l3lab/ntp-mathlib huggingface dataset:
theorem hofer {X : Type*} [MetricSpace X] [CompleteSpace X] (x : X) (ε : ℝ) (ε_pos : 0 < ε)
    {ϕ : X → ℝ} (cont : Continuous ϕ) (nonneg : ∀ y, 0 ≤ ϕ y) : ∃ ε' > 0, ∃ x' : X,
    ε' ≤ ε ∧ d x' x ≤ 2 * ε ∧ ε * ϕ x ≤ ε' * ϕ x' ∧ ∀ y, d x' y ≤ ε' → ϕ y ≤ 2 * ϕ x' := by
  by_contra H
  have reformulation : ∀ (x') (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x' := by
    intro x' k
    rw [div_mul_eq_mul_div, le_div_iff, mul_assoc, mul_le_mul_left ε_pos, mul_comm]
    positivity
  -- Now let's specialize to `ε/2^k`
  replace H :
    ∀ k : ℕ, ∀ x', d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x' → ∃ y, d x' y ≤ ε / 2 ^ k ∧ 2 * ϕ x' < ϕ y
  · intro k x'
    push_neg at H
    have := H (ε / 2 ^ k) (by simp [ε_pos]) x' (by simp [ε_pos.le, one_le_two])
    simpa [reformulation] using this
  clear reformulation
  haveI : Nonempty X := ⟨x⟩
  choose! F hF using H
  -- Use the axiom of choice
  -- Now define u by induction starting at x, with u_{n+1} = F(n, u_n)
  let u : ℕ → X := fun n => Nat.recOn n x F
  -- The properties of F translate to properties of u
  have hu :
    ∀ n,
      d (u n) x ≤ 2 * ε ∧ 2 ^ n * ϕ x ≤ ϕ (u n) →
        d (u n) (u <| n + 1) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u <| n + 1) := by
    intro n
    exact hF n (u n)
  clear hF
  -- Key properties of u, to be proven by induction
  have key : ∀ n, d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1)) := by
    intro n
    induction' n using Nat.case_strong_induction_on with n IH
    · specialize hu 0
      simpa [mul_nonneg_iff, zero_le_one, ε_pos.le, le_refl] using hu
    have A : d (u (n + 1)) x ≤ 2 * ε := by
      rw [dist_comm]
      let r := range (n + 1) -- range (n+1) = {0, ..., n}
      calc
        d (u 0) (u (n + 1)) ≤ ∑ i in r, d (u i) (u <| i + 1) := dist_le_range_sum_dist u (n + 1)
        _ ≤ ∑ i in r, ε / 2 ^ i :=
          (sum_le_sum fun i i_in => (IH i <| Nat.lt_succ_iff.mp <| Finset.mem_range.mp i_in).1)
        _ = ∑ i in r, (1 / 2) ^ i * ε := by
          congr with i
          field_simp
        _ = (∑ i in r, ((1 : ℝ) / 2) ^ i) * ε := Finset.sum_mul.symm
        _ ≤ 2 * ε := by gcongr; apply sum_geometric_two_le
    have B : 2 ^ (n + 1) * ϕ x ≤ ϕ (u (n + 1)) := by
      refine' @geom_le (ϕ ∘ u) _ zero_le_two (n + 1) fun m hm => _
      exact (IH _ <| Nat.lt_add_one_iff.1 hm).2.le
    exact hu (n + 1) ⟨A, B⟩
  

Proof state:
X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
this : Nonempty X
F : ℕ → X → X
u : ℕ → X := fun n => Nat.recOn n x F
hu :
  ∀ (n : ℕ), d (u n) x ≤ 2 * ε ∧ 2 ^ n * ϕ x ≤ ϕ (u n) → d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key : ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
⊢ False

