Theorem: BoxIntegral.Box.subbox_induction_on'
File: Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean

Proof steps:
state_before: ι : Type u_1
I✝ J : Box ι
p : Box ι → Prop
I : Box ι
H_ind : ∀ J ≤ I, (∀ (s : Set ι), p (J.splitCenterBox s)) → p J
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
⊢ p I
state_after: ι : Type u_1
I✝ J : Box ι
p : Box ι → Prop
I : Box ι
H_ind : ∀ J ≤ I, (∀ (s : Set ι), p (J.splitCenterBox s)) → p J
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
⊢ False
tactic: by_contra hpI
state_before: ι : Type u_1
I✝ J : Box ι
p : Box ι → Prop
I : Box ι
H_ind : ∀ J ≤ I, (∀ (s : Set ι), p (J.splitCenterBox s)) → p J
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
⊢ False
state_after: ι : Type u_1
I✝ J : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
H_ind : ∀ J ≤ I, ¬p J → ¬∀ (s : Set ι), p (J.splitCenterBox s)
⊢ False
tactic: replace H_ind := fun J hJ ↦ not_imp_not.2 (H_ind J hJ)
state_before: ι : Type u_1
I✝ J : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
H_ind : ∀ J ≤ I, ¬p J → ¬∀ (s : Set ι), p (J.splitCenterBox s)
⊢ False
state_after: ι : Type u_1
I✝ J : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
H_ind : ∀ J ≤ I, ¬p J → ∃ x, ¬p (J.splitCenterBox x)
⊢ False
tactic: simp only [exists_imp, not_forall] at H_ind
state_before: ι : Type u_1
I✝ J : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
H_ind : ∀ J ≤ I, ¬p J → ∃ x, ¬p (J.splitCenterBox x)
⊢ False
state_after: ι : Type u_1
I✝ J : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
s : Box ι → Set ι
hs : ∀ J ≤ I, ¬p J → ¬p (J.splitCenterBox (s J))
⊢ False
tactic: choose! s hs using H_ind
state_before: ι : Type u_1
I✝ J : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
s : Box ι → Set ι
hs : ∀ J ≤ I, ¬p J → ¬p (J.splitCenterBox (s J))
⊢ False
state_after: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
s : Box ι → Set ι
hs : ∀ J ≤ I, ¬p J → ¬p (J.splitCenterBox (s J))
J : ℕ → Box ι := fun m => (fun J => J.splitCenterBox (s J))^[m] I
⊢ False
tactic: set J : ℕ → Box ι := fun m ↦ (fun J ↦ splitCenterBox J (s J))^[m] I
state_before: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
s : Box ι → Set ι
hs : ∀ J ≤ I, ¬p J → ¬p (J.splitCenterBox (s J))
J : ℕ → Box ι := fun m => (fun J => J.splitCenterBox (s J))^[m] I
⊢ False
state_after: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
s : Box ι → Set ι
hs : ∀ J ≤ I, ¬p J → ¬p (J.splitCenterBox (s J))
J : ℕ → Box ι := fun m => (fun J => J.splitCenterBox (s J))^[m] I
J_succ : ∀ (m : ℕ), J (m + 1) = (J m).splitCenterBox (s (J m))
⊢ False
tactic: have J_succ : ∀ m, J (m + 1) = splitCenterBox (J m) (s <| J m) :=
    fun m ↦ iterate_succ_apply' _ _ _
state_before: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
s : Box ι → Set ι
hs : ∀ J ≤ I, ¬p J → ¬p (J.splitCenterBox (s J))
J : ℕ → Box ι := fun m => (fun J => J.splitCenterBox (s J))^[m] I
