url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenTerm_eq | [414, 1] | [420, 62] | simp only [unevenTerm, ←unevenSeries_apply, h] | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x : ℕ
h : unevenSeries' u r z1 = unevenSeries' u r1 z1
⊢ unevenTerm' u r z1 x = unevenTerm u z1 x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x : ℕ
h : unevenSeries' u r z1 = unevenSeries' u r1 z1
⊢ unevenTerm' u r z1 x = unevenTerm u z1 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenTerm_eq | [414, 1] | [420, 62] | rfl | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x : ℕ
p0 : HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u r z1) c0 (ENNReal.ofReal r)
⊢ r1 ≤ r1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x : ℕ
p0 : HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u r z1) c0 (ENNReal.ofReal r)
⊢ r1 ≤ r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_eq | [422, 1] | [426, 43] | intro z1s | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
⊢ z1 ∈ closedBall c1 r1 → unevenSeries' u r z1 = unevenSeries u z1 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
⊢ unevenSeries' u r z1 = unevenSeries u z1 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
⊢ z1 ∈ closedBall c1 r1 → unevenSeries' u r z1 = unevenSeries u z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_eq | [422, 1] | [426, 43] | funext | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
⊢ unevenSeries' u r z1 = unevenSeries u z1 | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x✝ : ℕ
⊢ unevenSeries' u r z1 x✝ = unevenSeries u z1 x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
⊢ unevenSeries' u r z1 = unevenSeries u z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_eq | [422, 1] | [426, 43] | simp_rw [unevenSeries, unevenSeries', unevenTerm_eq u rp rr1 z1s,
unevenTerm_eq u u.r1p (le_refl _) z1s] | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x✝ : ℕ
⊢ unevenSeries' u r z1 x✝ = unevenSeries u z1 x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x✝ : ℕ
⊢ unevenSeries' u r z1 x✝ = unevenSeries u z1 x✝
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_norm | [428, 1] | [430, 87] | rw [unevenSeries, unevenSeries', unevenTerm, ContinuousMultilinearMap.norm_mkPiRing] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
n : ℕ
⊢ ‖unevenSeries u z1 n‖ = ‖unevenTerm u z1 n‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
n : ℕ
⊢ ‖unevenSeries u z1 n‖ = ‖unevenTerm u z1 n‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | rcases (((isCompact_sphere _ _).prod (isCompact_closedBall _ _)).bddAbove_image
fc.norm).exists_ge 0 with ⟨b, bp, fb⟩ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
⊢ ∃ c a, c > 0 ∧ a > 0 ∧ ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 s, ‖unevenSeries u z1 n‖ ≤ c * a ^ n | case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ y ∈ (fun x => ‖f x‖) '' sphere c0 (r0 / 2) ×ˢ closedBall c1 s, y ≤ b
⊢ ∃ c a, c > 0 ∧ a > 0 ∧ ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 s, ‖unevenSeries u z1 n‖ ≤ c * a ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
⊢ ∃ c a, c > 0 ∧ a > 0 ∧ ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 s, ‖unevenSeries u z1 n‖ ≤ c * a ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | simp only [Set.forall_mem_image] at fb | case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ y ∈ (fun x => ‖f x‖) '' sphere c0 (r0 / 2) ×ˢ closedBall c1 s, y ≤ b
⊢ ∃ c a, c > 0 ∧ a > 0 ∧ ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 s, ‖unevenSeries u z1 n‖ ≤ c * a ^ n | case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
⊢ ∃ c a, c > 0 ∧ a > 0 ∧ ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 s, ‖unevenSeries u z1 n‖ ≤ c * a ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ y ∈ (fun x => ‖f x‖) '' sphere c0 (r0 / 2) ×ˢ closedBall c1 s, y ≤ b
⊢ ∃ c a, c > 0 ∧ a > 0 ∧ ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 s, ‖unevenSeries u z1 n‖ ≤ c * a ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | use b + 1, (r0 / 2)⁻¹, lt_of_le_of_lt bp (lt_add_one _), inv_pos.mpr (half_pos u.r0p) | case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
⊢ ∃ c a, c > 0 ∧ a > 0 ∧ ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 s, ‖unevenSeries u z1 n‖ ≤ c * a ^ n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
⊢ ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 s, ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
⊢ ∃ c a, c > 0 ∧ a > 0 ∧ ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 s, ‖unevenSeries u z1 n‖ ≤ c * a ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | intro n z1 z1s | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
⊢ ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 s, ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
⊢ ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 s, ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | have r0hp : r0 / 2 > 0 := by linarith [u.r0p] | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | have r0hr1 : r0 / 2 ≤ r1 := _root_.trans (by linarith [u.r0p]) u.r01 | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | set g := fun z0 ↦ f (z0, z1) | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | have gc : ContinuousOn g (sphere c0 (r0 / 2)) :=
ContinuousOn.comp fc (ContinuousOn.prod continuousOn_id continuousOn_const) fun z0 z0s ↦
Set.mk_mem_prod z0s z1s | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
gc : ContinuousOn g (sphere c0 (r0 / 2))
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | have gb : ∀ z0, z0 ∈ sphere c0 (r0 / 2) → ‖g z0‖ ≤ b := fun z0 z0s ↦ fb (Set.mk_mem_prod z0s z1s) | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
gc : ContinuousOn g (sphere c0 (r0 / 2))
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
gc : ContinuousOn g (sphere c0 (r0 / 2))
gb : ∀ z0 ∈ sphere c0 (r0 / 2), ‖g z0‖ ≤ b
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
gc : ContinuousOn g (sphere c0 (r0 / 2))
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | have cb := cauchy1_bound' r0hp b gc gb n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
gc : ContinuousOn g (sphere c0 (r0 / 2))
gb : ∀ z0 ∈ sphere c0 (r0 / 2), ‖g z0‖ ≤ b
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
