awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.Lipschitz (goal)
Goal:
α : Type u
β : Type v
γ : Type w
inst : PseudoEMetricSpace α
inst_1 : TopologicalSpace β
inst_2 : PseudoEMetricSpace γ
f : α × β → γ
s s' : Set α
t : Set β
hs' : s' ⊆ s
hss' : s ⊆ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s'
K : ℝ≥0
ha : ∀ a ∈ s', ContinuousOn.{v, w} (Y := γ) (fun (y : β) ↦ f (a, y)) t
hb : ∀ b ∈ t, LipschitzOnWith.{u, w} (β := γ) K (fun (x : α) ↦ f (x, b)) s
x : α
y : β
hx : x ∈ s
hy : y ∈ t
ε : ℝ≥0∞
ε0 : 0 < ε / 2
δ : ℝ≥0
δpos : 0 < (↑δ : ℝ≥0∞)
hδ : (↑δ : ℝ≥0∞) * (↑(3 * K) : ℝ≥0∞) < ε / 2
x' : α
hx' : x' ∈ s'
hxx' : EDist.edist x x' < (↑δ : ℝ≥0∞)
A : s ∩ Metric.eball x (↑δ : ℝ≥0∞) ∈ 𝓝[s] x
B : t ∩ {b : β | EDist.edist (f (x', b)) (f (x', y)) ≤ ε / 2} ∈ 𝓝[t] y
a : α
b : β
has : (a, b).1 ∈ s
hax : (a, b).1 ∈ Metric.eball x (↑δ : ℝ≥0∞)
hbt : (a, b).2 ∈ t
hby : Membership.mem (γ := Set β) {b : β | EDist.edist (f (x', b)) (f (x', y)) ≤ ε / 2} (a, b).2
edist :
  EDist.edist (f (a, b)) (f (x', b)) + EDist.edist (f (x', b)) (f (x', y)) + EDist.edist (f (x', y)) (f (x, y)) ≤
    (↑K : ℝ≥0∞) * ((↑δ : ℝ≥0∞) + (↑δ : ℝ≥0∞)) + ε / 2 + (↑K : ℝ≥0∞) * (↑δ : ℝ≥0∞)
edist₁ :
  (↑K : ℝ≥0∞) * ((↑δ : ℝ≥0∞) + (↑δ : ℝ≥0∞)) + ε / 2 + (↑K : ℝ≥0∞) * (↑δ : ℝ≥0∞) =
    (↑δ : ℝ≥0∞) * (↑(3 * K) : ℝ≥0∞) + ε / 2
⊢ f (a, b) ∈ Metric.closedEBall (f (x, y)) ε
have edist₂ : (↑δ : ℝ≥0∞) * (↑(3 * K) : ℝ≥0∞) + ε / 2 ≤ ε / 2 + ε / 2
Tactic: calc
    edist (f (a, b)) (f (x, y)) ≤ edist (f (a, b)) (f (x', b)) + edist (f (x', b)) (f (x', y)) +
        edist (f (x', y)) (f (x, y)) := edist_triangle4 _ _ _ _
    _ ≤ K * (δ + δ) + ε / 2 + K * δ := by
      gcongr
      · refine (hb b hbt).edist_le_mul_of_le has (hs' hx') ?_
        exact (edist_triangle _ _ _).trans (add_le_add (le_of_lt hax) hxx'.le)
      · exact hby
      · exact (hb y hy).edist_le_mul_of_le (hs' hx') hx ((edist_comm _ _).trans_le hxx'.le)
    _ = δ * ↑(3 * K) + ε / 2 := by push_cast; ring
    _ ≤ ε / 2 + ε / 2 := by gcongr
    _ = ε := ENNReal.add_halves _
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.Lipschitz (goal)
Goal:
α : Type u
β : Type v
γ : Type w
inst : PseudoEMetricSpace α
inst_1 : TopologicalSpace β
inst_2 : PseudoEMetricSpace γ
f : α × β → γ
K : ℝ≥0
s : Set α
hs : Dense s
ha : ∀ a ∈ s, Continuous[inst_1, PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] fun (y : β) ↦ f (a, y)
hb : ∀ (b : β), LipschitzWith.{u, w} (β := γ) K fun (x : α) ↦ f (x, b)
⊢ Continuous[instTopologicalSpaceProd, PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] f
have simp : ∀ (ha : ∀ a ∈ s, ContinuousOn.{v, w} (Y := γ) (fun (y : β) ↦ f (a, y)) univ)
  (hb : ∀ (b : β), LipschitzOnWith.{u, w} (β := γ) K (fun (x : α) ↦ f (x, b)) univ),
  ContinuousOn f (SProd.sprod (α := Set α) (β := Set β) univ univ)
Tactic: simp only [← continuousOn_univ, ← univ_prod_univ, ← lipschitzOnWith_univ] at *