J_succ : ∀ (m : ℕ), J (m + 1) = (J m).splitCenterBox (s (J m))
⊢ False
state_after: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
s : Box ι → Set ι
hs : ∀ J ≤ I, ¬p J → ¬p (J.splitCenterBox (s J))
J : ℕ → Box ι := fun m => (fun J => J.splitCenterBox (s J))^[m] I
J_succ : ∀ (m : ℕ), J (m + 1) = (J m).splitCenterBox (s (J m))
hJmono : Antitone J
⊢ False
tactic: have hJmono : Antitone J :=
    antitone_nat_of_succ_le fun n ↦ by simpa [J_succ] using splitCenterBox_le _ _
state_before: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
s : Box ι → Set ι
hs : ∀ J ≤ I, ¬p J → ¬p (J.splitCenterBox (s J))
J : ℕ → Box ι := fun m => (fun J => J.splitCenterBox (s J))^[m] I
J_succ : ∀ (m : ℕ), J (m + 1) = (J m).splitCenterBox (s (J m))
hJmono : Antitone J
⊢ False
state_after: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
s : Box ι → Set ι
hs : ∀ J ≤ I, ¬p J → ¬p (J.splitCenterBox (s J))
J : ℕ → Box ι := fun m => (fun J => J.splitCenterBox (s J))^[m] I
J_succ : ∀ (m : ℕ), J (m + 1) = (J m).splitCenterBox (s (J m))
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
⊢ False
tactic: have hJle : ∀ m, J m ≤ I := fun m ↦ hJmono (zero_le m)
state_before: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
s : Box ι → Set ι
hs : ∀ J ≤ I, ¬p J → ¬p (J.splitCenterBox (s J))
J : ℕ → Box ι := fun m => (fun J => J.splitCenterBox (s J))^[m] I
J_succ : ∀ (m : ℕ), J (m + 1) = (J m).splitCenterBox (s (J m))
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
⊢ False
state_after: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
s : Box ι → Set ι
hs : ∀ J ≤ I, ¬p J → ¬p (J.splitCenterBox (s J))
J : ℕ → Box ι := fun m => (fun J => J.splitCenterBox (s J))^[m] I
J_succ : ∀ (m : ℕ), J (m + 1) = (J m).splitCenterBox (s (J m))
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
⊢ False
tactic: have hJp : ∀ m, ¬p (J m) :=
    fun m ↦ Nat.recOn m hpI fun m ↦ by simpa only [J_succ] using hs (J m) (hJle m)
state_before: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
s : Box ι → Set ι
hs : ∀ J ≤ I, ¬p J → ¬p (J.splitCenterBox (s J))
J : ℕ → Box ι := fun m => (fun J => J.splitCenterBox (s J))^[m] I
J_succ : ∀ (m : ℕ), J (m + 1) = (J m).splitCenterBox (s (J m))
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
⊢ False
state_after: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
s : Box ι → Set ι
hs : ∀ J ≤ I, ¬p J → ¬p (J.splitCenterBox (s J))
J : ℕ → Box ι := fun m => (fun J => J.splitCenterBox (s J))^[m] I
J_succ : ∀ (m : ℕ), J (m + 1) = (J m).splitCenterBox (s (J m))
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
⊢ False
tactic: have h0 : J 0 = I := rfl
state_before: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
s : Box ι → Set ι
hs : ∀ J ≤ I, ¬p J → ¬p (J.splitCenterBox (s J))
J : ℕ → Box ι := fun m => (fun J => J.splitCenterBox (s J))^[m] I
J_succ : ∀ (m : ℕ), J (m + 1) = (J m).splitCenterBox (s (J m))