gc : ContinuousOn g (sphere c0 (r0 / 2))
gb : ∀ z0 ∈ sphere c0 (r0 / 2), ‖g z0‖ ≤ b
cb : ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c0, r0 / 2), (z - c0)⁻¹ ^ n • (z - c0)⁻¹ • g z‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
gc : ContinuousOn g (sphere c0 (r0 / 2))
gb : ∀ z0 ∈ sphere c0 (r0 / 2), ‖g z0‖ ≤ b
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | clear bp gc gb | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
gc : ContinuousOn g (sphere c0 (r0 / 2))
gb : ∀ z0 ∈ sphere c0 (r0 / 2), ‖g z0‖ ≤ b
cb : ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c0, r0 / 2), (z - c0)⁻¹ ^ n • (z - c0)⁻¹ • g z‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
cb : ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c0, r0 / 2), (z - c0)⁻¹ ^ n • (z - c0)⁻¹ • g z‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
gc : ContinuousOn g (sphere c0 (r0 / 2))
gb : ∀ z0 ∈ sphere c0 (r0 / 2), ‖g z0‖ ≤ b
cb : ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c0, r0 / 2), (z - c0)⁻¹ ^ n • (z - c0)⁻¹ • g z‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | have e : (2 * π * I : ℂ)⁻¹ • (∮ z0 in C(c0, r0 / 2), (z0 - c0)⁻¹ ^ n • (z0 - c0)⁻¹ • g z0) =
unevenTerm' u (r0 / 2) z1 n := rfl | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
cb : ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c0, r0 / 2), (z - c0)⁻¹ ^ n • (z - c0)⁻¹ • g z‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e✝ : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
cb : ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c0, r0 / 2), (z - c0)⁻¹ ^ n • (z - c0)⁻¹ • g z‖ ≤ b * (r0 / 2)⁻¹ ^ n
e : ((2 * ↑π * I)⁻¹ • ∮ (z0 : ℂ) in C(c0, r0 / 2), (z0 - c0)⁻¹ ^ n • (z0 - c0)⁻¹ • g z0) = unevenTerm' u (r0 / 2) z1 n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
cb : ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c0, r0 / 2), (z - c0)⁻¹ ^ n • (z - c0)⁻¹ • g z‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | rw [e] at cb | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e✝ : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
cb : ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c0, r0 / 2), (z - c0)⁻¹ ^ n • (z - c0)⁻¹ • g z‖ ≤ b * (r0 / 2)⁻¹ ^ n
e : ((2 * ↑π * I)⁻¹ • ∮ (z0 : ℂ) in C(c0, r0 / 2), (z0 - c0)⁻¹ ^ n • (z0 - c0)⁻¹ • g z0) = unevenTerm' u (r0 / 2) z1 n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e✝ : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
cb : ‖unevenTerm' u (r0 / 2) z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
e : ((2 * ↑π * I)⁻¹ • ∮ (z0 : ℂ) in C(c0, r0 / 2), (z0 - c0)⁻¹ ^ n • (z0 - c0)⁻¹ • g z0) = unevenTerm' u (r0 / 2) z1 n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e✝ : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
cb : ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c0, r0 / 2), (z - c0)⁻¹ ^ n • (z - c0)⁻¹ • g z‖ ≤ b * (r0 / 2)⁻¹ ^ n
e : ((2 * ↑π * I)⁻¹ • ∮ (z0 : ℂ) in C(c0, r0 / 2), (z0 - c0)⁻¹ ^ n • (z0 - c0)⁻¹ • g z0) = unevenTerm' u (r0 / 2) z1 n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | clear e g | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e✝ : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
cb : ‖unevenTerm' u (r0 / 2) z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
e : ((2 * ↑π * I)⁻¹ • ∮ (z0 : ℂ) in C(c0, r0 / 2), (z0 - c0)⁻¹ ^ n • (z0 - c0)⁻¹ • g z0) = unevenTerm' u (r0 / 2) z1 n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm' u (r0 / 2) z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e✝ : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
cb : ‖unevenTerm' u (r0 / 2) z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
e : ((2 * ↑π * I)⁻¹ • ∮ (z0 : ℂ) in C(c0, r0 / 2), (z0 - c0)⁻¹ ^ n • (z0 - c0)⁻¹ • g z0) = unevenTerm' u (r0 / 2) z1 n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | rw [unevenTerm_eq u r0hp r0hr1 (Metric.closedBall_subset_closedBall sr.le z1s)] at cb | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm' u (r0 / 2) z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm u z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm' u (r0 / 2) z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | rw [unevenSeries_norm u] | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm u z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm u z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenTerm u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm u z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | apply _root_.trans cb | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm u z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenTerm u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm u z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ b * (r0 / 2)⁻¹ ^ n ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm u z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenTerm u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | bound | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm u z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ b * (r0 / 2)⁻¹ ^ n ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm u z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ b * (r0 / 2)⁻¹ ^ n ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | suffices fa' : AnalyticOn ℂ f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s) by exact fa'.continuousOn | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ AnalyticOn ℂ f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | refine u.a.mono (Set.prod_mono ?_ ?_) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ AnalyticOn ℂ f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s) | case refine_1
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ sphere c0 (r0 / 2) ⊆ ball c0 r0
case refine_2
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ closedBall c1 s ⊆ ball c1 r1 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ AnalyticOn ℂ f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | exact fa'.continuousOn | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fa' : AnalyticOn ℂ f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
⊢ ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fa' : AnalyticOn ℂ f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