Next tactic:
cases' forall_and.mp key with key₁ key₂
theorem hofer {X : Type*} [MetricSpace X] [CompleteSpace X] (x : X) (ε : ℝ) (ε_pos : 0 < ε)
    {ϕ : X → ℝ} (cont : Continuous ϕ) (nonneg : ∀ y, 0 ≤ ϕ y) : ∃ ε' > 0, ∃ x' : X,
    ε' ≤ ε ∧ d x' x ≤ 2 * ε ∧ ε * ϕ x ≤ ε' * ϕ x' ∧ ∀ y, d x' y ≤ ε' → ϕ y ≤ 2 * ϕ x' := by
  by_contra H
  have reformulation : ∀ (x') (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x' := by
    intro x' k
    rw [div_mul_eq_mul_div, le_div_iff, mul_assoc, mul_le_mul_left ε_pos, mul_comm]
    positivity
  -- Now let's specialize to `ε/2^k`
  replace H :
    ∀ k : ℕ, ∀ x', d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x' → ∃ y, d x' y ≤ ε / 2 ^ k ∧ 2 * ϕ x' < ϕ y
  · intro k x'
    push_neg at H
    have := H (ε / 2 ^ k) (by simp [ε_pos]) x' (by simp [ε_pos.le, one_le_two])
    simpa [reformulation] using this
  clear reformulation
  haveI : Nonempty X := ⟨x⟩
  choose! F hF using H
  -- Use the axiom of choice
  -- Now define u by induction starting at x, with u_{n+1} = F(n, u_n)
  let u : ℕ → X := fun n => Nat.recOn n x F
  -- The properties of F translate to properties of u
  have hu :
    ∀ n,
      d (u n) x ≤ 2 * ε ∧ 2 ^ n * ϕ x ≤ ϕ (u n) →
        d (u n) (u <| n + 1) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u <| n + 1) := by
    intro n
    exact hF n (u n)
  clear hF
  -- Key properties of u, to be proven by induction
  have key : ∀ n, d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1)) := by
    intro n
    induction' n using Nat.case_strong_induction_on with n IH
    · specialize hu 0
      simpa [mul_nonneg_iff, zero_le_one, ε_pos.le, le_refl] using hu
    have A : d (u (n + 1)) x ≤ 2 * ε := by
      rw [dist_comm]
      let r := range (n + 1) -- range (n+1) = {0, ..., n}
      calc
        d (u 0) (u (n + 1)) ≤ ∑ i in r, d (u i) (u <| i + 1) := dist_le_range_sum_dist u (n + 1)
        _ ≤ ∑ i in r, ε / 2 ^ i :=
          (sum_le_sum fun i i_in => (IH i <| Nat.lt_succ_iff.mp <| Finset.mem_range.mp i_in).1)
        _ = ∑ i in r, (1 / 2) ^ i * ε := by
          congr with i
          field_simp
        _ = (∑ i in r, ((1 : ℝ) / 2) ^ i) * ε := Finset.sum_mul.symm
        _ ≤ 2 * ε := by gcongr; apply sum_geometric_two_le
    have B : 2 ^ (n + 1) * ϕ x ≤ ϕ (u (n + 1)) := by
      refine' @geom_le (ϕ ∘ u) _ zero_le_two (n + 1) fun m hm => _
      exact (IH _ <| Nat.lt_add_one_iff.1 hm).2.le
    exact hu (n + 1) ⟨A, B⟩
  cases' forall_and.mp key with key₁ key₂
  

Proof state:
case intro
X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
this : Nonempty X
F : ℕ → X → X
u : ℕ → X := fun n => Nat.recOn n x F
hu :
  ∀ (n : ℕ), d (u n) x ≤ 2 * ε ∧ 2 ^ n * ϕ x ≤ ϕ (u n) → d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key : ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / 2 ^ x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
⊢ False

