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 | 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:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.analyticAt | [574, 1] | [582, 64] | apply eventually_of_forall | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : 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
⊢ (𝓝 p).EventuallyEq (fun p => p.along0) ⇑(continuousLinearMap n) | case hp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : 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
⊢ ∀ (x : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E), (fun p => p.along0) x = (continuousLinearMap n) 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
⊢ (𝓝 p).EventuallyEq (fun p => p.along0) ⇑(continuousLinearMap n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.analyticAt | [574, 1] | [582, 64] | intro _ | case hp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : 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
⊢ ∀ (x : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E), (fun p => p.along0) x = (continuousLinearMap n) x | case hp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : 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 x✝ : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
⊢ (fun p => p.along0) x✝ = (continuousLinearMap n) x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : 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
⊢ ∀ (x : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E), (fun p => p.along0) x = (continuousLinearMap n) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Along0.analyticAt | [574, 1] | [582, 64] | rfl | case hp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : 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 x✝ : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
⊢ (fun p => p.along0) x✝ = (continuousLinearMap n) x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : 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 x✝ : ContinuousMultilinearMap ℂ (fun x => ℂ × ℂ) E
⊢ (fun p => p.along0) x✝ = (continuousLinearMap n) x✝
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | intro z1 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
n : ℕ
⊢ AnalyticOn ℂ (fun z1 => unevenSeries u z1 n) (ball c1 r1) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : 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
z1 : ℂ
z1s : z1 ∈ ball c1 r1
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) 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
n : ℕ
⊢ AnalyticOn ℂ (fun z1 => unevenSeries u z1 n) (ball c1 r1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | rcases u.a (c0, z1) (Set.mk_mem_prod (Metric.mem_ball_self u.r0p) z1s) with ⟨p, r, hp⟩ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : 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
z1 : ℂ
z1s : z1 ∈ ball c1 r1
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1 | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | have pa := (p.hasFPowerSeriesOnBall_changeOrigin n (lt_of_lt_of_le hp.r_pos hp.r_le)).analyticAt | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1 | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) 0
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1 | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | set g := fun w1 ↦ ((0 : ℂ), w1 - z1) | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) 0
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1 | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) 0
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1 | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) 0
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | have ga : AnalyticOn ℂ g univ := by
rw [analyticOn_univ_iff_differentiable]
exact (differentiable_const _).prod (differentiable_id.sub (differentiable_const _)) | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) 0
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1 | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) 0
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ga : AnalyticOn ℂ g univ
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1 | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) 0
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | have g0 : 0 = g z1 := by
simp only [Prod.ext_iff, Prod.fst_zero, Prod.snd_zero, sub_self, and_self_iff] | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) 0
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ga : AnalyticOn ℂ g univ
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1 | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) 0
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ga : AnalyticOn ℂ g univ
g0 : 0 = g z1
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1 | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) 0
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ga : AnalyticOn ℂ g univ
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | rw [g0] at pa | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) 0