⊢ ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | have rh : r0 / 2 < r0 := by linarith [u.r0p] | case refine_1
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ sphere c0 (r0 / 2) ⊆ ball c0 r0 | case refine_1
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
rh : r0 / 2 < r0
⊢ sphere c0 (r0 / 2) ⊆ ball c0 r0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ sphere c0 (r0 / 2) ⊆ ball c0 r0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | exact _root_.trans Metric.sphere_subset_closedBall (Metric.closedBall_subset_ball rh) | case refine_1
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
rh : r0 / 2 < r0
⊢ sphere c0 (r0 / 2) ⊆ ball c0 r0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
rh : r0 / 2 < r0
⊢ sphere c0 (r0 / 2) ⊆ ball c0 r0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | linarith [u.r0p] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ r0 / 2 < r0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ r0 / 2 < r0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | exact Metric.closedBall_subset_ball (by linarith [u.r1p]) | case refine_2
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ closedBall c1 s ⊆ ball c1 r1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ closedBall c1 s ⊆ ball c1 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | linarith [u.r1p] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ s < r1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ s < r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | linarith [u.r0p] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
⊢ r0 / 2 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
⊢ r0 / 2 > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | linarith [u.r0p] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
⊢ r0 / 2 ≤ r0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
⊢ r0 / 2 ≤ r0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | have h := (Uneven.has_series u u.r1p (le_refl _) z1s).r_le | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
h : ENNReal.ofReal r1 ≤ (unevenSeries' u r1 z1).radius
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | rw [FormalMultilinearSeries.radius, le_iSup_iff] at h | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
h : ENNReal.ofReal r1 ≤ (unevenSeries' u r1 z1).radius
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
h :
∀ (b : ℝ≥0∞),
(∀ (i : ℝ≥0), ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑i ^ n ≤ C), ↑i ≤ b) → ENNReal.ofReal r1 ≤ b
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
h : ENNReal.ofReal r1 ≤ (unevenSeries' u r1 z1).radius
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | have sr := not_le_of_lt ((ENNReal.ofReal_lt_ofReal_iff_of_nonneg sp.le).mpr sr1) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
h :
∀ (b : ℝ≥0∞),
(∀ (i : ℝ≥0), ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑i ^ n ≤ C), ↑i ≤ b) → ENNReal.ofReal r1 ≤ b
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
h :
∀ (b : ℝ≥0∞),
(∀ (i : ℝ≥0), ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑i ^ n ≤ C), ↑i ≤ b) → ENNReal.ofReal r1 ≤ b
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
h :
∀ (b : ℝ≥0∞),
(∀ (i : ℝ≥0), ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑i ^ n ≤ C), ↑i ≤ b) → ENNReal.ofReal r1 ≤ b
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | specialize h (ENNReal.ofReal s) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
h :
∀ (b : ℝ≥0∞),
(∀ (i : ℝ≥0), ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑i ^ n ≤ C), ↑i ≤ b) → ENNReal.ofReal r1 ≤ b
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
h :
(∀ (i : ℝ≥0), ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑i ^ n ≤ C), ↑i ≤ ENNReal.ofReal s) →
ENNReal.ofReal r1 ≤ ENNReal.ofReal s
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
h :
∀ (b : ℝ≥0∞),
(∀ (i : ℝ≥0), ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑i ^ n ≤ C), ↑i ≤ b) → ENNReal.ofReal r1 ≤ b
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | rw [imp_iff_not sr] at h | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
h :
(∀ (i : ℝ≥0), ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑i ^ n ≤ C), ↑i ≤ ENNReal.ofReal s) →
ENNReal.ofReal r1 ≤ ENNReal.ofReal s
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
h : ¬∀ (i : ℝ≥0), ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑i ^ n ≤ C), ↑i ≤ ENNReal.ofReal s
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
h :
(∀ (i : ℝ≥0), ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑i ^ n ≤ C), ↑i ≤ ENNReal.ofReal s) →
ENNReal.ofReal r1 ≤ ENNReal.ofReal s
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | simp only [not_forall, not_le, lt_iSup_iff] at h | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
h : ¬∀ (i : ℝ≥0), ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑i ^ n ≤ C), ↑i ≤ ENNReal.ofReal s
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
h : ∃ x i, ∃ (_ : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑x ^ n ≤ i), ENNReal.ofReal s < ↑x
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
h : ¬∀ (i : ℝ≥0), ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑i ^ n ≤ C), ↑i ≤ ENNReal.ofReal s
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | rcases h with ⟨t, c, th, st⟩ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
h : ∃ x i, ∃ (_ : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑x ^ n ≤ i), ENNReal.ofReal s < ↑x
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n | case intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : ENNReal.ofReal s < ↑t
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
h : ∃ x i, ∃ (_ : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑x ^ n ≤ i), ENNReal.ofReal s < ↑x
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | have st' : s < ↑t := by
rw [← ENNReal.ofReal_coe_nnreal, ENNReal.ofReal_lt_ofReal_iff_of_nonneg sp.le] at st
exact st | case intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : ENNReal.ofReal s < ↑t
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n | case intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : ENNReal.ofReal s < ↑t
