l3lab/ntp-mathlib huggingface dataset:
/-- The tangent cone of a product contains the tangent cone of its left factor. -/
theorem subset_tangentCone_prod_left {t : Set F} {y : F} (ht : y ∈ closure t) :
    LinearMap.inl 𝕜 E F '' tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) := by
  rintro _ ⟨v, ⟨c, d, hd, hc, hy⟩, rfl⟩
  have : ∀ n, ∃ d', y + d' ∈ t ∧ ‖c n • d'‖ < ((1 : ℝ) / 2) ^ n := by
    intro n
    rcases mem_closure_iff_nhds.1 ht _
        (eventually_nhds_norm_smul_sub_lt (c n) y (pow_pos one_half_pos n)) with
      ⟨z, hz, hzt⟩
    exact ⟨z - y, by 

Proof state:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y✝ : E
s t✝ : Set E
t : Set F
y : F
ht : y ∈ closure t
v : E
c : ℕ → 𝕜
d : ℕ → E
hd : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s
hc : Tendsto (fun n => ‖c n‖) atTop atTop
hy : Tendsto (fun n => c n • d n) atTop (𝓝 v)
n : ℕ
z : F
hz : z ∈ {x | (fun y_1 => ‖c n • (y_1 - y)‖ < (1 / 2) ^ n) x}
hzt : z ∈ t
⊢ y + (z - y) ∈ t

Next tactic:
simpa using hzt
/-- The tangent cone of a product contains the tangent cone of its left factor. -/
theorem subset_tangentCone_prod_left {t : Set F} {y : F} (ht : y ∈ closure t) :
    LinearMap.inl 𝕜 E F '' tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) := by
  rintro _ ⟨v, ⟨c, d, hd, hc, hy⟩, rfl⟩
  have : ∀ n, ∃ d', y + d' ∈ t ∧ ‖c n • d'‖ < ((1 : ℝ) / 2) ^ n := by
    intro n
    rcases mem_closure_iff_nhds.1 ht _
        (eventually_nhds_norm_smul_sub_lt (c n) y (pow_pos one_half_pos n)) with
      ⟨z, hz, hzt⟩
    exact ⟨z - y, by simpa using hzt, by 

Proof state:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y✝ : E
s t✝ : Set E
t : Set F
y : F
ht : y ∈ closure t
v : E
c : ℕ → 𝕜
d : ℕ → E
hd : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s
hc : Tendsto (fun n => ‖c n‖) atTop atTop
hy : Tendsto (fun n => c n • d n) atTop (𝓝 v)
n : ℕ
z : F
hz : z ∈ {x | (fun y_1 => ‖c n • (y_1 - y)‖ < (1 / 2) ^ n) x}
hzt : z ∈ t
⊢ ‖c n • (z - y)‖ < (1 / 2) ^ n

Next tactic:
simpa using hz
/-- The tangent cone of a product contains the tangent cone of its left factor. -/
theorem subset_tangentCone_prod_left {t : Set F} {y : F} (ht : y ∈ closure t) :
    LinearMap.inl 𝕜 E F '' tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) := by
  rintro _ ⟨v, ⟨c, d, hd, hc, hy⟩, rfl⟩
  have : ∀ n, ∃ d', y + d' ∈ t ∧ ‖c n • d'‖ < ((1 : ℝ) / 2) ^ n := by
    intro n
    rcases mem_closure_iff_nhds.1 ht _
        (eventually_nhds_norm_smul_sub_lt (c n) y (pow_pos one_half_pos n)) with
      ⟨z, hz, hzt⟩
    exact ⟨z - y, by simpa using hzt, by simpa using hz⟩
  

Proof state:
case intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y✝ : E
s t✝ : Set E
t : Set F
y : F
ht : y ∈ closure t
v : E
c : ℕ → 𝕜
d : ℕ → E
hd : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s
hc : Tendsto (fun n => ‖c n‖) atTop atTop
hy : Tendsto (fun n => c n • d n) atTop (𝓝 v)
this : ∀ (n : ℕ), ∃ d', y + d' ∈ t ∧ ‖c n • d'‖ < (1 / 2) ^ n
⊢ (LinearMap.inl 𝕜 E F) v ∈ tangentConeAt 𝕜 (s ×ˢ t) (x, y)