Cleared hypotheses: ha, hb
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (hypothesis)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a : ℝ≥0∞
t : T
V : Finset T
ha : 1 < a
c : ℝ≥0∞
⊢ ∃ (r : ℕ), 1 ≤ r ∧ (↑(#({x ∈ V | edist t x ≤ (↑r : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ r
have this : ∀ᶠ (a_1 : ℕ) in Filter.atTop, LT.lt (α := ℝ≥0∞) (↑(#V) : ℝ≥0∞) (((fun (x : ℝ) ↦ a ^ x) ∘ Nat.cast) a_1)
Tactic: have := ENNReal.tendsto_nhds_top_iff_nat.1
    ((ENNReal.tendsto_rpow_atTop_of_one_lt_base ha).comp tendsto_natCast_atTop_atTop) #V
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (hypothesis)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a : ℝ≥0∞
t : T
V : Finset T
ha : 1 < a
c : ℝ≥0∞
this : ∀ᶠ (a_1 : ℕ) in Filter.atTop, LT.lt (α := ℝ≥0∞) (↑(#V) : ℝ≥0∞) (((fun (x : ℝ) ↦ a ^ x) ∘ Nat.cast) a_1)
⊢ ∃ (r : ℕ), 1 ≤ r ∧ (↑(#({x ∈ V | edist t x ≤ (↑r : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ r
have this₁ : ∃ (a_1 : ℕ), ∀ (b : ℕ), a_1 ≤ b → (↑(#V) : ℝ≥0∞) < a ^ b
Tactic: simp only [Function.comp_apply, ENNReal.rpow_natCast, Filter.eventually_atTop, ge_iff_le] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a : ℝ≥0∞
t : T
V : Finset T
ha : 1 < a
c : ℝ≥0∞
this : ∃ (a_1 : ℕ), ∀ (b : ℕ), a_1 ≤ b → (↑(#V) : ℝ≥0∞) < a ^ b
⊢ ∃ (r : ℕ), 1 ≤ r ∧ (↑(#({x ∈ V | edist t x ≤ (↑r : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ r
have obtain : ∀ (r : ℕ) (hr : ∀ (b : ℕ), r ≤ b → (↑(#V) : ℝ≥0∞) < a ^ b),
  ∃ (r : ℕ), 1 ≤ r ∧ (↑(#({x ∈ V | edist t x ≤ (↑r : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ r
Tactic: obtain ⟨r, hr⟩ := this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
V : Finset T
t : T
ha : 1 < a
⊢ 1 ≤ logSizeRadius t V a c
have rw : 1 ≤ Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) sorry
Tactic: rw [logSizeRadius, dif_pos ha]
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
V : Finset T
t : T
ha : 1 < a
⊢ (↑(#({x ∈ V | edist t x ≤ (↑(logSizeRadius t V a c) : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ logSizeRadius t V a c
have rw : (↑(#({x ∈ V |
          edist t x ≤
            (↑(Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) sorry) :
                ℝ≥0∞) *
              c})) :
    ℝ≥0∞) ≤
  a ^ Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) sorry
Tactic: rw [logSizeRadius, dif_pos ha]
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
V : Finset T
t : T
ha : 1 < a
ht : t ∈ V
⊢ a ^ (logSizeRadius t V a c - 1) ≤
    (↑(#({x ∈ V | edist t x ≤ HSub.hSub (α := ℝ≥0∞) (↑(logSizeRadius t V a c) : ℝ≥0∞) 1 * c})) : ℝ≥0∞)
have pos : ∀ (h_one : logSizeRadius t V a c = 1),
  a ^ (logSizeRadius t V a c - 1) ≤
    (↑(#({x ∈ V | edist t x ≤ HSub.hSub (α := ℝ≥0∞) (↑(logSizeRadius t V a c) : ℝ≥0∞) 1 * c})) : ℝ≥0∞)
Tactic: by_cases h_one : logSizeRadius t V a c = 1
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
V : Finset T
