ⟨y⟩
    

Proof state:
α✝ : Type u_1
β : Type u_2
γ : Type u_3
α : Type u_4
P : α → Prop
r : α → α → Prop
h : ∀ (s : Finset α), (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y
this : Nonempty α
⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m < n → r (f m) (f n)

Next tactic:
choose! F hF using h
/-- Induction principle to build a sequence, by adding one point at a time satisfying a given
relation with respect to all the previously chosen points.

More precisely, Assume that, for any finite set `s`, one can find another point satisfying
some relation `r` with respect to all the points in `s`. Then one may construct a
function `f : ℕ → α` such that `r (f m) (f n)` holds whenever `m < n`.
We also ensure that all constructed points satisfy a given predicate `P`. -/
theorem exists_seq_of_forall_finset_exists {α : Type*} (P : α → Prop) (r : α → α → Prop)
    (h : ∀ s : Finset α, (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y) :
    ∃ f : ℕ → α, (∀ n, P (f n)) ∧ ∀ m n, m < n → r (f m) (f n) := by
  classical
    have : Nonempty α := by
      rcases h ∅ (by simp) with ⟨y, _⟩
      exact ⟨y⟩
    choose! F hF using h
    

Proof state:
α✝ : Type u_1
β : Type u_2
γ : Type u_3
α : Type u_4
P : α → Prop
r : α → α → Prop
this : Nonempty α
F : Finset α → α
hF : ∀ (s : Finset α), (∀ x ∈ s, P x) → P (F s) ∧ ∀ x ∈ s, r x (F s)
⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m < n → r (f m) (f n)

Next tactic:
have h' : ∀ s : Finset α, ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y := fun s => ⟨F s, hF s⟩
/-- Induction principle to build a sequence, by adding one point at a time satisfying a given
relation with respect to all the previously chosen points.

More precisely, Assume that, for any finite set `s`, one can find another point satisfying
some relation `r` with respect to all the points in `s`. Then one may construct a
function `f : ℕ → α` such that `r (f m) (f n)` holds whenever `m < n`.
We also ensure that all constructed points satisfy a given predicate `P`. -/
theorem exists_seq_of_forall_finset_exists {α : Type*} (P : α → Prop) (r : α → α → Prop)
    (h : ∀ s : Finset α, (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y) :
    ∃ f : ℕ → α, (∀ n, P (f n)) ∧ ∀ m n, m < n → r (f m) (f n) := by
  classical
    have : Nonempty α := by
      rcases h ∅ (by simp) with ⟨y, _⟩
      exact ⟨y⟩
    choose! F hF using h
    have h' : ∀ s : Finset α, ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y := fun s => ⟨F s, hF s⟩
    

Proof state:
α✝ : Type u_1
β : Type u_2
γ : Type u_3
α : Type u_4
P : α → Prop
r : α → α → Prop
this : Nonempty α
F : Finset α → α
hF : ∀ (s : Finset α), (∀ x ∈ s, P x) → P (F s) ∧ ∀ x ∈ s, r x (F s)
h' : ∀ (s : Finset α), ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y
⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m < n → r (f m) (f n)

Next tactic:
set f := seqOfForallFinsetExistsAux P r h' with hf
/-- Induction principle to build a sequence, by adding one point at a time satisfying a given
relation with respect to all the previously chosen points.

More precisely, Assume that, for any finite set `s`, one can find another point satisfying
some relation `r` with respect to all the points in `s`. Then one may construct a
function `f : ℕ → α` such that `r (f m) (f n)` holds whenever `m < n`.
We also ensure that all constructed points satisfy a given predicate `P`. -/
theorem exists_seq_of_forall_finset_exists {α : Type*} (P : α → Prop) (r : α → α → Prop)
    (h : ∀ s : Finset α, (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y) :
    ∃ f : ℕ → α, (∀ n, P (f n)) ∧ ∀ m n, m < n → r (f m) (f n) := by
  classical
    have : Nonempty α := by
      rcases h ∅ (by simp) with ⟨y, _⟩
      exact ⟨y⟩
    choose! F hF using h
    have h' : ∀ s : Finset α, ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y := fun s => ⟨F s, hF s⟩
    set f := seqOfForallFinsetExistsAux P r h' with hf
    