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | have cp : c ≥ 0 := ge_trans (th 0) (by bound) | case intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n | case intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | use max 1 c, lt_of_lt_of_le (by norm_num) (le_max_left 1 c) | case intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
⊢ ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ max 1 c * s⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
⊢ ∃ c > 0, ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ c * s⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | intro n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
⊢ ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ max 1 c * s⁻¹ ^ n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
⊢ ‖unevenSeries u z1 n‖ ≤ max 1 c * s⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
⊢ ∀ (n : ℕ), ‖unevenSeries u z1 n‖ ≤ max 1 c * s⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | specialize th n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
⊢ ‖unevenSeries u z1 n‖ ≤ max 1 c * s⁻¹ ^ n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
th : ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
⊢ ‖unevenSeries u z1 n‖ ≤ max 1 c * s⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
⊢ ‖unevenSeries u z1 n‖ ≤ max 1 c * s⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | rw [unevenSeries_eq u u.r1p (le_refl _) z1s] at th | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
th : ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
⊢ ‖unevenSeries u z1 n‖ ≤ max 1 c * s⁻¹ ^ n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
th : ‖unevenSeries u z1 n‖ * ↑t ^ n ≤ c
⊢ ‖unevenSeries u z1 n‖ ≤ max 1 c * s⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
th : ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
⊢ ‖unevenSeries u z1 n‖ ≤ max 1 c * s⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | generalize hy : ‖unevenSeries u z1 n‖ = y | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
th : ‖unevenSeries u z1 n‖ * ↑t ^ n ≤ c
⊢ ‖unevenSeries u z1 n‖ ≤ max 1 c * s⁻¹ ^ n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
th : ‖unevenSeries u z1 n‖ * ↑t ^ n ≤ c
y : ℝ
hy : ‖unevenSeries u z1 n‖ = y
⊢ y ≤ max 1 c * s⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
th : ‖unevenSeries u z1 n‖ * ↑t ^ n ≤ c
⊢ ‖unevenSeries u z1 n‖ ≤ max 1 c * s⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | rw [hy] at th | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
th : ‖unevenSeries u z1 n‖ * ↑t ^ n ≤ c
y : ℝ
hy : ‖unevenSeries u z1 n‖ = y
⊢ y ≤ max 1 c * s⁻¹ ^ n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
y : ℝ
th : y * ↑t ^ n ≤ c
hy : ‖unevenSeries u z1 n‖ = y
⊢ y ≤ max 1 c * s⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
th : ‖unevenSeries u z1 n‖ * ↑t ^ n ≤ c
y : ℝ
hy : ‖unevenSeries u z1 n‖ = y
⊢ y ≤ max 1 c * s⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | have tnz : (t : ℝ) ^ n ≠ 0 := pow_ne_zero _ (lt_trans sp st').ne' | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
y : ℝ
th : y * ↑t ^ n ≤ c
hy : ‖unevenSeries u z1 n‖ = y
⊢ y ≤ max 1 c * s⁻¹ ^ n | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
y : ℝ
th : y * ↑t ^ n ≤ c
hy : ‖unevenSeries u z1 n‖ = y
tnz : ↑t ^ n ≠ 0
⊢ y ≤ max 1 c * s⁻¹ ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
y : ℝ
th : y * ↑t ^ n ≤ c
hy : ‖unevenSeries u z1 n‖ = y
⊢ y ≤ max 1 c * s⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | calc
y = y * (↑t ^ n * (↑t ^ n)⁻¹) := by simp only [mul_inv_cancel tnz, mul_one]
_ = y * ↑t ^ n * (↑t ^ n)⁻¹ := by ring
_ ≤ c * (↑t ^ n)⁻¹ := by bound
_ ≤ c * (s ^ n)⁻¹ := by bound
_ = c * s⁻¹ ^ n := by simp only [inv_pow]
_ ≤ max 1 c * s⁻¹ ^ n := by bound | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
y : ℝ
th : y * ↑t ^ n ≤ c
hy : ‖unevenSeries u z1 n‖ = y
tnz : ↑t ^ n ≠ 0
⊢ y ≤ max 1 c * s⁻¹ ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
y : ℝ
th : y * ↑t ^ n ≤ c
hy : ‖unevenSeries u z1 n‖ = y
tnz : ↑t ^ n ≠ 0
⊢ y ≤ max 1 c * s⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | rw [← ENNReal.ofReal_coe_nnreal, ENNReal.ofReal_lt_ofReal_iff_of_nonneg sp.le] at st | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : ENNReal.ofReal s < ↑t
⊢ s < ↑t | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : s < ↑t
⊢ s < ↑t | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : ENNReal.ofReal s < ↑t
⊢ s < ↑t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | exact st | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : s < ↑t
⊢ s < ↑t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : s < ↑t
⊢ s < ↑t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | bound | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
⊢ ‖unevenSeries' u r1 z1 0‖ * ↑t ^ 0 ≥ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
⊢ ‖unevenSeries' u r1 z1 0‖ * ↑t ^ 0 ≥ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | norm_num | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
⊢ 0 < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
th : ∀ (n : ℕ), ‖unevenSeries' u r1 z1 n‖ * ↑t ^ n ≤ c
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
⊢ 0 < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | simp only [mul_inv_cancel tnz, mul_one] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
y : ℝ
th : y * ↑t ^ n ≤ c
hy : ‖unevenSeries u z1 n‖ = y
tnz : ↑t ^ n ≠ 0
⊢ y = y * (↑t ^ n * (↑t ^ n)⁻¹) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
y : ℝ
th : y * ↑t ^ n ≤ c
hy : ‖unevenSeries u z1 n‖ = y
tnz : ↑t ^ n ≠ 0
⊢ y = y * (↑t ^ n * (↑t ^ n)⁻¹)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | ring | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
y : ℝ
th : y * ↑t ^ n ≤ c
hy : ‖unevenSeries u z1 n‖ = y
tnz : ↑t ^ n ≠ 0
⊢ y * (↑t ^ n * (↑t ^ n)⁻¹) = y * ↑t ^ n * (↑t ^ n)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
y : ℝ
th : y * ↑t ^ n ≤ c
hy : ‖unevenSeries u z1 n‖ = y
tnz : ↑t ^ n ≠ 0
⊢ y * (↑t ^ n * (↑t ^ n)⁻¹) = y * ↑t ^ n * (↑t ^ n)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | bound | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
y : ℝ
th : y * ↑t ^ n ≤ c
hy : ‖unevenSeries u z1 n‖ = y
tnz : ↑t ^ n ≠ 0
⊢ y * ↑t ^ n * (↑t ^ n)⁻¹ ≤ c * (↑t ^ n)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
y : ℝ
th : y * ↑t ^ n ≤ c