Next tactic:
clear hu key
theorem hofer {X : Type*} [MetricSpace X] [CompleteSpace X] (x : X) (ε : ℝ) (ε_pos : 0 < ε)
    {ϕ : X → ℝ} (cont : Continuous ϕ) (nonneg : ∀ y, 0 ≤ ϕ y) : ∃ ε' > 0, ∃ x' : X,
    ε' ≤ ε ∧ d x' x ≤ 2 * ε ∧ ε * ϕ x ≤ ε' * ϕ x' ∧ ∀ y, d x' y ≤ ε' → ϕ y ≤ 2 * ϕ x' := by
  by_contra H
  have reformulation : ∀ (x') (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x' := by
    intro x' k
    rw [div_mul_eq_mul_div, le_div_iff, mul_assoc, mul_le_mul_left ε_pos, mul_comm]
    positivity
  -- Now let's specialize to `ε/2^k`
  replace H :
    ∀ k : ℕ, ∀ x', d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x' → ∃ y, d x' y ≤ ε / 2 ^ k ∧ 2 * ϕ x' < ϕ y
  · intro k x'
    push_neg at H
    have := H (ε / 2 ^ k) (by simp [ε_pos]) x' (by simp [ε_pos.le, one_le_two])
    simpa [reformulation] using this
  clear reformulation
  haveI : Nonempty X := ⟨x⟩
  choose! F hF using H
  -- Use the axiom of choice
  -- Now define u by induction starting at x, with u_{n+1} = F(n, u_n)
  let u : ℕ → X := fun n => Nat.recOn n x F
  -- The properties of F translate to properties of u
  have hu :
    ∀ n,
      d (u n) x ≤ 2 * ε ∧ 2 ^ n * ϕ x ≤ ϕ (u n) →
        d (u n) (u <| n + 1) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u <| n + 1) := by
    intro n
    exact hF n (u n)
  clear hF
  -- Key properties of u, to be proven by induction
  have key : ∀ n, d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1)) := by
    intro n
    induction' n using Nat.case_strong_induction_on with n IH
    · specialize hu 0
      simpa [mul_nonneg_iff, zero_le_one, ε_pos.le, le_refl] using hu
    have A : d (u (n + 1)) x ≤ 2 * ε := by
      rw [dist_comm]
      let r := range (n + 1) -- range (n+1) = {0, ..., n}
      calc
        d (u 0) (u (n + 1)) ≤ ∑ i in r, d (u i) (u <| i + 1) := dist_le_range_sum_dist u (n + 1)
        _ ≤ ∑ i in r, ε / 2 ^ i :=
          (sum_le_sum fun i i_in => (IH i <| Nat.lt_succ_iff.mp <| Finset.mem_range.mp i_in).1)
        _ = ∑ i in r, (1 / 2) ^ i * ε := by
          congr with i
          field_simp
        _ = (∑ i in r, ((1 : ℝ) / 2) ^ i) * ε := Finset.sum_mul.symm
        _ ≤ 2 * ε := by gcongr; apply sum_geometric_two_le
    have B : 2 ^ (n + 1) * ϕ x ≤ ϕ (u (n + 1)) := by
      refine' @geom_le (ϕ ∘ u) _ zero_le_two (n + 1) fun m hm => _
      exact (IH _ <| Nat.lt_add_one_iff.1 hm).2.le
    exact hu (n + 1) ⟨A, B⟩
  cases' forall_and.mp key with key₁ key₂
  clear hu key
  -- Hence u is Cauchy
  

Proof state:
case intro
X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
this : Nonempty X
F : ℕ → X → X
u : ℕ → X := fun n => Nat.recOn n x F
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / 2 ^ x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
⊢ False