Next tactic:
choose d' hd' using this
/-- The tangent cone of a product contains the tangent cone of its left factor. -/
theorem subset_tangentCone_prod_left {t : Set F} {y : F} (ht : y ∈ closure t) :
    LinearMap.inl 𝕜 E F '' tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) := by
  rintro _ ⟨v, ⟨c, d, hd, hc, hy⟩, rfl⟩
  have : ∀ n, ∃ d', y + d' ∈ t ∧ ‖c n • d'‖ < ((1 : ℝ) / 2) ^ n := by
    intro n
    rcases mem_closure_iff_nhds.1 ht _
        (eventually_nhds_norm_smul_sub_lt (c n) y (pow_pos one_half_pos n)) with
      ⟨z, hz, hzt⟩
    exact ⟨z - y, by simpa using hzt, by simpa using hz⟩
  choose d' hd' using this
  

Proof state:
case intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y✝ : E
s t✝ : Set E
t : Set F
y : F
ht : y ∈ closure t
v : E
c : ℕ → 𝕜
d : ℕ → E
hd : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s
hc : Tendsto (fun n => ‖c n‖) atTop atTop
hy : Tendsto (fun n => c n • d n) atTop (𝓝 v)
d' : ℕ → F
hd' : ∀ (n : ℕ), y + d' n ∈ t ∧ ‖c n • d' n‖ < (1 / 2) ^ n
⊢ (LinearMap.inl 𝕜 E F) v ∈ tangentConeAt 𝕜 (s ×ˢ t) (x, y)

Next tactic:
refine' ⟨c, fun n => (d n, d' n), _, hc, _⟩
/-- The tangent cone of a product contains the tangent cone of its left factor. -/
theorem subset_tangentCone_prod_left {t : Set F} {y : F} (ht : y ∈ closure t) :
    LinearMap.inl 𝕜 E F '' tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) := by
  rintro _ ⟨v, ⟨c, d, hd, hc, hy⟩, rfl⟩
  have : ∀ n, ∃ d', y + d' ∈ t ∧ ‖c n • d'‖ < ((1 : ℝ) / 2) ^ n := by
    intro n
    rcases mem_closure_iff_nhds.1 ht _
        (eventually_nhds_norm_smul_sub_lt (c n) y (pow_pos one_half_pos n)) with
      ⟨z, hz, hzt⟩
    exact ⟨z - y, by simpa using hzt, by simpa using hz⟩
  choose d' hd' using this
  refine' ⟨c, fun n => (d n, d' n), _, hc, _⟩
  

Proof state:
case intro.intro.intro.intro.intro.intro.refine'_1
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y✝ : E
s t✝ : Set E
t : Set F
y : F
ht : y ∈ closure t
v : E
c : ℕ → 𝕜
d : ℕ → E
hd : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s
hc : Tendsto (fun n => ‖c n‖) atTop atTop
hy : Tendsto (fun n => c n • d n) atTop (𝓝 v)
d' : ℕ → F
hd' : ∀ (n : ℕ), y + d' n ∈ t ∧ ‖c n • d' n‖ < (1 / 2) ^ n
⊢ ∀ᶠ (n : ℕ) in atTop, (x, y) + (fun n => (d n, d' n)) n ∈ s ×ˢ t
case intro.intro.intro.intro.intro.intro.refine'_2
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y✝ : E
s t✝ : Set E
t : Set F
y : F
ht : y ∈ closure t
v : E
c : ℕ → 𝕜
d : ℕ → E
hd : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s
hc : Tendsto (fun n => ‖c n‖) atTop atTop
hy : Tendsto (fun n => c n • d n) atTop (𝓝 v)
d' : ℕ → F
hd' : ∀ (n : ℕ), y + d' n ∈ t ∧ ‖c n • d' n‖ < (1 / 2) ^ n
⊢ Tendsto (fun n => c n • (fun n => (d n, d' n)) n) atTop (𝓝 ((LinearMap.inl 𝕜 E F) v))