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
⊢ False
state_after: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
s : Box ι → Set ι
hs : ∀ J ≤ I, ¬p J → ¬p (J.splitCenterBox (s J))
J : ℕ → Box ι
J_succ : ∀ (m : ℕ), J (m + 1) = (J m).splitCenterBox (s (J m))
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
⊢ False
tactic: clear_value J
state_before: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
s : Box ι → Set ι
hs : ∀ J ≤ I, ¬p J → ¬p (J.splitCenterBox (s J))
J : ℕ → Box ι
J_succ : ∀ (m : ℕ), J (m + 1) = (J m).splitCenterBox (s (J m))
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
⊢ False
state_after: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
⊢ False
tactic: clear hpI hs J_succ s
state_before: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
⊢ False
state_after: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
z : ι → ℝ := ⨆ m, (J m).lower
⊢ False
tactic: set z : ι → ℝ := ⨆ m, (J m).lower
state_before: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
z : ι → ℝ := ⨆ m, (J m).lower
⊢ False
state_after: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
z : ι → ℝ := ⨆ m, (J m).lower
hzJ : ∀ (m : ℕ), z ∈ Box.Icc (J m)
⊢ False
tactic: have hzJ : ∀ m, z ∈ Box.Icc (J m) :=
    mem_iInter.1 (ciSup_mem_iInter_Icc_of_antitone_Icc
      ((@Box.Icc ι).monotone.comp_antitone hJmono) fun m ↦ (J m).lower_le_upper)
state_before: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
z : ι → ℝ := ⨆ m, (J m).lower
hzJ : ∀ (m : ℕ), z ∈ Box.Icc (J m)
⊢ False
state_after: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
z : ι → ℝ := ⨆ m, (J m).lower
hzJ : ∀ (m : ℕ), z ∈ Box.Icc (J m)
hJl_mem : ∀ (m : ℕ), (J m).lower ∈ Box.Icc I
⊢ False
tactic: have hJl_mem : ∀ m, (J m).lower ∈ Box.Icc I := fun m ↦ le_iff_Icc.1 (hJle m) (J m).lower_mem_Icc
state_before: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
z : ι → ℝ := ⨆ m, (J m).lower
hzJ : ∀ (m : ℕ), z ∈ Box.Icc (J m)
hJl_mem : ∀ (m : ℕ), (J m).lower ∈ Box.Icc I
⊢ False
state_after: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
z : ι → ℝ := ⨆ m, (J m).lower
hzJ : ∀ (m : ℕ), z ∈ Box.Icc (J m)
hJl_mem : ∀ (m : ℕ), (J m).lower ∈ Box.Icc I
hJu_mem : ∀ (m : ℕ), (J m).upper ∈ Box.Icc I
⊢ False
tactic: have hJu_mem : ∀ m, (J m).upper ∈ Box.Icc I := fun m ↦ le_iff_Icc.1 (hJle m) (J m).upper_mem_Icc
state_before: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
z : ι → ℝ := ⨆ m, (J m).lower
hzJ : ∀ (m : ℕ), z ∈ Box.Icc (J m)
hJl_mem : ∀ (m : ℕ), (J m).lower ∈ Box.Icc I
hJu_mem : ∀ (m : ℕ), (J m).upper ∈ Box.Icc I
⊢ False
state_after: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
z : ι → ℝ := ⨆ m, (J m).lower
hzJ : ∀ (m : ℕ), z ∈ Box.Icc (J m)
hJl_mem : ∀ (m : ℕ), (J m).lower ∈ Box.Icc I
hJu_mem : ∀ (m : ℕ), (J m).upper ∈ Box.Icc I
hJlz : Tendsto (fun m => (J m).lower) atTop (𝓝 z)