t : T
ha : 1 < a
ht : t ∈ V
pos :
  ∀ (h_one : logSizeRadius t V a c = 1),
    a ^ (logSizeRadius t V a c - 1) ≤
      (↑(#({x ∈ V | edist t x ≤ HSub.hSub (α := ℝ≥0∞) (↑(logSizeRadius t V a c) : ℝ≥0∞) 1 * c})) : ℝ≥0∞)
⊢ a ^ (logSizeRadius t V a c - 1) ≤
    (↑(#({x ∈ V | edist t x ≤ HSub.hSub (α := ℝ≥0∞) (↑(logSizeRadius t V a c) : ℝ≥0∞) 1 * c})) : ℝ≥0∞)
have neg : ∀ (h_one : ¬logSizeRadius t V a c = 1),
  a ^ (logSizeRadius t V a c - 1) ≤
    (↑(#({x ∈ V | edist t x ≤ HSub.hSub (α := ℝ≥0∞) (↑(logSizeRadius t V a c) : ℝ≥0∞) 1 * c})) : ℝ≥0∞)
Tactic: by_cases h_one : logSizeRadius t V a c = 1
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
V : Finset T
t : T
ha : 1 < a
ht : t ∈ V
h_one : logSizeRadius t V a c = 1
⊢ a ^ (logSizeRadius t V a c - 1) ≤
    (↑(#({x ∈ V | edist t x ≤ HSub.hSub (α := ℝ≥0∞) (↑(logSizeRadius t V a c) : ℝ≥0∞) 1 * c})) : ℝ≥0∞)
have pos : {x ∈ V | edist t x = 0}.Nonempty
Tactic: simp only [h_one, tsub_self, pow_zero, Nat.cast_one, zero_mul, nonpos_iff_eq_zero,
      Nat.one_le_cast, Finset.one_le_card]
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
V : Finset T
t : T
ha : 1 < a
ht : t ∈ V
h_one : ¬logSizeRadius t V a c = 1
⊢ a ^ (logSizeRadius t V a c - 1) ≤
    (↑(#({x ∈ V | edist t x ≤ HSub.hSub (α := ℝ≥0∞) (↑(logSizeRadius t V a c) : ℝ≥0∞) 1 * c})) : ℝ≥0∞)
have neg : ∀ (h_one : ¬Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) sorry = 1),
  a ^ (Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) sorry - 1) ≤
    (↑(#({x ∈ V |
            edist t x ≤
              HSub.hSub (α := ℝ≥0∞)
                  (↑(Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n)
                        sorry) :
                    ℝ≥0∞)
                  1 *
                c})) :
      ℝ≥0∞)
Tactic: rw [logSizeRadius, dif_pos ha] at h_one ⊢
Cleared hypotheses: h_one
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (hypothesis)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
V : Finset T
t : T
ha : 1 < a
ht : t ∈ V
h_one : ¬Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) ⋯ = 1
⊢ a ^ (Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) ⋯ - 1) ≤
    (↑(#({x ∈ V |
            edist t x ≤
              HSub.hSub (α := ℝ≥0∞)
                  (↑(Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n)
                        ⋯) :
                    ℝ≥0∞)
                  1 *
                c})) :
      ℝ≥0∞)
have this : Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) sorry - 1 <
  Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) sorry
Tactic: have : Nat.find (exists_radius_le t V ha c) - 1 < Nat.find (exists_radius_le t V ha c) := by
    simp
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (hypothesis)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
V : Finset T
t : T
ha : 1 < a
ht : t ∈ V
h_one : ¬Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) ⋯ = 1