Next tactic:
show ∀ᶠ n in atTop, (x, y) + (d n, d' n) ∈ s ×ˢ t
/-- The tangent cone of a product contains the tangent cone of its left factor. -/
theorem subset_tangentCone_prod_left {t : Set F} {y : F} (ht : y ∈ closure t) :
    LinearMap.inl 𝕜 E F '' tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) := by
  rintro _ ⟨v, ⟨c, d, hd, hc, hy⟩, rfl⟩
  have : ∀ n, ∃ d', y + d' ∈ t ∧ ‖c n • d'‖ < ((1 : ℝ) / 2) ^ n := by
    intro n
    rcases mem_closure_iff_nhds.1 ht _
        (eventually_nhds_norm_smul_sub_lt (c n) y (pow_pos one_half_pos n)) with
      ⟨z, hz, hzt⟩
    exact ⟨z - y, by simpa using hzt, by simpa using hz⟩
  choose d' hd' using this
  refine' ⟨c, fun n => (d n, d' n), _, hc, _⟩
  show ∀ᶠ n in atTop, (x, y) + (d n, d' n) ∈ s ×ˢ t
  · 

Proof state:
case intro.intro.intro.intro.intro.intro.refine'_1
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y✝ : E
s t✝ : Set E
t : Set F
y : F
ht : y ∈ closure t
v : E
c : ℕ → 𝕜
d : ℕ → E
hd : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s
hc : Tendsto (fun n => ‖c n‖) atTop atTop
hy : Tendsto (fun n => c n • d n) atTop (𝓝 v)
d' : ℕ → F
hd' : ∀ (n : ℕ), y + d' n ∈ t ∧ ‖c n • d' n‖ < (1 / 2) ^ n
⊢ ∀ᶠ (n : ℕ) in atTop, (x, y) + (d n, d' n) ∈ s ×ˢ t

Next tactic:
filter_upwards [hd] with n hn
/-- The tangent cone of a product contains the tangent cone of its left factor. -/
theorem subset_tangentCone_prod_left {t : Set F} {y : F} (ht : y ∈ closure t) :
    LinearMap.inl 𝕜 E F '' tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) := by
  rintro _ ⟨v, ⟨c, d, hd, hc, hy⟩, rfl⟩
  have : ∀ n, ∃ d', y + d' ∈ t ∧ ‖c n • d'‖ < ((1 : ℝ) / 2) ^ n := by
    intro n
    rcases mem_closure_iff_nhds.1 ht _
        (eventually_nhds_norm_smul_sub_lt (c n) y (pow_pos one_half_pos n)) with
      ⟨z, hz, hzt⟩
    exact ⟨z - y, by simpa using hzt, by simpa using hz⟩
  choose d' hd' using this
  refine' ⟨c, fun n => (d n, d' n), _, hc, _⟩
  show ∀ᶠ n in atTop, (x, y) + (d n, d' n) ∈ s ×ˢ t
  · filter_upwards [hd] with n hn
    

Proof state:
case h
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y✝ : E
s t✝ : Set E
t : Set F
y : F
ht : y ∈ closure t
v : E
c : ℕ → 𝕜
d : ℕ → E
hd : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s
hc : Tendsto (fun n => ‖c n‖) atTop atTop
hy : Tendsto (fun n => c n • d n) atTop (𝓝 v)
d' : ℕ → F
hd' : ∀ (n : ℕ), y + d' n ∈ t ∧ ‖c n • d' n‖ < (1 / 2) ^ n
n : ℕ
hn : x + d n ∈ s
⊢ (x, y) + (d n, d' n) ∈ s ×ˢ t

Next tactic:
simp [hn, (hd' n).1]
/-- The tangent cone of a product contains the tangent cone of its left factor. -/
theorem subset_tangentCone_prod_left {t : Set F} {y : F} (ht : y ∈ closure t) :
    LinearMap.inl 𝕜 E F '' tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) := by
  rintro _ ⟨v, ⟨c, d, hd, hc, hy⟩, rfl⟩
  have : ∀ n, ∃ d', y + d' ∈ t ∧ ‖c n • d'‖ < ((1 : ℝ) / 2) ^ n := by
    intro n
    rcases mem_closure_iff_nhds.1 ht _
        (eventually_nhds_norm_smul_sub_lt (c n) y (pow_pos one_half_pos n)) with
      ⟨z, hz, hzt⟩
    exact ⟨z - y, by simpa using hzt, by simpa using hz⟩
  choose d' hd' using this
  refine' ⟨c, fun n => (d n, d' n), _, hc, _⟩
  show ∀ᶠ n in atTop, (x, y) + (d n, d' n) ∈ s ×ˢ t
  · filter_upwards [hd] with n hn
    simp [hn, (hd' n).1]
  · 