Proof state:
α✝ : Type u_1
β : Type u_2
γ : Type u_3
α : Type u_4
P : α → Prop
r : α → α → Prop
this : Nonempty α
F : Finset α → α
hF : ∀ (s : Finset α), (∀ x ∈ s, P x) → P (F s) ∧ ∀ x ∈ s, r x (F s)
h' : ∀ (s : Finset α), ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y
f : ℕ → α := seqOfForallFinsetExistsAux P r h'
hf : f = seqOfForallFinsetExistsAux P r h'
⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m < n → r (f m) (f n)

Next tactic:
have A : ∀ n : ℕ, P (f n) := by
      intro n
      induction' n using Nat.strong_induction_on with n IH
      have IH' : ∀ x : Fin n, P (f x) := fun n => IH n.1 n.2
      rw [hf, seqOfForallFinsetExistsAux]
      exact
        (Classical.choose_spec
            (h' (Finset.image (fun i : Fin n => f i) (Finset.univ : Finset (Fin n))))
            (by simp [IH'])).1
/-- Induction principle to build a sequence, by adding one point at a time satisfying a given
relation with respect to all the previously chosen points.

More precisely, Assume that, for any finite set `s`, one can find another point satisfying
some relation `r` with respect to all the points in `s`. Then one may construct a
function `f : ℕ → α` such that `r (f m) (f n)` holds whenever `m < n`.
We also ensure that all constructed points satisfy a given predicate `P`. -/
theorem exists_seq_of_forall_finset_exists {α : Type*} (P : α → Prop) (r : α → α → Prop)
    (h : ∀ s : Finset α, (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y) :
    ∃ f : ℕ → α, (∀ n, P (f n)) ∧ ∀ m n, m < n → r (f m) (f n) := by
  classical
    have : Nonempty α := by
      rcases h ∅ (by simp) with ⟨y, _⟩
      exact ⟨y⟩
    choose! F hF using h
    have h' : ∀ s : Finset α, ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y := fun s => ⟨F s, hF s⟩
    set f := seqOfForallFinsetExistsAux P r h' with hf
    have A : ∀ n : ℕ, P (f n) := by
      

Proof state:
α✝ : Type u_1
β : Type u_2
γ : Type u_3
α : Type u_4
P : α → Prop
r : α → α → Prop
this : Nonempty α
F : Finset α → α
hF : ∀ (s : Finset α), (∀ x ∈ s, P x) → P (F s) ∧ ∀ x ∈ s, r x (F s)
h' : ∀ (s : Finset α), ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y
f : ℕ → α := seqOfForallFinsetExistsAux P r h'
hf : f = seqOfForallFinsetExistsAux P r h'
⊢ ∀ (n : ℕ), P (f n)

Next tactic:
intro n
/-- Induction principle to build a sequence, by adding one point at a time satisfying a given
relation with respect to all the previously chosen points.

More precisely, Assume that, for any finite set `s`, one can find another point satisfying
some relation `r` with respect to all the points in `s`. Then one may construct a
function `f : ℕ → α` such that `r (f m) (f n)` holds whenever `m < n`.
We also ensure that all constructed points satisfy a given predicate `P`. -/
theorem exists_seq_of_forall_finset_exists {α : Type*} (P : α → Prop) (r : α → α → Prop)
    (h : ∀ s : Finset α, (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y) :
    ∃ f : ℕ → α, (∀ n, P (f n)) ∧ ∀ m n, m < n → r (f m) (f n) := by
  classical
    have : Nonempty α := by
      rcases h ∅ (by simp) with ⟨y, _⟩
      exact ⟨y⟩
    choose! F hF using h
    have h' : ∀ s : Finset α, ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y := fun s => ⟨F s, hF s⟩
    set f := seqOfForallFinsetExistsAux P r h' with hf
    have A : ∀ n : ℕ, P (f n) := by
      intro n
      