Next tactic:
have cauchy_u : CauchySeq u := by
    refine' cauchySeq_of_le_geometric _ ε one_half_lt_one fun n => _
    simpa only [one_div, inv_pow] using key₁ n
theorem hofer {X : Type*} [MetricSpace X] [CompleteSpace X] (x : X) (ε : ℝ) (ε_pos : 0 < ε)
    {ϕ : X → ℝ} (cont : Continuous ϕ) (nonneg : ∀ y, 0 ≤ ϕ y) : ∃ ε' > 0, ∃ x' : X,
    ε' ≤ ε ∧ d x' x ≤ 2 * ε ∧ ε * ϕ x ≤ ε' * ϕ x' ∧ ∀ y, d x' y ≤ ε' → ϕ y ≤ 2 * ϕ x' := by
  by_contra H
  have reformulation : ∀ (x') (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x' := by
    intro x' k
    rw [div_mul_eq_mul_div, le_div_iff, mul_assoc, mul_le_mul_left ε_pos, mul_comm]
    positivity
  -- Now let's specialize to `ε/2^k`
  replace H :
    ∀ k : ℕ, ∀ x', d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x' → ∃ y, d x' y ≤ ε / 2 ^ k ∧ 2 * ϕ x' < ϕ y
  · intro k x'
    push_neg at H
    have := H (ε / 2 ^ k) (by simp [ε_pos]) x' (by simp [ε_pos.le, one_le_two])
    simpa [reformulation] using this
  clear reformulation
  haveI : Nonempty X := ⟨x⟩
  choose! F hF using H
  -- Use the axiom of choice
  -- Now define u by induction starting at x, with u_{n+1} = F(n, u_n)
  let u : ℕ → X := fun n => Nat.recOn n x F
  -- The properties of F translate to properties of u
  have hu :
    ∀ n,
      d (u n) x ≤ 2 * ε ∧ 2 ^ n * ϕ x ≤ ϕ (u n) →
        d (u n) (u <| n + 1) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u <| n + 1) := by
    intro n
    exact hF n (u n)
  clear hF
  -- Key properties of u, to be proven by induction
  have key : ∀ n, d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1)) := by
    intro n
    induction' n using Nat.case_strong_induction_on with n IH
    · specialize hu 0
      simpa [mul_nonneg_iff, zero_le_one, ε_pos.le, le_refl] using hu
    have A : d (u (n + 1)) x ≤ 2 * ε := by
      rw [dist_comm]
      let r := range (n + 1) -- range (n+1) = {0, ..., n}
      calc
        d (u 0) (u (n + 1)) ≤ ∑ i in r, d (u i) (u <| i + 1) := dist_le_range_sum_dist u (n + 1)
        _ ≤ ∑ i in r, ε / 2 ^ i :=
          (sum_le_sum fun i i_in => (IH i <| Nat.lt_succ_iff.mp <| Finset.mem_range.mp i_in).1)
        _ = ∑ i in r, (1 / 2) ^ i * ε := by
          congr with i
          field_simp
        _ = (∑ i in r, ((1 : ℝ) / 2) ^ i) * ε := Finset.sum_mul.symm
        _ ≤ 2 * ε := by gcongr; apply sum_geometric_two_le
    have B : 2 ^ (n + 1) * ϕ x ≤ ϕ (u (n + 1)) := by
      refine' @geom_le (ϕ ∘ u) _ zero_le_two (n + 1) fun m hm => _
      exact (IH _ <| Nat.lt_add_one_iff.1 hm).2.le
    exact hu (n + 1) ⟨A, B⟩
  cases' forall_and.mp key with key₁ key₂
  clear hu key
  -- Hence u is Cauchy
  have cauchy_u : CauchySeq u := by
    

Proof state:
X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
this : Nonempty X
F : ℕ → X → X
u : ℕ → X := fun n => Nat.recOn n x F
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / 2 ^ x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
⊢ CauchySeq u