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ga : AnalyticOn ℂ g univ
g0 : 0 = g z1
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1 | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) (g z1)
ga : AnalyticOn ℂ g univ
g0 : 0 = g z1
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1 | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) 0
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ga : AnalyticOn ℂ g univ
g0 : 0 = g z1
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | have ta := pa.comp (ga z1 (Set.mem_univ _)) | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) (g z1)
ga : AnalyticOn ℂ g univ
g0 : 0 = g z1
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1 | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) (g z1)
ga : AnalyticOn ℂ g univ
g0 : 0 = g z1
ta : AnalyticAt ℂ ((fun x => p.changeOrigin x n) ∘ g) z1
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1 | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) (g z1)
ga : AnalyticOn ℂ g univ
g0 : 0 = g z1
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | simp_rw [Function.comp] at ta | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) (g z1)
ga : AnalyticOn ℂ g univ
g0 : 0 = g z1
ta : AnalyticAt ℂ ((fun x => p.changeOrigin x n) ∘ g) z1
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1 | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) (g z1)
ga : AnalyticOn ℂ g univ
g0 : 0 = g z1
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1 | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) (g z1)
ga : AnalyticOn ℂ g univ
g0 : 0 = g z1
ta : AnalyticAt ℂ ((fun x => p.changeOrigin x n) ∘ g) z1
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | clear pa ga g0 | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) (g z1)
ga : AnalyticOn ℂ g univ
g0 : 0 = g z1
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1 | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1 | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) (g z1)
ga : AnalyticOn ℂ g univ
g0 : 0 = g z1
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | rw [analyticAt_congr pu] | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
pu : ∀ᶠ (w1 : ℂ) in 𝓝 z1, unevenSeries u w1 n = (p.changeOrigin (g w1)).along0 n
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1 | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
pu : ∀ᶠ (w1 : ℂ) in 𝓝 z1, unevenSeries u w1 n = (p.changeOrigin (g w1)).along0 n
⊢ AnalyticAt ℂ (fun x => (p.changeOrigin (g x)).along0 n) z1 | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
pu : ∀ᶠ (w1 : ℂ) in 𝓝 z1, unevenSeries u w1 n = (p.changeOrigin (g w1)).along0 n
⊢ AnalyticAt ℂ (fun z1 => unevenSeries u z1 n) z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | exact (Along0.analyticAt _).comp ta | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
pu : ∀ᶠ (w1 : ℂ) in 𝓝 z1, unevenSeries u w1 n = (p.changeOrigin (g w1)).along0 n
⊢ AnalyticAt ℂ (fun x => (p.changeOrigin (g x)).along0 n) z1 | no goals | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
pu : ∀ᶠ (w1 : ℂ) in 𝓝 z1, unevenSeries u w1 n = (p.changeOrigin (g w1)).along0 n
⊢ AnalyticAt ℂ (fun x => (p.changeOrigin (g x)).along0 n) z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | rw [analyticOn_univ_iff_differentiable] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : 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
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) 0
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
⊢ AnalyticOn ℂ g univ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : 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
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) 0
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
⊢ Differentiable ℂ g | 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
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) 0
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
⊢ AnalyticOn ℂ g univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | exact (differentiable_const _).prod (differentiable_id.sub (differentiable_const _)) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : 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
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) 0
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
⊢ Differentiable ℂ g | 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
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) 0
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
⊢ Differentiable ℂ g
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | simp only [Prod.ext_iff, Prod.fst_zero, Prod.snd_zero, sub_self, and_self_iff] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : 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
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) 0
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ga : AnalyticOn ℂ g univ
⊢ 0 = g z1 | 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