Proof state:
α✝ : Type u_1
β : Type u_2
γ : Type u_3
α : Type u_4
P : α → Prop
r : α → α → Prop
this : Nonempty α
F : Finset α → α
hF : ∀ (s : Finset α), (∀ x ∈ s, P x) → P (F s) ∧ ∀ x ∈ s, r x (F s)
h' : ∀ (s : Finset α), ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y
f : ℕ → α := seqOfForallFinsetExistsAux P r h'
hf : f = seqOfForallFinsetExistsAux P r h'
n : ℕ
⊢ P (f n)

Next tactic:
induction' n using Nat.strong_induction_on with n IH
/-- Induction principle to build a sequence, by adding one point at a time satisfying a given
relation with respect to all the previously chosen points.

More precisely, Assume that, for any finite set `s`, one can find another point satisfying
some relation `r` with respect to all the points in `s`. Then one may construct a
function `f : ℕ → α` such that `r (f m) (f n)` holds whenever `m < n`.
We also ensure that all constructed points satisfy a given predicate `P`. -/
theorem exists_seq_of_forall_finset_exists {α : Type*} (P : α → Prop) (r : α → α → Prop)
    (h : ∀ s : Finset α, (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y) :
    ∃ f : ℕ → α, (∀ n, P (f n)) ∧ ∀ m n, m < n → r (f m) (f n) := by
  classical
    have : Nonempty α := by
      rcases h ∅ (by simp) with ⟨y, _⟩
      exact ⟨y⟩
    choose! F hF using h
    have h' : ∀ s : Finset α, ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y := fun s => ⟨F s, hF s⟩
    set f := seqOfForallFinsetExistsAux P r h' with hf
    have A : ∀ n : ℕ, P (f n) := by
      intro n
      induction' n using Nat.strong_induction_on with n IH
      

Proof state:
case h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
α : Type u_4
P : α → Prop
r : α → α → Prop
this : Nonempty α
F : Finset α → α
hF : ∀ (s : Finset α), (∀ x ∈ s, P x) → P (F s) ∧ ∀ x ∈ s, r x (F s)
h' : ∀ (s : Finset α), ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y
f : ℕ → α := seqOfForallFinsetExistsAux P r h'
hf : f = seqOfForallFinsetExistsAux P r h'
n : ℕ
IH : ∀ m < n, P (f m)
⊢ P (f n)

Next tactic:
have IH' : ∀ x : Fin n, P (f x) := fun n => IH n.1 n.2
/-- Induction principle to build a sequence, by adding one point at a time satisfying a given
relation with respect to all the previously chosen points.

More precisely, Assume that, for any finite set `s`, one can find another point satisfying
some relation `r` with respect to all the points in `s`. Then one may construct a
function `f : ℕ → α` such that `r (f m) (f n)` holds whenever `m < n`.
We also ensure that all constructed points satisfy a given predicate `P`. -/
theorem exists_seq_of_forall_finset_exists {α : Type*} (P : α → Prop) (r : α → α → Prop)
    (h : ∀ s : Finset α, (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y) :
    ∃ f : ℕ → α, (∀ n, P (f n)) ∧ ∀ m n, m < n → r (f m) (f n) := by
  classical
    have : Nonempty α := by
      rcases h ∅ (by simp) with ⟨y, _⟩
      exact ⟨y⟩
    choose! F hF using h
    have h' : ∀ s : Finset α, ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y := fun s => ⟨F s, hF s⟩
    set f := seqOfForallFinsetExistsAux P r h' with hf
    have A : ∀ n : ℕ, P (f n) := by
      intro n
      induction' n using Nat.strong_induction_on with n IH
      have IH' : ∀ x : Fin n, P (f x) := fun n => IH n.1 n.2
      