hy : ‖unevenSeries u z1 n‖ = y
tnz : ↑t ^ n ≠ 0
⊢ y * ↑t ^ n * (↑t ^ n)⁻¹ ≤ c * (↑t ^ n)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | bound | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
y : ℝ
th : y * ↑t ^ n ≤ c
hy : ‖unevenSeries u z1 n‖ = y
tnz : ↑t ^ n ≠ 0
⊢ c * (↑t ^ n)⁻¹ ≤ c * (s ^ n)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
y : ℝ
th : y * ↑t ^ n ≤ c
hy : ‖unevenSeries u z1 n‖ = y
tnz : ↑t ^ n ≠ 0
⊢ c * (↑t ^ n)⁻¹ ≤ c * (s ^ n)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | simp only [inv_pow] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
y : ℝ
th : y * ↑t ^ n ≤ c
hy : ‖unevenSeries u z1 n‖ = y
tnz : ↑t ^ n ≠ 0
⊢ c * (s ^ n)⁻¹ = c * s⁻¹ ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
y : ℝ
th : y * ↑t ^ n ≤ c
hy : ‖unevenSeries u z1 n‖ = y
tnz : ↑t ^ n ≠ 0
⊢ c * (s ^ n)⁻¹ = c * s⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_nonuniform_bound | [462, 1] | [484, 38] | bound | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
y : ℝ
th : y * ↑t ^ n ≤ c
hy : ‖unevenSeries u z1 n‖ = y
tnz : ↑t ^ n ≠ 0
⊢ c * s⁻¹ ^ n ≤ max 1 c * s⁻¹ ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr1 : s < r1
z1s : z1 ∈ closedBall c1 r1
sr : ¬ENNReal.ofReal r1 ≤ ENNReal.ofReal s
t : ℝ≥0
c : ℝ
st : ENNReal.ofReal s < ↑t
st' : s < ↑t
cp : c ≥ 0
n : ℕ
y : ℝ
th : y * ↑t ^ n ≤ c
hy : ‖unevenSeries u z1 n‖ = y
tnz : ↑t ^ n ≠ 0
⊢ c * s⁻¹ ^ n ≤ max 1 c * s⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.norm | [498, 1] | [511, 47] | have pp : 0 ≤ ‖p‖ := by bound | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
⊢ ‖p.along0‖ ≤ ‖p‖ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
⊢ ‖p.along0‖ ≤ ‖p‖ | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
⊢ ‖p.along0‖ ≤ ‖p‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.norm | [498, 1] | [511, 47] | apply p.along0.opNorm_le_bound pp | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
⊢ ‖p.along0‖ ≤ ‖p‖ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
⊢ ∀ (m : Fin n → ℂ), ‖p.along0 m‖ ≤ ‖p‖ * Finset.univ.prod fun i => ‖m i‖ | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
⊢ ‖p.along0‖ ≤ ‖p‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.norm | [498, 1] | [511, 47] | intro m | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
⊢ ∀ (m : Fin n → ℂ), ‖p.along0 m‖ ≤ ‖p‖ * Finset.univ.prod fun i => ‖m i‖ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
m : Fin n → ℂ
⊢ ‖p.along0 m‖ ≤ ‖p‖ * Finset.univ.prod fun i => ‖m i‖ | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
⊢ ∀ (m : Fin n → ℂ), ‖p.along0 m‖ ≤ ‖p‖ * Finset.univ.prod fun i => ‖m i‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.norm | [498, 1] | [511, 47] | simp only [ContinuousMultilinearMap.along0,
ContinuousMultilinearMap.compContinuousLinearMap_apply, Complex.norm_eq_abs] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
m : Fin n → ℂ
⊢ ‖p.along0 m‖ ≤ ‖p‖ * Finset.univ.prod fun i => ‖m i‖ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
m : Fin n → ℂ
⊢ ‖p fun i => idZeroLm (m i)‖ ≤ ‖p‖ * Finset.univ.prod fun x => Complex.abs (m x) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
m : Fin n → ℂ
⊢ ‖p.along0 m‖ ≤ ‖p‖ * Finset.univ.prod fun i => ‖m i‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.norm | [498, 1] | [511, 47] | have e : ∀ i : Fin n, abs (m i) = ‖idZeroLm (m i)‖ := by
intro i
simp only [idZeroLm, ContinuousLinearMap.prod_apply, ContinuousLinearMap.coe_id', id_eq,
ContinuousLinearMap.zero_apply, Prod.norm_def, Complex.norm_eq_abs, norm_zero, ge_iff_le,
apply_nonneg, max_eq_left] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
m : Fin n → ℂ
⊢ ‖p fun i => idZeroLm (m i)‖ ≤ ‖p‖ * Finset.univ.prod fun x => Complex.abs (m x) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e✝ : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
m : Fin n → ℂ
e : ∀ (i : Fin n), Complex.abs (m i) = ‖idZeroLm (m i)‖
⊢ ‖p fun i => idZeroLm (m i)‖ ≤ ‖p‖ * Finset.univ.prod fun x => Complex.abs (m x) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
m : Fin n → ℂ
⊢ ‖p fun i => idZeroLm (m i)‖ ≤ ‖p‖ * Finset.univ.prod fun x => Complex.abs (m x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.norm | [498, 1] | [511, 47] | simp_rw [e] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e✝ : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
m : Fin n → ℂ
e : ∀ (i : Fin n), Complex.abs (m i) = ‖idZeroLm (m i)‖
⊢ ‖p fun i => idZeroLm (m i)‖ ≤ ‖p‖ * Finset.univ.prod fun x => Complex.abs (m x) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e✝ : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
m : Fin n → ℂ
e : ∀ (i : Fin n), Complex.abs (m i) = ‖idZeroLm (m i)‖
⊢ ‖p fun i => idZeroLm (m i)‖ ≤ ‖p‖ * Finset.univ.prod fun x => ‖idZeroLm (m x)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e✝ : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
m : Fin n → ℂ
e : ∀ (i : Fin n), Complex.abs (m i) = ‖idZeroLm (m i)‖
⊢ ‖p fun i => idZeroLm (m i)‖ ≤ ‖p‖ * Finset.univ.prod fun x => Complex.abs (m x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.norm | [498, 1] | [511, 47] | exact ContinuousMultilinearMap.le_opNorm p _ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e✝ : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
m : Fin n → ℂ
e : ∀ (i : Fin n), Complex.abs (m i) = ‖idZeroLm (m i)‖
⊢ ‖p fun i => idZeroLm (m i)‖ ≤ ‖p‖ * Finset.univ.prod fun x => ‖idZeroLm (m x)‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e✝ : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
m : Fin n → ℂ
e : ∀ (i : Fin n), Complex.abs (m i) = ‖idZeroLm (m i)‖
⊢ ‖p fun i => idZeroLm (m i)‖ ≤ ‖p‖ * Finset.univ.prod fun x => ‖idZeroLm (m x)‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.norm | [498, 1] | [511, 47] | bound | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
⊢ 0 ≤ ‖p‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
⊢ 0 ≤ ‖p‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.norm | [498, 1] | [511, 47] | intro i | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
m : Fin n → ℂ
⊢ ∀ (i : Fin n), Complex.abs (m i) = ‖idZeroLm (m i)‖ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
m : Fin n → ℂ
i : Fin n
⊢ Complex.abs (m i) = ‖idZeroLm (m i)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