Proof state:
case intro.intro.intro.intro.intro.intro.refine'_2
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y✝ : E
s t✝ : Set E
t : Set F
y : F
ht : y ∈ closure t
v : E
c : ℕ → 𝕜
d : ℕ → E
hd : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s
hc : Tendsto (fun n => ‖c n‖) atTop atTop
hy : Tendsto (fun n => c n • d n) atTop (𝓝 v)
d' : ℕ → F
hd' : ∀ (n : ℕ), y + d' n ∈ t ∧ ‖c n • d' n‖ < (1 / 2) ^ n
⊢ Tendsto (fun n => c n • (fun n => (d n, d' n)) n) atTop (𝓝 ((LinearMap.inl 𝕜 E F) v))

Next tactic:
apply Tendsto.prod_mk_nhds hy _
/-- The tangent cone of a product contains the tangent cone of its left factor. -/
theorem subset_tangentCone_prod_left {t : Set F} {y : F} (ht : y ∈ closure t) :
    LinearMap.inl 𝕜 E F '' tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) := by
  rintro _ ⟨v, ⟨c, d, hd, hc, hy⟩, rfl⟩
  have : ∀ n, ∃ d', y + d' ∈ t ∧ ‖c n • d'‖ < ((1 : ℝ) / 2) ^ n := by
    intro n
    rcases mem_closure_iff_nhds.1 ht _
        (eventually_nhds_norm_smul_sub_lt (c n) y (pow_pos one_half_pos n)) with
      ⟨z, hz, hzt⟩
    exact ⟨z - y, by simpa using hzt, by simpa using hz⟩
  choose d' hd' using this
  refine' ⟨c, fun n => (d n, d' n), _, hc, _⟩
  show ∀ᶠ n in atTop, (x, y) + (d n, d' n) ∈ s ×ˢ t
  · filter_upwards [hd] with n hn
    simp [hn, (hd' n).1]
  · apply Tendsto.prod_mk_nhds hy _
    

Proof state:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y✝ : E
s t✝ : Set E
t : Set F
y : F
ht : y ∈ closure t
v : E
c : ℕ → 𝕜
d : ℕ → E
hd : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s
hc : Tendsto (fun n => ‖c n‖) atTop atTop
hy : Tendsto (fun n => c n • d n) atTop (𝓝 v)
d' : ℕ → F
hd' : ∀ (n : ℕ), y + d' n ∈ t ∧ ‖c n • d' n‖ < (1 / 2) ^ n
⊢ Tendsto (fun c_1 => c c_1 • ((fun n => (d n, d' n)) c_1).2) atTop (𝓝 (0 v))

Next tactic:
refine' squeeze_zero_norm (fun n => (hd' n).2.le) _
/-- The tangent cone of a product contains the tangent cone of its left factor. -/
theorem subset_tangentCone_prod_left {t : Set F} {y : F} (ht : y ∈ closure t) :
    LinearMap.inl 𝕜 E F '' tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) := by
  rintro _ ⟨v, ⟨c, d, hd, hc, hy⟩, rfl⟩
  have : ∀ n, ∃ d', y + d' ∈ t ∧ ‖c n • d'‖ < ((1 : ℝ) / 2) ^ n := by
    intro n
    rcases mem_closure_iff_nhds.1 ht _
        (eventually_nhds_norm_smul_sub_lt (c n) y (pow_pos one_half_pos n)) with
      ⟨z, hz, hzt⟩
    exact ⟨z - y, by simpa using hzt, by simpa using hz⟩
  choose d' hd' using this
  refine' ⟨c, fun n => (d n, d' n), _, hc, _⟩
  show ∀ᶠ n in atTop, (x, y) + (d n, d' n) ∈ s ×ˢ t
  · filter_upwards [hd] with n hn
    simp [hn, (hd' n).1]
  · apply Tendsto.prod_mk_nhds hy _
    refine' squeeze_zero_norm (fun n => (hd' n).2.le) _
    