⊢ False
tactic: have hJlz : Tendsto (fun m ↦ (J m).lower) atTop (𝓝 z) :=
    tendsto_atTop_ciSup (antitone_lower.comp hJmono) ⟨I.upper, fun x ⟨m, hm⟩ ↦ hm ▸ (hJl_mem m).2⟩
state_before: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
z : ι → ℝ := ⨆ m, (J m).lower
hzJ : ∀ (m : ℕ), z ∈ Box.Icc (J m)
hJl_mem : ∀ (m : ℕ), (J m).lower ∈ Box.Icc I
hJu_mem : ∀ (m : ℕ), (J m).upper ∈ Box.Icc I
hJlz : Tendsto (fun m => (J m).lower) atTop (𝓝 z)
⊢ False
state_after: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
z : ι → ℝ := ⨆ m, (J m).lower
hzJ : ∀ (m : ℕ), z ∈ Box.Icc (J m)
hJl_mem : ∀ (m : ℕ), (J m).lower ∈ Box.Icc I
hJu_mem : ∀ (m : ℕ), (J m).upper ∈ Box.Icc I
hJlz : Tendsto (fun m => (J m).lower) atTop (𝓝 z)
hJuz : Tendsto (fun m => (J m).upper) atTop (𝓝 z)
⊢ False
tactic: have hJuz : Tendsto (fun m ↦ (J m).upper) atTop (𝓝 z) := by
    suffices Tendsto (fun m ↦ (J m).upper - (J m).lower) atTop (𝓝 0) by simpa using hJlz.add this
    refine tendsto_pi_nhds.2 fun i ↦ ?_
    simpa [hJsub] using
      tendsto_const_nhds.div_atTop (tendsto_pow_atTop_atTop_of_one_lt _root_.one_lt_two)
state_before: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
z : ι → ℝ := ⨆ m, (J m).lower
hzJ : ∀ (m : ℕ), z ∈ Box.Icc (J m)
hJl_mem : ∀ (m : ℕ), (J m).lower ∈ Box.Icc I
hJu_mem : ∀ (m : ℕ), (J m).upper ∈ Box.Icc I
hJlz : Tendsto (fun m => (J m).lower) atTop (𝓝 z)
hJuz : Tendsto (fun m => (J m).upper) atTop (𝓝 z)
⊢ False
state_after: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
z : ι → ℝ := ⨆ m, (J m).lower
hzJ : ∀ (m : ℕ), z ∈ Box.Icc (J m)
hJl_mem : ∀ (m : ℕ), (J m).lower ∈ Box.Icc I
hJu_mem : ∀ (m : ℕ), (J m).upper ∈ Box.Icc I
hJuz : Tendsto (fun m => (J m).upper) atTop (𝓝 z)
hJlz : Tendsto (fun m => (J m).lower) atTop (𝓝[Set.Icc I.lower I.upper] z)
⊢ False
tactic: replace hJlz : Tendsto (fun m ↦ (J m).lower) atTop (𝓝[Icc I.lower I.upper] z) :=
    tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ hJlz (eventually_of_forall hJl_mem)
state_before: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
z : ι → ℝ := ⨆ m, (J m).lower
hzJ : ∀ (m : ℕ), z ∈ Box.Icc (J m)
hJl_mem : ∀ (m : ℕ), (J m).lower ∈ Box.Icc I
hJu_mem : ∀ (m : ℕ), (J m).upper ∈ Box.Icc I
hJuz : Tendsto (fun m => (J m).upper) atTop (𝓝 z)
hJlz : Tendsto (fun m => (J m).lower) atTop (𝓝[Set.Icc I.lower I.upper] z)
⊢ False
state_after: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
z : ι → ℝ := ⨆ m, (J m).lower
hzJ : ∀ (m : ℕ), z ∈ Box.Icc (J m)
hJl_mem : ∀ (m : ℕ), (J m).lower ∈ Box.Icc I
hJu_mem : ∀ (m : ℕ), (J m).upper ∈ Box.Icc I
hJlz : Tendsto (fun m => (J m).lower) atTop (𝓝[Set.Icc I.lower I.upper] z)
hJuz : Tendsto (fun m => (J m).upper) atTop (𝓝[Set.Icc I.lower I.upper] z)
⊢ False
tactic: replace hJuz : Tendsto (fun m ↦ (J m).upper) atTop (𝓝[Icc I.lower I.upper] z) :=
    tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ hJuz (eventually_of_forall hJu_mem)