m : Fin n → ℂ
⊢ ∀ (i : Fin n), Complex.abs (m i) = ‖idZeroLm (m i)‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.norm | [498, 1] | [511, 47] | simp only [idZeroLm, ContinuousLinearMap.prod_apply, ContinuousLinearMap.coe_id', id_eq,
ContinuousLinearMap.zero_apply, Prod.norm_def, Complex.norm_eq_abs, norm_zero, ge_iff_le,
apply_nonneg, max_eq_left] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
m : Fin n → ℂ
i : Fin n
⊢ Complex.abs (m i) = ‖idZeroLm (m i)‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
pp : 0 ≤ ‖p‖
m : Fin n → ℂ
i : Fin n
⊢ Complex.abs (m i) = ‖idZeroLm (m i)‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.radius | [541, 1] | [549, 9] | simp_rw [FormalMultilinearSeries.radius] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
⊢ p.radius ≤ p.along0.radius | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
⊢ ⨆ r, ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C), ↑r ≤ ⨆ r, ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C), ↑r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
⊢ p.radius ≤ p.along0.radius
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.radius | [541, 1] | [549, 9] | refine iSup_mono ?_ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
⊢ ⨆ r, ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C), ↑r ≤ ⨆ r, ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C), ↑r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
⊢ ∀ (i : ℝ≥0), ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖p n‖ * ↑i ^ n ≤ C), ↑i ≤ ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑i ^ n ≤ C), ↑i | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
⊢ ⨆ r, ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C), ↑r ≤ ⨆ r, ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C), ↑r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.radius | [541, 1] | [549, 9] | intro r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
⊢ ∀ (i : ℝ≥0), ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖p n‖ * ↑i ^ n ≤ C), ↑i ≤ ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑i ^ n ≤ C), ↑i | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
⊢ ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C), ↑r ≤ ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C), ↑r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
⊢ ∀ (i : ℝ≥0), ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖p n‖ * ↑i ^ n ≤ C), ↑i ≤ ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑i ^ n ≤ C), ↑i
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.radius | [541, 1] | [549, 9] | refine iSup_mono ?_ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
⊢ ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C), ↑r ≤ ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C), ↑r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
⊢ ∀ (i : ℝ), ⨆ (_ : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ i), ↑r ≤ ⨆ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ i), ↑r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
⊢ ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C), ↑r ≤ ⨆ C, ⨆ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C), ↑r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.radius | [541, 1] | [549, 9] | intro C | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
⊢ ∀ (i : ℝ), ⨆ (_ : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ i), ↑r ≤ ⨆ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ i), ↑r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
⊢ ⨆ (_ : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C), ↑r ≤ ⨆ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C), ↑r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
⊢ ∀ (i : ℝ), ⨆ (_ : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ i), ↑r ≤ ⨆ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ i), ↑r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.radius | [541, 1] | [549, 9] | refine iSup_mono' ?_ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
⊢ ⨆ (_ : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C), ↑r ≤ ⨆ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C), ↑r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
⊢ (∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C) → ∃ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C), ↑r ≤ ↑r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
⊢ ⨆ (_ : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C), ↑r ≤ ⨆ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C), ↑r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.radius | [541, 1] | [549, 9] | intro h | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
⊢ (∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C) → ∃ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C), ↑r ≤ ↑r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
⊢ ∃ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C), ↑r ≤ ↑r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
⊢ (∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C) → ∃ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C), ↑r ≤ ↑r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.radius | [541, 1] | [549, 9] | have h' : ∀ n, ‖p.along0 n‖ * (r:ℝ)^n ≤ C := by
intro n; refine le_trans ?_ (h n); apply mul_le_mul_of_nonneg_right
exact Along0.norm (p n); bound | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
⊢ ∃ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C), ↑r ≤ ↑r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
h' : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C
⊢ ∃ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C), ↑r ≤ ↑r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
⊢ ∃ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C), ↑r ≤ ↑r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.radius | [541, 1] | [549, 9] | use h' | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
h' : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C
⊢ ∃ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C), ↑r ≤ ↑r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
h' : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C