this :
  Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) ⋯ - 1 <
    Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) ⋯
⊢ a ^ (Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) ⋯ - 1) ≤
    (↑(#({x ∈ V |
            edist t x ≤
              HSub.hSub (α := ℝ≥0∞)
                  (↑(Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n)
                        ⋯) :
                    ℝ≥0∞)
                  1 *
                c})) :
      ℝ≥0∞)
have h : ¬(1 ≤ Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) sorry - 1 ∧
    (↑(#({x ∈ V |
              edist t x ≤
                (↑(Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n)
                          sorry -
                        1) :
                    ℝ≥0∞) *
                  c})) :
        ℝ≥0∞) ≤
      a ^ (Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) sorry - 1))
Tactic: have h := Nat.find_min (exists_radius_le t V ha c) this
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (hypothesis)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
V : Finset T
t : T
ha : 1 < a
ht : t ∈ V
h_one : ¬Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) ⋯ = 1
this :
  Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) ⋯ - 1 <
    Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) ⋯
h :
  ¬(1 ≤ Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) ⋯ - 1 ∧
      (↑(#({x ∈ V |
                edist t x ≤
                  (↑(Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n)
                            ⋯ -
                          1) :
                      ℝ≥0∞) *
                    c})) :
          ℝ≥0∞) ≤
        a ^ (Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) ⋯ - 1))
⊢ a ^ (Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) ⋯ - 1) ≤
    (↑(#({x ∈ V |
            edist t x ≤
              HSub.hSub (α := ℝ≥0∞)
                  (↑(Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n)
                        ⋯) :
                    ℝ≥0∞)
                  1 *
                c})) :
      ℝ≥0∞)
have h₁ : 1 ≤ Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) sorry - 1 →
  a ^ (Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) sorry - 1) <
    (↑(#({x ∈ V |
            edist t x ≤
              HSub.hSub (α := ℝ≥0∞)
                  (↑(Nat.find (p := fun (n : ℕ) ↦ 1 ≤ n ∧ (↑(#({x ∈ V | edist t x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n)
                        sorry) :
                    ℝ≥0∞)
                  1 *
                c})) :
      ℝ≥0∞)
Tactic: simp only [ENNReal.natCast_sub, Nat.cast_one, not_and, not_le] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
hJ : J.Nonempty
i : ℕ
⊢ (logSizeBallSeq J hJ a c i).finset ⊆ J
have apply : (fun (i : ℕ) ↦ (logSizeBallSeq J hJ a c i).finset) 0 ⊆ J
Tactic: apply subset_trans <| antitone_logSizeBallSeq_add_one_subset hJ zero_le