Proof state:
case h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
α : Type u_4
P : α → Prop
r : α → α → Prop
this : Nonempty α
F : Finset α → α
hF : ∀ (s : Finset α), (∀ x ∈ s, P x) → P (F s) ∧ ∀ x ∈ s, r x (F s)
h' : ∀ (s : Finset α), ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y
f : ℕ → α := seqOfForallFinsetExistsAux P r h'
hf : f = seqOfForallFinsetExistsAux P r h'
n : ℕ
IH : ∀ m < n, P (f m)
IH' : ∀ (x : Fin n), P (f ↑x)
⊢ P (f n)

Next tactic:
rw [hf, seqOfForallFinsetExistsAux]
/-- Induction principle to build a sequence, by adding one point at a time satisfying a given
relation with respect to all the previously chosen points.

More precisely, Assume that, for any finite set `s`, one can find another point satisfying
some relation `r` with respect to all the points in `s`. Then one may construct a
function `f : ℕ → α` such that `r (f m) (f n)` holds whenever `m < n`.
We also ensure that all constructed points satisfy a given predicate `P`. -/
theorem exists_seq_of_forall_finset_exists {α : Type*} (P : α → Prop) (r : α → α → Prop)
    (h : ∀ s : Finset α, (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y) :
    ∃ f : ℕ → α, (∀ n, P (f n)) ∧ ∀ m n, m < n → r (f m) (f n) := by
  classical
    have : Nonempty α := by
      rcases h ∅ (by simp) with ⟨y, _⟩
      exact ⟨y⟩
    choose! F hF using h
    have h' : ∀ s : Finset α, ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y := fun s => ⟨F s, hF s⟩
    set f := seqOfForallFinsetExistsAux P r h' with hf
    have A : ∀ n : ℕ, P (f n) := by
      intro n
      induction' n using Nat.strong_induction_on with n IH
      have IH' : ∀ x : Fin n, P (f x) := fun n => IH n.1 n.2
      rw [hf, seqOfForallFinsetExistsAux]
      

Proof state:
case h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
α : Type u_4
P : α → Prop
r : α → α → Prop
this : Nonempty α
F : Finset α → α
hF : ∀ (s : Finset α), (∀ x ∈ s, P x) → P (F s) ∧ ∀ x ∈ s, r x (F s)
h' : ∀ (s : Finset α), ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y
f : ℕ → α := seqOfForallFinsetExistsAux P r h'
hf : f = seqOfForallFinsetExistsAux P r h'
n : ℕ
IH : ∀ m < n, P (f m)
IH' : ∀ (x : Fin n), P (f ↑x)
⊢ P
    (Classical.choose
      (_ :
        ∃ y,
          (∀ x ∈ image (fun i => seqOfForallFinsetExistsAux P r h' ↑i) univ, P x) →
            P y ∧ ∀ x ∈ image (fun i => seqOfForallFinsetExistsAux P r h' ↑i) univ, r x y))

Next tactic:
exact
        (Classical.choose_spec
            (h' (Finset.image (fun i : Fin n => f i) (Finset.univ : Finset (Fin n))))
            (by simp [IH'])).1
/-- Induction principle to build a sequence, by adding one point at a time satisfying a given
relation with respect to all the previously chosen points.

More precisely, Assume that, for any finite set `s`, one can find another point satisfying
some relation `r` with respect to all the points in `s`. Then one may construct a
function `f : ℕ → α` such that `r (f m) (f n)` holds whenever `m < n`.
We also ensure that all constructed points satisfy a given predicate `P`. -/
theorem exists_seq_of_forall_finset_exists {α : Type*} (P : α → Prop) (r : α → α → Prop)
    (h : ∀ s : Finset α, (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y) :
    ∃ f : ℕ → α, (∀ n, P (f n)) ∧ ∀ m n, m < n → r (f m) (f n) := by
  classical
    have : Nonempty α := by
      rcases h ∅ (by simp) with ⟨y, _⟩
      exact ⟨y⟩
    choose! F hF using h
    have h' : ∀ s : Finset α, ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y := fun s => ⟨F s, hF s⟩
    set f := seqOfForallFinsetExistsAux P r h' with hf
    have A : ∀ n : ℕ, P (f n) := by
      intro n
      induction' n using Nat.strong_induction_on with n IH
      have IH' : ∀ x : Fin n, P (f x) := fun n => IH n.1 n.2
      rw [hf, seqOfForallFinsetExistsAux]
      exact
        (Classical.choose_spec
            (h' (Finset.image (fun i : Fin n => f i) (Finset.univ : Finset (Fin n))))
            (by 