Next tactic:
refine' cauchySeq_of_le_geometric _ ε one_half_lt_one fun n => _
theorem hofer {X : Type*} [MetricSpace X] [CompleteSpace X] (x : X) (ε : ℝ) (ε_pos : 0 < ε)
    {ϕ : X → ℝ} (cont : Continuous ϕ) (nonneg : ∀ y, 0 ≤ ϕ y) : ∃ ε' > 0, ∃ x' : X,
    ε' ≤ ε ∧ d x' x ≤ 2 * ε ∧ ε * ϕ x ≤ ε' * ϕ x' ∧ ∀ y, d x' y ≤ ε' → ϕ y ≤ 2 * ϕ x' := by
  by_contra H
  have reformulation : ∀ (x') (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x' := by
    intro x' k
    rw [div_mul_eq_mul_div, le_div_iff, mul_assoc, mul_le_mul_left ε_pos, mul_comm]
    positivity
  -- Now let's specialize to `ε/2^k`
  replace H :
    ∀ k : ℕ, ∀ x', d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x' → ∃ y, d x' y ≤ ε / 2 ^ k ∧ 2 * ϕ x' < ϕ y
  · intro k x'
    push_neg at H
    have := H (ε / 2 ^ k) (by simp [ε_pos]) x' (by simp [ε_pos.le, one_le_two])
    simpa [reformulation] using this
  clear reformulation
  haveI : Nonempty X := ⟨x⟩
  choose! F hF using H
  -- Use the axiom of choice
  -- Now define u by induction starting at x, with u_{n+1} = F(n, u_n)
  let u : ℕ → X := fun n => Nat.recOn n x F
  -- The properties of F translate to properties of u
  have hu :
    ∀ n,
      d (u n) x ≤ 2 * ε ∧ 2 ^ n * ϕ x ≤ ϕ (u n) →
        d (u n) (u <| n + 1) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u <| n + 1) := by
    intro n
    exact hF n (u n)
  clear hF
  -- Key properties of u, to be proven by induction
  have key : ∀ n, d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1)) := by
    intro n
    induction' n using Nat.case_strong_induction_on with n IH
    · specialize hu 0
      simpa [mul_nonneg_iff, zero_le_one, ε_pos.le, le_refl] using hu
    have A : d (u (n + 1)) x ≤ 2 * ε := by
      rw [dist_comm]
      let r := range (n + 1) -- range (n+1) = {0, ..., n}
      calc
        d (u 0) (u (n + 1)) ≤ ∑ i in r, d (u i) (u <| i + 1) := dist_le_range_sum_dist u (n + 1)
        _ ≤ ∑ i in r, ε / 2 ^ i :=
          (sum_le_sum fun i i_in => (IH i <| Nat.lt_succ_iff.mp <| Finset.mem_range.mp i_in).1)
        _ = ∑ i in r, (1 / 2) ^ i * ε := by
          congr with i
          field_simp
        _ = (∑ i in r, ((1 : ℝ) / 2) ^ i) * ε := Finset.sum_mul.symm
        _ ≤ 2 * ε := by gcongr; apply sum_geometric_two_le
    have B : 2 ^ (n + 1) * ϕ x ≤ ϕ (u (n + 1)) := by
      refine' @geom_le (ϕ ∘ u) _ zero_le_two (n + 1) fun m hm => _
      exact (IH _ <| Nat.lt_add_one_iff.1 hm).2.le
    exact hu (n + 1) ⟨A, B⟩
  cases' forall_and.mp key with key₁ key₂
  clear hu key
  -- Hence u is Cauchy
  have cauchy_u : CauchySeq u := by
    refine' cauchySeq_of_le_geometric _ ε one_half_lt_one fun n => _
    

Proof state:
X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
this : Nonempty X
F : ℕ → X → X
u : ℕ → X := fun n => Nat.recOn n x F
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / 2 ^ x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
n : ℕ
⊢ d (u n) (u (n + 1)) ≤ ε * (1 / 2) ^ n