⊢ ∃ (_ : ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C), ↑r ≤ ↑r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.radius | [541, 1] | [549, 9] | intro n | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
⊢ ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
n : ℕ
⊢ ‖p.along0 n‖ * ↑r ^ n ≤ C | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
⊢ ∀ (n : ℕ), ‖p.along0 n‖ * ↑r ^ n ≤ C
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.radius | [541, 1] | [549, 9] | refine le_trans ?_ (h n) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
n : ℕ
⊢ ‖p.along0 n‖ * ↑r ^ n ≤ C | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
n : ℕ
⊢ ‖p.along0 n‖ * ↑r ^ n ≤ ‖p n‖ * ↑r ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
n : ℕ
⊢ ‖p.along0 n‖ * ↑r ^ n ≤ C
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.radius | [541, 1] | [549, 9] | apply mul_le_mul_of_nonneg_right | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
n : ℕ
⊢ ‖p.along0 n‖ * ↑r ^ n ≤ ‖p n‖ * ↑r ^ n | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
n : ℕ
⊢ ‖p.along0 n‖ ≤ ‖p n‖
case a0
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
n : ℕ
⊢ 0 ≤ ↑r ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
n : ℕ
⊢ ‖p.along0 n‖ * ↑r ^ n ≤ ‖p n‖ * ↑r ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.radius | [541, 1] | [549, 9] | exact Along0.norm (p n) | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
n : ℕ
⊢ ‖p.along0 n‖ ≤ ‖p n‖
case a0
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
n : ℕ
⊢ 0 ≤ ↑r ^ n | case a0
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
n : ℕ
⊢ 0 ≤ ↑r ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
n : ℕ
⊢ ‖p.along0 n‖ ≤ ‖p n‖
case a0
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
n : ℕ
⊢ 0 ≤ ↑r ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.radius | [541, 1] | [549, 9] | bound | case a0
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
n : ℕ
⊢ 0 ≤ ↑r ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a0
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
n : ℕ
⊢ 0 ≤ ↑r ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | HasFPowerSeriesAt.along0 | [553, 1] | [571, 59] | rcases fp with ⟨r, fpr⟩ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
fp : HasFPowerSeriesAt f p (c0, c1)
⊢ HasFPowerSeriesAt (fun z0 => f (z0, c1)) p.along0 c0 | case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
⊢ HasFPowerSeriesAt (fun z0 => f (z0, c1)) p.along0 c0 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
fp : HasFPowerSeriesAt f p (c0, c1)
⊢ HasFPowerSeriesAt (fun z0 => f (z0, c1)) p.along0 c0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | HasFPowerSeriesAt.along0 | [553, 1] | [571, 59] | suffices h : HasFPowerSeriesOnBall (fun z0 ↦ f (z0, c1)) p.along0 c0 r by
exact h.hasFPowerSeriesAt | case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
⊢ HasFPowerSeriesAt (fun z0 => f (z0, c1)) p.along0 c0 | case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
⊢ HasFPowerSeriesOnBall (fun z0 => f (z0, c1)) p.along0 c0 r | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
⊢ HasFPowerSeriesAt (fun z0 => f (z0, c1)) p.along0 c0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | HasFPowerSeriesAt.along0 | [553, 1] | [571, 59] | refine
{ r_le := le_trans fpr.r_le (Along0.radius p)
r_pos := fpr.r_pos
hasSum := ?_ } | case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
⊢ HasFPowerSeriesOnBall (fun z0 => f (z0, c1)) p.along0 c0 r | case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
⊢ ∀ {y : ℂ}, y ∈ EMetric.ball 0 r → HasSum (fun n => (p.along0 n) fun x => y) (f (c0 + y, c1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
⊢ HasFPowerSeriesOnBall (fun z0 => f (z0, c1)) p.along0 c0 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | HasFPowerSeriesAt.along0 | [553, 1] | [571, 59] | exact h.hasFPowerSeriesAt | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
h : HasFPowerSeriesOnBall (fun z0 => f (z0, c1)) p.along0 c0 r
⊢ HasFPowerSeriesAt (fun z0 => f (z0, c1)) p.along0 c0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
h : HasFPowerSeriesOnBall (fun z0 => f (z0, c1)) p.along0 c0 r
⊢ HasFPowerSeriesAt (fun z0 => f (z0, c1)) p.along0 c0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | HasFPowerSeriesAt.along0 | [553, 1] | [571, 59] | intro w0 w0r | case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
⊢ ∀ {y : ℂ}, y ∈ EMetric.ball 0 r → HasSum (fun n => (p.along0 n) fun x => y) (f (c0 + y, c1)) | case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0✝ w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
w0 : ℂ
w0r : w0 ∈ EMetric.ball 0 r
⊢ HasSum (fun n => (p.along0 n) fun x => w0) (f (c0 + w0, c1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
⊢ ∀ {y : ℂ}, y ∈ EMetric.ball 0 r → HasSum (fun n => (p.along0 n) fun x => y) (f (c0 + y, c1))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | HasFPowerSeriesAt.along0 | [553, 1] | [571, 59] | simp_rw [FormalMultilinearSeries.along0, ContinuousMultilinearMap.along0, idZeroLm] | case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0✝ w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
w0 : ℂ
w0r : w0 ∈ EMetric.ball 0 r
⊢ HasSum (fun n => (p.along0 n) fun x => w0) (f (c0 + w0, c1)) | case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0✝ w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
w0 : ℂ
w0r : w0 ∈ EMetric.ball 0 r
⊢ HasSum (fun n => ((p n).compContinuousLinearMap fun x => (ContinuousLinearMap.id ℂ ℂ).prod 0) fun x => w0)
(f (c0 + w0, c1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0✝ w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
w0 : ℂ
w0r : w0 ∈ EMetric.ball 0 r
⊢ HasSum (fun n => (p.along0 n) fun x => w0) (f (c0 + w0, c1))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | HasFPowerSeriesAt.along0 | [553, 1] | [571, 59] | simp only [ContinuousMultilinearMap.compContinuousLinearMap_apply,
ContinuousLinearMap.prod_apply, ContinuousLinearMap.coe_id', id_eq,
ContinuousLinearMap.zero_apply] | case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0✝ w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
w0 : ℂ
w0r : w0 ∈ EMetric.ball 0 r
⊢ HasSum (fun n => ((p n).compContinuousLinearMap fun x => (ContinuousLinearMap.id ℂ ℂ).prod 0) fun x => w0)