Proof state:
α✝ : Type u_1
β : Type u_2
γ : Type u_3
α : Type u_4
P : α → Prop
r : α → α → Prop
this : Nonempty α
F : Finset α → α
hF : ∀ (s : Finset α), (∀ x ∈ s, P x) → P (F s) ∧ ∀ x ∈ s, r x (F s)
h' : ∀ (s : Finset α), ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y
f : ℕ → α := seqOfForallFinsetExistsAux P r h'
hf : f = seqOfForallFinsetExistsAux P r h'
n : ℕ
IH : ∀ m < n, P (f m)
IH' : ∀ (x : Fin n), P (f ↑x)
⊢ ∀ x ∈ image (fun i => f ↑i) univ, P x

Next tactic:
simp [IH']
/-- Induction principle to build a sequence, by adding one point at a time satisfying a given
relation with respect to all the previously chosen points.

More precisely, Assume that, for any finite set `s`, one can find another point satisfying
some relation `r` with respect to all the points in `s`. Then one may construct a
function `f : ℕ → α` such that `r (f m) (f n)` holds whenever `m < n`.
We also ensure that all constructed points satisfy a given predicate `P`. -/
theorem exists_seq_of_forall_finset_exists {α : Type*} (P : α → Prop) (r : α → α → Prop)
    (h : ∀ s : Finset α, (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y) :
    ∃ f : ℕ → α, (∀ n, P (f n)) ∧ ∀ m n, m < n → r (f m) (f n) := by
  classical
    have : Nonempty α := by
      rcases h ∅ (by simp) with ⟨y, _⟩
      exact ⟨y⟩
    choose! F hF using h
    have h' : ∀ s : Finset α, ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y := fun s => ⟨F s, hF s⟩
    set f := seqOfForallFinsetExistsAux P r h' with hf
    have A : ∀ n : ℕ, P (f n) := by
      intro n
      induction' n using Nat.strong_induction_on with n IH
      have IH' : ∀ x : Fin n, P (f x) := fun n => IH n.1 n.2
      rw [hf, seqOfForallFinsetExistsAux]
      exact
        (Classical.choose_spec
            (h' (Finset.image (fun i : Fin n => f i) (Finset.univ : Finset (Fin n))))
            (by simp [IH'])).1
    

Proof state:
α✝ : Type u_1
β : Type u_2
γ : Type u_3
α : Type u_4
P : α → Prop
r : α → α → Prop
this : Nonempty α
F : Finset α → α
hF : ∀ (s : Finset α), (∀ x ∈ s, P x) → P (F s) ∧ ∀ x ∈ s, r x (F s)
h' : ∀ (s : Finset α), ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y
f : ℕ → α := seqOfForallFinsetExistsAux P r h'
hf : f = seqOfForallFinsetExistsAux P r h'
A : ∀ (n : ℕ), P (f n)
⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m < n → r (f m) (f n)

Next tactic:
refine' ⟨f, A, fun m n hmn => _⟩
/-- Induction principle to build a sequence, by adding one point at a time satisfying a given
relation with respect to all the previously chosen points.