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
pa : AnalyticAt ℂ (fun x => p.changeOrigin x n) 0
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ga : AnalyticOn ℂ g univ
⊢ 0 = g z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | rw [eventually_nhds_iff] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : 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
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
⊢ ∀ᶠ (w1 : ℂ) in 𝓝 z1, unevenSeries u w1 n = (p.changeOrigin (g w1)).along0 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
⊢ ∃ t, (∀ x ∈ t, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧ IsOpen t ∧ z1 ∈ 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
⊢ ∀ᶠ (w1 : ℂ) in 𝓝 z1, unevenSeries u w1 n = (p.changeOrigin (g w1)).along0 n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | set s' := r1 - dist z1 c1 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : 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
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
⊢ ∃ t, (∀ x ∈ t, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧ IsOpen t ∧ z1 ∈ 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
⊢ ∃ t, (∀ x ∈ t, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧ IsOpen t ∧ z1 ∈ 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
⊢ ∃ t, (∀ x ∈ t, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧ IsOpen t ∧ z1 ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | set s := min r (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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
⊢ ∃ t, (∀ x ∈ t, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧ IsOpen t ∧ z1 ∈ 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
⊢ ∃ t, (∀ x ∈ t, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧ IsOpen t ∧ z1 ∈ 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
⊢ ∃ t, (∀ x ∈ t, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧ IsOpen t ∧ z1 ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | have s'p : s' > 0 := by simp only [Metric.mem_ball] at z1s; 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
⊢ ∃ t, (∀ x ∈ t, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧ IsOpen t ∧ z1 ∈ 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
⊢ ∃ t, (∀ x ∈ t, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧ IsOpen t ∧ z1 ∈ 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
⊢ ∃ t, (∀ x ∈ t, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧ IsOpen t ∧ z1 ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | have sp : s > 0 := 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
u : Uneven f c0 c1 r0 r1
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
⊢ ∃ t, (∀ x ∈ t, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧ IsOpen t ∧ z1 ∈ 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
sp : s > 0
⊢ ∃ t, (∀ x ∈ t, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧ IsOpen t ∧ z1 ∈ 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
⊢ ∃ t, (∀ x ∈ t, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧ IsOpen t ∧ z1 ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | have sr : s ≤ r := 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
u : Uneven f c0 c1 r0 r1
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
sp : s > 0
⊢ ∃ t, (∀ x ∈ t, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧ IsOpen t ∧ z1 ∈ 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
sp : s > 0
sr : s ≤ r
⊢ ∃ t, (∀ x ∈ t, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧ IsOpen t ∧ z1 ∈ 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
sp : s > 0
⊢ ∃ t, (∀ x ∈ t, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧ IsOpen t ∧ z1 ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | have sb : EMetric.ball z1 s ⊆ ball c1 r1 := by
rw [Set.subset_def]; intro x xs
simp only [Metric.mem_ball, EMetric.mem_ball, lt_min_iff, edist_lt_ofReal, s] at xs z1s ⊢
calc dist x c1
_ ≤ dist x z1 + dist z1 c1 := by bound
_ < s' + dist z1 c1 := (add_lt_add_right xs.right _)
_ = r1 - dist z1 c1 + dist z1 c1 := rfl
_ = r1 := by ring_nf | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : 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
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
sp : s > 0
sr : s ≤ r
⊢ ∃ t, (∀ x ∈ t, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧ IsOpen t ∧ z1 ∈ 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
sp : s > 0
sr : s ≤ r
sb : EMetric.ball z1 s ⊆ ball c1 r1
⊢ ∃ t, (∀ x ∈ t, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧ IsOpen t ∧ z1 ∈ 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
sp : s > 0
sr : s ≤ r
⊢ ∃ t, (∀ x ∈ t, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧ IsOpen t ∧ z1 ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | use EMetric.ball z1 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
sp : s > 0
sr : s ≤ r
sb : EMetric.ball z1 s ⊆ ball c1 r1