Next tactic:
simpa only [one_div, inv_pow] using key₁ n
theorem hofer {X : Type*} [MetricSpace X] [CompleteSpace X] (x : X) (ε : ℝ) (ε_pos : 0 < ε)
    {ϕ : X → ℝ} (cont : Continuous ϕ) (nonneg : ∀ y, 0 ≤ ϕ y) : ∃ ε' > 0, ∃ x' : X,
    ε' ≤ ε ∧ d x' x ≤ 2 * ε ∧ ε * ϕ x ≤ ε' * ϕ x' ∧ ∀ y, d x' y ≤ ε' → ϕ y ≤ 2 * ϕ x' := by
  by_contra H
  have reformulation : ∀ (x') (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x' := by
    intro x' k
    rw [div_mul_eq_mul_div, le_div_iff, mul_assoc, mul_le_mul_left ε_pos, mul_comm]
    positivity
  -- Now let's specialize to `ε/2^k`
  replace H :
    ∀ k : ℕ, ∀ x', d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x' → ∃ y, d x' y ≤ ε / 2 ^ k ∧ 2 * ϕ x' < ϕ y
  · intro k x'
    push_neg at H
    have := H (ε / 2 ^ k) (by simp [ε_pos]) x' (by simp [ε_pos.le, one_le_two])
    simpa [reformulation] using this
  clear reformulation
  haveI : Nonempty X := ⟨x⟩
  choose! F hF using H
  -- Use the axiom of choice
  -- Now define u by induction starting at x, with u_{n+1} = F(n, u_n)
  let u : ℕ → X := fun n => Nat.recOn n x F
  -- The properties of F translate to properties of u
  have hu :
    ∀ n,
      d (u n) x ≤ 2 * ε ∧ 2 ^ n * ϕ x ≤ ϕ (u n) →
        d (u n) (u <| n + 1) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u <| n + 1) := by
    intro n
    exact hF n (u n)
  clear hF
  -- Key properties of u, to be proven by induction
  have key : ∀ n, d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1)) := by
    intro n
    induction' n using Nat.case_strong_induction_on with n IH
    · specialize hu 0
      simpa [mul_nonneg_iff, zero_le_one, ε_pos.le, le_refl] using hu
    have A : d (u (n + 1)) x ≤ 2 * ε := by
      rw [dist_comm]
      let r := range (n + 1) -- range (n+1) = {0, ..., n}
      calc
        d (u 0) (u (n + 1)) ≤ ∑ i in r, d (u i) (u <| i + 1) := dist_le_range_sum_dist u (n + 1)
        _ ≤ ∑ i in r, ε / 2 ^ i :=
          (sum_le_sum fun i i_in => (IH i <| Nat.lt_succ_iff.mp <| Finset.mem_range.mp i_in).1)
        _ = ∑ i in r, (1 / 2) ^ i * ε := by
          congr with i
          field_simp
        _ = (∑ i in r, ((1 : ℝ) / 2) ^ i) * ε := Finset.sum_mul.symm
        _ ≤ 2 * ε := by gcongr; apply sum_geometric_two_le
    have B : 2 ^ (n + 1) * ϕ x ≤ ϕ (u (n + 1)) := by
      refine' @geom_le (ϕ ∘ u) _ zero_le_two (n + 1) fun m hm => _
      exact (IH _ <| Nat.lt_add_one_iff.1 hm).2.le
    exact hu (n + 1) ⟨A, B⟩
  cases' forall_and.mp key with key₁ key₂
  clear hu key
  -- Hence u is Cauchy
  have cauchy_u : CauchySeq u := by
    refine' cauchySeq_of_le_geometric _ ε one_half_lt_one fun n => _
    simpa only [one_div, inv_pow] using key₁ n
  -- So u converges to some y
  

Proof state:
case intro
X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
this : Nonempty X
F : ℕ → X → X
u : ℕ → X := fun n => Nat.recOn n x F
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / 2 ^ x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
cauchy_u : CauchySeq u
⊢ False

Next tactic:
obtain ⟨y, limy⟩ : ∃ y, Tendsto u atTop (𝓝 y)
theorem actionDiagonalSucc_hom_apply {G : Type u} [Group G] {n : ℕ} (f : Fin (n + 1) → G) :
    (actionDiagonalSucc G n).hom.hom f = (f 0, fun i => (f (Fin.castSucc i))⁻¹ * f i.succ) := by
  

Proof state:
k G✝ : Type u
inst✝² : CommRing k
n✝ : ℕ
inst✝¹ : Group G✝
G : Type u
inst✝ : Group G
n : ℕ
f : Fin (n + 1) → G
⊢ Hom.hom (actionDiagonalSucc G n).hom f = (f 0, fun i => (f (Fin.castSucc i))⁻¹ * f (Fin.succ i))

Next tactic:
induction' n with n hn
theorem actionDiagonalSucc_hom_apply {G : Type u} [Group G] {n : ℕ} (f : Fin (n + 1) → G) :
    (actionDiagonalSucc G n).hom.hom f = (f 0, fun i => (f (Fin.castSucc i))⁻¹ * f i.succ) := by
  induction' n with n hn
  · 

Proof state:
case zero
k G✝ : Type u
inst✝² : CommRing k
n : ℕ
inst✝¹ : Group G✝
G : Type u
inst✝ : Group G
f : Fin (Nat.zero + 1) → G
⊢ Hom.hom (actionDiagonalSucc G Nat.zero).hom f = (f 0, fun i => (f (Fin.castSucc i))⁻¹ * f (Fin.succ i))