Proof state:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y✝ : E
s t✝ : Set E
t : Set F
y : F
ht : y ∈ closure t
v : E
c : ℕ → 𝕜
d : ℕ → E
hd : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s
hc : Tendsto (fun n => ‖c n‖) atTop atTop
hy : Tendsto (fun n => c n • d n) atTop (𝓝 v)
d' : ℕ → F
hd' : ∀ (n : ℕ), y + d' n ∈ t ∧ ‖c n • d' n‖ < (1 / 2) ^ n
⊢ Tendsto (fun n => (1 / 2) ^ n) atTop (𝓝 0)

Next tactic:
exact tendsto_pow_atTop_nhds_0_of_lt_1 one_half_pos.le one_half_lt_one
/-- The tangent cone of a product contains the tangent cone of its right factor. -/
theorem subset_tangentCone_prod_right {t : Set F} {y : F} (hs : x ∈ closure s) :
    LinearMap.inr 𝕜 E F '' tangentConeAt 𝕜 t y ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) := by
  

Proof state:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y✝ : E
s t✝ : Set E
t : Set F
y : F
hs : x ∈ closure s
⊢ ⇑(LinearMap.inr 𝕜 E F) '' tangentConeAt 𝕜 t y ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y)

Next tactic:
rintro _ ⟨w, ⟨c, d, hd, hc, hy⟩, rfl⟩
/-- The tangent cone of a product contains the tangent cone of its right factor. -/
theorem subset_tangentCone_prod_right {t : Set F} {y : F} (hs : x ∈ closure s) :
    LinearMap.inr 𝕜 E F '' tangentConeAt 𝕜 t y ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) := by
  rintro _ ⟨w, ⟨c, d, hd, hc, hy⟩, rfl⟩
  

Proof state:
case intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y✝ : E
s t✝ : Set E
t : Set F
y : F
hs : x ∈ closure s
w : F
c : ℕ → 𝕜
d : ℕ → F
hd : ∀ᶠ (n : ℕ) in atTop, y + d n ∈ t
hc : Tendsto (fun n => ‖c n‖) atTop atTop
hy : Tendsto (fun n => c n • d n) atTop (𝓝 w)
⊢ (LinearMap.inr 𝕜 E F) w ∈ tangentConeAt 𝕜 (s ×ˢ t) (x, y)

Next tactic:
have : ∀ n, ∃ d', x + d' ∈ s ∧ ‖c n • d'‖ < ((1 : ℝ) / 2) ^ n := by
    intro n
    rcases mem_closure_iff_nhds.1 hs _
        (eventually_nhds_norm_smul_sub_lt (c n) x (pow_pos one_half_pos n)) with
      ⟨z, hz, hzs⟩
    exact ⟨z - x, by simpa using hzs, by simpa using hz⟩
/-- The tangent cone of a product contains the tangent cone of its right factor. -/
theorem subset_tangentCone_prod_right {t : Set F} {y : F} (hs : x ∈ closure s) :
    LinearMap.inr 𝕜 E F '' tangentConeAt 𝕜 t y ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) := by
  rintro _ ⟨w, ⟨c, d, hd, hc, hy⟩, rfl⟩
  have : ∀ n, ∃ d', x + d' ∈ s ∧ ‖c n • d'‖ < ((1 : ℝ) / 2) ^ n := by
    

Proof state:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y✝ : E
s t✝ : Set E
t : Set F
y : F
hs : x ∈ closure s
w : F
c : ℕ → 𝕜
d : ℕ → F
hd : ∀ᶠ (n : ℕ) in atTop, y + d n ∈ t
hc : Tendsto (fun n => ‖c n‖) atTop atTop
hy : Tendsto (fun n => c n • d n) atTop (𝓝 w)
⊢ ∀ (n : ℕ), ∃ d', x + d' ∈ s ∧ ‖c n • d'‖ < (1 / 2) ^ n

Next tactic:
intro n
/-- The tangent cone of a product contains the tangent cone of its right factor. -/
theorem subset_tangentCone_prod_right {t : Set F} {y : F} (hs : x ∈ closure s) :
    LinearMap.inr 𝕜 E F '' tangentConeAt 𝕜 t y ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) := by
  rintro _ ⟨w, ⟨c, d, hd, hc, hy⟩, rfl⟩
  have : ∀ n, ∃ d', x + d' ∈ s ∧ ‖c n • d'‖ < ((1 : ℝ) / 2) ^ n := by
    intro n
    