⊢ ∃ t, (∀ x ∈ t, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧ IsOpen t ∧ z1 ∈ t | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
sp : s > 0
sr : s ≤ r
sb : EMetric.ball z1 s ⊆ ball c1 r1
⊢ (∀ x ∈ EMetric.ball z1 s, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧
IsOpen (EMetric.ball z1 s) ∧ z1 ∈ EMetric.ball z1 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
sp : s > 0
sr : s ≤ r
sb : EMetric.ball z1 s ⊆ ball c1 r1
⊢ ∃ t, (∀ x ∈ t, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧ IsOpen t ∧ z1 ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | refine ⟨?_, EMetric.isOpen_ball, EMetric.mem_ball_self sp⟩ | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
sp : s > 0
sr : s ≤ r
sb : EMetric.ball z1 s ⊆ ball c1 r1
⊢ (∀ x ∈ EMetric.ball z1 s, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧
IsOpen (EMetric.ball z1 s) ∧ z1 ∈ EMetric.ball z1 s | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
sp : s > 0
sr : s ≤ r
sb : EMetric.ball z1 s ⊆ ball c1 r1
⊢ ∀ x ∈ EMetric.ball z1 s, unevenSeries u x n = (p.changeOrigin (g x)).along0 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
u : Uneven f c0 c1 r0 r1
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
sp : s > 0
sr : s ≤ r
sb : EMetric.ball z1 s ⊆ ball c1 r1
⊢ (∀ x ∈ EMetric.ball z1 s, unevenSeries u x n = (p.changeOrigin (g x)).along0 n) ∧
IsOpen (EMetric.ball z1 s) ∧ z1 ∈ EMetric.ball z1 s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | intro w1 w1s | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
sp : s > 0
sr : s ≤ r
sb : EMetric.ball z1 s ⊆ ball c1 r1
⊢ ∀ x ∈ EMetric.ball z1 s, unevenSeries u x n = (p.changeOrigin (g x)).along0 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
u : Uneven f c0 c1 r0 r1
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
sp : s > 0
sr : s ≤ r
sb : EMetric.ball z1 s ⊆ ball c1 r1
w1 : ℂ
w1s : w1 ∈ EMetric.ball z1 s
⊢ unevenSeries u w1 n = (p.changeOrigin (g w1)).along0 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
u : Uneven f c0 c1 r0 r1
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
sp : s > 0
sr : s ≤ r
sb : EMetric.ball z1 s ⊆ ball c1 r1
⊢ ∀ x ∈ EMetric.ball z1 s, unevenSeries u x n = (p.changeOrigin (g x)).along0 n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | have p0 : HasFPowerSeriesAt (fun z0 ↦ f (z0, w1)) (unevenSeries u w1) c0 := by
have w1c : w1 ∈ closedBall c1 r1 := mem_open_closed (sb w1s)
refine (Uneven.has_series u u.r1p (le_refl _) w1c).hasFPowerSeriesAt | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
sp : s > 0
sr : s ≤ r
sb : EMetric.ball z1 s ⊆ ball c1 r1
w1 : ℂ
w1s : w1 ∈ EMetric.ball z1 s
⊢ unevenSeries u w1 n = (p.changeOrigin (g w1)).along0 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
u : Uneven f c0 c1 r0 r1
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
sp : s > 0
sr : s ≤ r
sb : EMetric.ball z1 s ⊆ ball c1 r1
w1 : ℂ
w1s : w1 ∈ EMetric.ball z1 s
p0 : HasFPowerSeriesAt (fun z0 => f (z0, w1)) (unevenSeries u w1) c0
⊢ unevenSeries u w1 n = (p.changeOrigin (g w1)).along0 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
u : Uneven f c0 c1 r0 r1
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
sp : s > 0
sr : s ≤ r
sb : EMetric.ball z1 s ⊆ ball c1 r1
w1 : ℂ
w1s : w1 ∈ EMetric.ball z1 s
⊢ unevenSeries u w1 n = (p.changeOrigin (g w1)).along0 n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | rw [HasFPowerSeriesAt.eq_formalMultilinearSeries p0 p1] | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
sp : s > 0
sr : s ≤ r
sb : EMetric.ball z1 s ⊆ ball c1 r1
w1 : ℂ
w1s : w1 ∈ EMetric.ball z1 s
p0 : HasFPowerSeriesAt (fun z0 => f (z0, w1)) (unevenSeries u w1) c0
p1 : HasFPowerSeriesAt (fun z0 => f (z0, w1)) (p.changeOrigin (g w1)).along0 c0
⊢ unevenSeries u w1 n = (p.changeOrigin (g w1)).along0 n | 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
s'p : s' > 0
sp : s > 0
sr : s ≤ r
sb : EMetric.ball z1 s ⊆ ball c1 r1
w1 : ℂ
w1s : w1 ∈ EMetric.ball z1 s
p0 : HasFPowerSeriesAt (fun z0 => f (z0, w1)) (unevenSeries u w1) c0
p1 : HasFPowerSeriesAt (fun z0 => f (z0, w1)) (p.changeOrigin (g w1)).along0 c0
⊢ unevenSeries u w1 n = (p.changeOrigin (g w1)).along0 n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_analytic | [585, 1] | [631, 38] | simp only [Metric.mem_ball] at 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
n : ℕ
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
⊢ s' > 0 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : 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
z1 : ℂ
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
z1s : dist z1 c1 < r1
⊢ s' > 0 | 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
z1 : ℂ
z1s : z1 ∈ ball c1 r1
p : FormalMultilinearSeries ℂ (ℂ × ℂ) E
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p (c0, z1) r
g : ℂ → ℂ × ℂ := fun w1 => (0, w1 - z1)
ta : AnalyticAt ℂ (fun x => p.changeOrigin (g x) n) z1
s' : ℝ := r1 - dist z1 c1
s : ℝ≥0∞ := min r (ENNReal.ofReal s')
⊢ s' > 0
TACTIC:
|
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