Next tactic:
exact Prod.ext rfl (funext fun x => Fin.elim0 x)
theorem actionDiagonalSucc_hom_apply {G : Type u} [Group G] {n : ℕ} (f : Fin (n + 1) → G) :
    (actionDiagonalSucc G n).hom.hom f = (f 0, fun i => (f (Fin.castSucc i))⁻¹ * f i.succ) := by
  induction' n with n hn
  · exact Prod.ext rfl (funext fun x => Fin.elim0 x)
  · 

Proof state:
case succ
k G✝ : Type u
inst✝² : CommRing k
n✝ : ℕ
inst✝¹ : Group G✝
G : Type u
inst✝ : Group G
n : ℕ
hn :
  ∀ (f : Fin (n + 1) → G),
    Hom.hom (actionDiagonalSucc G n).hom f = (f 0, fun i => (f (Fin.castSucc i))⁻¹ * f (Fin.succ i))
f : Fin (Nat.succ n + 1) → G
⊢ Hom.hom (actionDiagonalSucc G (Nat.succ n)).hom f = (f 0, fun i => (f (Fin.castSucc i))⁻¹ * f (Fin.succ i))

Next tactic:
refine' Prod.ext rfl (funext fun x => _)
theorem actionDiagonalSucc_hom_apply {G : Type u} [Group G] {n : ℕ} (f : Fin (n + 1) → G) :
    (actionDiagonalSucc G n).hom.hom f = (f 0, fun i => (f (Fin.castSucc i))⁻¹ * f i.succ) := by
  induction' n with n hn
  · exact Prod.ext rfl (funext fun x => Fin.elim0 x)
  · refine' Prod.ext rfl (funext fun x => _)
/- Porting note: broken proof was
    · dsimp only [actionDiagonalSucc]
      simp only [Iso.trans_hom, comp_hom, types_comp_apply, diagonalSucc_hom_hom,
        leftRegularTensorIso_hom_hom, tensorIso_hom, mkIso_hom_hom, Equiv.toIso_hom,
        Action.tensorHom, Equiv.piFinSuccAbove_symm_apply, tensor_apply, types_id_apply,
        tensor_rho, MonoidHom.one_apply, End.one_def, hn fun j : Fin (n + 1) => f j.succ,
        Fin.insertNth_zero']
      refine' Fin.cases (Fin.cons_zero _ _) (fun i => _) x
      · simp only [Fin.cons_succ, mul_left_inj, inv_inj, Fin.castSucc_fin_succ] -/
    · 

Proof state:
case succ
k G✝ : Type u
inst✝² : CommRing k
n✝ : ℕ
inst✝¹ : Group G✝
G : Type u
inst✝ : Group G
n : ℕ
hn :
  ∀ (f : Fin (n + 1) → G),
    Hom.hom (actionDiagonalSucc G n).hom f = (f 0, fun i => (f (Fin.castSucc i))⁻¹ * f (Fin.succ i))
f : Fin (Nat.succ n + 1) → G
x : Fin (Nat.succ n)
⊢ (Hom.hom (actionDiagonalSucc G (Nat.succ n)).hom f).2 x = (f 0, fun i => (f (Fin.castSucc i))⁻¹ * f (Fin.succ i)).2 x

Next tactic:
dsimp [actionDiagonalSucc]
theorem actionDiagonalSucc_hom_apply {G : Type u} [Group G] {n : ℕ} (f : Fin (n + 1) → G) :
    (actionDiagonalSucc G n).hom.hom f = (f 0, fun i => (f (Fin.castSucc i))⁻¹ * f i.succ) := by
  induction' n with n hn
  · exact Prod.ext rfl (funext fun x => Fin.elim0 x)
  · refine' Prod.ext rfl (funext fun x => _)
/- Porting note: broken proof was
    · dsimp only [actionDiagonalSucc]
      simp only [Iso.trans_hom, comp_hom, types_comp_apply, diagonalSucc_hom_hom,
        leftRegularTensorIso_hom_hom, tensorIso_hom, mkIso_hom_hom, Equiv.toIso_hom,
        Action.tensorHom, Equiv.piFinSuccAbove_symm_apply, tensor_apply, types_id_apply,
        tensor_rho, MonoidHom.one_apply, End.one_def, hn fun j : Fin (n + 1) => f j.succ,
        Fin.insertNth_zero']
      refine' Fin.cases (Fin.cons_zero _ _) (fun i => _) x
      · simp only [Fin.cons_succ, mul_left_inj, inv_inj, Fin.castSucc_fin_succ] -/
    · dsimp [actionDiagonalSucc]
      