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
n : ℕ
J : Finset T
inst_1 : DecidableEq T
hJ : J.Nonempty
ha : 1 < a
hn : 1 ≤ n
hJ_card : (↑(#J) : ℝ≥0∞) ≤ a ^ n
i : ℕ
⊢ (logSizeBallSeq J hJ a c 0).radius ≤ n
have simp : Nat.find (p := fun (n : ℕ) ↦
    1 ≤ n ∧ (↑(#({x ∈ J | edist (Exists.choose (p := Membership.mem J) hJ) x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞) ≤ a ^ n) sorry ≤
  n
Tactic: simp only [radius_logSizeBallSeq_zero, logSizeRadius, ha, ↓reduceDIte]
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
n : ℕ
J : Finset T
inst_1 : DecidableEq T
hJ : J.Nonempty
ha : 1 < a
hn : 1 ≤ n
hJ_card : (↑(#J) : ℝ≥0∞) ≤ a ^ n
i : ℕ
⊢ LE.le (α := ℝ≥0∞) (↑(#({x ∈ J | edist (Exists.choose (p := Membership.mem J) hJ) x ≤ (↑n : ℝ≥0∞) * c})) : ℝ≥0∞)
    (↑(#J) : ℝ≥0∞)
have a₁ : {x ∈ J | edist (Exists.choose (p := Membership.mem J) hJ) x ≤ (↑n : ℝ≥0∞) * c} ⊆ J
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
n : ℕ
J : Finset T
inst_1 : DecidableEq T
hJ : J.Nonempty
ha : 1 < a
hn : 1 ≤ n
hJ_card : (↑(#J) : ℝ≥0∞) ≤ a ^ n
i_1 i : ℕ
⊢ (logSizeBallSeq J hJ a c (i + 1)).radius ≤ n
have simp : Nat.find (p := fun (n : ℕ) ↦
    1 ≤ n ∧
      (↑(#({x ∈ (logSizeBallSeq J hJ a c (i + 1)).finset |
                edist (logSizeBallSeq J hJ a c (i + 1)).point x ≤ (↑n : ℝ≥0∞) * c})) :
          ℝ≥0∞) ≤
        a ^ n)
    sorry ≤
  n
Tactic: simp only [radius_logSizeBallSeq_add_one, logSizeRadius, ha, ↓reduceDIte]
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
n : ℕ
J : Finset T
inst_1 : DecidableEq T
hJ : J.Nonempty
ha : 1 < a
hn : 1 ≤ n
hJ_card : (↑(#J) : ℝ≥0∞) ≤ a ^ n
i_1 i : ℕ
⊢ Nat.find (p := fun (n : ℕ) ↦
      1 ≤ n ∧
        (↑(#({x ∈ (logSizeBallSeq J hJ a c (i + 1)).finset |
                  edist (logSizeBallSeq J hJ a c (i + 1)).point x ≤ (↑n : ℝ≥0∞) * c})) :
            ℝ≥0∞) ≤
          a ^ n)
      ⋯ ≤
    n
have refine : LE.le (α := ℝ≥0∞)
  (↑(#({x ∈ (logSizeBallSeq J hJ a c (i + 1)).finset |
          edist (logSizeBallSeq J hJ a c (i + 1)).point x ≤ (↑n : ℝ≥0∞) * c})) :
    ℝ≥0∞)
  (↑(#J) : ℝ≥0∞)
Tactic: refine Nat.find_min' _ ⟨hn, le_trans ?_ hJ_card⟩
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
n : ℕ
J : Finset T
inst_1 : DecidableEq T
hJ : J.Nonempty
ha : 1 < a
hn : 1 ≤ n
hJ_card : (↑(#J) : ℝ≥0∞) ≤ a ^ n
i_1 i : ℕ
⊢ LE.le (α := ℝ≥0∞)
    (↑(#({x ∈ (logSizeBallSeq J hJ a c (i + 1)).finset |
            edist (logSizeBallSeq J hJ a c (i + 1)).point x ≤ (↑n : ℝ≥0∞) * c})) :
      ℝ≥0∞)
    (↑(#J) : ℝ≥0∞)
have a₁ : {x ∈ (logSizeBallSeq J hJ a c (i + 1)).finset | edist (logSizeBallSeq J hJ a c (i + 1)).point x ≤ (↑n : ℝ≥0∞) * c} ⊆ J
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
hJ : J.Nonempty
i : ℕ
ih : (logSizeBallSeq J hJ a c i).point ∈ J
⊢ (logSizeBallSeq J hJ a c (i + 1)).point ∈ J
have pos : ∀ (h : (logSizeBallSeq J hJ a c (i + 1)).finset.Nonempty), (logSizeBallSeq J hJ a c (i + 1)).point ∈ J