(f (c0 + w0, c1)) | case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0✝ w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
w0 : ℂ
w0r : w0 ∈ EMetric.ball 0 r
⊢ HasSum (fun n => (p n) fun i => (w0, 0)) (f (c0 + w0, c1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0✝ w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
w0 : ℂ
w0r : w0 ∈ EMetric.ball 0 r
⊢ HasSum (fun n => ((p n).compContinuousLinearMap fun x => (ContinuousLinearMap.id ℂ ℂ).prod 0) fun x => w0)
(f (c0 + w0, c1))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | HasFPowerSeriesAt.along0 | [553, 1] | [571, 59] | have w01r : (w0, (0 : ℂ)) ∈ EMetric.ball (0 : ℂ × ℂ) r := by
simpa only [Prod.edist_eq, EMetric.mem_ball, Prod.fst_zero, Prod.snd_zero, edist_self,
ENNReal.max_zero_right] using w0r | case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0✝ w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
w0 : ℂ
w0r : w0 ∈ EMetric.ball 0 r
⊢ HasSum (fun n => (p n) fun i => (w0, 0)) (f (c0 + w0, c1)) | case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0✝ w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
w0 : ℂ
w0r : w0 ∈ EMetric.ball 0 r
w01r : (w0, 0) ∈ EMetric.ball 0 r
⊢ HasSum (fun n => (p n) fun i => (w0, 0)) (f (c0 + w0, c1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0✝ w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
w0 : ℂ
w0r : w0 ∈ EMetric.ball 0 r
⊢ HasSum (fun n => (p n) fun i => (w0, 0)) (f (c0 + w0, c1))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | HasFPowerSeriesAt.along0 | [553, 1] | [571, 59] | convert fpr.hasSum w01r | case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0✝ w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
w0 : ℂ
w0r : w0 ∈ EMetric.ball 0 r
w01r : (w0, 0) ∈ EMetric.ball 0 r
⊢ HasSum (fun n => (p n) fun i => (w0, 0)) (f (c0 + w0, c1)) | case h.e'_6.h.e'_1.h.e'_4
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0✝ w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
w0 : ℂ
w0r : w0 ∈ EMetric.ball 0 r
w01r : (w0, 0) ∈ EMetric.ball 0 r
⊢ c1 = ((c0, c1) + (w0, 0)).2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0✝ w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
w0 : ℂ
w0r : w0 ∈ EMetric.ball 0 r
w01r : (w0, 0) ∈ EMetric.ball 0 r
⊢ HasSum (fun n => (p n) fun i => (w0, 0)) (f (c0 + w0, c1))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | HasFPowerSeriesAt.along0 | [553, 1] | [571, 59] | rw [Prod.mk_add_mk, add_zero] | case h.e'_6.h.e'_1.h.e'_4
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0✝ w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
w0 : ℂ
w0r : w0 ∈ EMetric.ball 0 r
w01r : (w0, 0) ∈ EMetric.ball 0 r
⊢ c1 = ((c0, c1) + (w0, 0)).2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_6.h.e'_1.h.e'_4
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0✝ w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
w0 : ℂ
w0r : w0 ∈ EMetric.ball 0 r
w01r : (w0, 0) ∈ EMetric.ball 0 r
⊢ c1 = ((c0, c1) + (w0, 0)).2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | HasFPowerSeriesAt.along0 | [553, 1] | [571, 59] | simpa only [Prod.edist_eq, EMetric.mem_ball, Prod.fst_zero, Prod.snd_zero, edist_self,
ENNReal.max_zero_right] using w0r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0✝ w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
w0 : ℂ
w0r : w0 ∈ EMetric.ball 0 r
⊢ (w0, 0) ∈ EMetric.ball 0 r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0✝ w1 : ℂ
r✝ r0 r1 b e : ℝ
f : ℂ × ℂ → E
c0 c1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
fpr : HasFPowerSeriesOnBall f p (c0, c1) r
w0 : ℂ
w0r : w0 ∈ EMetric.ball 0 r
⊢ (w0, 0) ∈ EMetric.ball 0 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.analyticAt | [574, 1] | [582, 64] | intro p | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
⊢ ∀ {p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E}, AnalyticAt ℂ (fun p => p.along0) p | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
⊢ AnalyticAt ℂ (fun p => p.along0) p | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
⊢ ∀ {p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E}, AnalyticAt ℂ (fun p => p.along0) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.analyticAt | [574, 1] | [582, 64] | have e : (fun p : ContinuousMultilinearMap ℂ (fun _ : Fin n ↦ ℂ × ℂ) E => p.along0) =ᶠ[𝓝 p]
(Along0.continuousLinearMap n : (ContinuousMultilinearMap ℂ (fun _ : Fin n ↦ ℂ × ℂ) E →
ContinuousMultilinearMap ℂ (fun _ : Fin n ↦ ℂ) E)) := by
apply eventually_of_forall; intro _; rfl | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
⊢ AnalyticAt ℂ (fun p => p.along0) p | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e✝ : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
e : (𝓝 p).EventuallyEq (fun p => p.along0) ⇑(continuousLinearMap n)
⊢ AnalyticAt ℂ (fun p => p.along0) p | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
⊢ AnalyticAt ℂ (fun p => p.along0) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.analyticAt | [574, 1] | [582, 64] | rw [analyticAt_congr e] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e✝ : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
e : (𝓝 p).EventuallyEq (fun p => p.along0) ⇑(continuousLinearMap n)
⊢ AnalyticAt ℂ (fun p => p.along0) p | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e✝ : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
e : (𝓝 p).EventuallyEq (fun p => p.along0) ⇑(continuousLinearMap n)
⊢ AnalyticAt ℂ (⇑(continuousLinearMap n)) p | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e✝ : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
e : (𝓝 p).EventuallyEq (fun p => p.along0) ⇑(continuousLinearMap n)
⊢ AnalyticAt ℂ (fun p => p.along0) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.analyticAt | [574, 1] | [582, 64] | apply ContinuousLinearMap.analyticAt | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e✝ : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
e : (𝓝 p).EventuallyEq (fun p => p.along0) ⇑(continuousLinearMap n)
⊢ AnalyticAt ℂ (⇑(continuousLinearMap n)) p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e✝ : ℝ
n : ℕ
p : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
e : (𝓝 p).EventuallyEq (fun p => p.along0) ⇑(continuousLinearMap n)
⊢ AnalyticAt ℂ (⇑(continuousLinearMap n)) p
TACTIC:
|
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