Proof state:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y✝ : E
s t✝ : Set E
t : Set F
y : F
hs : x ∈ closure s
w : F
c : ℕ → 𝕜
d : ℕ → F
hd : ∀ᶠ (n : ℕ) in atTop, y + d n ∈ t
hc : Tendsto (fun n => ‖c n‖) atTop atTop
hy : Tendsto (fun n => c n • d n) atTop (𝓝 w)
n : ℕ
⊢ ∃ d', x + d' ∈ s ∧ ‖c n • d'‖ < (1 / 2) ^ n

Next tactic:
rcases mem_closure_iff_nhds.1 hs _
        (eventually_nhds_norm_smul_sub_lt (c n) x (pow_pos one_half_pos n)) with
      ⟨z, hz, hzs⟩
/-- The tangent cone of a product contains the tangent cone of its right factor. -/
theorem subset_tangentCone_prod_right {t : Set F} {y : F} (hs : x ∈ closure s) :
    LinearMap.inr 𝕜 E F '' tangentConeAt 𝕜 t y ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) := by
  rintro _ ⟨w, ⟨c, d, hd, hc, hy⟩, rfl⟩
  have : ∀ n, ∃ d', x + d' ∈ s ∧ ‖c n • d'‖ < ((1 : ℝ) / 2) ^ n := by
    intro n
    rcases mem_closure_iff_nhds.1 hs _
        (eventually_nhds_norm_smul_sub_lt (c n) x (pow_pos one_half_pos n)) with
      ⟨z, hz, hzs⟩
    

Proof state:
case intro.intro
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y✝ : E
s t✝ : Set E
t : Set F
y : F
hs : x ∈ closure s
w : F
c : ℕ → 𝕜
d : ℕ → F
hd : ∀ᶠ (n : ℕ) in atTop, y + d n ∈ t
hc : Tendsto (fun n => ‖c n‖) atTop atTop
hy : Tendsto (fun n => c n • d n) atTop (𝓝 w)
n : ℕ
z : E
hz : z ∈ {x_1 | (fun y => ‖c n • (y - x)‖ < (1 / 2) ^ n) x_1}
hzs : z ∈ s
⊢ ∃ d', x + d' ∈ s ∧ ‖c n • d'‖ < (1 / 2) ^ n

Next tactic:
exact ⟨z - x, by simpa using hzs, by simpa using hz⟩
/-- The tangent cone of a product contains the tangent cone of its right factor. -/
theorem subset_tangentCone_prod_right {t : Set F} {y : F} (hs : x ∈ closure s) :
    LinearMap.inr 𝕜 E F '' tangentConeAt 𝕜 t y ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) := by
  rintro _ ⟨w, ⟨c, d, hd, hc, hy⟩, rfl⟩
  have : ∀ n, ∃ d', x + d' ∈ s ∧ ‖c n • d'‖ < ((1 : ℝ) / 2) ^ n := by
    intro n
    rcases mem_closure_iff_nhds.1 hs _
        (eventually_nhds_norm_smul_sub_lt (c n) x (pow_pos one_half_pos n)) with
      ⟨z, hz, hzs⟩
    exact ⟨z - x, by 

Proof state:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y✝ : E
s t✝ : Set E
t : Set F
y : F
hs : x ∈ closure s
w : F
c : ℕ → 𝕜
d : ℕ → F
hd : ∀ᶠ (n : ℕ) in atTop, y + d n ∈ t
hc : Tendsto (fun n => ‖c n‖) atTop atTop
hy : Tendsto (fun n => c n • d n) atTop (𝓝 w)
n : ℕ
z : E
hz : z ∈ {x_1 | (fun y => ‖c n • (y - x)‖ < (1 / 2) ^ n) x_1}
hzs : z ∈ s
⊢ x + (z - x) ∈ s

Next tactic:
simpa using hzs
/-- The tangent cone of a product contains the tangent cone of its right factor. -/
theorem subset_tangentCone_prod_right {t : Set F} {y : F} (hs : x ∈ closure s) :
    LinearMap.inr 𝕜 E F '' tangentConeAt 𝕜 t y ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) := by
  rintro _ ⟨w, ⟨c, d, hd, hc, hy⟩, rfl⟩
  have : ∀ n, ∃ d', x + d' ∈ s ∧ ‖c n • d'‖ < ((1 : ℝ) / 2) ^ n := by
    intro n
    rcases mem_closure_iff_nhds.1 hs _
        (eventually_nhds_norm_smul_sub_lt (c n) x (pow_pos one_half_pos n)) with
      ⟨z, hz, hzs⟩
    exact ⟨z - x, by simpa using hzs, by 