More precisely, Assume that, for any finite set `s`, one can find another point satisfying
some relation `r` with respect to all the points in `s`. Then one may construct a
function `f : ℕ → α` such that `r (f m) (f n)` holds whenever `m < n`.
We also ensure that all constructed points satisfy a given predicate `P`. -/
theorem exists_seq_of_forall_finset_exists {α : Type*} (P : α → Prop) (r : α → α → Prop)
    (h : ∀ s : Finset α, (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y) :
    ∃ f : ℕ → α, (∀ n, P (f n)) ∧ ∀ m n, m < n → r (f m) (f n) := by
  classical
    have : Nonempty α := by
      rcases h ∅ (by simp) with ⟨y, _⟩
      exact ⟨y⟩
    choose! F hF using h
    have h' : ∀ s : Finset α, ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y := fun s => ⟨F s, hF s⟩
    set f := seqOfForallFinsetExistsAux P r h' with hf
    have A : ∀ n : ℕ, P (f n) := by
      intro n
      induction' n using Nat.strong_induction_on with n IH
      have IH' : ∀ x : Fin n, P (f x) := fun n => IH n.1 n.2
      rw [hf, seqOfForallFinsetExistsAux]
      exact
        (Classical.choose_spec
            (h' (Finset.image (fun i : Fin n => f i) (Finset.univ : Finset (Fin n))))
            (by simp [IH'])).1
    refine' ⟨f, A, fun m n hmn => _⟩
    

Proof state:
α✝ : Type u_1
β : Type u_2
γ : Type u_3
α : Type u_4
P : α → Prop
r : α → α → Prop
this : Nonempty α
F : Finset α → α
hF : ∀ (s : Finset α), (∀ x ∈ s, P x) → P (F s) ∧ ∀ x ∈ s, r x (F s)
h' : ∀ (s : Finset α), ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y
f : ℕ → α := seqOfForallFinsetExistsAux P r h'
hf : f = seqOfForallFinsetExistsAux P r h'
A : ∀ (n : ℕ), P (f n)
m n : ℕ
hmn : m < n
⊢ r (f m) (f n)

Next tactic:
conv_rhs => rw [hf]
/-- Induction principle to build a sequence, by adding one point at a time satisfying a given
relation with respect to all the previously chosen points.

More precisely, Assume that, for any finite set `s`, one can find another point satisfying
some relation `r` with respect to all the points in `s`. Then one may construct a
function `f : ℕ → α` such that `r (f m) (f n)` holds whenever `m < n`.
We also ensure that all constructed points satisfy a given predicate `P`. -/
theorem exists_seq_of_forall_finset_exists {α : Type*} (P : α → Prop) (r : α → α → Prop)
    (h : ∀ s : Finset α, (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y) :
    ∃ f : ℕ → α, (∀ n, P (f n)) ∧ ∀ m n, m < n → r (f m) (f n) := by
  classical
    have : Nonempty α := by
      rcases h ∅ (by simp) with ⟨y, _⟩
      exact ⟨y⟩
    choose! F hF using h
    have h' : ∀ s : Finset α, ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y := fun s => ⟨F s, hF s⟩
    set f := seqOfForallFinsetExistsAux P r h' with hf
    have A : ∀ n : ℕ, P (f n) := by
      intro n
      induction' n using Nat.strong_induction_on with n IH
      have IH' : ∀ x : Fin n, P (f x) := fun n => IH n.1 n.2
      rw [hf, seqOfForallFinsetExistsAux]
      exact
        (Classical.choose_spec
            (h' (Finset.image (fun i : Fin n => f i) (Finset.univ : Finset (Fin n))))
            (by simp [IH'])).1
    refine' ⟨f, A, fun m n hmn => _⟩
    conv_rhs => 

Proof state:
α✝ : Type u_1
β : Type u_2
γ : Type u_3
α : Type u_4
P : α → Prop
r : α → α → Prop
this : Nonempty α
F : Finset α → α
hF : ∀ (s : Finset α), (∀ x ∈ s, P x) → P (F s) ∧ ∀ x ∈ s, r x (F s)
h' : ∀ (s : Finset α), ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y
f : ℕ → α := seqOfForallFinsetExistsAux P r h'
hf : f = seqOfForallFinsetExistsAux P r h'
A : ∀ (n : ℕ), P (f n)
m n : ℕ
hmn : m < n
| f n

Next tactic:
rw [hf]