Proof state:
case succ
k G✝ : Type u
inst✝² : CommRing k
n✝ : ℕ
inst✝¹ : Group G✝
G : Type u
inst✝ : Group G
n : ℕ
hn :
  ∀ (f : Fin (n + 1) → G),
    Hom.hom (actionDiagonalSucc G n).hom f = (f 0, fun i => (f (Fin.castSucc i))⁻¹ * f (Fin.succ i))
f : Fin (Nat.succ n + 1) → G
x : Fin (Nat.succ n)
⊢ (Equiv.piFinSuccAbove (fun x => G) 0).symm
      ((FunctorCategoryEquivalence.inverse.obj
            (FunctorCategoryEquivalence.functor.obj (leftRegular G) ⊗
              FunctorCategoryEquivalence.functor.obj { V := Fin n → G, ρ := 1 })).ρ
        ((Equiv.piFinSuccAbove (fun a => G) 0) f).1⁻¹
        (Hom.hom (actionDiagonalSucc G n).hom ((Equiv.piFinSuccAbove (fun a => G) 0) f).2))
      x =
    (f (Fin.castSucc x))⁻¹ * f (Fin.succ x)

Next tactic:
erw [hn (fun (j : Fin (n + 1)) => f j.succ)]
theorem actionDiagonalSucc_hom_apply {G : Type u} [Group G] {n : ℕ} (f : Fin (n + 1) → G) :
    (actionDiagonalSucc G n).hom.hom f = (f 0, fun i => (f (Fin.castSucc i))⁻¹ * f i.succ) := by
  induction' n with n hn
  · exact Prod.ext rfl (funext fun x => Fin.elim0 x)
  · refine' Prod.ext rfl (funext fun x => _)
/- Porting note: broken proof was
    · dsimp only [actionDiagonalSucc]
      simp only [Iso.trans_hom, comp_hom, types_comp_apply, diagonalSucc_hom_hom,
        leftRegularTensorIso_hom_hom, tensorIso_hom, mkIso_hom_hom, Equiv.toIso_hom,
        Action.tensorHom, Equiv.piFinSuccAbove_symm_apply, tensor_apply, types_id_apply,
        tensor_rho, MonoidHom.one_apply, End.one_def, hn fun j : Fin (n + 1) => f j.succ,
        Fin.insertNth_zero']
      refine' Fin.cases (Fin.cons_zero _ _) (fun i => _) x
      · simp only [Fin.cons_succ, mul_left_inj, inv_inj, Fin.castSucc_fin_succ] -/
    · dsimp [actionDiagonalSucc]
      erw [hn (fun (j : Fin (n + 1)) => f j.succ)]
      

Proof state:
case succ
k G✝ : Type u
inst✝² : CommRing k
n✝ : ℕ
inst✝¹ : Group G✝
G : Type u
inst✝ : Group G
n : ℕ
hn :
  ∀ (f : Fin (n + 1) → G),
    Hom.hom (actionDiagonalSucc G n).hom f = (f 0, fun i => (f (Fin.castSucc i))⁻¹ * f (Fin.succ i))
f : Fin (Nat.succ n + 1) → G
x : Fin (Nat.succ n)
⊢ (Equiv.piFinSuccAbove (fun x => G) 0).symm
      ((FunctorCategoryEquivalence.inverse.obj
            (FunctorCategoryEquivalence.functor.obj (leftRegular G) ⊗
              FunctorCategoryEquivalence.functor.obj { V := Fin n → G, ρ := 1 })).ρ
        ((Equiv.piFinSuccAbove (fun a => G) 0) f).1⁻¹
        (f (Fin.succ 0), fun i => (f (Fin.succ (Fin.castSucc i)))⁻¹ * f (Fin.succ (Fin.succ i))))
      x =
    (f (Fin.castSucc x))⁻¹ * f (Fin.succ x)

Next tactic:
exact Fin.cases rfl (fun i => rfl) x