Proof state:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y✝ : E
s t✝ : Set E
t : Set F
y : F
hs : x ∈ closure s
w : F
c : ℕ → 𝕜
d : ℕ → F
hd : ∀ᶠ (n : ℕ) in atTop, y + d n ∈ t
hc : Tendsto (fun n => ‖c n‖) atTop atTop
hy : Tendsto (fun n => c n • d n) atTop (𝓝 w)
n : ℕ
z : E
hz : z ∈ {x_1 | (fun y => ‖c n • (y - x)‖ < (1 / 2) ^ n) x_1}
hzs : z ∈ s
⊢ ‖c n • (z - x)‖ < (1 / 2) ^ n

Next tactic:
simpa using hz
/-- The tangent cone of a product contains the tangent cone of its right factor. -/
theorem subset_tangentCone_prod_right {t : Set F} {y : F} (hs : x ∈ closure s) :
    LinearMap.inr 𝕜 E F '' tangentConeAt 𝕜 t y ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) := by
  rintro _ ⟨w, ⟨c, d, hd, hc, hy⟩, rfl⟩
  have : ∀ n, ∃ d', x + d' ∈ s ∧ ‖c n • d'‖ < ((1 : ℝ) / 2) ^ n := by
    intro n
    rcases mem_closure_iff_nhds.1 hs _
        (eventually_nhds_norm_smul_sub_lt (c n) x (pow_pos one_half_pos n)) with
      ⟨z, hz, hzs⟩
    exact ⟨z - x, by simpa using hzs, by simpa using hz⟩
  

Proof state:
case intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y✝ : E
s t✝ : Set E
t : Set F
y : F
hs : x ∈ closure s
w : F
c : ℕ → 𝕜
d : ℕ → F
hd : ∀ᶠ (n : ℕ) in atTop, y + d n ∈ t
hc : Tendsto (fun n => ‖c n‖) atTop atTop
hy : Tendsto (fun n => c n • d n) atTop (𝓝 w)
this : ∀ (n : ℕ), ∃ d', x + d' ∈ s ∧ ‖c n • d'‖ < (1 / 2) ^ n
⊢ (LinearMap.inr 𝕜 E F) w ∈ tangentConeAt 𝕜 (s ×ˢ t) (x, y)

Next tactic:
choose d' hd' using this
/-- The tangent cone of a product contains the tangent cone of its right factor. -/
theorem subset_tangentCone_prod_right {t : Set F} {y : F} (hs : x ∈ closure s) :
    LinearMap.inr 𝕜 E F '' tangentConeAt 𝕜 t y ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) := by
  rintro _ ⟨w, ⟨c, d, hd, hc, hy⟩, rfl⟩
  have : ∀ n, ∃ d', x + d' ∈ s ∧ ‖c n • d'‖ < ((1 : ℝ) / 2) ^ n := by
    intro n
    rcases mem_closure_iff_nhds.1 hs _
        (eventually_nhds_norm_smul_sub_lt (c n) x (pow_pos one_half_pos n)) with
      ⟨z, hz, hzs⟩
    exact ⟨z - x, by simpa using hzs, by simpa using hz⟩
  choose d' hd' using this
  

Proof state:
case intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y✝ : E
s t✝ : Set E
t : Set F
y : F
hs : x ∈ closure s
w : F
c : ℕ → 𝕜
d : ℕ → F
hd : ∀ᶠ (n : ℕ) in atTop, y + d n ∈ t
hc : Tendsto (fun n => ‖c n‖) atTop atTop
hy : Tendsto (fun n => c n • d n) atTop (𝓝 w)
d' : ℕ → E
hd' : ∀ (n : ℕ), x + d' n ∈ s ∧ ‖c n • d' n‖ < (1 / 2) ^ n
⊢ (LinearMap.inr 𝕜 E F) w ∈ tangentConeAt 𝕜 (s ×ˢ t) (x, y)

Next tactic:
refine' ⟨c, fun n => (d' n, d n), _, hc, _⟩