state_before: ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
z : ι → ℝ := ⨆ m, (J m).lower
hzJ : ∀ (m : ℕ), z ∈ Box.Icc (J m)
hJl_mem : ∀ (m : ℕ), (J m).lower ∈ Box.Icc I
hJu_mem : ∀ (m : ℕ), (J m).upper ∈ Box.Icc I
hJlz : Tendsto (fun m => (J m).lower) atTop (𝓝[Set.Icc I.lower I.upper] z)
hJuz : Tendsto (fun m => (J m).upper) atTop (𝓝[Set.Icc I.lower I.upper] z)
⊢ False
state_after: case intro.intro
ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
z : ι → ℝ := ⨆ m, (J m).lower
hzJ : ∀ (m : ℕ), z ∈ Box.Icc (J m)
hJl_mem : ∀ (m : ℕ), (J m).lower ∈ Box.Icc I
hJu_mem : ∀ (m : ℕ), (J m).upper ∈ Box.Icc I
hJlz : Tendsto (fun m => (J m).lower) atTop (𝓝[Set.Icc I.lower I.upper] z)
hJuz : Tendsto (fun m => (J m).upper) atTop (𝓝[Set.Icc I.lower I.upper] z)
U : Set (ι → ℝ)
hUz : U ∈ 𝓝[Box.Icc I] z
hU :
  ∀ J ≤ I,
    ∀ (m : ℕ),
      z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
⊢ False
tactic: rcases H_nhds z (h0 ▸ hzJ 0) with ⟨U, hUz, hU⟩
state_before: case intro.intro
ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
z : ι → ℝ := ⨆ m, (J m).lower
hzJ : ∀ (m : ℕ), z ∈ Box.Icc (J m)
hJl_mem : ∀ (m : ℕ), (J m).lower ∈ Box.Icc I
hJu_mem : ∀ (m : ℕ), (J m).upper ∈ Box.Icc I
hJlz : Tendsto (fun m => (J m).lower) atTop (𝓝[Set.Icc I.lower I.upper] z)
hJuz : Tendsto (fun m => (J m).upper) atTop (𝓝[Set.Icc I.lower I.upper] z)
U : Set (ι → ℝ)
hUz : U ∈ 𝓝[Box.Icc I] z
hU :
  ∀ J ≤ I,
    ∀ (m : ℕ),
      z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
⊢ False
state_after: case intro.intro.intro
ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
z : ι → ℝ := ⨆ m, (J m).lower
hzJ : ∀ (m : ℕ), z ∈ Box.Icc (J m)
hJl_mem : ∀ (m : ℕ), (J m).lower ∈ Box.Icc I
hJu_mem : ∀ (m : ℕ), (J m).upper ∈ Box.Icc I
hJlz : Tendsto (fun m => (J m).lower) atTop (𝓝[Set.Icc I.lower I.upper] z)
hJuz : Tendsto (fun m => (J m).upper) atTop (𝓝[Set.Icc I.lower I.upper] z)
U : Set (ι → ℝ)
hUz : U ∈ 𝓝[Box.Icc I] z
hU :
  ∀ J ≤ I,
    ∀ (m : ℕ),
      z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
m : ℕ
hUm : Set.Icc (J m).lower (J m).upper ∈ 𝒫 U
⊢ False
tactic: rcases (tendsto_lift'.1 (hJlz.Icc hJuz) U hUz).exists with ⟨m, hUm⟩
state_before: case intro.intro.intro
ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
z : ι → ℝ := ⨆ m, (J m).lower
hzJ : ∀ (m : ℕ), z ∈ Box.Icc (J m)
hJl_mem : ∀ (m : ℕ), (J m).lower ∈ Box.Icc I
hJu_mem : ∀ (m : ℕ), (J m).upper ∈ Box.Icc I
hJlz : Tendsto (fun m => (J m).lower) atTop (𝓝[Set.Icc I.lower I.upper] z)
hJuz : Tendsto (fun m => (J m).upper) atTop (𝓝[Set.Icc I.lower I.upper] z)
U : Set (ι → ℝ)
hUz : U ∈ 𝓝[Box.Icc I] z
hU :
  ∀ J ≤ I,
    ∀ (m : ℕ),
      z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
m : ℕ
hUm : Set.Icc (J m).lower (J m).upper ∈ 𝒫 U
⊢ False
state_after: no goals
tactic: exact hJp m (hU (J m) (hJle m) m (hzJ m) hUm (hJsub m))