Tactic: by_cases h : (logSizeBallSeq J hJ a c (i + 1)).finset.Nonempty
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
hJ : J.Nonempty
i : ℕ
ih : (logSizeBallSeq J hJ a c i).point ∈ J
pos : ∀ (h : (logSizeBallSeq J hJ a c (i + 1)).finset.Nonempty), (logSizeBallSeq J hJ a c (i + 1)).point ∈ J
⊢ (logSizeBallSeq J hJ a c (i + 1)).point ∈ J
have neg : ∀ (h : ¬(logSizeBallSeq J hJ a c (i + 1)).finset.Nonempty), (logSizeBallSeq J hJ a c (i + 1)).point ∈ J
Tactic: by_cases h : (logSizeBallSeq J hJ a c (i + 1)).finset.Nonempty
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
hJ : J.Nonempty
i : ℕ
ih : (logSizeBallSeq J hJ a c i).point ∈ J
h : (logSizeBallSeq J hJ a c (i + 1)).finset.Nonempty
⊢ (logSizeBallSeq J hJ a c (i + 1)).point ∈ J
have pos : (logSizeBallSeq J hJ a c (i + 1)).finset ⊆ J
Tactic: refine Finset.mem_of_subset ?_ (point_mem_finset_logSizeBallSeq hJ (i + 1) h)
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
hJ : J.Nonempty
i : ℕ
⊢ (logSizeBallSeq J hJ a c i).point ∈ J
have zero : (logSizeBallSeq J hJ a c 0).point ∈ J
Tactic: induction i with
  | zero => exact point_mem_finset_logSizeBallSeq hJ 0 hJ
  | succ i ih =>
    by_cases h : (logSizeBallSeq J hJ a c (i + 1)).finset.Nonempty
    · refine Finset.mem_of_subset ?_ (point_mem_finset_logSizeBallSeq hJ (i + 1) h)
      apply finset_logSizeBallSeq_subset_logSizeBallSeq_init
    simp [point_logSizeBallSeq_add_one, ih, h]
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
hJ : J.Nonempty
i : ℕ
zero : (logSizeBallSeq J hJ a c 0).point ∈ J
⊢ (logSizeBallSeq J hJ a c i).point ∈ J
have succ : ∀ (i : ℕ) (ih : (logSizeBallSeq J hJ a c i).point ∈ J), (logSizeBallSeq J hJ a c (i + 1)).point ∈ J
Tactic: induction i with
  | zero => exact point_mem_finset_logSizeBallSeq hJ 0 hJ
  | succ i ih =>
    by_cases h : (logSizeBallSeq J hJ a c (i + 1)).finset.Nonempty
    · refine Finset.mem_of_subset ?_ (point_mem_finset_logSizeBallSeq hJ (i + 1) h)
      apply finset_logSizeBallSeq_subset_logSizeBallSeq_init
    simp [point_logSizeBallSeq_add_one, ih, h]
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
hJ : J.Nonempty
i : ℕ
h : (logSizeBallSeq J hJ a c i).finset.Nonempty
⊢ (logSizeBallSeq J hJ a c (i + 1)).finset ⊂ (logSizeBallSeq J hJ a c i).finset
have h₁ : (logSizeBallSeq J hJ a c (i + 1)).finset ⊆ (logSizeBallSeq J hJ a c i).finset
Tactic: apply ssubset_of_subset_not_subset
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
hJ : J.Nonempty
i : ℕ
h : (logSizeBallSeq J hJ a c i).finset.Nonempty
h₁ : (logSizeBallSeq J hJ a c (i + 1)).finset ⊆ (logSizeBallSeq J hJ a c i).finset
⊢ (logSizeBallSeq J hJ a c (i + 1)).finset ⊂ (logSizeBallSeq J hJ a c i).finset
have h₂ : ¬(logSizeBallSeq J hJ a c i).finset ⊆ (logSizeBallSeq J hJ a c (i + 1)).finset
Tactic: apply ssubset_of_subset_not_subset
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
hJ : J.Nonempty
i : ℕ
h : (logSizeBallSeq J hJ a c i).finset.Nonempty
⊢ ¬(logSizeBallSeq J hJ a c i).finset ⊆ (logSizeBallSeq J hJ a c (i + 1)).finset
have h₂ : (logSizeBallSeq J hJ a c i).point ∈ (↑(logSizeBallSeq J hJ a c i).finset : Set T)
Tactic: refine Set.not_subset.mpr ⟨(logSizeBallSeq J hJ a c i).point, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
hJ : J.Nonempty
i : ℕ
h : (logSizeBallSeq J hJ a c i).finset.Nonempty
h₂ : (logSizeBallSeq J hJ a c i).point ∈ (↑(logSizeBallSeq J hJ a c i).finset : Set T)
⊢ ¬(logSizeBallSeq J hJ a c i).finset ⊆ (logSizeBallSeq J hJ a c (i + 1)).finset
have h₃ : (logSizeBallSeq J hJ a c i).point ∉ (↑(logSizeBallSeq J hJ a c (i + 1)).finset : Set T)
Tactic: refine Set.not_subset.mpr ⟨(logSizeBallSeq J hJ a c i).point, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
hJ : J.Nonempty
i : ℕ
ih : #(logSizeBallSeq J hJ a c i).finset ≤ #J - i
⊢ #(logSizeBallSeq J hJ a c (i + 1)).finset ≤ #J - (i + 1)
have pos : ∀ (h : (logSizeBallSeq J hJ a c i).finset.Nonempty), #(logSizeBallSeq J hJ a c (i + 1)).finset ≤ #J - (i + 1)
Tactic: by_cases h : (logSizeBallSeq J hJ a c i).finset.Nonempty
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
hJ : J.Nonempty
i : ℕ
ih : #(logSizeBallSeq J hJ a c i).finset ≤ #J - i
pos : ∀ (h : (logSizeBallSeq J hJ a c i).finset.Nonempty), #(logSizeBallSeq J hJ a c (i + 1)).finset ≤ #J - (i + 1)
⊢ #(logSizeBallSeq J hJ a c (i + 1)).finset ≤ #J - (i + 1)
have neg : ∀ (h : ¬(logSizeBallSeq J hJ a c i).finset.Nonempty), #(logSizeBallSeq J hJ a c (i + 1)).finset ≤ #J - (i + 1)
Tactic: by_cases h : (logSizeBallSeq J hJ a c i).finset.Nonempty
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (hypothesis)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
hJ : J.Nonempty
i : ℕ
ih : #(logSizeBallSeq J hJ a c i).finset ≤ #J - i
h : (logSizeBallSeq J hJ a c i).finset.Nonempty
⊢ #(logSizeBallSeq J hJ a c (i + 1)).finset ≤ #J - (i + 1)
have this : #(logSizeBallSeq J hJ a c (i + 1)).finset < #(logSizeBallSeq J hJ a c i).finset
Tactic: have := card_finset_logSizeBallSeq_add_one_lt hJ i h
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
hJ : J.Nonempty
i : ℕ
ih : #(logSizeBallSeq J hJ a c i).finset ≤ #J - i
h : ¬(logSizeBallSeq J hJ a c i).finset.Nonempty
⊢ #(logSizeBallSeq J hJ a c (i + 1)).finset ≤ #J - (i + 1)
have neg : #(logSizeBallSeq J hJ a c i).finset ≤ #J - (i + 1)
Tactic: apply le_trans <| Finset.card_le_card (finset_logSizeBallSeq_add_one_subset hJ i)
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
hJ : J.Nonempty
i : ℕ
ih : #(logSizeBallSeq J hJ a c i).finset ≤ #J - i
h : ¬(logSizeBallSeq J hJ a c i).finset.Nonempty
⊢ #(logSizeBallSeq J hJ a c i).finset ≤ #J - (i + 1)
have neg : #(logSizeBallSeq J hJ a c i).finset = 0
Tactic: suffices #(logSizeBallSeq J hJ a c i).finset = 0 by simp [this]